libMesh::FESubdivision Class Reference

#include <fe.h>

Inheritance diagram for libMesh::FESubdivision:

Public Types
typedef FEGenericBase< typename FEOutputType< T >::type >::OutputShape	OutputShape
typedef TensorTools::IncrementRank< OutputShape >::type	OutputGradient
typedef TensorTools::IncrementRank< OutputGradient >::type	OutputTensor
typedef TensorTools::DecrementRank< OutputShape >::type	OutputDivergence
typedef TensorTools::MakeNumber< OutputShape >::type	OutputNumber
typedef TensorTools::IncrementRank< OutputNumber >::type	OutputNumberGradient
typedef TensorTools::IncrementRank< OutputNumberGradient >::type	OutputNumberTensor
typedef TensorTools::DecrementRank< OutputNumber >::type	OutputNumberDivergence

Public Member Functions
	FESubdivision (const FEType &fet)
virtual void	reinit (const Elem *elem, const std::vector< Point > *const pts=nullptr, const std::vector< Real > *const weights=nullptr) override
virtual void	reinit (const Elem *, const unsigned int, const Real=TOLERANCE, const std::vector< Point > *const =nullptr, const std::vector< Real > *const =nullptr) override
virtual void	attach_quadrature_rule (QBase *q) override
virtual void	init_shape_functions (const std::vector< Point > &qp, const Elem *elem) override
Real	shape (const ElemType, const Order, const unsigned int libmesh_dbg_var(i), const Point &)
Real	shape (const Elem *, const Order, const unsigned int libmesh_dbg_var(i), const Point &)
Real	shape (const ElemType, const Order order, const unsigned int i, const Point &p)
Real	shape (const Elem *elem, const Order order, const unsigned int i, const Point &p)
Real	shape (const ElemType, const Order, const unsigned int, const Point &)
Real	shape (const Elem *elem, const Order order, const unsigned int i, const Point &p)
Real	shape (const ElemType, const Order, const unsigned int, const Point &)
Real	shape (const Elem *elem, const Order order, const unsigned int i, const Point &p)
Real	shape (const ElemType, const Order, const unsigned int libmesh_dbg_var(i), const Point &)
Real	shape (const Elem *, const Order, const unsigned int libmesh_dbg_var(i), const Point &)
Real	shape (const ElemType, const Order, const unsigned int, const Point &)
Real	shape (const Elem *elem, const Order order, const unsigned int i, const Point &p)
Real	shape (const ElemType, const Order, const unsigned int, const Point &)
Real	shape (const Elem *elem, const Order order, const unsigned int i, const Point &p)
Real	shape (const ElemType, const Order, const unsigned int, const Point &)
Real	shape (const Elem *libmesh_dbg_var(elem), const Order, const unsigned int, const Point &)
Real	shape (const ElemType, const Order, const unsigned int libmesh_dbg_var(i), const Point &)
Real	shape (const Elem *, const Order, const unsigned int libmesh_dbg_var(i), const Point &)
Real	shape (const ElemType, const Order, const unsigned int, const Point &)
Real	shape (const Elem *elem, const Order libmesh_dbg_var(order), const unsigned int i, const Point &p)
Real	shape (const ElemType, const Order, const unsigned int, const Point &)
Real	shape (const Elem *elem, const Order order, const unsigned int i, const Point &p)
Real	shape (const ElemType, const Order, const unsigned int, const Point &)
Real	shape (const Elem *elem, const Order order, const unsigned int i, const Point &p)
Real	shape (const ElemType, const Order, const unsigned int libmesh_dbg_var(i), const Point &)
Real	shape (const Elem *, const Order, const unsigned int libmesh_dbg_var(i), const Point &)
Real	shape (const ElemType, const Order libmesh_dbg_var(order), const unsigned int i, const Point &p)
Real	shape (const Elem *elem, const Order order, const unsigned int i, const Point &p)
Real	shape (const ElemType, const Order, const unsigned int, const Point &)
Real	shape (const Elem *elem, const Order order, const unsigned int i, const Point &p)
Real	shape (const ElemType, const Order, const unsigned int, const Point &)
Real	shape (const Elem *elem, const Order order, const unsigned int i, const Point &p)
Real	shape (const ElemType, const Order, const unsigned int libmesh_dbg_var(i), const Point &)
Real	shape (const Elem *, const Order, const unsigned int libmesh_dbg_var(i), const Point &)
Real	shape (const ElemType, const Order libmesh_dbg_var(order), const unsigned int i, const Point &p)
Real	shape (const Elem *elem, const Order order, const unsigned int i, const Point &p)
Real	shape (const ElemType, const Order, const unsigned int, const Point &)
Real	shape (const Elem *elem, const Order order, const unsigned int i, const Point &p)
Real	shape (const ElemType, const Order, const unsigned int, const Point &)
Real	shape (const Elem *elem, const Order order, const unsigned int i, const Point &p)
Real	shape (const ElemType, const Order, const unsigned int libmesh_dbg_var(i), const Point &)
Real	shape (const Elem *, const Order, const unsigned int libmesh_dbg_var(i), const Point &)
Real	shape (const ElemType, const Order order, const unsigned int i, const Point &p)
Real	shape (const Elem *elem, const Order order, const unsigned int i, const Point &p)
Real	shape (const ElemType type, const Order order, const unsigned int i, const Point &p)
Real	shape (const Elem *elem, const Order order, const unsigned int i, const Point &p)
Real	shape (const ElemType type, const Order order, const unsigned int i, const Point &p)
Real	shape (const Elem *elem, const Order order, const unsigned int i, const Point &p)
Real	shape (const ElemType, const Order, const unsigned int libmesh_dbg_var(i), const Point &)
Real	shape (const Elem *, const Order, const unsigned int libmesh_dbg_var(i), const Point &)
Real	shape (const ElemType, const Order order, const unsigned int i, const Point &p)
Real	shape (const Elem *elem, const Order order, const unsigned int i, const Point &p)
Real	shape (const ElemType type, const Order order, const unsigned int i, const Point &p)
Real	shape (const Elem *elem, const Order order, const unsigned int i, const Point &p)
Real	shape (const ElemType type, const Order order, const unsigned int i, const Point &p)
Real	shape (const Elem *elem, const Order order, const unsigned int i, const Point &p)
RealGradient	shape (const ElemType type, const Order order, const unsigned int i, const Point &p)
RealGradient	shape (const ElemType type, const Order order, const unsigned int i, const Point &p)
RealGradient	shape (const ElemType type, const Order order, const unsigned int i, const Point &p)
RealGradient	shape (const ElemType type, const Order order, const unsigned int i, const Point &p)
RealGradient	shape (const Elem *elem, const Order order, const unsigned int i, const Point &p)
RealGradient	shape (const Elem *elem, const Order order, const unsigned int i, const Point &p)
RealGradient	shape (const Elem *elem, const Order order, const unsigned int i, const Point &p)
RealGradient	shape (const Elem *elem, const Order order, const unsigned int i, const Point &p)
Real	shape (const ElemType, const Order, const unsigned int libmesh_dbg_var(i), const Point &)
Real	shape (const Elem *, const Order, const unsigned int libmesh_dbg_var(i), const Point &)
Real	shape (const ElemType, const Order libmesh_dbg_var(order), const unsigned int i, const Point &p)
Real	shape (const Elem *elem, const Order order, const unsigned int i, const Point &p)
Real	shape (const ElemType, const Order libmesh_dbg_var(order), const unsigned int i, const Point &p)
Real	shape (const Elem *elem, const Order order, const unsigned int i, const Point &p)
Real	shape (const ElemType, const Order libmesh_dbg_var(order), const unsigned int i, const Point &p)
Real	shape (const Elem *elem, const Order order, const unsigned int i, const Point &p)
RealGradient	shape (const ElemType, const Order, const unsigned int, const Point &)
RealGradient	shape (const Elem *, const Order, const unsigned int, const Point &)
RealGradient	shape (const ElemType, const Order, const unsigned int, const Point &)
RealGradient	shape (const Elem *, const Order, const unsigned int, const Point &)
RealGradient	shape (const ElemType, const Order, const unsigned int, const Point &)
RealGradient	shape (const Elem *elem, const Order order, const unsigned int i, const Point &p)
RealGradient	shape (const ElemType, const Order, const unsigned int, const Point &)
RealGradient	shape (const Elem *elem, const Order order, const unsigned int i, const Point &p)
Real	shape (const ElemType, const Order, const unsigned int, const Point &)
Real	shape (const Elem *, const Order, const unsigned int, const Point &)
Real	shape (const ElemType, const Order, const unsigned int, const Point &)
Real	shape (const Elem *, const Order, const unsigned int, const Point &)
Real	shape (const ElemType, const Order, const unsigned int, const Point &)
Real	shape (const Elem *, const Order, const unsigned int, const Point &)
Real	shape (const ElemType, const Order, const unsigned int, const Point &)
Real	shape (const Elem *, const Order, const unsigned int, const Point &)
Real	shape (const ElemType type, const Order order, const unsigned int i, const Point &p)
Real	shape (const Elem *elem, const Order order, const unsigned int i, const Point &p)
Real	shape (const ElemType, const Order, const unsigned int libmesh_dbg_var(i), const Point &)
Real	shape (const Elem *, const Order, const unsigned int libmesh_dbg_var(i), const Point &)
Real	shape (const ElemType, const Order libmesh_dbg_var(order), const unsigned int i, const Point &p)
Real	shape (const Elem *elem, const Order order, const unsigned int i, const Point &p)
Real	shape (const ElemType, const Order, const unsigned int, const Point &)
Real	shape (const Elem *elem, const Order order, const unsigned int i, const Point &p)
Real	shape (const ElemType, const Order, const unsigned int, const Point &)
Real	shape (const Elem *, const Order, const unsigned int, const Point &)
Real	shape (const ElemType, const Order, const unsigned int libmesh_dbg_var(i), const Point &)
Real	shape (const Elem *, const Order, const unsigned int libmesh_dbg_var(i), const Point &)
Real	shape (const ElemType, const Order, const unsigned int, const Point &)
Real	shape (const Elem *elem, const Order libmesh_dbg_var(order), const unsigned int i, const Point &point_in)
Real	shape (const ElemType, const Order, const unsigned int, const Point &)
Real	shape (const Elem *elem, const Order libmesh_dbg_var(order), const unsigned int i, const Point &point_in)
Real	shape (const ElemType, const Order, const unsigned int, const Point &)
Real	shape (const Elem *elem, const Order libmesh_dbg_var(order), const unsigned int i, const Point &point_in)
Real	shape (const Elem *elem, const Order order, const unsigned int i, const Point &p)
Real	shape_deriv (const ElemType, const Order, const unsigned int, const unsigned int, const Point &)
Real	shape_deriv (const Elem *, const Order, const unsigned int, const unsigned int, const Point &)
Real	shape_deriv (const ElemType, const Order order, const unsigned int i, const unsigned int libmesh_dbg_var(j), const Point &p)
Real	shape_deriv (const Elem *elem, const Order order, const unsigned int i, const unsigned int j, const Point &p)
Real	shape_deriv (const ElemType, const Order, const unsigned int, const unsigned int, const Point &)
Real	shape_deriv (const Elem *elem, const Order order, const unsigned int i, const unsigned int j, const Point &p)
Real	shape_deriv (const ElemType, const Order, const unsigned int, const unsigned int, const Point &)
Real	shape_deriv (const Elem *elem, const Order order, const unsigned int i, const unsigned int j, const Point &p)
Real	shape_deriv (const ElemType, const Order, const unsigned int, const unsigned int, const Point &)
Real	shape_deriv (const Elem *, const Order, const unsigned int, const unsigned int, const Point &)
Real	shape_deriv (const ElemType, const Order, const unsigned int, const unsigned int, const Point &)
Real	shape_deriv (const Elem *elem, const Order order, const unsigned int i, const unsigned int j, const Point &p)
Real	shape_deriv (const ElemType, const Order, const unsigned int, const unsigned int, const Point &)
Real	shape_deriv (const Elem *elem, const Order order, const unsigned int i, const unsigned int j, const Point &p)
Real	shape_deriv (const ElemType, const Order, const unsigned int, const unsigned int, const Point &)
Real	shape_deriv (const Elem *libmesh_dbg_var(elem), const Order, const unsigned int, const unsigned int, const Point &)
Real	shape_deriv (const ElemType, const Order, const unsigned int, const unsigned int, const Point &)
Real	shape_deriv (const Elem *, const Order, const unsigned int, const unsigned int, const Point &)
Real	shape_deriv (const ElemType, const Order, const unsigned int, const unsigned int, const Point &)
Real	shape_deriv (const Elem *elem, const Order libmesh_dbg_var(order), const unsigned int i, const unsigned int, const Point &p)
Real	shape_deriv (const ElemType, const Order, const unsigned int, const unsigned int, const Point &)
Real	shape_deriv (const Elem *elem, const Order order, const unsigned int i, const unsigned int j, const Point &p)
Real	shape_deriv (const ElemType, const Order, const unsigned int, const unsigned int, const Point &)
Real	shape_deriv (const Elem *elem, const Order order, const unsigned int i, const unsigned int j, const Point &p)
Real	shape_deriv (const ElemType, const Order, const unsigned int, const unsigned int, const Point &)
Real	shape_deriv (const Elem *, const Order, const unsigned int, const unsigned int, const Point &)
Real	shape_deriv (const ElemType, const Order libmesh_dbg_var(order), const unsigned int i, const unsigned int libmesh_dbg_var(j), const Point &p)
Real	shape_deriv (const Elem *elem, const Order order, const unsigned int i, const unsigned int j, const Point &p)
Real	shape_deriv (const ElemType, const Order, const unsigned int, const unsigned int, const Point &)
Real	shape_deriv (const Elem *elem, const Order order, const unsigned int i, const unsigned int j, const Point &p)
Real	shape_deriv (const ElemType, const Order, const unsigned int, const unsigned int, const Point &)
Real	shape_deriv (const Elem *elem, const Order order, const unsigned int i, const unsigned int j, const Point &p)
Real	shape_deriv (const ElemType, const Order, const unsigned int, const unsigned int, const Point &)
Real	shape_deriv (const Elem *, const Order, const unsigned int, const unsigned int, const Point &)
Real	shape_deriv (const ElemType, const Order libmesh_dbg_var(order), const unsigned int i, const unsigned int libmesh_dbg_var(j), const Point &p)
Real	shape_deriv (const Elem *elem, const Order order, const unsigned int i, const unsigned int j, const Point &p)
Real	shape_deriv (const ElemType, const Order, const unsigned int, const unsigned int, const Point &)
Real	shape_deriv (const Elem *elem, const Order order, const unsigned int i, const unsigned int j, const Point &p)
Real	shape_deriv (const ElemType, const Order, const unsigned int, const unsigned int, const Point &)
Real	shape_deriv (const Elem *elem, const Order order, const unsigned int i, const unsigned int j, const Point &p)
Real	shape_deriv (const ElemType, const Order, const unsigned int, const unsigned int, const Point &)
Real	shape_deriv (const Elem *, const Order, const unsigned int, const unsigned int, const Point &)
Real	shape_deriv (const ElemType, const Order order, const unsigned int i, const unsigned int libmesh_dbg_var(j), const Point &p)
Real	shape_deriv (const Elem *elem, const Order order, const unsigned int i, const unsigned int j, const Point &p)
Real	shape_deriv (const ElemType type, const Order order, const unsigned int i, const unsigned int j, const Point &p)
Real	shape_deriv (const Elem *elem, const Order order, const unsigned int i, const unsigned int j, const Point &p)
Real	shape_deriv (const ElemType type, const Order order, const unsigned int i, const unsigned int j, const Point &p)
Real	shape_deriv (const Elem *elem, const Order order, const unsigned int i, const unsigned int j, const Point &p)
Real	shape_deriv (const ElemType, const Order, const unsigned int, const unsigned int, const Point &)
Real	shape_deriv (const Elem *, const Order, const unsigned int, const unsigned int, const Point &)
Real	shape_deriv (const ElemType, const Order order, const unsigned int i, const unsigned int libmesh_dbg_var(j), const Point &p)
Real	shape_deriv (const Elem *elem, const Order order, const unsigned int i, const unsigned int j, const Point &p)
Real	shape_deriv (const ElemType type, const Order order, const unsigned int i, const unsigned int j, const Point &p)
Real	shape_deriv (const Elem *elem, const Order order, const unsigned int i, const unsigned int j, const Point &p)
Real	shape_deriv (const ElemType type, const Order order, const unsigned int i, const unsigned int j, const Point &p)
Real	shape_deriv (const Elem *elem, const Order order, const unsigned int i, const unsigned int j, const Point &p)
RealGradient	shape_deriv (const ElemType type, const Order order, const unsigned int i, const unsigned int j, const Point &p)
RealGradient	shape_deriv (const ElemType type, const Order order, const unsigned int i, const unsigned int j, const Point &p)
RealGradient	shape_deriv (const ElemType type, const Order order, const unsigned int i, const unsigned int j, const Point &p)
RealGradient	shape_deriv (const ElemType type, const Order order, const unsigned int i, const unsigned int j, const Point &p)
RealGradient	shape_deriv (const Elem *elem, const Order order, const unsigned int i, const unsigned int j, const Point &p)
RealGradient	shape_deriv (const Elem *elem, const Order order, const unsigned int i, const unsigned int j, const Point &p)
RealGradient	shape_deriv (const Elem *elem, const Order order, const unsigned int i, const unsigned int j, const Point &p)
RealGradient	shape_deriv (const Elem *elem, const Order order, const unsigned int i, const unsigned int j, const Point &p)
Real	shape_deriv (const ElemType, const Order, const unsigned int, const unsigned int, const Point &)
Real	shape_deriv (const Elem *, const Order, const unsigned int, const unsigned int, const Point &)
Real	shape_deriv (const ElemType, const Order libmesh_dbg_var(order), const unsigned int i, const unsigned int libmesh_dbg_var(j), const Point &p)
Real	shape_deriv (const Elem *elem, const Order order, const unsigned int i, const unsigned int j, const Point &p)
Real	shape_deriv (const ElemType, const Order libmesh_dbg_var(order), const unsigned int i, const unsigned int j, const Point &p)
Real	shape_deriv (const Elem *elem, const Order order, const unsigned int i, const unsigned int j, const Point &p)
Real	shape_deriv (const ElemType, const Order libmesh_dbg_var(order), const unsigned int i, const unsigned int j, const Point &p)
Real	shape_deriv (const Elem *elem, const Order order, const unsigned int i, const unsigned int j, const Point &p)
RealGradient	shape_deriv (const ElemType, const Order, const unsigned int, const unsigned int, const Point &)
RealGradient	shape_deriv (const Elem *, const Order, const unsigned int, const unsigned int, const Point &)
RealGradient	shape_deriv (const ElemType, const Order, const unsigned int, const unsigned int, const Point &)
RealGradient	shape_deriv (const Elem *, const Order, const unsigned int, const unsigned int, const Point &)
RealGradient	shape_deriv (const ElemType, const Order, const unsigned int, const unsigned int, const Point &)
RealGradient	shape_deriv (const Elem *elem, const Order order, const unsigned int i, const unsigned int j, const Point &)
RealGradient	shape_deriv (const ElemType, const Order, const unsigned int, const unsigned int, const Point &)
RealGradient	shape_deriv (const Elem *elem, const Order order, const unsigned int i, const unsigned int j, const Point &p)
Real	shape_deriv (const ElemType, const Order, const unsigned int, const unsigned int, const Point &)
Real	shape_deriv (const Elem *, const Order, const unsigned int, const unsigned int, const Point &)
Real	shape_deriv (const ElemType, const Order, const unsigned int, const unsigned int, const Point &)
Real	shape_deriv (const Elem *, const Order, const unsigned int, const unsigned int, const Point &)
Real	shape_deriv (const ElemType, const Order, const unsigned int, const unsigned int, const Point &)
Real	shape_deriv (const Elem *, const Order, const unsigned int, const unsigned int, const Point &)
Real	shape_deriv (const ElemType, const Order, const unsigned int, const unsigned int, const Point &)
Real	shape_deriv (const Elem *, const Order, const unsigned int, const unsigned int, const Point &)
Real	shape_deriv (const ElemType type, const Order order, const unsigned int i, const unsigned int j, const Point &p)
Real	shape_deriv (const Elem *elem, const Order order, const unsigned int i, const unsigned int j, const Point &p)
Real	shape_deriv (const ElemType, const Order, const unsigned int, const unsigned int, const Point &)
Real	shape_deriv (const Elem *, const Order, const unsigned int, const unsigned int, const Point &)
Real	shape_deriv (const ElemType, const Order libmesh_dbg_var(order), const unsigned int i, const unsigned int libmesh_dbg_var(j), const Point &p)
Real	shape_deriv (const Elem *elem, const Order order, const unsigned int i, const unsigned int j, const Point &p)
Real	shape_deriv (const ElemType, const Order, const unsigned int, const unsigned int, const Point &)
Real	shape_deriv (const Elem *elem, const Order order, const unsigned int i, const unsigned int j, const Point &p)
Real	shape_deriv (const ElemType, const Order, const unsigned int, const unsigned int, const Point &)
Real	shape_deriv (const Elem *, const Order, const unsigned int, const unsigned int, const Point &)
Real	shape_deriv (const ElemType, const Order, const unsigned int, const unsigned int, const Point &)
Real	shape_deriv (const Elem *, const Order, const unsigned int, const unsigned int, const Point &)
Real	shape_deriv (const ElemType, const Order, const unsigned int, const unsigned int, const Point &)
Real	shape_deriv (const Elem *elem, const Order libmesh_dbg_var(order), const unsigned int i, const unsigned int libmesh_dbg_var(j), const Point &point_in)
Real	shape_deriv (const ElemType, const Order, const unsigned int, const unsigned int, const Point &)
Real	shape_deriv (const Elem *elem, const Order libmesh_dbg_var(order), const unsigned int i, const unsigned int j, const Point &point_in)
Real	shape_deriv (const ElemType, const Order, const unsigned int, const unsigned int, const Point &)
Real	shape_deriv (const Elem *elem, const Order libmesh_dbg_var(order), const unsigned int i, const unsigned int j, const Point &point_in)
Real	shape_deriv (const Elem *elem, const Order order, const unsigned int i, const unsigned int j, const Point &p)
Real	shape_second_deriv (const ElemType, const Order, const unsigned int, const unsigned int, const Point &)
Real	shape_second_deriv (const Elem *, const Order, const unsigned int, const unsigned int, const Point &)
Real	shape_second_deriv (const ElemType, const Order, const unsigned int, const unsigned int, const Point &)
Real	shape_second_deriv (const Elem *, const Order, const unsigned int, const unsigned int, const Point &)
Real	shape_second_deriv (const ElemType, const Order, const unsigned int, const unsigned int, const Point &)
Real	shape_second_deriv (const Elem *, const Order, const unsigned int, const unsigned int, const Point &)
Real	shape_second_deriv (const ElemType, const Order, const unsigned int, const unsigned int, const Point &)
Real	shape_second_deriv (const Elem *, const Order, const unsigned int, const unsigned int, const Point &)
Real	shape_second_deriv (const ElemType, const Order, const unsigned int, const unsigned int, const Point &)
Real	shape_second_deriv (const Elem *, const Order, const unsigned int, const unsigned int, const Point &)
Real	shape_second_deriv (const Elem *elem, const Order order, const unsigned int i, const unsigned int j, const Point &p)
Real	shape_second_deriv (const Elem *elem, const Order order, const unsigned int i, const unsigned int j, const Point &p)
Real	shape_second_deriv (const Elem *libmesh_dbg_var(elem), const Order, const unsigned int, const unsigned int, const Point &)
Real	shape_second_deriv (const ElemType, const Order, const unsigned int, const unsigned int, const Point &)
Real	shape_second_deriv (const Elem *, const Order, const unsigned int, const unsigned int, const Point &)
Real	shape_second_deriv (const Elem *elem, const Order libmesh_dbg_var(order), const unsigned int i, const unsigned int, const Point &p)
Real	shape_second_deriv (const Elem *elem, const Order order, const unsigned int i, const unsigned int j, const Point &p)
Real	shape_second_deriv (const Elem *elem, const Order order, const unsigned int i, const unsigned int j, const Point &p)
Real	shape_second_deriv (const ElemType, const Order, const unsigned int, const unsigned int, const Point &)
Real	shape_second_deriv (const Elem *, const Order, const unsigned int, const unsigned int, const Point &)
Real	shape_second_deriv (const ElemType, const Order libmesh_dbg_var(order), const unsigned int i, const unsigned int libmesh_dbg_var(j), const Point &p)
Real	shape_second_deriv (const Elem *elem, const Order order, const unsigned int i, const unsigned int j, const Point &p)
Real	shape_second_deriv (const ElemType, const Order, const unsigned int, const unsigned int, const Point &)
Real	shape_second_deriv (const Elem *elem, const Order order, const unsigned int i, const unsigned int j, const Point &p)
Real	shape_second_deriv (const ElemType, const Order, const unsigned int, const unsigned int, const Point &)
Real	shape_second_deriv (const Elem *elem, const Order order, const unsigned int i, const unsigned int j, const Point &p)
Real	shape_second_deriv (const ElemType, const Order, const unsigned int, const unsigned int, const Point &)
Real	shape_second_deriv (const Elem *, const Order, const unsigned int, const unsigned int, const Point &)
Real	shape_second_deriv (const ElemType, const Order libmesh_dbg_var(order), const unsigned int i, const unsigned int libmesh_dbg_var(j), const Point &p)
Real	shape_second_deriv (const Elem *elem, const Order order, const unsigned int i, const unsigned int j, const Point &p)
Real	shape_second_deriv (const ElemType, const Order, const unsigned int, const unsigned int, const Point &)
Real	shape_second_deriv (const Elem *elem, const Order order, const unsigned int i, const unsigned int j, const Point &p)
Real	shape_second_deriv (const ElemType, const Order, const unsigned int, const unsigned int, const Point &)
Real	shape_second_deriv (const Elem *elem, const Order order, const unsigned int i, const unsigned int j, const Point &p)
Real	shape_second_deriv (const ElemType, const Order, const unsigned int, const unsigned int, const Point &)
Real	shape_second_deriv (const Elem *, const Order, const unsigned int, const unsigned int, const Point &)
Real	shape_second_deriv (const ElemType, const Order order, const unsigned int i, const unsigned int libmesh_dbg_var(j), const Point &p)
Real	shape_second_deriv (const Elem *elem, const Order order, const unsigned int i, const unsigned int j, const Point &p)
Real	shape_second_deriv (const ElemType type, const Order order, const unsigned int i, const unsigned int j, const Point &p)
Real	shape_second_deriv (const Elem *elem, const Order order, const unsigned int i, const unsigned int j, const Point &p)
Real	shape_second_deriv (const ElemType type, const Order order, const unsigned int i, const unsigned int j, const Point &p)
Real	shape_second_deriv (const Elem *elem, const Order order, const unsigned int i, const unsigned int j, const Point &p)
Real	shape_second_deriv (const ElemType, const Order, const unsigned int, const unsigned int, const Point &)
Real	shape_second_deriv (const Elem *, const Order, const unsigned int, const unsigned int, const Point &)
Real	shape_second_deriv (const ElemType, const Order order, const unsigned int i, const unsigned int libmesh_dbg_var(j), const Point &p)
Real	shape_second_deriv (const Elem *elem, const Order order, const unsigned int i, const unsigned int j, const Point &p)
Real	shape_second_deriv (const ElemType type, const Order order, const unsigned int i, const unsigned int j, const Point &p)
Real	shape_second_deriv (const Elem *elem, const Order order, const unsigned int i, const unsigned int j, const Point &p)
Real	shape_second_deriv (const ElemType type, const Order order, const unsigned int i, const unsigned int j, const Point &p)
Real	shape_second_deriv (const Elem *elem, const Order order, const unsigned int i, const unsigned int j, const Point &p)
RealGradient	shape_second_deriv (const ElemType type, const Order order, const unsigned int i, const unsigned int j, const Point &p)
RealGradient	shape_second_deriv (const ElemType type, const Order order, const unsigned int i, const unsigned int j, const Point &p)
RealGradient	shape_second_deriv (const ElemType type, const Order order, const unsigned int i, const unsigned int j, const Point &p)
RealGradient	shape_second_deriv (const ElemType type, const Order order, const unsigned int i, const unsigned int j, const Point &p)
RealGradient	shape_second_deriv (const Elem *elem, const Order order, const unsigned int i, const unsigned int j, const Point &p)
RealGradient	shape_second_deriv (const Elem *elem, const Order order, const unsigned int i, const unsigned int j, const Point &p)
RealGradient	shape_second_deriv (const Elem *elem, const Order order, const unsigned int i, const unsigned int j, const Point &p)
RealGradient	shape_second_deriv (const Elem *elem, const Order order, const unsigned int i, const unsigned int j, const Point &p)
Real	shape_second_deriv (const ElemType, const Order, const unsigned int, const unsigned int, const Point &)
Real	shape_second_deriv (const Elem *, const Order, const unsigned int, const unsigned int, const Point &)
Real	shape_second_deriv (const ElemType, const Order libmesh_dbg_var(order), const unsigned int i, const unsigned int libmesh_dbg_var(j), const Point &p)
Real	shape_second_deriv (const Elem *elem, const Order order, const unsigned int i, const unsigned int j, const Point &p)
Real	shape_second_deriv (const ElemType, const Order libmesh_dbg_var(order), const unsigned int i, const unsigned int j, const Point &p)
Real	shape_second_deriv (const Elem *elem, const Order order, const unsigned int i, const unsigned int j, const Point &p)
Real	shape_second_deriv (const ElemType, const Order libmesh_dbg_var(order), const unsigned int i, const unsigned int j, const Point &p)
Real	shape_second_deriv (const Elem *elem, const Order order, const unsigned int i, const unsigned int j, const Point &p)
RealGradient	shape_second_deriv (const ElemType, const Order, const unsigned int, const unsigned int, const Point &)
RealGradient	shape_second_deriv (const Elem *, const Order, const unsigned int, const unsigned int, const Point &)
RealGradient	shape_second_deriv (const ElemType, const Order, const unsigned int, const unsigned int, const Point &)
RealGradient	shape_second_deriv (const Elem *, const Order, const unsigned int, const unsigned int, const Point &)
RealGradient	shape_second_deriv (const ElemType, const Order, const unsigned int, const unsigned int, const Point &)
RealGradient	shape_second_deriv (const Elem *elem, const Order order, const unsigned int libmesh_dbg_var(i), const unsigned int libmesh_dbg_var(j), const Point &)
RealGradient	shape_second_deriv (const ElemType, const Order, const unsigned int, const unsigned int, const Point &)
RealGradient	shape_second_deriv (const Elem *elem, const Order order, const unsigned int i, const unsigned int j, const Point &libmesh_dbg_var(p))
Real	shape_second_deriv (const ElemType, const Order, const unsigned int, const unsigned int, const Point &)
Real	shape_second_deriv (const Elem *, const Order, const unsigned int, const unsigned int, const Point &)
Real	shape_second_deriv (const ElemType, const Order, const unsigned int, const unsigned int, const Point &)
Real	shape_second_deriv (const Elem *, const Order, const unsigned int, const unsigned int, const Point &)
Real	shape_second_deriv (const ElemType, const Order, const unsigned int, const unsigned int, const Point &)
Real	shape_second_deriv (const Elem *, const Order, const unsigned int, const unsigned int, const Point &)
Real	shape_second_deriv (const ElemType, const Order, const unsigned int, const unsigned int, const Point &)
Real	shape_second_deriv (const Elem *, const Order, const unsigned int, const unsigned int, const Point &)
Real	shape_second_deriv (const ElemType type, const Order order, const unsigned int i, const unsigned int j, const Point &p)
Real	shape_second_deriv (const Elem *elem, const Order order, const unsigned int i, const unsigned int j, const Point &p)
Real	shape_second_deriv (const ElemType, const Order, const unsigned int, const unsigned int, const Point &)
Real	shape_second_deriv (const Elem *, const Order, const unsigned int, const unsigned int, const Point &)
Real	shape_second_deriv (const ElemType, const Order, const unsigned int, const unsigned int, const Point &)
Real	shape_second_deriv (const Elem *, const Order, const unsigned int, const unsigned int, const Point &)
Real	shape_second_deriv (const ElemType, const Order, const unsigned int, const unsigned int, const Point &)
Real	shape_second_deriv (const Elem *, const Order, const unsigned int, const unsigned int, const Point &)
Real	shape_second_deriv (const ElemType, const Order, const unsigned int, const unsigned int, const Point &)
Real	shape_second_deriv (const Elem *, const Order, const unsigned int, const unsigned int, const Point &)
Real	shape_second_deriv (const ElemType, const Order, const unsigned int, const unsigned int, const Point &)
Real	shape_second_deriv (const Elem *, const Order, const unsigned int, const unsigned int, const Point &)
Real	shape_second_deriv (const ElemType, const Order, const unsigned int, const unsigned int, const Point &)
Real	shape_second_deriv (const Elem *elem, const Order libmesh_dbg_var(order), const unsigned int i, const unsigned int libmesh_dbg_var(j), const Point &point_in)
Real	shape_second_deriv (const ElemType, const Order, const unsigned int, const unsigned int, const Point &)
Real	shape_second_deriv (const Elem *elem, const Order libmesh_dbg_var(order), const unsigned int i, const unsigned int j, const Point &point_in)
Real	shape_second_deriv (const ElemType, const Order, const unsigned int, const unsigned int, const Point &)
Real	shape_second_deriv (const Elem *elem, const Order libmesh_dbg_var(order), const unsigned int i, const unsigned int j, const Point &point_in)
Real	shape_second_deriv (const Elem *elem, const Order order, const unsigned int i, const unsigned int j, const Point &p)
void	nodal_soln (const Elem *elem, const Order order, const std::vector< Number > &elem_soln, std::vector< Number > &nodal_soln)
void	nodal_soln (const Elem *elem, const Order order, const std::vector< Number > &elem_soln, std::vector< Number > &nodal_soln)
void	nodal_soln (const Elem *elem, const Order order, const std::vector< Number > &elem_soln, std::vector< Number > &nodal_soln)
void	nodal_soln (const Elem *elem, const Order order, const std::vector< Number > &elem_soln, std::vector< Number > &nodal_soln)
void	nodal_soln (const Elem *elem, const Order order, const std::vector< Number > &elem_soln, std::vector< Number > &nodal_soln)
void	nodal_soln (const Elem *elem, const Order order, const std::vector< Number > &elem_soln, std::vector< Number > &nodal_soln)
void	nodal_soln (const Elem *elem, const Order order, const std::vector< Number > &elem_soln, std::vector< Number > &nodal_soln)
void	nodal_soln (const Elem *elem, const Order order, const std::vector< Number > &elem_soln, std::vector< Number > &nodal_soln)
void	nodal_soln (const Elem *elem, const Order order, const std::vector< Number > &elem_soln, std::vector< Number > &nodal_soln)
void	nodal_soln (const Elem *elem, const Order order, const std::vector< Number > &elem_soln, std::vector< Number > &nodal_soln)
void	nodal_soln (const Elem *elem, const Order order, const std::vector< Number > &elem_soln, std::vector< Number > &nodal_soln)
void	nodal_soln (const Elem *elem, const Order order, const std::vector< Number > &elem_soln, std::vector< Number > &nodal_soln)
void	nodal_soln (const Elem *elem, const Order order, const std::vector< Number > &elem_soln, std::vector< Number > &nodal_soln)
void	nodal_soln (const Elem *elem, const Order order, const std::vector< Number > &elem_soln, std::vector< Number > &nodal_soln)
void	nodal_soln (const Elem *elem, const Order order, const std::vector< Number > &elem_soln, std::vector< Number > &nodal_soln)
void	nodal_soln (const Elem *elem, const Order order, const std::vector< Number > &elem_soln, std::vector< Number > &nodal_soln)
void	nodal_soln (const Elem *elem, const Order order, const std::vector< Number > &elem_soln, std::vector< Number > &nodal_soln)
void	nodal_soln (const Elem *elem, const Order order, const std::vector< Number > &elem_soln, std::vector< Number > &nodal_soln)
void	nodal_soln (const Elem *elem, const Order order, const std::vector< Number > &elem_soln, std::vector< Number > &nodal_soln)
void	nodal_soln (const Elem *elem, const Order order, const std::vector< Number > &elem_soln, std::vector< Number > &nodal_soln)
void	nodal_soln (const Elem *elem, const Order order, const std::vector< Number > &elem_soln, std::vector< Number > &nodal_soln)
void	nodal_soln (const Elem *elem, const Order order, const std::vector< Number > &elem_soln, std::vector< Number > &nodal_soln)
void	nodal_soln (const Elem *elem, const Order order, const std::vector< Number > &elem_soln, std::vector< Number > &nodal_soln)
void	nodal_soln (const Elem *elem, const Order order, const std::vector< Number > &elem_soln, std::vector< Number > &nodal_soln)
void	nodal_soln (const Elem *elem, const Order order, const std::vector< Number > &elem_soln, std::vector< Number > &nodal_soln)
void	nodal_soln (const Elem *elem, const Order order, const std::vector< Number > &elem_soln, std::vector< Number > &nodal_soln)
void	nodal_soln (const Elem *elem, const Order order, const std::vector< Number > &elem_soln, std::vector< Number > &nodal_soln)
void	nodal_soln (const Elem *elem, const Order order, const std::vector< Number > &elem_soln, std::vector< Number > &nodal_soln)
void	nodal_soln (const Elem *elem, const Order order, const std::vector< Number > &elem_soln, std::vector< Number > &nodal_soln)
void	nodal_soln (const Elem *elem, const Order order, const std::vector< Number > &elem_soln, std::vector< Number > &nodal_soln)
void	nodal_soln (const Elem *elem, const Order order, const std::vector< Number > &elem_soln, std::vector< Number > &nodal_soln)
void	nodal_soln (const Elem *elem, const Order order, const std::vector< Number > &elem_soln, std::vector< Number > &nodal_soln)
void	nodal_soln (const Elem *elem, const Order order, const std::vector< Number > &elem_soln, std::vector< Number > &nodal_soln)
void	nodal_soln (const Elem *elem, const Order order, const std::vector< Number > &elem_soln, std::vector< Number > &nodal_soln)
void	nodal_soln (const Elem *elem, const Order order, const std::vector< Number > &elem_soln, std::vector< Number > &nodal_soln)
void	nodal_soln (const Elem *elem, const Order order, const std::vector< Number > &elem_soln, std::vector< Number > &nodal_soln)
void	nodal_soln (const Elem *, const Order, const std::vector< Number > &, std::vector< Number > &)
void	nodal_soln (const Elem *, const Order, const std::vector< Number > &, std::vector< Number > &)
void	nodal_soln (const Elem *elem, const Order order, const std::vector< Number > &elem_soln, std::vector< Number > &nodal_soln)
void	nodal_soln (const Elem *elem, const Order order, const std::vector< Number > &elem_soln, std::vector< Number > &nodal_soln)
void	nodal_soln (const Elem *elem, const Order order, const std::vector< Number > &elem_soln, std::vector< Number > &nodal_soln)
void	nodal_soln (const Elem *elem, const Order order, const std::vector< Number > &elem_soln, std::vector< Number > &nodal_soln)
void	nodal_soln (const Elem *elem, const Order order, const std::vector< Number > &elem_soln, std::vector< Number > &nodal_soln)
void	nodal_soln (const Elem *elem, const Order order, const std::vector< Number > &elem_soln, std::vector< Number > &nodal_soln)
void	nodal_soln (const Elem *elem, const Order, const std::vector< Number > &elem_soln, std::vector< Number > &nodal_soln)
void	nodal_soln (const Elem *elem, const Order order, const std::vector< Number > &elem_soln, std::vector< Number > &nodal_soln)
void	nodal_soln (const Elem *elem, const Order order, const std::vector< Number > &elem_soln, std::vector< Number > &nodal_soln)
void	nodal_soln (const Elem *elem, const Order order, const std::vector< Number > &elem_soln, std::vector< Number > &nodal_soln)
void	nodal_soln (const Elem *elem, const Order order, const std::vector< Number > &elem_soln, std::vector< Number > &nodal_soln)
void	nodal_soln (const Elem *elem, const Order order, const std::vector< Number > &elem_soln, std::vector< Number > &nodal_soln)
void	nodal_soln (const Elem *elem, const Order order, const std::vector< Number > &elem_soln, std::vector< Number > &nodal_soln)
void	nodal_soln (const Elem *elem, const Order order, const std::vector< Number > &elem_soln, std::vector< Number > &nodal_soln)
void	nodal_soln (const Elem *elem, const Order order, const std::vector< Number > &elem_soln, std::vector< Number > &nodal_soln)
virtual unsigned int	n_shape_functions () const override
unsigned int	n_dofs (const ElemType t, const Order o)
unsigned int	n_dofs (const ElemType t, const Order o)
unsigned int	n_dofs (const ElemType t, const Order o)
unsigned int	n_dofs (const ElemType t, const Order o)
unsigned int	n_dofs (const ElemType t, const Order o)
unsigned int	n_dofs (const ElemType t, const Order o)
unsigned int	n_dofs (const ElemType t, const Order o)
unsigned int	n_dofs (const ElemType t, const Order o)
unsigned int	n_dofs (const ElemType t, const Order o)
unsigned int	n_dofs (const ElemType t, const Order o)
unsigned int	n_dofs (const ElemType t, const Order o)
unsigned int	n_dofs (const ElemType t, const Order o)
unsigned int	n_dofs (const ElemType t, const Order o)
unsigned int	n_dofs (const ElemType t, const Order o)
unsigned int	n_dofs (const ElemType t, const Order o)
unsigned int	n_dofs (const ElemType t, const Order o)
unsigned int	n_dofs (const ElemType t, const Order o)
unsigned int	n_dofs (const ElemType t, const Order o)
unsigned int	n_dofs (const ElemType t, const Order o)
unsigned int	n_dofs (const ElemType t, const Order o)
unsigned int	n_dofs (const ElemType t, const Order o)
unsigned int	n_dofs (const ElemType t, const Order o)
unsigned int	n_dofs (const ElemType t, const Order o)
unsigned int	n_dofs (const ElemType t, const Order o)
unsigned int	n_dofs (const ElemType t, const Order o)
unsigned int	n_dofs (const ElemType t, const Order o)
unsigned int	n_dofs (const ElemType t, const Order o)
unsigned int	n_dofs (const ElemType t, const Order o)
unsigned int	n_dofs (const ElemType t, const Order o)
unsigned int	n_dofs (const ElemType t, const Order o)
unsigned int	n_dofs (const ElemType t, const Order o)
unsigned int	n_dofs (const ElemType t, const Order o)
unsigned int	n_dofs (const ElemType t, const Order o)
unsigned int	n_dofs (const ElemType t, const Order o)
unsigned int	n_dofs (const ElemType t, const Order o)
unsigned int	n_dofs (const ElemType t, const Order o)
unsigned int	n_dofs (const ElemType, const Order)
unsigned int	n_dofs (const ElemType, const Order)
unsigned int	n_dofs (const ElemType t, const Order o)
unsigned int	n_dofs (const ElemType t, const Order o)
unsigned int	n_dofs (const ElemType, const Order o)
unsigned int	n_dofs (const ElemType, const Order o)
unsigned int	n_dofs (const ElemType, const Order o)
unsigned int	n_dofs (const ElemType, const Order o)
unsigned int	n_dofs (const ElemType, const Order)
unsigned int	n_dofs (const ElemType t, const Order o)
unsigned int	n_dofs (const ElemType t, const Order o)
unsigned int	n_dofs (const ElemType t, const Order o)
unsigned int	n_dofs (const ElemType t, const Order o)
unsigned int	n_dofs (const ElemType t, const Order o)
unsigned int	n_dofs (const ElemType t, const Order o)
unsigned int	n_dofs (const ElemType t, const Order o)
unsigned int	n_dofs (const ElemType t, const Order o)
unsigned int	n_dofs_at_node (const ElemType t, const Order o, const unsigned int n)
unsigned int	n_dofs_at_node (const ElemType t, const Order o, const unsigned int n)
unsigned int	n_dofs_at_node (const ElemType t, const Order o, const unsigned int n)
unsigned int	n_dofs_at_node (const ElemType t, const Order o, const unsigned int n)
unsigned int	n_dofs_at_node (const ElemType t, const Order o, const unsigned int n)
unsigned int	n_dofs_at_node (const ElemType t, const Order o, const unsigned int n)
unsigned int	n_dofs_at_node (const ElemType t, const Order o, const unsigned int n)
unsigned int	n_dofs_at_node (const ElemType t, const Order o, const unsigned int n)
unsigned int	n_dofs_at_node (const ElemType t, const Order o, const unsigned int n)
unsigned int	n_dofs_at_node (const ElemType t, const Order o, const unsigned int n)
unsigned int	n_dofs_at_node (const ElemType t, const Order o, const unsigned int n)
unsigned int	n_dofs_at_node (const ElemType t, const Order o, const unsigned int n)
unsigned int	n_dofs_at_node (const ElemType t, const Order o, const unsigned int n)
unsigned int	n_dofs_at_node (const ElemType t, const Order o, const unsigned int n)
unsigned int	n_dofs_at_node (const ElemType t, const Order o, const unsigned int n)
unsigned int	n_dofs_at_node (const ElemType t, const Order o, const unsigned int n)
unsigned int	n_dofs_at_node (const ElemType, const Order, const unsigned int)
unsigned int	n_dofs_at_node (const ElemType, const Order, const unsigned int)
unsigned int	n_dofs_at_node (const ElemType, const Order, const unsigned int)
unsigned int	n_dofs_at_node (const ElemType, const Order, const unsigned int)
unsigned int	n_dofs_at_node (const ElemType, const Order, const unsigned int)
unsigned int	n_dofs_at_node (const ElemType, const Order, const unsigned int)
unsigned int	n_dofs_at_node (const ElemType, const Order, const unsigned int)
unsigned int	n_dofs_at_node (const ElemType, const Order, const unsigned int)
unsigned int	n_dofs_at_node (const ElemType t, const Order o, const unsigned int n)
unsigned int	n_dofs_at_node (const ElemType t, const Order o, const unsigned int n)
unsigned int	n_dofs_at_node (const ElemType t, const Order o, const unsigned int n)
unsigned int	n_dofs_at_node (const ElemType t, const Order o, const unsigned int n)
unsigned int	n_dofs_at_node (const ElemType t, const Order o, const unsigned int n)
unsigned int	n_dofs_at_node (const ElemType t, const Order o, const unsigned int n)
unsigned int	n_dofs_at_node (const ElemType t, const Order o, const unsigned int n)
unsigned int	n_dofs_at_node (const ElemType t, const Order o, const unsigned int n)
unsigned int	n_dofs_at_node (const ElemType, const Order, const unsigned int)
unsigned int	n_dofs_at_node (const ElemType, const Order, const unsigned int)
unsigned int	n_dofs_at_node (const ElemType, const Order, const unsigned int)
unsigned int	n_dofs_at_node (const ElemType, const Order, const unsigned int)
unsigned int	n_dofs_at_node (const ElemType, const Order, const unsigned int)
unsigned int	n_dofs_at_node (const ElemType, const Order, const unsigned int)
unsigned int	n_dofs_at_node (const ElemType t, const Order o, const unsigned int n)
unsigned int	n_dofs_at_node (const ElemType t, const Order o, const unsigned int n)
unsigned int	n_dofs_at_node (const ElemType, const Order, const unsigned int)
unsigned int	n_dofs_at_node (const ElemType, const Order, const unsigned int)
unsigned int	n_dofs_at_node (const ElemType, const Order, const unsigned int)
unsigned int	n_dofs_at_node (const ElemType, const Order, const unsigned int)
unsigned int	n_dofs_at_node (const ElemType, const Order, const unsigned int)
unsigned int	n_dofs_at_node (const ElemType t, const Order o, const unsigned int n)
unsigned int	n_dofs_at_node (const ElemType t, const Order o, const unsigned int n)
unsigned int	n_dofs_at_node (const ElemType t, const Order o, const unsigned int n)
unsigned int	n_dofs_at_node (const ElemType t, const Order o, const unsigned int n)
unsigned int	n_dofs_at_node (const ElemType, const Order, const unsigned int)
unsigned int	n_dofs_at_node (const ElemType, const Order, const unsigned int)
unsigned int	n_dofs_at_node (const ElemType, const Order, const unsigned int)
unsigned int	n_dofs_at_node (const ElemType, const Order, const unsigned int)
unsigned int	n_dofs_per_elem (const ElemType t, const Order o)
unsigned int	n_dofs_per_elem (const ElemType t, const Order o)
unsigned int	n_dofs_per_elem (const ElemType t, const Order o)
unsigned int	n_dofs_per_elem (const ElemType t, const Order o)
unsigned int	n_dofs_per_elem (const ElemType t, const Order o)
unsigned int	n_dofs_per_elem (const ElemType t, const Order o)
unsigned int	n_dofs_per_elem (const ElemType t, const Order o)
unsigned int	n_dofs_per_elem (const ElemType t, const Order o)
unsigned int	n_dofs_per_elem (const ElemType t, const Order o)
unsigned int	n_dofs_per_elem (const ElemType t, const Order o)
unsigned int	n_dofs_per_elem (const ElemType t, const Order o)
unsigned int	n_dofs_per_elem (const ElemType t, const Order o)
unsigned int	n_dofs_per_elem (const ElemType t, const Order o)
unsigned int	n_dofs_per_elem (const ElemType t, const Order o)
unsigned int	n_dofs_per_elem (const ElemType t, const Order o)
unsigned int	n_dofs_per_elem (const ElemType t, const Order o)
unsigned int	n_dofs_per_elem (const ElemType t, const Order o)
unsigned int	n_dofs_per_elem (const ElemType t, const Order o)
unsigned int	n_dofs_per_elem (const ElemType t, const Order o)
unsigned int	n_dofs_per_elem (const ElemType t, const Order o)
unsigned int	n_dofs_per_elem (const ElemType t, const Order o)
unsigned int	n_dofs_per_elem (const ElemType t, const Order o)
unsigned int	n_dofs_per_elem (const ElemType t, const Order o)
unsigned int	n_dofs_per_elem (const ElemType t, const Order o)
unsigned int	n_dofs_per_elem (const ElemType, const Order)
unsigned int	n_dofs_per_elem (const ElemType, const Order)
unsigned int	n_dofs_per_elem (const ElemType, const Order)
unsigned int	n_dofs_per_elem (const ElemType, const Order)
unsigned int	n_dofs_per_elem (const ElemType, const Order)
unsigned int	n_dofs_per_elem (const ElemType, const Order)
unsigned int	n_dofs_per_elem (const ElemType, const Order)
unsigned int	n_dofs_per_elem (const ElemType, const Order)
unsigned int	n_dofs_per_elem (const ElemType t, const Order o)
unsigned int	n_dofs_per_elem (const ElemType t, const Order o)
unsigned int	n_dofs_per_elem (const ElemType t, const Order o)
unsigned int	n_dofs_per_elem (const ElemType t, const Order o)
unsigned int	n_dofs_per_elem (const ElemType, const Order)
unsigned int	n_dofs_per_elem (const ElemType, const Order)
unsigned int	n_dofs_per_elem (const ElemType, const Order)
unsigned int	n_dofs_per_elem (const ElemType, const Order)
unsigned int	n_dofs_per_elem (const ElemType, const Order)
unsigned int	n_dofs_per_elem (const ElemType, const Order)
unsigned int	n_dofs_per_elem (const ElemType, const Order)
unsigned int	n_dofs_per_elem (const ElemType, const Order)
unsigned int	n_dofs_per_elem (const ElemType, const Order)
unsigned int	n_dofs_per_elem (const ElemType t, const Order o)
unsigned int	n_dofs_per_elem (const ElemType t, const Order o)
unsigned int	n_dofs_per_elem (const ElemType t, const Order o)
unsigned int	n_dofs_per_elem (const ElemType t, const Order o)
unsigned int	n_dofs_per_elem (const ElemType t, const Order o)
unsigned int	n_dofs_per_elem (const ElemType t, const Order o)
unsigned int	n_dofs_per_elem (const ElemType t, const Order o)
unsigned int	n_dofs_per_elem (const ElemType t, const Order o)
virtual FEContinuity	get_continuity () const override
FEContinuity	get_continuity () const
FEContinuity	get_continuity () const
FEContinuity	get_continuity () const
FEContinuity	get_continuity () const
FEContinuity	get_continuity () const
FEContinuity	get_continuity () const
FEContinuity	get_continuity () const
FEContinuity	get_continuity () const
FEContinuity	get_continuity () const
FEContinuity	get_continuity () const
FEContinuity	get_continuity () const
FEContinuity	get_continuity () const
FEContinuity	get_continuity () const
FEContinuity	get_continuity () const
FEContinuity	get_continuity () const
FEContinuity	get_continuity () const
FEContinuity	get_continuity () const
FEContinuity	get_continuity () const
FEContinuity	get_continuity () const
FEContinuity	get_continuity () const
FEContinuity	get_continuity () const
FEContinuity	get_continuity () const
FEContinuity	get_continuity () const
FEContinuity	get_continuity () const
FEContinuity	get_continuity () const
FEContinuity	get_continuity () const
FEContinuity	get_continuity () const
FEContinuity	get_continuity () const
FEContinuity	get_continuity () const
FEContinuity	get_continuity () const
FEContinuity	get_continuity () const
FEContinuity	get_continuity () const
FEContinuity	get_continuity () const
FEContinuity	get_continuity () const
FEContinuity	get_continuity () const
FEContinuity	get_continuity () const
FEContinuity	get_continuity () const
FEContinuity	get_continuity () const
FEContinuity	get_continuity () const
FEContinuity	get_continuity () const
FEContinuity	get_continuity () const
FEContinuity	get_continuity () const
FEContinuity	get_continuity () const
FEContinuity	get_continuity () const
FEContinuity	get_continuity () const
FEContinuity	get_continuity () const
FEContinuity	get_continuity () const
FEContinuity	get_continuity () const
FEContinuity	get_continuity () const
FEContinuity	get_continuity () const
FEContinuity	get_continuity () const
FEContinuity	get_continuity () const
FEContinuity	get_continuity () const
virtual bool	is_hierarchic () const override
bool	is_hierarchic () const
bool	is_hierarchic () const
bool	is_hierarchic () const
bool	is_hierarchic () const
bool	is_hierarchic () const
bool	is_hierarchic () const
bool	is_hierarchic () const
bool	is_hierarchic () const
bool	is_hierarchic () const
bool	is_hierarchic () const
bool	is_hierarchic () const
bool	is_hierarchic () const
bool	is_hierarchic () const
bool	is_hierarchic () const
bool	is_hierarchic () const
bool	is_hierarchic () const
bool	is_hierarchic () const
bool	is_hierarchic () const
bool	is_hierarchic () const
bool	is_hierarchic () const
bool	is_hierarchic () const
bool	is_hierarchic () const
bool	is_hierarchic () const
bool	is_hierarchic () const
bool	is_hierarchic () const
bool	is_hierarchic () const
bool	is_hierarchic () const
bool	is_hierarchic () const
bool	is_hierarchic () const
bool	is_hierarchic () const
bool	is_hierarchic () const
bool	is_hierarchic () const
bool	is_hierarchic () const
bool	is_hierarchic () const
bool	is_hierarchic () const
bool	is_hierarchic () const
bool	is_hierarchic () const
bool	is_hierarchic () const
bool	is_hierarchic () const
bool	is_hierarchic () const
bool	is_hierarchic () const
bool	is_hierarchic () const
bool	is_hierarchic () const
bool	is_hierarchic () const
bool	is_hierarchic () const
bool	is_hierarchic () const
bool	is_hierarchic () const
bool	is_hierarchic () const
bool	is_hierarchic () const
bool	is_hierarchic () const
bool	is_hierarchic () const
bool	is_hierarchic () const
bool	is_hierarchic () const
void	dofs_on_side (const Elem *const, const Order, unsigned int, std::vector< unsigned int > &di)
void	dofs_on_edge (const Elem *const, const Order, unsigned int, std::vector< unsigned int > &di)
Point	inverse_map (const Elem *, const Point &, const Real, const bool)
void	inverse_map (const Elem *, const std::vector< Point > &, std::vector< Point > &, Real, bool)
virtual void	edge_reinit (const Elem *elem, const unsigned int edge, const Real tolerance=TOLERANCE, const std::vector< Point > *const pts=nullptr, const std::vector< Real > *const weights=nullptr) override
void	edge_reinit (Elem const *, unsigned int, Real, const std::vector< Point > *const, const std::vector< Real > *const)
virtual void	side_map (const Elem *elem, const Elem *side, const unsigned int s, const std::vector< Point > &reference_side_points, std::vector< Point > &reference_points) override
void	side_map (const Elem *, const Elem *, const unsigned int, const std::vector< Point > &, std::vector< Point > &)
virtual unsigned int	n_quadrature_points () const override
void	compute_constraints (DofConstraints &constraints, DofMap &dof_map, const unsigned int variable_number, const Elem *elem)
void	compute_constraints (DofConstraints &constraints, DofMap &dof_map, const unsigned int variable_number, const Elem *elem)
void	compute_constraints (DofConstraints &constraints, DofMap &dof_map, const unsigned int variable_number, const Elem *elem)
void	compute_constraints (DofConstraints &constraints, DofMap &dof_map, const unsigned int variable_number, const Elem *elem)
void	compute_constraints (DofConstraints &constraints, DofMap &dof_map, const unsigned int variable_number, const Elem *elem)
void	compute_constraints (DofConstraints &constraints, DofMap &dof_map, const unsigned int variable_number, const Elem *elem)
void	compute_constraints (DofConstraints &constraints, DofMap &dof_map, const unsigned int variable_number, const Elem *elem)
void	compute_constraints (DofConstraints &constraints, DofMap &dof_map, const unsigned int variable_number, const Elem *elem)
void	compute_constraints (DofConstraints &, DofMap &, const unsigned int, const Elem *)
void	compute_constraints (DofConstraints &, DofMap &, const unsigned int, const Elem *)
void	compute_constraints (DofConstraints &, DofMap &, const unsigned int, const Elem *)
void	compute_constraints (DofConstraints &, DofMap &, const unsigned int, const Elem *)
void	compute_constraints (DofConstraints &constraints, DofMap &dof_map, const unsigned int variable_number, const Elem *elem)
void	compute_constraints (DofConstraints &constraints, DofMap &dof_map, const unsigned int variable_number, const Elem *elem)
void	compute_constraints (DofConstraints &constraints, DofMap &dof_map, const unsigned int variable_number, const Elem *elem)
void	compute_constraints (DofConstraints &constraints, DofMap &dof_map, const unsigned int variable_number, const Elem *elem)
void	compute_constraints (DofConstraints &, DofMap &, const unsigned int, const Elem *)
void	compute_constraints (DofConstraints &, DofMap &, const unsigned int, const Elem *)
void	compute_constraints (DofConstraints &, DofMap &, const unsigned int, const Elem *)
void	compute_constraints (DofConstraints &, DofMap &, const unsigned int, const Elem *)
void	compute_constraints (DofConstraints &constraints, DofMap &dof_map, const unsigned int variable_number, const Elem *elem)
void	compute_constraints (DofConstraints &constraints, DofMap &dof_map, const unsigned int variable_number, const Elem *elem)
void	compute_constraints (DofConstraints &, DofMap &, const unsigned int, const Elem *)
void	compute_constraints (DofConstraints &, DofMap &, const unsigned int, const Elem *)
void	compute_constraints (DofConstraints &constraints, DofMap &dof_map, const unsigned int variable_number, const Elem *elem)
void	compute_constraints (DofConstraints &constraints, DofMap &dof_map, const unsigned int variable_number, const Elem *elem)
void	compute_constraints (DofConstraints &, DofMap &, const unsigned int, const Elem *)
void	compute_constraints (DofConstraints &, DofMap &, const unsigned int, const Elem *)
virtual bool	shapes_need_reinit () const override
bool	shapes_need_reinit () const
bool	shapes_need_reinit () const
bool	shapes_need_reinit () const
bool	shapes_need_reinit () const
bool	shapes_need_reinit () const
bool	shapes_need_reinit () const
bool	shapes_need_reinit () const
bool	shapes_need_reinit () const
bool	shapes_need_reinit () const
bool	shapes_need_reinit () const
bool	shapes_need_reinit () const
bool	shapes_need_reinit () const
bool	shapes_need_reinit () const
bool	shapes_need_reinit () const
bool	shapes_need_reinit () const
bool	shapes_need_reinit () const
bool	shapes_need_reinit () const
bool	shapes_need_reinit () const
bool	shapes_need_reinit () const
bool	shapes_need_reinit () const
bool	shapes_need_reinit () const
bool	shapes_need_reinit () const
bool	shapes_need_reinit () const
bool	shapes_need_reinit () const
bool	shapes_need_reinit () const
bool	shapes_need_reinit () const
bool	shapes_need_reinit () const
bool	shapes_need_reinit () const
bool	shapes_need_reinit () const
bool	shapes_need_reinit () const
bool	shapes_need_reinit () const
bool	shapes_need_reinit () const
bool	shapes_need_reinit () const
bool	shapes_need_reinit () const
bool	shapes_need_reinit () const
bool	shapes_need_reinit () const
bool	shapes_need_reinit () const
bool	shapes_need_reinit () const
bool	shapes_need_reinit () const
bool	shapes_need_reinit () const
bool	shapes_need_reinit () const
bool	shapes_need_reinit () const
bool	shapes_need_reinit () const
bool	shapes_need_reinit () const
bool	shapes_need_reinit () const
bool	shapes_need_reinit () const
bool	shapes_need_reinit () const
bool	shapes_need_reinit () const
bool	shapes_need_reinit () const
bool	shapes_need_reinit () const
bool	shapes_need_reinit () const
bool	shapes_need_reinit () const
bool	shapes_need_reinit () const
std::unique_ptr< FEGenericBase< Real > >	build (const unsigned int dim, const FEType &fet)
std::unique_ptr< FEGenericBase< RealGradient > >	build (const unsigned int dim, const FEType &fet)
std::unique_ptr< FEGenericBase< Real > >	build_InfFE (const unsigned int dim, const FEType &fet)
std::unique_ptr< FEGenericBase< RealGradient > >	build_InfFE (const unsigned int, const FEType &)
const std::vector< std::vector< OutputShape > > &	get_phi () const
const std::vector< std::vector< OutputGradient > > &	get_dphi () const
const std::vector< std::vector< OutputShape > > &	get_curl_phi () const
const std::vector< std::vector< OutputDivergence > > &	get_div_phi () const
const std::vector< std::vector< OutputShape > > &	get_dphidx () const
const std::vector< std::vector< OutputShape > > &	get_dphidy () const
const std::vector< std::vector< OutputShape > > &	get_dphidz () const
const std::vector< std::vector< OutputShape > > &	get_dphidxi () const
const std::vector< std::vector< OutputShape > > &	get_dphideta () const
const std::vector< std::vector< OutputShape > > &	get_dphidzeta () const
const std::vector< std::vector< OutputTensor > > &	get_d2phi () const
const std::vector< std::vector< OutputShape > > &	get_d2phidx2 () const
const std::vector< std::vector< OutputShape > > &	get_d2phidxdy () const
const std::vector< std::vector< OutputShape > > &	get_d2phidxdz () const
const std::vector< std::vector< OutputShape > > &	get_d2phidy2 () const
const std::vector< std::vector< OutputShape > > &	get_d2phidydz () const
const std::vector< std::vector< OutputShape > > &	get_d2phidz2 () const
const std::vector< std::vector< OutputShape > > &	get_d2phidxi2 () const
const std::vector< std::vector< OutputShape > > &	get_d2phidxideta () const
const std::vector< std::vector< OutputShape > > &	get_d2phidxidzeta () const
const std::vector< std::vector< OutputShape > > &	get_d2phideta2 () const
const std::vector< std::vector< OutputShape > > &	get_d2phidetadzeta () const
const std::vector< std::vector< OutputShape > > &	get_d2phidzeta2 () const
const std::vector< OutputGradient > &	get_dphase () const
const std::vector< Real > &	get_Sobolev_weight () const
const std::vector< RealGradient > &	get_Sobolev_dweight () const
void	print_phi (std::ostream &os) const
void	print_dphi (std::ostream &os) const
void	print_d2phi (std::ostream &os) const
unsigned int	get_dim () const
const std::vector< Point > &	get_xyz () const
const std::vector< Real > &	get_JxW () const
const std::vector< RealGradient > &	get_dxyzdxi () const
const std::vector< RealGradient > &	get_dxyzdeta () const
const std::vector< RealGradient > &	get_dxyzdzeta () const
const std::vector< RealGradient > &	get_d2xyzdxi2 () const
const std::vector< RealGradient > &	get_d2xyzdeta2 () const
const std::vector< RealGradient > &	get_d2xyzdzeta2 () const
const std::vector< RealGradient > &	get_d2xyzdxideta () const
const std::vector< RealGradient > &	get_d2xyzdxidzeta () const
const std::vector< RealGradient > &	get_d2xyzdetadzeta () const
const std::vector< Real > &	get_dxidx () const
const std::vector< Real > &	get_dxidy () const
const std::vector< Real > &	get_dxidz () const
const std::vector< Real > &	get_detadx () const
const std::vector< Real > &	get_detady () const
const std::vector< Real > &	get_detadz () const
const std::vector< Real > &	get_dzetadx () const
const std::vector< Real > &	get_dzetady () const
const std::vector< Real > &	get_dzetadz () const
const std::vector< std::vector< Point > > &	get_tangents () const
const std::vector< Point > &	get_normals () const
const std::vector< Real > &	get_curvatures () const
ElemType	get_type () const
unsigned int	get_p_level () const
FEType	get_fe_type () const
Order	get_order () const
void	set_fe_order (int new_order)
FEFamily	get_family () const
const FEMap &	get_fe_map () const
void	print_JxW (std::ostream &os) const
void	print_xyz (std::ostream &os) const
void	print_info (std::ostream &os) const

Static Public Member Functions
static Real	regular_shape (const unsigned int i, const Real v, const Real w)
static Real	regular_shape_deriv (const unsigned int i, const unsigned int j, const Real v, const Real w)
static Real	regular_shape_second_deriv (const unsigned int i, const unsigned int j, const Real v, const Real w)
static void	loop_subdivision_mask (std::vector< Real > &weights, const unsigned int valence)
static void	init_subdivision_matrix (DenseMatrix< Real > &A, unsigned int valence)
static OutputShape	shape (const ElemType t, const Order o, const unsigned int i, const Point &p)
static OutputShape	shape (const Elem *elem, const Order o, const unsigned int i, const Point &p)
static OutputShape	shape_deriv (const ElemType t, const Order o, const unsigned int i, const unsigned int j, const Point &p)
static OutputShape	shape_deriv (const Elem *elem, const Order o, const unsigned int i, const unsigned int j, const Point &p)
static OutputShape	shape_second_deriv (const ElemType t, const Order o, const unsigned int i, const unsigned int j, const Point &p)
static OutputShape	shape_second_deriv (const Elem *elem, const Order o, const unsigned int i, const unsigned int j, const Point &p)
static void	nodal_soln (const Elem *elem, const Order o, const std::vector< Number > &elem_soln, std::vector< Number > &nodal_soln)
static unsigned int	n_shape_functions (const ElemType t, const Order o)
static unsigned int	n_dofs (const ElemType t, const Order o)
static unsigned int	n_dofs_at_node (const ElemType t, const Order o, const unsigned int n)
static unsigned int	n_dofs_per_elem (const ElemType t, const Order o)
static void	dofs_on_side (const Elem *const elem, const Order o, unsigned int s, std::vector< unsigned int > &di)
static void	dofs_on_edge (const Elem *const elem, const Order o, unsigned int e, std::vector< unsigned int > &di)
static Point	inverse_map (const Elem *elem, const Point &p, const Real tolerance=TOLERANCE, const bool secure=true)
static void	inverse_map (const Elem *elem, const std::vector< Point > &physical_points, std::vector< Point > &reference_points, const Real tolerance=TOLERANCE, const bool secure=true)
static void	compute_constraints (DofConstraints &constraints, DofMap &dof_map, const unsigned int variable_number, const Elem *elem)
static Point	map (const Elem *elem, const Point &reference_point)
static Point	map_xi (const Elem *elem, const Point &reference_point)
static Point	map_eta (const Elem *elem, const Point &reference_point)
static Point	map_zeta (const Elem *elem, const Point &reference_point)
static std::unique_ptr< FEGenericBase >	build (const unsigned int dim, const FEType &type)
static std::unique_ptr< FEGenericBase >	build_InfFE (const unsigned int dim, const FEType &type)
static void	compute_proj_constraints (DofConstraints &constraints, DofMap &dof_map, const unsigned int variable_number, const Elem *elem)
static void	coarsened_dof_values (const NumericVector< Number > &global_vector, const DofMap &dof_map, const Elem *coarse_elem, DenseVector< Number > &coarse_dofs, const unsigned int var, const bool use_old_dof_indices=false)
static void	coarsened_dof_values (const NumericVector< Number > &global_vector, const DofMap &dof_map, const Elem *coarse_elem, DenseVector< Number > &coarse_dofs, const bool use_old_dof_indices=false)
static void	compute_periodic_constraints (DofConstraints &constraints, DofMap &dof_map, const PeriodicBoundaries &boundaries, const MeshBase &mesh, const PointLocatorBase *point_locator, const unsigned int variable_number, const Elem *elem)
static bool	on_reference_element (const Point &p, const ElemType t, const Real eps=TOLERANCE)
static void	get_refspace_nodes (const ElemType t, std::vector< Point > &nodes)
static void	compute_node_constraints (NodeConstraints &constraints, const Elem *elem)
static void	compute_periodic_node_constraints (NodeConstraints &constraints, const PeriodicBoundaries &boundaries, const MeshBase &mesh, const PointLocatorBase *point_locator, const Elem *elem)
static void	print_info (std::ostream &out=libMesh::out)
static std::string	get_info ()
static unsigned int	n_objects ()
static void	enable_print_counter_info ()
static void	disable_print_counter_info ()

Protected Types
typedef std::map< std::string, std::pair< unsigned int, unsigned int > >	Counts

Protected Member Functions
virtual void	init_base_shape_functions (const std::vector< Point > &qp, const Elem *e) override
void	determine_calculations ()
virtual void	compute_shape_functions (const Elem *elem, const std::vector< Point > &qp)
void	increment_constructor_count (const std::string &name)
void	increment_destructor_count (const std::string &name)

Protected Attributes
std::vector< Point >	cached_nodes
ElemType	last_side
unsigned int	last_edge
std::unique_ptr< FETransformationBase< FEOutputType< T >::type > >	_fe_trans
std::vector< std::vector< OutputShape > >	phi
std::vector< std::vector< OutputGradient > >	dphi
std::vector< std::vector< OutputShape > >	curl_phi
std::vector< std::vector< OutputDivergence > >	div_phi
std::vector< std::vector< OutputShape > >	dphidxi
std::vector< std::vector< OutputShape > >	dphideta
std::vector< std::vector< OutputShape > >	dphidzeta
std::vector< std::vector< OutputShape > >	dphidx
std::vector< std::vector< OutputShape > >	dphidy
std::vector< std::vector< OutputShape > >	dphidz
std::vector< std::vector< OutputTensor > >	d2phi
std::vector< std::vector< OutputShape > >	d2phidxi2
std::vector< std::vector< OutputShape > >	d2phidxideta
std::vector< std::vector< OutputShape > >	d2phidxidzeta
std::vector< std::vector< OutputShape > >	d2phideta2
std::vector< std::vector< OutputShape > >	d2phidetadzeta
std::vector< std::vector< OutputShape > >	d2phidzeta2
std::vector< std::vector< OutputShape > >	d2phidx2
std::vector< std::vector< OutputShape > >	d2phidxdy
std::vector< std::vector< OutputShape > >	d2phidxdz
std::vector< std::vector< OutputShape > >	d2phidy2
std::vector< std::vector< OutputShape > >	d2phidydz
std::vector< std::vector< OutputShape > >	d2phidz2
std::vector< OutputGradient >	dphase
std::vector< RealGradient >	dweight
std::vector< Real >	weight
std::unique_ptr< FEMap >	_fe_map
const unsigned int	dim
bool	calculations_started
bool	calculate_phi
bool	calculate_dphi
bool	calculate_d2phi
bool	calculate_curl_phi
bool	calculate_div_phi
bool	calculate_dphiref
FEType	fe_type
ElemType	elem_type
unsigned int	_p_level
QBase *	qrule
bool	shapes_on_quadrature

Static Protected Attributes
static Counts	_counts
static Threads::atomic< unsigned int >	_n_objects
static Threads::spin_mutex	_mutex
static bool	_enable_print_counter = true

Detailed Description

Definition at line 572 of file fe.h.

Member Typedef Documentation

Counts

typedef std::map<std::string, std::pair<unsigned int, unsigned int> > libMesh::ReferenceCounter::Counts

protectedinherited

Data structure to log the information. The log is identified by the class name.

Definition at line 117 of file reference_counter.h.

OutputDivergence

typedef TensorTools::DecrementRank<OutputShape>::type libMesh::FEGenericBase< FEOutputType< T >::type >::OutputDivergence

inherited

Definition at line 122 of file fe_base.h.

OutputGradient

typedef TensorTools::IncrementRank<OutputShape>::type libMesh::FEGenericBase< FEOutputType< T >::type >::OutputGradient

inherited

Definition at line 120 of file fe_base.h.

OutputNumber

typedef TensorTools::MakeNumber<OutputShape>::type libMesh::FEGenericBase< FEOutputType< T >::type >::OutputNumber

inherited

Definition at line 123 of file fe_base.h.

OutputNumberDivergence

typedef TensorTools::DecrementRank<OutputNumber>::type libMesh::FEGenericBase< FEOutputType< T >::type >::OutputNumberDivergence

inherited

Definition at line 126 of file fe_base.h.

OutputNumberGradient

typedef TensorTools::IncrementRank<OutputNumber>::type libMesh::FEGenericBase< FEOutputType< T >::type >::OutputNumberGradient

inherited

Definition at line 124 of file fe_base.h.

OutputNumberTensor

typedef TensorTools::IncrementRank<OutputNumberGradient>::type libMesh::FEGenericBase< FEOutputType< T >::type >::OutputNumberTensor

inherited

Definition at line 125 of file fe_base.h.

OutputShape

typedef FEGenericBase<typename FEOutputType<T>::type>::OutputShape libMesh::FE< Dim, T >::OutputShape

inherited

Definition at line 101 of file fe.h.

OutputTensor

typedef TensorTools::IncrementRank<OutputGradient>::type libMesh::FEGenericBase< FEOutputType< T >::type >::OutputTensor

inherited

Definition at line 121 of file fe_base.h.

Constructor & Destructor Documentation

FESubdivision()

libMesh::FESubdivision::FESubdivision

(

const FEType &

fet

)

Constructor. Creates a subdivision surface finite element. Currently only supported for two-dimensional meshes in three-dimensional space.

Definition at line 33 of file fe_subdivision_2D.C.

                                               :
  FE<2,SUBDIVISION>(fet)
{
  // Only 2D meshes in 3D space are supported
  libmesh_assert_equal_to(LIBMESH_DIM, 3);
}

Member Function Documentation

attach_quadrature_rule()

void libMesh::FESubdivision::attach_quadrature_rule ( QBase *  q )

overridevirtual

Provides the class with the quadrature rule, which provides the locations (on a reference element) where the shape functions are to be calculated.

Reimplemented from libMesh::FE< 2, SUBDIVISION >.

Definition at line 648 of file fe_subdivision_2D.C.

References libMesh::FEAbstract::elem_type, libMesh::INVALID_ELEM, and libMesh::FEAbstract::qrule.

{
  libmesh_assert(q);

  qrule = q;
  // make sure we don't cache results from a previous quadrature rule
  elem_type = INVALID_ELEM;
  return;
}

build() [1/3]

static std::unique_ptr<FEGenericBase> libMesh::FEGenericBase< FEOutputType< T >::type >::build ( const unsigned int  dim, const FEType &  type  )

staticinherited

Builds a specific finite element type. A std::unique_ptr<FEGenericBase> is returned to prevent a memory leak. This way the user need not remember to delete the object.

The build call will fail if the OutputType of this class is not compatible with the output required for the requested type

build() [2/3]

std::unique_ptr< FEGenericBase< Real > > libMesh::FEGenericBase< Real >::build ( const unsigned int  dim, const FEType &  fet  )

inherited

Definition at line 182 of file fe_base.C.

References libMesh::BERNSTEIN, libMesh::CLOUGH, libMesh::FEType::family, libMesh::HERMITE, libMesh::HIERARCHIC, libMesh::L2_HIERARCHIC, libMesh::L2_LAGRANGE, libMesh::LAGRANGE, libMesh::MONOMIAL, libMesh::SCALAR, libMesh::SUBDIVISION, libMesh::SZABAB, and libMesh::XYZ.

{
  switch (dim)
    {
      // 0D
    case 0:
      {
        switch (fet.family)
          {
          case CLOUGH:
            return libmesh_make_unique<FE<0,CLOUGH>>(fet);

          case HERMITE:
            return libmesh_make_unique<FE<0,HERMITE>>(fet);

          case LAGRANGE:
            return libmesh_make_unique<FE<0,LAGRANGE>>(fet);

          case L2_LAGRANGE:
            return libmesh_make_unique<FE<0,L2_LAGRANGE>>(fet);

          case HIERARCHIC:
            return libmesh_make_unique<FE<0,HIERARCHIC>>(fet);

          case L2_HIERARCHIC:
            return libmesh_make_unique<FE<0,L2_HIERARCHIC>>(fet);

          case MONOMIAL:
            return libmesh_make_unique<FE<0,MONOMIAL>>(fet);

#ifdef LIBMESH_ENABLE_HIGHER_ORDER_SHAPES
          case SZABAB:
            return libmesh_make_unique<FE<0,SZABAB>>(fet);

          case BERNSTEIN:
            return libmesh_make_unique<FE<0,BERNSTEIN>>(fet);
#endif

          case XYZ:
            return libmesh_make_unique<FEXYZ<0>>(fet);

          case SCALAR:
            return libmesh_make_unique<FEScalar<0>>(fet);

          default:
            libmesh_error_msg("ERROR: Bad FEType.family= " << fet.family);
          }
      }
      // 1D
    case 1:
      {
        switch (fet.family)
          {
          case CLOUGH:
            return libmesh_make_unique<FE<1,CLOUGH>>(fet);

          case HERMITE:
            return libmesh_make_unique<FE<1,HERMITE>>(fet);

          case LAGRANGE:
            return libmesh_make_unique<FE<1,LAGRANGE>>(fet);

          case L2_LAGRANGE:
            return libmesh_make_unique<FE<1,L2_LAGRANGE>>(fet);

          case HIERARCHIC:
            return libmesh_make_unique<FE<1,HIERARCHIC>>(fet);

          case L2_HIERARCHIC:
            return libmesh_make_unique<FE<1,L2_HIERARCHIC>>(fet);

          case MONOMIAL:
            return libmesh_make_unique<FE<1,MONOMIAL>>(fet);

#ifdef LIBMESH_ENABLE_HIGHER_ORDER_SHAPES
          case SZABAB:
            return libmesh_make_unique<FE<1,SZABAB>>(fet);

          case BERNSTEIN:
            return libmesh_make_unique<FE<1,BERNSTEIN>>(fet);
#endif

          case XYZ:
            return libmesh_make_unique<FEXYZ<1>>(fet);

          case SCALAR:
            return libmesh_make_unique<FEScalar<1>>(fet);

          default:
            libmesh_error_msg("ERROR: Bad FEType.family= " << fet.family);
          }
      }


      // 2D
    case 2:
      {
        switch (fet.family)
          {
          case CLOUGH:
            return libmesh_make_unique<FE<2,CLOUGH>>(fet);

          case HERMITE:
            return libmesh_make_unique<FE<2,HERMITE>>(fet);

          case LAGRANGE:
            return libmesh_make_unique<FE<2,LAGRANGE>>(fet);

          case L2_LAGRANGE:
            return libmesh_make_unique<FE<2,L2_LAGRANGE>>(fet);

          case HIERARCHIC:
            return libmesh_make_unique<FE<2,HIERARCHIC>>(fet);

          case L2_HIERARCHIC:
            return libmesh_make_unique<FE<2,L2_HIERARCHIC>>(fet);

          case MONOMIAL:
            return libmesh_make_unique<FE<2,MONOMIAL>>(fet);

#ifdef LIBMESH_ENABLE_HIGHER_ORDER_SHAPES
          case SZABAB:
            return libmesh_make_unique<FE<2,SZABAB>>(fet);

          case BERNSTEIN:
            return libmesh_make_unique<FE<2,BERNSTEIN>>(fet);
#endif

          case XYZ:
            return libmesh_make_unique<FEXYZ<2>>(fet);

          case SCALAR:
            return libmesh_make_unique<FEScalar<2>>(fet);

          case SUBDIVISION:
            return libmesh_make_unique<FESubdivision>(fet);

          default:
            libmesh_error_msg("ERROR: Bad FEType.family= " << fet.family);
          }
      }


      // 3D
    case 3:
      {
        switch (fet.family)
          {
          case CLOUGH:
            libmesh_error_msg("ERROR: Clough-Tocher elements currently only support 1D and 2D");

          case HERMITE:
            return libmesh_make_unique<FE<3,HERMITE>>(fet);

          case LAGRANGE:
            return libmesh_make_unique<FE<3,LAGRANGE>>(fet);

          case L2_LAGRANGE:
            return libmesh_make_unique<FE<3,L2_LAGRANGE>>(fet);

          case HIERARCHIC:
            return libmesh_make_unique<FE<3,HIERARCHIC>>(fet);

          case L2_HIERARCHIC:
            return libmesh_make_unique<FE<3,L2_HIERARCHIC>>(fet);

          case MONOMIAL:
            return libmesh_make_unique<FE<3,MONOMIAL>>(fet);

#ifdef LIBMESH_ENABLE_HIGHER_ORDER_SHAPES
          case SZABAB:
            return libmesh_make_unique<FE<3,SZABAB>>(fet);

          case BERNSTEIN:
            return libmesh_make_unique<FE<3,BERNSTEIN>>(fet);
#endif

          case XYZ:
            return libmesh_make_unique<FEXYZ<3>>(fet);

          case SCALAR:
            return libmesh_make_unique<FEScalar<3>>(fet);

          default:
            libmesh_error_msg("ERROR: Bad FEType.family= " << fet.family);
          }
      }

    default:
      libmesh_error_msg("Invalid dimension dim = " << dim);
    }
}

build() [3/3]

std::unique_ptr< FEGenericBase< RealGradient > > libMesh::FEGenericBase< RealGradient >::build ( const unsigned int  dim, const FEType &  fet  )

inherited

Definition at line 380 of file fe_base.C.

References libMesh::FEType::family, libMesh::LAGRANGE_VEC, and libMesh::NEDELEC_ONE.

{
  switch (dim)
    {
      // 0D
    case 0:
      {
        switch (fet.family)
          {
          case LAGRANGE_VEC:
            return libmesh_make_unique<FELagrangeVec<0>>(fet);

          default:
            libmesh_error_msg("ERROR: Bad FEType.family= " << fet.family);
          }
      }
    case 1:
      {
        switch (fet.family)
          {
          case LAGRANGE_VEC:
            return libmesh_make_unique<FELagrangeVec<1>>(fet);

          default:
            libmesh_error_msg("ERROR: Bad FEType.family= " << fet.family);
          }
      }
    case 2:
      {
        switch (fet.family)
          {
          case LAGRANGE_VEC:
            return libmesh_make_unique<FELagrangeVec<2>>(fet);

          case NEDELEC_ONE:
            return libmesh_make_unique<FENedelecOne<2>>(fet);

          default:
            libmesh_error_msg("ERROR: Bad FEType.family= " << fet.family);
          }
      }
    case 3:
      {
        switch (fet.family)
          {
          case LAGRANGE_VEC:
            return libmesh_make_unique<FELagrangeVec<3>>(fet);

          case NEDELEC_ONE:
            return libmesh_make_unique<FENedelecOne<3>>(fet);

          default:
            libmesh_error_msg("ERROR: Bad FEType.family= " << fet.family);
          }
      }

    default:
      libmesh_error_msg("Invalid dimension dim = " << dim);
    } // switch(dim)
}

build_InfFE() [1/3]

static std::unique_ptr<FEGenericBase> libMesh::FEGenericBase< FEOutputType< T >::type >::build_InfFE ( const unsigned int  dim, const FEType &  type  )

staticinherited

Builds a specific infinite element type. A std::unique_ptr<FEGenericBase> is returned to prevent a memory leak. This way the user need not remember to delete the object.

The build call will fail if the OutputShape of this class is not compatible with the output required for the requested type

build_InfFE() [2/3]

std::unique_ptr< FEGenericBase< Real > > libMesh::FEGenericBase< Real >::build_InfFE ( const unsigned int  dim, const FEType &  fet  )

inherited

Definition at line 453 of file fe_base.C.

References libMesh::CARTESIAN, libMesh::FEType::inf_map, libMesh::INFINITE_MAP, libMesh::JACOBI_20_00, libMesh::JACOBI_30_00, libMesh::LAGRANGE, libMesh::LEGENDRE, and libMesh::FEType::radial_family.

{
  switch (dim)
    {

      // 1D
    case 1:
      {
        switch (fet.radial_family)
          {
          case INFINITE_MAP:
            libmesh_error_msg("ERROR: Can't build an infinite element with FEFamily = " << fet.radial_family);

          case JACOBI_20_00:
            {
              switch (fet.inf_map)
                {
                case CARTESIAN:
                  return libmesh_make_unique<InfFE<1,JACOBI_20_00,CARTESIAN>>(fet);

                default:
                  libmesh_error_msg("ERROR: Can't build an infinite element with InfMapType = " << fet.inf_map);
                }
            }

          case JACOBI_30_00:
            {
              switch (fet.inf_map)
                {
                case CARTESIAN:
                  return libmesh_make_unique<InfFE<1,JACOBI_30_00,CARTESIAN>>(fet);

                default:
                  libmesh_error_msg("ERROR: Can't build an infinite element with InfMapType = " << fet.inf_map);
                }
            }

          case LEGENDRE:
            {
              switch (fet.inf_map)
                {
                case CARTESIAN:
                  return libmesh_make_unique<InfFE<1,LEGENDRE,CARTESIAN>>(fet);

                default:
                  libmesh_error_msg("ERROR: Can't build an infinite element with InfMapType = " << fet.inf_map);
                }
            }

          case LAGRANGE:
            {
              switch (fet.inf_map)
                {
                case CARTESIAN:
                  return libmesh_make_unique<InfFE<1,LAGRANGE,CARTESIAN>>(fet);

                default:
                  libmesh_error_msg("ERROR: Can't build an infinite element with InfMapType = " << fet.inf_map);
                }
            }

          default:
            libmesh_error_msg("ERROR: Bad FEType.radial_family= " << fet.radial_family);
          }
      }




      // 2D
    case 2:
      {
        switch (fet.radial_family)
          {
          case INFINITE_MAP:
            libmesh_error_msg("ERROR: Can't build an infinite element with FEFamily = " << fet.radial_family);

          case JACOBI_20_00:
            {
              switch (fet.inf_map)
                {
                case CARTESIAN:
                  return libmesh_make_unique<InfFE<2,JACOBI_20_00,CARTESIAN>>(fet);

                default:
                  libmesh_error_msg("ERROR: Don't build an infinite element with InfMapType = " << fet.inf_map);
                }
            }

          case JACOBI_30_00:
            {
              switch (fet.inf_map)
                {
                case CARTESIAN:
                  return libmesh_make_unique<InfFE<2,JACOBI_30_00,CARTESIAN>>(fet);

                default:
                  libmesh_error_msg("ERROR: Don't build an infinite element with InfMapType = " << fet.inf_map);
                }
            }

          case LEGENDRE:
            {
              switch (fet.inf_map)
                {
                case CARTESIAN:
                  return libmesh_make_unique<InfFE<2,LEGENDRE,CARTESIAN>>(fet);

                default:
                  libmesh_error_msg("ERROR: Don't build an infinite element with InfMapType = " << fet.inf_map);
                }
            }

          case LAGRANGE:
            {
              switch (fet.inf_map)
                {
                case CARTESIAN:
                  return libmesh_make_unique<InfFE<2,LAGRANGE,CARTESIAN>>(fet);

                default:
                  libmesh_error_msg("ERROR: Don't build an infinite element with InfMapType = " << fet.inf_map);
                }
            }

          default:
            libmesh_error_msg("ERROR: Bad FEType.radial_family= " << fet.radial_family);
          }
      }




      // 3D
    case 3:
      {
        switch (fet.radial_family)
          {
          case INFINITE_MAP:
            libmesh_error_msg("ERROR: Don't build an infinite element with FEFamily = " << fet.radial_family);

          case JACOBI_20_00:
            {
              switch (fet.inf_map)
                {
                case CARTESIAN:
                  return libmesh_make_unique<InfFE<3,JACOBI_20_00,CARTESIAN>>(fet);

                default:
                  libmesh_error_msg("ERROR: Don't build an infinite element with InfMapType = " << fet.inf_map);
                }
            }

          case JACOBI_30_00:
            {
              switch (fet.inf_map)
                {
                case CARTESIAN:
                  return libmesh_make_unique<InfFE<3,JACOBI_30_00,CARTESIAN>>(fet);

                default:
                  libmesh_error_msg("ERROR: Don't build an infinite element with InfMapType = " << fet.inf_map);
                }
            }

          case LEGENDRE:
            {
              switch (fet.inf_map)
                {
                case CARTESIAN:
                  return libmesh_make_unique<InfFE<3,LEGENDRE,CARTESIAN>>(fet);

                default:
                  libmesh_error_msg("ERROR: Don't build an infinite element with InfMapType = " << fet.inf_map);
                }
            }

          case LAGRANGE:
            {
              switch (fet.inf_map)
                {
                case CARTESIAN:
                  return libmesh_make_unique<InfFE<3,LAGRANGE,CARTESIAN>>(fet);

                default:
                  libmesh_error_msg("ERROR: Don't build an infinite element with InfMapType = " << fet.inf_map);
                }
            }

          default:
            libmesh_error_msg("ERROR: Bad FEType.radial_family= " << fet.radial_family);
          }
      }

    default:
      libmesh_error_msg("Invalid dimension dim = " << dim);
    }
}

build_InfFE() [3/3]

std::unique_ptr< FEGenericBase< RealGradient > > libMesh::FEGenericBase< RealGradient >::build_InfFE ( const unsigned  int, const FEType &    )

inherited

Definition at line 657 of file fe_base.C.

{
  // No vector types defined... YET.
  libmesh_not_implemented();
  return std::unique_ptr<FEVectorBase>();
}

coarsened_dof_values() [1/2]

void libMesh::FEGenericBase< FEOutputType< T >::type >::coarsened_dof_values ( const NumericVector< Number > &  global_vector, const DofMap &  dof_map, const Elem *  coarse_elem, DenseVector< Number > &  coarse_dofs, const unsigned int  var, const bool  use_old_dof_indices = false  )

staticinherited

Creates a local projection on coarse_elem, based on the DoF values in global_vector for it's children. Computes a vector of coefficients corresponding to dof_indices for only the single given var

Definition at line 791 of file fe_base.C.

References std::abs(), libMesh::C_ONE, libMesh::Elem::child_ptr(), libMesh::Elem::child_ref_range(), libMesh::DenseMatrix< T >::cholesky_solve(), libMesh::FEType::default_quadrature_rule(), libMesh::Elem::dim(), libMesh::DISCONTINUOUS, libMesh::DofMap::dof_indices(), libMesh::FEInterface::dofs_on_edge(), libMesh::FEInterface::dofs_on_side(), libMesh::Elem::edge_index_range(), libMesh::TensorTools::inner_product(), libMesh::FEInterface::inverse_map(), libMesh::Elem::is_child_on_edge(), libMesh::Elem::is_child_on_side(), libMesh::Elem::is_vertex(), libMesh::Elem::max_descendant_p_level(), libMesh::Elem::n_children(), libMesh::FEInterface::n_dofs(), libMesh::FEInterface::n_dofs_at_node(), n_nodes, libMesh::Elem::n_nodes(), libMesh::DofMap::old_dof_indices(), libMesh::FEType::order, libMesh::Elem::p_level(), libMesh::DenseVector< T >::resize(), libMesh::DenseMatrix< T >::resize(), libMesh::Elem::side_index_range(), libMesh::TOLERANCE, libMesh::Elem::type(), libMesh::DofMap::variable_type(), libMesh::DenseMatrix< T >::zero(), libMesh::DenseVector< T >::zero(), and libMesh::zero.

{
  // Side/edge local DOF indices
  std::vector<unsigned int> new_side_dofs, old_side_dofs;

  // FIXME: what about 2D shells in 3D space?
  unsigned int dim = elem->dim();

  // Cache n_children(); it's a virtual call but it's const.
  const unsigned int n_children = elem->n_children();

  // We use local FE objects for now
  // FIXME: we should use more, external objects instead for efficiency
  const FEType & base_fe_type = dof_map.variable_type(var);
  std::unique_ptr<FEGenericBase<OutputShape>> fe
    (FEGenericBase<OutputShape>::build(dim, base_fe_type));
  std::unique_ptr<FEGenericBase<OutputShape>> fe_coarse
    (FEGenericBase<OutputShape>::build(dim, base_fe_type));

  std::unique_ptr<QBase> qrule     (base_fe_type.default_quadrature_rule(dim));
  std::unique_ptr<QBase> qedgerule (base_fe_type.default_quadrature_rule(1));
  std::unique_ptr<QBase> qsiderule (base_fe_type.default_quadrature_rule(dim-1));
  std::vector<Point> coarse_qpoints;

  // The values of the shape functions at the quadrature
  // points
  const std::vector<std::vector<OutputShape>> & phi_values =
    fe->get_phi();
  const std::vector<std::vector<OutputShape>> & phi_coarse =
    fe_coarse->get_phi();

  // The gradients of the shape functions at the quadrature
  // points on the child element.
  const std::vector<std::vector<OutputGradient>> * dphi_values =
    nullptr;
  const std::vector<std::vector<OutputGradient>> * dphi_coarse =
    nullptr;

  const FEContinuity cont = fe->get_continuity();

  if (cont == C_ONE)
    {
      const std::vector<std::vector<OutputGradient>> &
        ref_dphi_values = fe->get_dphi();
      dphi_values = &ref_dphi_values;
      const std::vector<std::vector<OutputGradient>> &
        ref_dphi_coarse = fe_coarse->get_dphi();
      dphi_coarse = &ref_dphi_coarse;
    }

  // The Jacobian * quadrature weight at the quadrature points
  const std::vector<Real> & JxW =
    fe->get_JxW();

  // The XYZ locations of the quadrature points on the
  // child element
  const std::vector<Point> & xyz_values =
    fe->get_xyz();



  FEType fe_type = base_fe_type, temp_fe_type;
  const ElemType elem_type = elem->type();
  fe_type.order = static_cast<Order>(fe_type.order +
                                     elem->max_descendant_p_level());

  // Number of nodes on parent element
  const unsigned int n_nodes = elem->n_nodes();

  // Number of dofs on parent element
  const unsigned int new_n_dofs =
    FEInterface::n_dofs(dim, fe_type, elem_type);

  // Fixed vs. free DoFs on edge/face projections
  std::vector<char> dof_is_fixed(new_n_dofs, false); // bools
  std::vector<int> free_dof(new_n_dofs, 0);

  DenseMatrix<Real> Ke;
  DenseVector<Number> Fe;
  Ue.resize(new_n_dofs); Ue.zero();


  // When coarsening, in general, we need a series of
  // projections to ensure a unique and continuous
  // solution.  We start by interpolating nodes, then
  // hold those fixed and project edges, then
  // hold those fixed and project faces, then
  // hold those fixed and project interiors

  // Copy node values first
  {
    std::vector<dof_id_type> node_dof_indices;
    if (use_old_dof_indices)
      dof_map.old_dof_indices (elem, node_dof_indices, var);
    else
      dof_map.dof_indices (elem, node_dof_indices, var);

    unsigned int current_dof = 0;
    for (unsigned int n=0; n!= n_nodes; ++n)
      {
        // FIXME: this should go through the DofMap,
        // not duplicate dof_indices code badly!
        const unsigned int my_nc =
          FEInterface::n_dofs_at_node (dim, fe_type,
                                       elem_type, n);
        if (!elem->is_vertex(n))
          {
            current_dof += my_nc;
            continue;
          }

        temp_fe_type = base_fe_type;
        // We're assuming here that child n shares vertex n,
        // which is wrong on non-simplices right now
        // ... but this code isn't necessary except on elements
        // where p refinement creates more vertex dofs; we have
        // no such elements yet.
        /*
          if (elem->child_ptr(n)->p_level() < elem->p_level())
          {
          temp_fe_type.order =
          static_cast<Order>(temp_fe_type.order +
          elem->child_ptr(n)->p_level());
          }
        */
        const unsigned int nc =
          FEInterface::n_dofs_at_node (dim, temp_fe_type,
                                       elem_type, n);
        for (unsigned int i=0; i!= nc; ++i)
          {
            Ue(current_dof) =
              old_vector(node_dof_indices[current_dof]);
            dof_is_fixed[current_dof] = true;
            current_dof++;
          }
      }
  }

  // In 3D, project any edge values next
  if (dim > 2 && cont != DISCONTINUOUS)
    for (auto e : elem->edge_index_range())
      {
        FEInterface::dofs_on_edge(elem, dim, fe_type,
                                  e, new_side_dofs);

        const unsigned int n_new_side_dofs =
          cast_int<unsigned int>(new_side_dofs.size());

        // Some edge dofs are on nodes and already
        // fixed, others are free to calculate
        unsigned int free_dofs = 0;
        for (unsigned int i=0; i != n_new_side_dofs; ++i)
          if (!dof_is_fixed[new_side_dofs[i]])
            free_dof[free_dofs++] = i;
        Ke.resize (free_dofs, free_dofs); Ke.zero();
        Fe.resize (free_dofs); Fe.zero();
        // The new edge coefficients
        DenseVector<Number> Uedge(free_dofs);

        // Add projection terms from each child sharing
        // this edge
        for (unsigned int c=0; c != n_children; ++c)
          {
            if (!elem->is_child_on_edge(c,e))
              continue;
            const Elem * child = elem->child_ptr(c);

            std::vector<dof_id_type> child_dof_indices;
            if (use_old_dof_indices)
              dof_map.old_dof_indices (child,
                                       child_dof_indices, var);
            else
              dof_map.dof_indices (child,
                                   child_dof_indices, var);
            const unsigned int child_n_dofs =
              cast_int<unsigned int>
              (child_dof_indices.size());

            temp_fe_type = base_fe_type;
            temp_fe_type.order =
              static_cast<Order>(temp_fe_type.order +
                                 child->p_level());

            FEInterface::dofs_on_edge(child, dim,
                                      temp_fe_type, e, old_side_dofs);

            // Initialize both child and parent FE data
            // on the child's edge
            fe->attach_quadrature_rule (qedgerule.get());
            fe->edge_reinit (child, e);
            const unsigned int n_qp = qedgerule->n_points();

            FEInterface::inverse_map (dim, fe_type, elem,
                                      xyz_values, coarse_qpoints);

            fe_coarse->reinit(elem, &coarse_qpoints);

            // Loop over the quadrature points
            for (unsigned int qp=0; qp<n_qp; qp++)
              {
                // solution value at the quadrature point
                OutputNumber fineval = libMesh::zero;
                // solution grad at the quadrature point
                OutputNumberGradient finegrad;

                // Sum the solution values * the DOF
                // values at the quadrature point to
                // get the solution value and gradient.
                for (unsigned int i=0; i<child_n_dofs;
                     i++)
                  {
                    fineval +=
                      (old_vector(child_dof_indices[i])*
                       phi_values[i][qp]);
                    if (cont == C_ONE)
                      finegrad += (*dphi_values)[i][qp] *
                        old_vector(child_dof_indices[i]);
                  }

                // Form edge projection matrix
                for (unsigned int sidei=0, freei=0; sidei != n_new_side_dofs; ++sidei)
                  {
                    unsigned int i = new_side_dofs[sidei];
                    // fixed DoFs aren't test functions
                    if (dof_is_fixed[i])
                      continue;
                    for (unsigned int sidej=0, freej=0; sidej != n_new_side_dofs; ++sidej)
                      {
                        unsigned int j =
                          new_side_dofs[sidej];
                        if (dof_is_fixed[j])
                          Fe(freei) -=
                            TensorTools::inner_product(phi_coarse[i][qp],
                                                       phi_coarse[j][qp]) *
                            JxW[qp] * Ue(j);
                        else
                          Ke(freei,freej) +=
                            TensorTools::inner_product(phi_coarse[i][qp],
                                                       phi_coarse[j][qp]) *
                            JxW[qp];
                        if (cont == C_ONE)
                          {
                            if (dof_is_fixed[j])
                              Fe(freei) -=
                                TensorTools::inner_product((*dphi_coarse)[i][qp],
                                                           (*dphi_coarse)[j][qp]) *
                                JxW[qp] * Ue(j);
                            else
                              Ke(freei,freej) +=
                                TensorTools::inner_product((*dphi_coarse)[i][qp],
                                                           (*dphi_coarse)[j][qp]) *
                                JxW[qp];
                          }
                        if (!dof_is_fixed[j])
                          freej++;
                      }
                    Fe(freei) += TensorTools::inner_product(phi_coarse[i][qp],
                                                            fineval) * JxW[qp];
                    if (cont == C_ONE)
                      Fe(freei) +=
                        TensorTools::inner_product(finegrad, (*dphi_coarse)[i][qp]) * JxW[qp];
                    freei++;
                  }
              }
          }
        Ke.cholesky_solve(Fe, Uedge);

        // Transfer new edge solutions to element
        for (unsigned int i=0; i != free_dofs; ++i)
          {
            Number & ui = Ue(new_side_dofs[free_dof[i]]);
            libmesh_assert(std::abs(ui) < TOLERANCE ||
                           std::abs(ui - Uedge(i)) < TOLERANCE);
            ui = Uedge(i);
            dof_is_fixed[new_side_dofs[free_dof[i]]] = true;
          }
      }

  // Project any side values (edges in 2D, faces in 3D)
  if (dim > 1 && cont != DISCONTINUOUS)
    for (auto s : elem->side_index_range())
      {
        FEInterface::dofs_on_side(elem, dim, fe_type,
                                  s, new_side_dofs);

        const unsigned int n_new_side_dofs =
          cast_int<unsigned int>(new_side_dofs.size());

        // Some side dofs are on nodes/edges and already
        // fixed, others are free to calculate
        unsigned int free_dofs = 0;
        for (unsigned int i=0; i != n_new_side_dofs; ++i)
          if (!dof_is_fixed[new_side_dofs[i]])
            free_dof[free_dofs++] = i;
        Ke.resize (free_dofs, free_dofs); Ke.zero();
        Fe.resize (free_dofs); Fe.zero();
        // The new side coefficients
        DenseVector<Number> Uside(free_dofs);

        // Add projection terms from each child sharing
        // this side
        for (unsigned int c=0; c != n_children; ++c)
          {
            if (!elem->is_child_on_side(c,s))
              continue;
            const Elem * child = elem->child_ptr(c);

            std::vector<dof_id_type> child_dof_indices;
            if (use_old_dof_indices)
              dof_map.old_dof_indices (child,
                                       child_dof_indices, var);
            else
              dof_map.dof_indices (child,
                                   child_dof_indices, var);
            const unsigned int child_n_dofs =
              cast_int<unsigned int>
              (child_dof_indices.size());

            temp_fe_type = base_fe_type;
            temp_fe_type.order =
              static_cast<Order>(temp_fe_type.order +
                                 child->p_level());

            FEInterface::dofs_on_side(child, dim,
                                      temp_fe_type, s, old_side_dofs);

            // Initialize both child and parent FE data
            // on the child's side
            fe->attach_quadrature_rule (qsiderule.get());
            fe->reinit (child, s);
            const unsigned int n_qp = qsiderule->n_points();

            FEInterface::inverse_map (dim, fe_type, elem,
                                      xyz_values, coarse_qpoints);

            fe_coarse->reinit(elem, &coarse_qpoints);

            // Loop over the quadrature points
            for (unsigned int qp=0; qp<n_qp; qp++)
              {
                // solution value at the quadrature point
                OutputNumber fineval = libMesh::zero;
                // solution grad at the quadrature point
                OutputNumberGradient finegrad;

                // Sum the solution values * the DOF
                // values at the quadrature point to
                // get the solution value and gradient.
                for (unsigned int i=0; i<child_n_dofs;
                     i++)
                  {
                    fineval +=
                      old_vector(child_dof_indices[i]) *
                      phi_values[i][qp];
                    if (cont == C_ONE)
                      finegrad += (*dphi_values)[i][qp] *
                        old_vector(child_dof_indices[i]);
                  }

                // Form side projection matrix
                for (unsigned int sidei=0, freei=0; sidei != n_new_side_dofs; ++sidei)
                  {
                    unsigned int i = new_side_dofs[sidei];
                    // fixed DoFs aren't test functions
                    if (dof_is_fixed[i])
                      continue;
                    for (unsigned int sidej=0, freej=0; sidej != n_new_side_dofs; ++sidej)
                      {
                        unsigned int j =
                          new_side_dofs[sidej];
                        if (dof_is_fixed[j])
                          Fe(freei) -=
                            TensorTools::inner_product(phi_coarse[i][qp],
                                                       phi_coarse[j][qp]) *
                            JxW[qp] * Ue(j);
                        else
                          Ke(freei,freej) +=
                            TensorTools::inner_product(phi_coarse[i][qp],
                                                       phi_coarse[j][qp]) *
                            JxW[qp];
                        if (cont == C_ONE)
                          {
                            if (dof_is_fixed[j])
                              Fe(freei) -=
                                TensorTools::inner_product((*dphi_coarse)[i][qp],
                                                           (*dphi_coarse)[j][qp]) *
                                JxW[qp] * Ue(j);
                            else
                              Ke(freei,freej) +=
                                TensorTools::inner_product((*dphi_coarse)[i][qp],
                                                           (*dphi_coarse)[j][qp]) *
                                JxW[qp];
                          }
                        if (!dof_is_fixed[j])
                          freej++;
                      }
                    Fe(freei) += TensorTools::inner_product(fineval, phi_coarse[i][qp]) * JxW[qp];
                    if (cont == C_ONE)
                      Fe(freei) +=
                        TensorTools::inner_product(finegrad, (*dphi_coarse)[i][qp]) * JxW[qp];
                    freei++;
                  }
              }
          }
        Ke.cholesky_solve(Fe, Uside);

        // Transfer new side solutions to element
        for (unsigned int i=0; i != free_dofs; ++i)
          {
            Number & ui = Ue(new_side_dofs[free_dof[i]]);
            libmesh_assert(std::abs(ui) < TOLERANCE ||
                           std::abs(ui - Uside(i)) < TOLERANCE);
            ui = Uside(i);
            dof_is_fixed[new_side_dofs[free_dof[i]]] = true;
          }
      }

  // Project the interior values, finally

  // Some interior dofs are on nodes/edges/sides and
  // already fixed, others are free to calculate
  unsigned int free_dofs = 0;
  for (unsigned int i=0; i != new_n_dofs; ++i)
    if (!dof_is_fixed[i])
      free_dof[free_dofs++] = i;
  Ke.resize (free_dofs, free_dofs); Ke.zero();
  Fe.resize (free_dofs); Fe.zero();
  // The new interior coefficients
  DenseVector<Number> Uint(free_dofs);

  // Add projection terms from each child
  for (auto & child : elem->child_ref_range())
    {
      std::vector<dof_id_type> child_dof_indices;
      if (use_old_dof_indices)
        dof_map.old_dof_indices (&child,
                                 child_dof_indices, var);
      else
        dof_map.dof_indices (&child,
                             child_dof_indices, var);
      const unsigned int child_n_dofs =
        cast_int<unsigned int>
        (child_dof_indices.size());

      // Initialize both child and parent FE data
      // on the child's quadrature points
      fe->attach_quadrature_rule (qrule.get());
      fe->reinit (&child);
      const unsigned int n_qp = qrule->n_points();

      FEInterface::inverse_map (dim, fe_type, elem,
                                xyz_values, coarse_qpoints);

      fe_coarse->reinit(elem, &coarse_qpoints);

      // Loop over the quadrature points
      for (unsigned int qp=0; qp<n_qp; qp++)
        {
          // solution value at the quadrature point
          OutputNumber fineval = libMesh::zero;
          // solution grad at the quadrature point
          OutputNumberGradient finegrad;

          // Sum the solution values * the DOF
          // values at the quadrature point to
          // get the solution value and gradient.
          for (unsigned int i=0; i<child_n_dofs; i++)
            {
              fineval +=
                (old_vector(child_dof_indices[i]) *
                 phi_values[i][qp]);
              if (cont == C_ONE)
                finegrad += (*dphi_values)[i][qp] *
                  old_vector(child_dof_indices[i]);
            }

          // Form interior projection matrix
          for (unsigned int i=0, freei=0;
               i != new_n_dofs; ++i)
            {
              // fixed DoFs aren't test functions
              if (dof_is_fixed[i])
                continue;
              for (unsigned int j=0, freej=0; j !=
                     new_n_dofs; ++j)
                {
                  if (dof_is_fixed[j])
                    Fe(freei) -=
                      TensorTools::inner_product(phi_coarse[i][qp],
                                                 phi_coarse[j][qp]) *
                      JxW[qp] * Ue(j);
                  else
                    Ke(freei,freej) +=
                      TensorTools::inner_product(phi_coarse[i][qp],
                                                 phi_coarse[j][qp]) *
                      JxW[qp];
                  if (cont == C_ONE)
                    {
                      if (dof_is_fixed[j])
                        Fe(freei) -=
                          TensorTools::inner_product((*dphi_coarse)[i][qp],
                                                     (*dphi_coarse)[j][qp]) *
                          JxW[qp] * Ue(j);
                      else
                        Ke(freei,freej) +=
                          TensorTools::inner_product((*dphi_coarse)[i][qp],
                                                     (*dphi_coarse)[j][qp]) *
                          JxW[qp];
                    }
                  if (!dof_is_fixed[j])
                    freej++;
                }
              Fe(freei) += TensorTools::inner_product(phi_coarse[i][qp], fineval) *
                JxW[qp];
              if (cont == C_ONE)
                Fe(freei) += TensorTools::inner_product(finegrad, (*dphi_coarse)[i][qp]) * JxW[qp];
              freei++;
            }
        }
    }
  Ke.cholesky_solve(Fe, Uint);

  // Transfer new interior solutions to element
  for (unsigned int i=0; i != free_dofs; ++i)
    {
      Number & ui = Ue(free_dof[i]);
      libmesh_assert(std::abs(ui) < TOLERANCE ||
                     std::abs(ui - Uint(i)) < TOLERANCE);
      ui = Uint(i);
      // We should be fixing all dofs by now; no need to keep track of
      // that unless we're debugging
#ifndef NDEBUG
      dof_is_fixed[free_dof[i]] = true;
#endif
    }

#ifndef NDEBUG
  // Make sure every DoF got reached!
  for (unsigned int i=0; i != new_n_dofs; ++i)
    libmesh_assert(dof_is_fixed[i]);
#endif
}

coarsened_dof_values() [2/2]

void libMesh::FEGenericBase< FEOutputType< T >::type >::coarsened_dof_values ( const NumericVector< Number > &  global_vector, const DofMap &  dof_map, const Elem *  coarse_elem, DenseVector< Number > &  coarse_dofs, const bool  use_old_dof_indices = false  )

staticinherited

Creates a local projection on coarse_elem, based on the DoF values in global_vector for it's children. Computes a vector of coefficients corresponding to all dof_indices.

Definition at line 1343 of file fe_base.C.

References libMesh::DenseVector< T >::append(), libMesh::DofMap::n_variables(), and libMesh::DenseVector< T >::resize().

{
  Ue.resize(0);

  for (unsigned int v=0; v != dof_map.n_variables(); ++v)
    {
      DenseVector<Number> Usub;

      coarsened_dof_values(old_vector, dof_map, elem, Usub,
                           use_old_dof_indices);

      Ue.append (Usub);
    }
}

compute_constraints() [1/29]

void libMesh::FE< 2, SCALAR >::compute_constraints ( DofConstraints &  , DofMap &  , const unsigned  int, const Elem *    )

inherited

Definition at line 122 of file fe_scalar.C.

{ }

compute_constraints() [2/29]

void libMesh::FE< 3, SCALAR >::compute_constraints ( DofConstraints &  , DofMap &  , const unsigned  int, const Elem *    )

inherited

Definition at line 129 of file fe_scalar.C.

{ }

compute_constraints() [3/29]

void libMesh::FE< 2, L2_HIERARCHIC >::compute_constraints ( DofConstraints &  , DofMap &  , const unsigned  int, const Elem *    )

inherited

Definition at line 207 of file fe_l2_hierarchic.C.

{ }

compute_constraints() [4/29]

void libMesh::FE< 3, L2_HIERARCHIC >::compute_constraints ( DofConstraints &  , DofMap &  , const unsigned  int, const Elem *    )

inherited

Definition at line 214 of file fe_l2_hierarchic.C.

{ }

compute_constraints() [5/29]

void libMesh::FE< 2, CLOUGH >::compute_constraints ( DofConstraints &  constraints, DofMap &  dof_map, const unsigned int  variable_number, const Elem *  elem  )

inherited

Definition at line 304 of file fe_clough.C.

{ compute_proj_constraints(constraints, dof_map, variable_number, elem); }

compute_constraints() [6/29]

void libMesh::FE< 3, CLOUGH >::compute_constraints ( DofConstraints &  constraints, DofMap &  dof_map, const unsigned int  variable_number, const Elem *  elem  )

inherited

Definition at line 311 of file fe_clough.C.

{ compute_proj_constraints(constraints, dof_map, variable_number, elem); }

compute_constraints() [7/29]

void libMesh::FE< 2, HERMITE >::compute_constraints ( DofConstraints &  constraints, DofMap &  dof_map, const unsigned int  variable_number, const Elem *  elem  )

inherited

Definition at line 355 of file fe_hermite.C.

{ compute_proj_constraints(constraints, dof_map, variable_number, elem); }

compute_constraints() [8/29]

void libMesh::FE< 3, HERMITE >::compute_constraints ( DofConstraints &  constraints, DofMap &  dof_map, const unsigned int  variable_number, const Elem *  elem  )

inherited

Definition at line 362 of file fe_hermite.C.

{ compute_proj_constraints(constraints, dof_map, variable_number, elem); }

compute_constraints() [9/29]

void libMesh::FE< 2, HIERARCHIC >::compute_constraints ( DofConstraints &  constraints, DofMap &  dof_map, const unsigned int  variable_number, const Elem *  elem  )

inherited

Definition at line 383 of file fe_hierarchic.C.

{ compute_proj_constraints(constraints, dof_map, variable_number, elem); }

compute_constraints() [10/29]

void libMesh::FE< 3, HIERARCHIC >::compute_constraints ( DofConstraints &  constraints, DofMap &  dof_map, const unsigned int  variable_number, const Elem *  elem  )

inherited

Definition at line 390 of file fe_hierarchic.C.

{ compute_proj_constraints(constraints, dof_map, variable_number, elem); }

compute_constraints() [11/29]

static void libMesh::FE< Dim, T >::compute_constraints ( DofConstraints &  constraints, DofMap &  dof_map, const unsigned int  variable_number, const Elem *  elem  )

staticinherited

Computes the constraint matrix contributions (for non-conforming adapted meshes) corresponding to variable number var_number, using element-specific optimizations if possible.

compute_constraints() [12/29]

void libMesh::FE< 2, MONOMIAL >::compute_constraints ( DofConstraints &  , DofMap &  , const unsigned  int, const Elem *    )

inherited

Definition at line 419 of file fe_monomial.C.

{}

compute_constraints() [13/29]

void libMesh::FE< 3, MONOMIAL >::compute_constraints ( DofConstraints &  , DofMap &  , const unsigned  int, const Elem *    )

inherited

Definition at line 420 of file fe_monomial.C.

{}

compute_constraints() [14/29]

void libMesh::FE< 2, BERNSTEIN >::compute_constraints ( DofConstraints &  constraints, DofMap &  dof_map, const unsigned int  variable_number, const Elem *  elem  )

inherited

Definition at line 450 of file fe_bernstein.C.

{ compute_proj_constraints(constraints, dof_map, variable_number, elem); }

compute_constraints() [15/29]

void libMesh::FE< 3, BERNSTEIN >::compute_constraints ( DofConstraints &  constraints, DofMap &  dof_map, const unsigned int  variable_number, const Elem *  elem  )

inherited

Definition at line 457 of file fe_bernstein.C.

{ compute_proj_constraints(constraints, dof_map, variable_number, elem); }

compute_constraints() [16/29]

void libMesh::FE< 2, L2_LAGRANGE >::compute_constraints ( DofConstraints &  , DofMap &  , const unsigned  int, const Elem *    )

inherited

Definition at line 495 of file fe_l2_lagrange.C.

{ }

compute_constraints() [17/29]

void libMesh::FE< 3, L2_LAGRANGE >::compute_constraints ( DofConstraints &  , DofMap &  , const unsigned  int, const Elem *    )

inherited

Definition at line 502 of file fe_l2_lagrange.C.

{ }

compute_constraints() [18/29]

void libMesh::FE< 0, NEDELEC_ONE >::compute_constraints ( DofConstraints &  , DofMap &  , const unsigned  int, const Elem *    )

inherited

Definition at line 563 of file fe_nedelec_one.C.

{ NEDELEC_LOW_D_ERROR_MESSAGE }

compute_constraints() [19/29]

void libMesh::FE< 1, NEDELEC_ONE >::compute_constraints ( DofConstraints &  , DofMap &  , const unsigned  int, const Elem *    )

inherited

Definition at line 570 of file fe_nedelec_one.C.

{ NEDELEC_LOW_D_ERROR_MESSAGE }

compute_constraints() [20/29]

void libMesh::FE< 2, NEDELEC_ONE >::compute_constraints ( DofConstraints &  constraints, DofMap &  dof_map, const unsigned int  variable_number, const Elem *  elem  )

inherited

Definition at line 577 of file fe_nedelec_one.C.

{ nedelec_one_compute_constraints(constraints, dof_map, variable_number, elem, /*Dim=*/2); }

compute_constraints() [21/29]

void libMesh::FE< 3, NEDELEC_ONE >::compute_constraints ( DofConstraints &  constraints, DofMap &  dof_map, const unsigned int  variable_number, const Elem *  elem  )

inherited

Definition at line 584 of file fe_nedelec_one.C.

{ nedelec_one_compute_constraints(constraints, dof_map, variable_number, elem, /*Dim=*/3); }

compute_constraints() [22/29]

void libMesh::FE< 2, LAGRANGE >::compute_constraints ( DofConstraints &  constraints, DofMap &  dof_map, const unsigned int  variable_number, const Elem *  elem  )

inherited

Definition at line 900 of file fe_lagrange.C.

{ lagrange_compute_constraints(constraints, dof_map, variable_number, elem, /*Dim=*/2); }

compute_constraints() [23/29]

void libMesh::FE< 3, LAGRANGE >::compute_constraints ( DofConstraints &  constraints, DofMap &  dof_map, const unsigned int  variable_number, const Elem *  elem  )

inherited

Definition at line 907 of file fe_lagrange.C.

{ lagrange_compute_constraints(constraints, dof_map, variable_number, elem, /*Dim=*/3); }

compute_constraints() [24/29]

void libMesh::FE< 2, XYZ >::compute_constraints ( DofConstraints &  , DofMap &  , const unsigned  int, const Elem *    )

inherited

Definition at line 942 of file fe_xyz.C.

{}

compute_constraints() [25/29]

void libMesh::FE< 3, XYZ >::compute_constraints ( DofConstraints &  , DofMap &  , const unsigned  int, const Elem *    )

inherited

Definition at line 943 of file fe_xyz.C.

{}

compute_constraints() [26/29]

void libMesh::FE< 2, LAGRANGE_VEC >::compute_constraints ( DofConstraints &  constraints, DofMap &  dof_map, const unsigned int  variable_number, const Elem *  elem  )

inherited

Definition at line 950 of file fe_lagrange_vec.C.

{ //libmesh_not_implemented();
  FEVectorBase::compute_proj_constraints(constraints, dof_map, variable_number, elem);
}

compute_constraints() [27/29]

void libMesh::FE< 3, LAGRANGE_VEC >::compute_constraints ( DofConstraints &  constraints, DofMap &  dof_map, const unsigned int  variable_number, const Elem *  elem  )

inherited

Definition at line 959 of file fe_lagrange_vec.C.

{ //libmesh_not_implemented();
  FEVectorBase::compute_proj_constraints(constraints, dof_map, variable_number, elem);
}

compute_constraints() [28/29]

void libMesh::FE< 2, SZABAB >::compute_constraints ( DofConstraints &  constraints, DofMap &  dof_map, const unsigned int  variable_number, const Elem *  elem  )

inherited

Definition at line 1287 of file fe_szabab.C.

{ compute_proj_constraints(constraints, dof_map, variable_number, elem); }

compute_constraints() [29/29]

void libMesh::FE< 3, SZABAB >::compute_constraints ( DofConstraints &  constraints, DofMap &  dof_map, const unsigned int  variable_number, const Elem *  elem  )

inherited

Definition at line 1294 of file fe_szabab.C.

{ compute_proj_constraints(constraints, dof_map, variable_number, elem); }

compute_node_constraints()

void libMesh::FEAbstract::compute_node_constraints ( NodeConstraints &  constraints, const Elem *  elem  )

staticinherited

Computes the nodal constraint contributions (for non-conforming adapted meshes), using Lagrange geometry

Definition at line 820 of file fe_abstract.C.

References std::abs(), libMesh::Elem::build_side_ptr(), libMesh::Elem::default_order(), libMesh::Elem::dim(), libMesh::FEAbstract::fe_type, libMesh::FEInterface::inverse_map(), libMesh::LAGRANGE, libMesh::Elem::level(), libMesh::FEInterface::n_dofs(), libMesh::Elem::neighbor_ptr(), libMesh::Elem::parent(), libMesh::Real, libMesh::remote_elem, libMesh::FEInterface::shape(), libMesh::Elem::side_index_range(), libMesh::Threads::spin_mtx, and libMesh::Elem::subactive().

{
  libmesh_assert(elem);

  const unsigned int Dim = elem->dim();

  // Only constrain elements in 2,3D.
  if (Dim == 1)
    return;

  // Only constrain active and ancestor elements
  if (elem->subactive())
    return;

  // We currently always use LAGRANGE mappings for geometry
  const FEType fe_type(elem->default_order(), LAGRANGE);

  // Pull objects out of the loop to reduce heap operations
  std::vector<const Node *> my_nodes, parent_nodes;
  std::unique_ptr<const Elem> my_side, parent_side;

  // Look at the element faces.  Check to see if we need to
  // build constraints.
  for (auto s : elem->side_index_range())
    if (elem->neighbor_ptr(s) != nullptr &&
        elem->neighbor_ptr(s) != remote_elem)
      if (elem->neighbor_ptr(s)->level() < elem->level()) // constrain dofs shared between
        {                                                 // this element and ones coarser
          // than this element.
          // Get pointers to the elements of interest and its parent.
          const Elem * parent = elem->parent();

          // This can't happen...  Only level-0 elements have nullptr
          // parents, and no level-0 elements can be at a higher
          // level than their neighbors!
          libmesh_assert(parent);

          elem->build_side_ptr(my_side, s);
          parent->build_side_ptr(parent_side, s);

          const unsigned int n_side_nodes = my_side->n_nodes();

          my_nodes.clear();
          my_nodes.reserve (n_side_nodes);
          parent_nodes.clear();
          parent_nodes.reserve (n_side_nodes);

          for (unsigned int n=0; n != n_side_nodes; ++n)
            my_nodes.push_back(my_side->node_ptr(n));

          for (unsigned int n=0; n != n_side_nodes; ++n)
            parent_nodes.push_back(parent_side->node_ptr(n));

          for (unsigned int my_side_n=0;
               my_side_n < n_side_nodes;
               my_side_n++)
            {
              libmesh_assert_less (my_side_n, FEInterface::n_dofs(Dim-1, fe_type, my_side->type()));

              const Node * my_node = my_nodes[my_side_n];

              // The support point of the DOF
              const Point & support_point = *my_node;

              // Figure out where my node lies on their reference element.
              const Point mapped_point = FEInterface::inverse_map(Dim-1, fe_type,
                                                                  parent_side.get(),
                                                                  support_point);

              // Compute the parent's side shape function values.
              for (unsigned int their_side_n=0;
                   their_side_n < n_side_nodes;
                   their_side_n++)
                {
                  libmesh_assert_less (their_side_n, FEInterface::n_dofs(Dim-1, fe_type, parent_side->type()));

                  const Node * their_node = parent_nodes[their_side_n];
                  libmesh_assert(their_node);

                  const Real their_value = FEInterface::shape(Dim-1,
                                                              fe_type,
                                                              parent_side->type(),
                                                              their_side_n,
                                                              mapped_point);

                  const Real their_mag = std::abs(their_value);
#ifdef DEBUG
                  // Protect for the case u_i ~= u_j,
                  // in which case i better equal j.
                  if (their_mag > 0.999)
                    {
                      libmesh_assert_equal_to (my_node, their_node);
                      libmesh_assert_less (std::abs(their_value - 1.), 0.001);
                    }
                  else
#endif
                    // To make nodal constraints useful for constructing
                    // sparsity patterns faster, we need to get EVERY
                    // POSSIBLE constraint coupling identified, even if
                    // there is no coupling in the isoparametric
                    // Lagrange case.
                    if (their_mag < 1.e-5)
                      {
                        // since we may be running this method concurrently
                        // on multiple threads we need to acquire a lock
                        // before modifying the shared constraint_row object.
                        Threads::spin_mutex::scoped_lock lock(Threads::spin_mtx);

                        // A reference to the constraint row.
                        NodeConstraintRow & constraint_row = constraints[my_node].first;

                        constraint_row.insert(std::make_pair (their_node,
                                                              0.));
                      }
                  // To get nodal coordinate constraints right, only
                  // add non-zero and non-identity values for Lagrange
                  // basis functions.
                    else // (1.e-5 <= their_mag <= .999)
                      {
                        // since we may be running this method concurrently
                        // on multiple threads we need to acquire a lock
                        // before modifying the shared constraint_row object.
                        Threads::spin_mutex::scoped_lock lock(Threads::spin_mtx);

                        // A reference to the constraint row.
                        NodeConstraintRow & constraint_row = constraints[my_node].first;

                        constraint_row.insert(std::make_pair (their_node,
                                                              their_value));
                      }
                }
            }
        }
}

compute_periodic_constraints()

void libMesh::FEGenericBase< FEOutputType< T >::type >::compute_periodic_constraints ( DofConstraints &  constraints, DofMap &  dof_map, const PeriodicBoundaries &  boundaries, const MeshBase &  mesh, const PointLocatorBase *  point_locator, const unsigned int  variable_number, const Elem *  elem  )

staticinherited

Computes the constraint matrix contributions (for meshes with periodic boundary conditions) corresponding to variable number var_number, using generic projections.

Definition at line 1650 of file fe_base.C.

References std::abs(), libMesh::TypeVector< T >::absolute_fuzzy_equals(), libMesh::Elem::active(), libMesh::PeriodicBoundaries::boundary(), libMesh::BoundaryInfo::boundary_ids(), libMesh::C_ONE, libMesh::C_ZERO, libMesh::DenseMatrix< T >::cholesky_solve(), libMesh::DofMap::constrain_p_dofs(), libMesh::FEType::default_quadrature_order(), libMesh::Elem::dim(), libMesh::DISCONTINUOUS, libMesh::DofMap::dof_indices(), libMesh::DofObject::dof_number(), libMesh::FEInterface::dofs_on_side(), libMesh::MeshBase::get_boundary_info(), libMesh::PeriodicBoundaryBase::get_corresponding_pos(), libMesh::PeriodicBoundaryBase::get_transformation_matrix(), libMesh::PeriodicBoundaryBase::get_variables(), libMesh::PeriodicBoundaryBase::has_transformation_matrix(), libMesh::Elem::hmin(), libMesh::DofObject::id(), libMesh::index_range(), libMesh::TensorTools::inner_product(), libMesh::DofObject::invalid_id, libMesh::invalid_uint, libMesh::FEInterface::inverse_map(), libMesh::DofMap::is_constrained_dof(), libMesh::Elem::is_edge(), libMesh::Elem::is_face(), libMesh::PeriodicBoundaryBase::is_my_variable(), libMesh::Elem::is_node_on_edge(), libMesh::Elem::is_node_on_side(), libMesh::Elem::is_vertex(), libMesh::Elem::level(), mesh, std::min(), libMesh::Elem::min_p_level_by_neighbor(), libMesh::DofObject::n_comp(), libMesh::Elem::n_edges(), libMesh::Elem::n_nodes(), libMesh::Elem::n_sides(), libMesh::PeriodicBoundaries::neighbor(), libMesh::Elem::neighbor_ptr(), libMesh::Elem::node_ptr(), libMesh::Elem::node_ref(), libMesh::Elem::p_level(), libMesh::PeriodicBoundaryBase::pairedboundary, libMesh::Real, libMesh::DenseVector< T >::resize(), libMesh::DenseMatrix< T >::resize(), libMesh::BoundaryInfo::side_with_boundary_id(), libMesh::Threads::spin_mtx, swap(), libMesh::DofMap::sys_number(), libMesh::TOLERANCE, and libMesh::DofMap::variable_type().

{
  // Only bother if we truly have periodic boundaries
  if (boundaries.empty())
    return;

  libmesh_assert(elem);

  // Only constrain active elements with this method
  if (!elem->active())
    return;

  const unsigned int Dim = elem->dim();

  // We need sys_number and variable_number for DofObject methods
  // later
  const unsigned int sys_number = dof_map.sys_number();

  const FEType & base_fe_type = dof_map.variable_type(variable_number);

  // Construct FE objects for this element and its pseudo-neighbors.
  std::unique_ptr<FEGenericBase<OutputShape>> my_fe
    (FEGenericBase<OutputShape>::build(Dim, base_fe_type));
  const FEContinuity cont = my_fe->get_continuity();

  // We don't need to constrain discontinuous elements
  if (cont == DISCONTINUOUS)
    return;
  libmesh_assert (cont == C_ZERO || cont == C_ONE);

  // We'll use element size to generate relative tolerances later
  const Real primary_hmin = elem->hmin();

  std::unique_ptr<FEGenericBase<OutputShape>> neigh_fe
    (FEGenericBase<OutputShape>::build(Dim, base_fe_type));

  QGauss my_qface(Dim-1, base_fe_type.default_quadrature_order());
  my_fe->attach_quadrature_rule (&my_qface);
  std::vector<Point> neigh_qface;

  const std::vector<Real> & JxW = my_fe->get_JxW();
  const std::vector<Point> & q_point = my_fe->get_xyz();
  const std::vector<std::vector<OutputShape>> & phi = my_fe->get_phi();
  const std::vector<std::vector<OutputShape>> & neigh_phi =
    neigh_fe->get_phi();
  const std::vector<Point> * face_normals = nullptr;
  const std::vector<std::vector<OutputGradient>> * dphi = nullptr;
  const std::vector<std::vector<OutputGradient>> * neigh_dphi = nullptr;
  std::vector<dof_id_type> my_dof_indices, neigh_dof_indices;
  std::vector<unsigned int> my_side_dofs, neigh_side_dofs;

  if (cont != C_ZERO)
    {
      const std::vector<Point> & ref_face_normals =
        my_fe->get_normals();
      face_normals = &ref_face_normals;
      const std::vector<std::vector<OutputGradient>> & ref_dphi =
        my_fe->get_dphi();
      dphi = &ref_dphi;
      const std::vector<std::vector<OutputGradient>> & ref_neigh_dphi =
        neigh_fe->get_dphi();
      neigh_dphi = &ref_neigh_dphi;
    }

  DenseMatrix<Real> Ke;
  DenseVector<Real> Fe;
  std::vector<DenseVector<Real>> Ue;

  // Container to catch the boundary ids that BoundaryInfo hands us.
  std::vector<boundary_id_type> bc_ids;

  // Look at the element faces.  Check to see if we need to
  // build constraints.
  const unsigned short int max_ns = elem->n_sides();
  for (unsigned short int s = 0; s != max_ns; ++s)
    {
      if (elem->neighbor_ptr(s))
        continue;

      mesh.get_boundary_info().boundary_ids (elem, s, bc_ids);

      for (const auto & boundary_id : bc_ids)
        {
          const PeriodicBoundaryBase * periodic = boundaries.boundary(boundary_id);
          if (periodic && periodic->is_my_variable(variable_number))
            {
              libmesh_assert(point_locator);

              // Get pointers to the element's neighbor.
              const Elem * neigh = boundaries.neighbor(boundary_id, *point_locator, elem, s);

              if (neigh == nullptr)
                libmesh_error_msg("PeriodicBoundaries point locator object returned nullptr!");

              // periodic (and possibly h refinement) constraints:
              // constrain dofs shared between
              // this element and ones as coarse
              // as or coarser than this element.
              if (neigh->level() <= elem->level())
                {
                  unsigned int s_neigh =
                    mesh.get_boundary_info().side_with_boundary_id(neigh, periodic->pairedboundary);
                  libmesh_assert_not_equal_to (s_neigh, libMesh::invalid_uint);

#ifdef LIBMESH_ENABLE_AMR
                  // Find the minimum p level; we build the h constraint
                  // matrix with this and then constrain away all higher p
                  // DoFs.
                  libmesh_assert(neigh->active());
                  const unsigned int min_p_level =
                    std::min(elem->p_level(), neigh->p_level());

                  // we may need to make the FE objects reinit with the
                  // minimum shared p_level
                  // FIXME - I hate using const_cast<> and avoiding
                  // accessor functions; there's got to be a
                  // better way to do this!
                  const unsigned int old_elem_level = elem->p_level();
                  if (old_elem_level != min_p_level)
                    (const_cast<Elem *>(elem))->hack_p_level(min_p_level);
                  const unsigned int old_neigh_level = neigh->p_level();
                  if (old_neigh_level != min_p_level)
                    (const_cast<Elem *>(neigh))->hack_p_level(min_p_level);
#endif // #ifdef LIBMESH_ENABLE_AMR

                  // We can do a projection with a single integration,
                  // due to the assumption of nested finite element
                  // subspaces.
                  // FIXME: it might be more efficient to do nodes,
                  // then edges, then side, to reduce the size of the
                  // Cholesky factorization(s)
                  my_fe->reinit(elem, s);

                  dof_map.dof_indices (elem, my_dof_indices,
                                       variable_number);
                  dof_map.dof_indices (neigh, neigh_dof_indices,
                                       variable_number);

                  // We use neigh_dof_indices_all_variables in the case that the
                  // periodic boundary condition involves mappings between multiple
                  // variables.
                  std::vector<std::vector<dof_id_type>> neigh_dof_indices_all_variables;
                  if(periodic->has_transformation_matrix())
                    {
                      const std::set<unsigned int> & variables = periodic->get_variables();
                      neigh_dof_indices_all_variables.resize(variables.size());
                      unsigned int index = 0;
                      for(unsigned int var : variables)
                        {
                          dof_map.dof_indices (neigh, neigh_dof_indices_all_variables[index],
                                               var);
                          index++;
                        }
                    }

                  const unsigned int n_qp = my_qface.n_points();

                  // Translate the quadrature points over to the
                  // neighbor's boundary
                  std::vector<Point> neigh_point(q_point.size());
                  for (auto i : index_range(neigh_point))
                    neigh_point[i] = periodic->get_corresponding_pos(q_point[i]);

                  FEInterface::inverse_map (Dim, base_fe_type, neigh,
                                            neigh_point, neigh_qface);

                  neigh_fe->reinit(neigh, &neigh_qface);

                  // We're only concerned with DOFs whose values (and/or first
                  // derivatives for C1 elements) are supported on side nodes
                  FEInterface::dofs_on_side(elem, Dim, base_fe_type, s, my_side_dofs);
                  FEInterface::dofs_on_side(neigh, Dim, base_fe_type, s_neigh, neigh_side_dofs);

                  // We're done with functions that examine Elem::p_level(),
                  // so let's unhack those levels
#ifdef LIBMESH_ENABLE_AMR
                  if (elem->p_level() != old_elem_level)
                    (const_cast<Elem *>(elem))->hack_p_level(old_elem_level);
                  if (neigh->p_level() != old_neigh_level)
                    (const_cast<Elem *>(neigh))->hack_p_level(old_neigh_level);
#endif // #ifdef LIBMESH_ENABLE_AMR

                  const unsigned int n_side_dofs =
                    cast_int<unsigned int>
                    (my_side_dofs.size());
                  libmesh_assert_equal_to (n_side_dofs, neigh_side_dofs.size());

                  Ke.resize (n_side_dofs, n_side_dofs);
                  Ue.resize(n_side_dofs);

                  // Form the projection matrix, (inner product of fine basis
                  // functions against fine test functions)
                  for (unsigned int is = 0; is != n_side_dofs; ++is)
                    {
                      const unsigned int i = my_side_dofs[is];
                      for (unsigned int js = 0; js != n_side_dofs; ++js)
                        {
                          const unsigned int j = my_side_dofs[js];
                          for (unsigned int qp = 0; qp != n_qp; ++qp)
                            {
                              Ke(is,js) += JxW[qp] *
                                TensorTools::inner_product(phi[i][qp],
                                                           phi[j][qp]);
                              if (cont != C_ZERO)
                                Ke(is,js) += JxW[qp] *
                                  TensorTools::inner_product((*dphi)[i][qp] *
                                                             (*face_normals)[qp],
                                                             (*dphi)[j][qp] *
                                                             (*face_normals)[qp]);
                            }
                        }
                    }

                  // Form the right hand sides, (inner product of coarse basis
                  // functions against fine test functions)
                  for (unsigned int is = 0; is != n_side_dofs; ++is)
                    {
                      const unsigned int i = neigh_side_dofs[is];
                      Fe.resize (n_side_dofs);
                      for (unsigned int js = 0; js != n_side_dofs; ++js)
                        {
                          const unsigned int j = my_side_dofs[js];
                          for (unsigned int qp = 0; qp != n_qp; ++qp)
                            {
                              Fe(js) += JxW[qp] *
                                TensorTools::inner_product(neigh_phi[i][qp],
                                                           phi[j][qp]);
                              if (cont != C_ZERO)
                                Fe(js) += JxW[qp] *
                                  TensorTools::inner_product((*neigh_dphi)[i][qp] *
                                                             (*face_normals)[qp],
                                                             (*dphi)[j][qp] *
                                                             (*face_normals)[qp]);
                            }
                        }
                      Ke.cholesky_solve(Fe, Ue[is]);
                    }

                  // Make sure we're not adding recursive constraints
                  // due to the redundancy in the way we add periodic
                  // boundary constraints
                  //
                  // In order for this to work while threaded or on
                  // distributed meshes, we need a rigorous way to
                  // avoid recursive constraints.  Here it is:
                  //
                  // For vertex DoFs, if there is a "prior" element
                  // (i.e. a coarser element or an equally refined
                  // element with a lower id) on this boundary which
                  // contains the vertex point, then we will avoid
                  // generating constraints; the prior element (or
                  // something prior to it) may do so.  If we are the
                  // most prior (or "primary") element on this
                  // boundary sharing this point, then we look at the
                  // boundary periodic to us, we find the primary
                  // element there, and if that primary is coarser or
                  // equal-but-lower-id, then our vertex dofs are
                  // constrained in terms of that element.
                  //
                  // For edge DoFs, if there is a coarser element
                  // on this boundary sharing this edge, then we will
                  // avoid generating constraints (we will be
                  // constrained indirectly via AMR constraints
                  // connecting us to the coarser element's DoFs).  If
                  // we are the coarsest element sharing this edge,
                  // then we generate constraints if and only if we
                  // are finer than the coarsest element on the
                  // boundary periodic to us sharing the corresponding
                  // periodic edge, or if we are at equal level but
                  // our edge nodes have higher ids than the periodic
                  // edge nodes (sorted from highest to lowest, then
                  // compared lexicographically)
                  //
                  // For face DoFs, we generate constraints if we are
                  // finer than our periodic neighbor, or if we are at
                  // equal level but our element id is higher than its
                  // element id.
                  //
                  // If the primary neighbor is also the current elem
                  // (a 1-element-thick mesh) then we choose which
                  // vertex dofs to constrain via lexicographic
                  // ordering on point locations

                  // FIXME: This code doesn't yet properly handle
                  // cases where multiple different periodic BCs
                  // intersect.
                  std::set<dof_id_type> my_constrained_dofs;

                  // Container to catch boundary IDs handed back by BoundaryInfo.
                  std::vector<boundary_id_type> new_bc_ids;

                  for (unsigned int n = 0; n != elem->n_nodes(); ++n)
                    {
                      if (!elem->is_node_on_side(n,s))
                        continue;

                      const Node & my_node = elem->node_ref(n);

                      if (elem->is_vertex(n))
                        {
                          // Find all boundary ids that include this
                          // point and have periodic boundary
                          // conditions for this variable
                          std::set<boundary_id_type> point_bcids;

                          for (unsigned int new_s = 0;
                               new_s != max_ns; ++new_s)
                            {
                              if (!elem->is_node_on_side(n,new_s))
                                continue;

                              mesh.get_boundary_info().boundary_ids (elem, s, new_bc_ids);

                              for (const auto & new_boundary_id : new_bc_ids)
                                {
                                  const PeriodicBoundaryBase * new_periodic = boundaries.boundary(new_boundary_id);
                                  if (new_periodic && new_periodic->is_my_variable(variable_number))
                                    point_bcids.insert(new_boundary_id);
                                }
                            }

                          // See if this vertex has point neighbors to
                          // defer to
                          if (primary_boundary_point_neighbor
                              (elem, my_node, mesh.get_boundary_info(), point_bcids)
                              != elem)
                            continue;

                          // Find the complementary boundary id set
                          std::set<boundary_id_type> point_pairedids;
                          for (const auto & new_boundary_id : point_bcids)
                            {
                              const PeriodicBoundaryBase * new_periodic = boundaries.boundary(new_boundary_id);
                              point_pairedids.insert(new_periodic->pairedboundary);
                            }

                          // What do we want to constrain against?
                          const Elem * primary_elem = nullptr;
                          const Elem * main_neigh = nullptr;
                          Point main_pt = my_node,
                            primary_pt = my_node;

                          for (const auto & new_boundary_id : point_bcids)
                            {
                              // Find the corresponding periodic point and
                              // its primary neighbor
                              const PeriodicBoundaryBase * new_periodic = boundaries.boundary(new_boundary_id);

                              const Point neigh_pt =
                                new_periodic->get_corresponding_pos(my_node);

                              // If the point is getting constrained
                              // to itself by this PBC then we don't
                              // generate any constraints
                              if (neigh_pt.absolute_fuzzy_equals
                                  (my_node, primary_hmin*TOLERANCE))
                                continue;

                              // Otherwise we'll have a constraint in
                              // one direction or another
                              if (!primary_elem)
                                primary_elem = elem;

                              const Elem * primary_neigh =
                                primary_boundary_point_neighbor(neigh, neigh_pt,
                                                                mesh.get_boundary_info(),
                                                                point_pairedids);

                              libmesh_assert(primary_neigh);

                              if (new_boundary_id == boundary_id)
                                {
                                  main_neigh = primary_neigh;
                                  main_pt = neigh_pt;
                                }

                              // Finer elements will get constrained in
                              // terms of coarser neighbors, not the
                              // other way around
                              if ((primary_neigh->level() > primary_elem->level()) ||

                                  // For equal-level elements, the one with
                                  // higher id gets constrained in terms of
                                  // the one with lower id
                                  (primary_neigh->level() == primary_elem->level() &&
                                   primary_neigh->id() > primary_elem->id()) ||

                                  // On a one-element-thick mesh, we compare
                                  // points to see what side gets constrained
                                  (primary_neigh == primary_elem &&
                                   (neigh_pt > primary_pt)))
                                continue;

                              primary_elem = primary_neigh;
                              primary_pt = neigh_pt;
                            }

                          if (!primary_elem ||
                              primary_elem != main_neigh ||
                              primary_pt != main_pt)
                            continue;
                        }
                      else if (elem->is_edge(n))
                        {
                          // Find which edge we're on
                          unsigned int e=0;
                          for (; e != elem->n_edges(); ++e)
                            {
                              if (elem->is_node_on_edge(n,e))
                                break;
                            }
                          libmesh_assert_less (e, elem->n_edges());

                          // Find the edge end nodes
                          const Node
                            * e1 = nullptr,
                            * e2 = nullptr;
                          for (unsigned int nn = 0; nn != elem->n_nodes(); ++nn)
                            {
                              if (nn == n)
                                continue;

                              if (elem->is_node_on_edge(nn, e))
                                {
                                  if (e1 == nullptr)
                                    {
                                      e1 = elem->node_ptr(nn);
                                    }
                                  else
                                    {
                                      e2 = elem->node_ptr(nn);
                                      break;
                                    }
                                }
                            }
                          libmesh_assert (e1 && e2);

                          // Find all boundary ids that include this
                          // edge and have periodic boundary
                          // conditions for this variable
                          std::set<boundary_id_type> edge_bcids;

                          for (unsigned int new_s = 0;
                               new_s != max_ns; ++new_s)
                            {
                              if (!elem->is_node_on_side(n,new_s))
                                continue;

                              // We're reusing the new_bc_ids vector created outside the loop over nodes.
                              mesh.get_boundary_info().boundary_ids (elem, s, new_bc_ids);

                              for (const auto & new_boundary_id : new_bc_ids)
                                {
                                  const PeriodicBoundaryBase * new_periodic = boundaries.boundary(new_boundary_id);
                                  if (new_periodic && new_periodic->is_my_variable(variable_number))
                                    edge_bcids.insert(new_boundary_id);
                                }
                            }


                          // See if this edge has neighbors to defer to
                          if (primary_boundary_edge_neighbor
                              (elem, *e1, *e2, mesh.get_boundary_info(), edge_bcids)
                              != elem)
                            continue;

                          // Find the complementary boundary id set
                          std::set<boundary_id_type> edge_pairedids;
                          for (const auto & new_boundary_id : edge_bcids)
                            {
                              const PeriodicBoundaryBase * new_periodic = boundaries.boundary(new_boundary_id);
                              edge_pairedids.insert(new_periodic->pairedboundary);
                            }

                          // What do we want to constrain against?
                          const Elem * primary_elem = nullptr;
                          const Elem * main_neigh = nullptr;
                          Point main_pt1 = *e1,
                            main_pt2 = *e2,
                            primary_pt1 = *e1,
                            primary_pt2 = *e2;

                          for (const auto & new_boundary_id : edge_bcids)
                            {
                              // Find the corresponding periodic edge and
                              // its primary neighbor
                              const PeriodicBoundaryBase * new_periodic = boundaries.boundary(new_boundary_id);

                              Point neigh_pt1 = new_periodic->get_corresponding_pos(*e1),
                                neigh_pt2 = new_periodic->get_corresponding_pos(*e2);

                              // If the edge is getting constrained
                              // to itself by this PBC then we don't
                              // generate any constraints
                              if (neigh_pt1.absolute_fuzzy_equals
                                  (*e1, primary_hmin*TOLERANCE) &&
                                  neigh_pt2.absolute_fuzzy_equals
                                  (*e2, primary_hmin*TOLERANCE))
                                continue;

                              // Otherwise we'll have a constraint in
                              // one direction or another
                              if (!primary_elem)
                                primary_elem = elem;

                              const Elem * primary_neigh = primary_boundary_edge_neighbor
                                (neigh, neigh_pt1, neigh_pt2,
                                 mesh.get_boundary_info(), edge_pairedids);

                              libmesh_assert(primary_neigh);

                              if (new_boundary_id == boundary_id)
                                {
                                  main_neigh = primary_neigh;
                                  main_pt1 = neigh_pt1;
                                  main_pt2 = neigh_pt2;
                                }

                              // If we have a one-element thick mesh,
                              // we'll need to sort our points to get a
                              // consistent ordering rule
                              //
                              // Use >= in this test to make sure that,
                              // for angular constraints, no node gets
                              // constrained to itself.
                              if (primary_neigh == primary_elem)
                                {
                                  if (primary_pt1 > primary_pt2)
                                    std::swap(primary_pt1, primary_pt2);
                                  if (neigh_pt1 > neigh_pt2)
                                    std::swap(neigh_pt1, neigh_pt2);

                                  if (neigh_pt2 >= primary_pt2)
                                    continue;
                                }

                              // Otherwise:
                              // Finer elements will get constrained in
                              // terms of coarser ones, not the other way
                              // around
                              if ((primary_neigh->level() > primary_elem->level()) ||

                                  // For equal-level elements, the one with
                                  // higher id gets constrained in terms of
                                  // the one with lower id
                                  (primary_neigh->level() == primary_elem->level() &&
                                   primary_neigh->id() > primary_elem->id()))
                                continue;

                              primary_elem = primary_neigh;
                              primary_pt1 = neigh_pt1;
                              primary_pt2 = neigh_pt2;
                            }

                          if (!primary_elem ||
                              primary_elem != main_neigh ||
                              primary_pt1 != main_pt1 ||
                              primary_pt2 != main_pt2)
                            continue;
                        }
                      else if (elem->is_face(n))
                        {
                          // If we have a one-element thick mesh,
                          // use the ordering of the face node and its
                          // periodic counterpart to determine what
                          // gets constrained
                          if (neigh == elem)
                            {
                              const Point neigh_pt =
                                periodic->get_corresponding_pos(my_node);
                              if (neigh_pt > my_node)
                                continue;
                            }

                          // Otherwise:
                          // Finer elements will get constrained in
                          // terms of coarser ones, not the other way
                          // around
                          if ((neigh->level() > elem->level()) ||

                              // For equal-level elements, the one with
                              // higher id gets constrained in terms of
                              // the one with lower id
                              (neigh->level() == elem->level() &&
                               neigh->id() > elem->id()))
                            continue;
                        }

                      // If we made it here without hitting a continue
                      // statement, then we're at a node whose dofs
                      // should be constrained by this element's
                      // calculations.
                      const unsigned int n_comp =
                        my_node.n_comp(sys_number, variable_number);

                      for (unsigned int i=0; i != n_comp; ++i)
                        my_constrained_dofs.insert
                          (my_node.dof_number
                           (sys_number, variable_number, i));
                    }

                  // FIXME: old code for disambiguating periodic BCs:
                  // this is not threadsafe nor safe to run on a
                  // non-serialized mesh.
                  /*
                    std::vector<bool> recursive_constraint(n_side_dofs, false);

                    for (unsigned int is = 0; is != n_side_dofs; ++is)
                    {
                    const unsigned int i = neigh_side_dofs[is];
                    const dof_id_type their_dof_g = neigh_dof_indices[i];
                    libmesh_assert_not_equal_to (their_dof_g, DofObject::invalid_id);

                    {
                    Threads::spin_mutex::scoped_lock lock(Threads::spin_mtx);

                    if (!dof_map.is_constrained_dof(their_dof_g))
                    continue;
                    }

                    DofConstraintRow & their_constraint_row =
                    constraints[their_dof_g].first;

                    for (unsigned int js = 0; js != n_side_dofs; ++js)
                    {
                    const unsigned int j = my_side_dofs[js];
                    const dof_id_type my_dof_g = my_dof_indices[j];
                    libmesh_assert_not_equal_to (my_dof_g, DofObject::invalid_id);

                    if (their_constraint_row.count(my_dof_g))
                    recursive_constraint[js] = true;
                    }
                    }
                  */

                  for (unsigned int js = 0; js != n_side_dofs; ++js)
                    {
                      // FIXME: old code path
                      // if (recursive_constraint[js])
                      //  continue;

                      const unsigned int j = my_side_dofs[js];
                      const dof_id_type my_dof_g = my_dof_indices[j];
                      libmesh_assert_not_equal_to (my_dof_g, DofObject::invalid_id);

                      // FIXME: new code path
                      if (!my_constrained_dofs.count(my_dof_g))
                        continue;

                      DofConstraintRow * constraint_row;

                      // we may be running constraint methods concurrently
                      // on multiple threads, so we need a lock to
                      // ensure that this constraint is "ours"
                      {
                        Threads::spin_mutex::scoped_lock lock(Threads::spin_mtx);

                        if (dof_map.is_constrained_dof(my_dof_g))
                          continue;

                        constraint_row = &(constraints[my_dof_g]);
                        libmesh_assert(constraint_row->empty());
                      }

                      for (unsigned int is = 0; is != n_side_dofs; ++is)
                        {
                          const unsigned int i = neigh_side_dofs[is];
                          const dof_id_type their_dof_g = neigh_dof_indices[i];
                          libmesh_assert_not_equal_to (their_dof_g, DofObject::invalid_id);

                          // Periodic constraints should never be
                          // self-constraints
                          // libmesh_assert_not_equal_to (their_dof_g, my_dof_g);

                          const Real their_dof_value = Ue[is](js);

                          if (their_dof_g == my_dof_g)
                            {
                              libmesh_assert_less (std::abs(their_dof_value-1.), 1.e-5);
                              for (unsigned int k = 0; k != n_side_dofs; ++k)
                                libmesh_assert(k == is || std::abs(Ue[k](js)) < 1.e-5);
                              continue;
                            }

                          if (std::abs(their_dof_value) < 10*TOLERANCE)
                            continue;

                          if(!periodic->has_transformation_matrix())
                            {
                              constraint_row->insert(std::make_pair(their_dof_g,
                                                                    their_dof_value));
                            }
                          else
                            {
                              // In this case the current variable is constrained in terms of other variables.
                              // We assume that all variables in this constraint have the same FE type (this
                              // is asserted below), and hence we can create the constraint row contribution
                              // by multiplying their_dof_value by the corresponding row of the transformation
                              // matrix.

                              const std::set<unsigned int> & variables = periodic->get_variables();
                              neigh_dof_indices_all_variables.resize(variables.size());
                              unsigned int index = 0;
                              for(unsigned int other_var : variables)
                                {
                                  libmesh_assert_msg(base_fe_type == dof_map.variable_type(other_var), "FE types must match for all variables involved in constraint");

                                  Real var_weighting = periodic->get_transformation_matrix()(variable_number, other_var);
                                  constraint_row->insert(std::make_pair(neigh_dof_indices_all_variables[index][i],
                                                                        var_weighting*their_dof_value));
                                  index++;
                                }
                            }

                        }
                    }
                }
              // p refinement constraints:
              // constrain dofs shared between
              // active elements and neighbors with
              // lower polynomial degrees
#ifdef LIBMESH_ENABLE_AMR
              const unsigned int min_p_level =
                neigh->min_p_level_by_neighbor(elem, elem->p_level());
              if (min_p_level < elem->p_level())
                {
                  // Adaptive p refinement of non-hierarchic bases will
                  // require more coding
                  libmesh_assert(my_fe->is_hierarchic());
                  dof_map.constrain_p_dofs(variable_number, elem,
                                           s, min_p_level);
                }
#endif // #ifdef LIBMESH_ENABLE_AMR
            }
        }
    }
}

compute_periodic_node_constraints()

void libMesh::FEAbstract::compute_periodic_node_constraints ( NodeConstraints &  constraints, const PeriodicBoundaries &  boundaries, const MeshBase &  mesh, const PointLocatorBase *  point_locator, const Elem *  elem  )

staticinherited

Computes the node position constraint equation contributions (for meshes with periodic boundary conditions)

Definition at line 965 of file fe_abstract.C.

References libMesh::Elem::active(), libMesh::PeriodicBoundaries::boundary(), libMesh::Elem::build_side_ptr(), libMesh::Elem::default_order(), libMesh::Elem::dim(), libMesh::FEAbstract::fe_type, libMesh::PeriodicBoundaryBase::get_corresponding_pos(), libMesh::invalid_uint, libMesh::FEInterface::inverse_map(), libMesh::LAGRANGE, libMesh::Elem::level(), mesh, libMesh::FEInterface::n_dofs(), libMesh::PeriodicBoundaries::neighbor(), libMesh::Elem::neighbor_ptr(), libMesh::PeriodicBoundaryBase::pairedboundary, libMesh::Real, libMesh::FEInterface::shape(), libMesh::Elem::side_index_range(), and libMesh::Threads::spin_mtx.

{
  // Only bother if we truly have periodic boundaries
  if (boundaries.empty())
    return;

  libmesh_assert(elem);

  // Only constrain active elements with this method
  if (!elem->active())
    return;

  const unsigned int Dim = elem->dim();

  // We currently always use LAGRANGE mappings for geometry
  const FEType fe_type(elem->default_order(), LAGRANGE);

  // Pull objects out of the loop to reduce heap operations
  std::vector<const Node *> my_nodes, neigh_nodes;
  std::unique_ptr<const Elem> my_side, neigh_side;

  // Look at the element faces.  Check to see if we need to
  // build constraints.
  std::vector<boundary_id_type> bc_ids;
  for (auto s : elem->side_index_range())
    {
      if (elem->neighbor_ptr(s))
        continue;

      mesh.get_boundary_info().boundary_ids (elem, s, bc_ids);
      for (const auto & boundary_id : bc_ids)
        {
          const PeriodicBoundaryBase * periodic = boundaries.boundary(boundary_id);
          if (periodic)
            {
              libmesh_assert(point_locator);

              // Get pointers to the element's neighbor.
              const Elem * neigh = boundaries.neighbor(boundary_id, *point_locator, elem, s);

              // h refinement constraints:
              // constrain dofs shared between
              // this element and ones as coarse
              // as or coarser than this element.
              if (neigh->level() <= elem->level())
                {
                  unsigned int s_neigh =
                    mesh.get_boundary_info().side_with_boundary_id(neigh, periodic->pairedboundary);
                  libmesh_assert_not_equal_to (s_neigh, libMesh::invalid_uint);

#ifdef LIBMESH_ENABLE_AMR
                  libmesh_assert(neigh->active());
#endif // #ifdef LIBMESH_ENABLE_AMR

                  elem->build_side_ptr(my_side, s);
                  neigh->build_side_ptr(neigh_side, s_neigh);

                  const unsigned int n_side_nodes = my_side->n_nodes();

                  my_nodes.clear();
                  my_nodes.reserve (n_side_nodes);
                  neigh_nodes.clear();
                  neigh_nodes.reserve (n_side_nodes);

                  for (unsigned int n=0; n != n_side_nodes; ++n)
                    my_nodes.push_back(my_side->node_ptr(n));

                  for (unsigned int n=0; n != n_side_nodes; ++n)
                    neigh_nodes.push_back(neigh_side->node_ptr(n));

                  // Make sure we're not adding recursive constraints
                  // due to the redundancy in the way we add periodic
                  // boundary constraints, or adding constraints to
                  // nodes that already have AMR constraints
                  std::vector<bool> skip_constraint(n_side_nodes, false);

                  for (unsigned int my_side_n=0;
                       my_side_n < n_side_nodes;
                       my_side_n++)
                    {
                      libmesh_assert_less (my_side_n, FEInterface::n_dofs(Dim-1, fe_type, my_side->type()));

                      const Node * my_node = my_nodes[my_side_n];

                      // Figure out where my node lies on their reference element.
                      const Point neigh_point = periodic->get_corresponding_pos(*my_node);

                      const Point mapped_point = FEInterface::inverse_map(Dim-1, fe_type,
                                                                          neigh_side.get(),
                                                                          neigh_point);

                      // If we've already got a constraint on this
                      // node, then the periodic constraint is
                      // redundant
                      {
                        Threads::spin_mutex::scoped_lock lock(Threads::spin_mtx);

                        if (constraints.count(my_node))
                          {
                            skip_constraint[my_side_n] = true;
                            continue;
                          }
                      }

                      // Compute the neighbors's side shape function values.
                      for (unsigned int their_side_n=0;
                           their_side_n < n_side_nodes;
                           their_side_n++)
                        {
                          libmesh_assert_less (their_side_n, FEInterface::n_dofs(Dim-1, fe_type, neigh_side->type()));

                          const Node * their_node = neigh_nodes[their_side_n];

                          // If there's a constraint on an opposing node,
                          // we need to see if it's constrained by
                          // *our side* making any periodic constraint
                          // on us recursive
                          {
                            Threads::spin_mutex::scoped_lock lock(Threads::spin_mtx);

                            if (!constraints.count(their_node))
                              continue;

                            const NodeConstraintRow & their_constraint_row =
                              constraints[their_node].first;

                            for (unsigned int orig_side_n=0;
                                 orig_side_n < n_side_nodes;
                                 orig_side_n++)
                              {
                                libmesh_assert_less (orig_side_n, FEInterface::n_dofs(Dim-1, fe_type, my_side->type()));

                                const Node * orig_node = my_nodes[orig_side_n];

                                if (their_constraint_row.count(orig_node))
                                  skip_constraint[orig_side_n] = true;
                              }
                          }
                        }
                    }
                  for (unsigned int my_side_n=0;
                       my_side_n < n_side_nodes;
                       my_side_n++)
                    {
                      libmesh_assert_less (my_side_n, FEInterface::n_dofs(Dim-1, fe_type, my_side->type()));

                      if (skip_constraint[my_side_n])
                        continue;

                      const Node * my_node = my_nodes[my_side_n];

                      // Figure out where my node lies on their reference element.
                      const Point neigh_point = periodic->get_corresponding_pos(*my_node);

                      // Figure out where my node lies on their reference element.
                      const Point mapped_point = FEInterface::inverse_map(Dim-1, fe_type,
                                                                          neigh_side.get(),
                                                                          neigh_point);

                      for (unsigned int their_side_n=0;
                           their_side_n < n_side_nodes;
                           their_side_n++)
                        {
                          libmesh_assert_less (their_side_n, FEInterface::n_dofs(Dim-1, fe_type, neigh_side->type()));

                          const Node * their_node = neigh_nodes[their_side_n];
                          libmesh_assert(their_node);

                          const Real their_value = FEInterface::shape(Dim-1,
                                                                      fe_type,
                                                                      neigh_side->type(),
                                                                      their_side_n,
                                                                      mapped_point);

                          // since we may be running this method concurrently
                          // on multiple threads we need to acquire a lock
                          // before modifying the shared constraint_row object.
                          {
                            Threads::spin_mutex::scoped_lock lock(Threads::spin_mtx);

                            NodeConstraintRow & constraint_row =
                              constraints[my_node].first;

                            constraint_row.insert(std::make_pair(their_node,
                                                                 their_value));
                          }
                        }
                    }
                }
            }
        }
    }
}

compute_proj_constraints()

void libMesh::FEGenericBase< FEOutputType< T >::type >::compute_proj_constraints ( DofConstraints &  constraints, DofMap &  dof_map, const unsigned int  variable_number, const Elem *  elem  )

staticinherited

Computes the constraint matrix contributions (for non-conforming adapted meshes) corresponding to variable number var_number, using generic projections.

Definition at line 1366 of file fe_base.C.

References std::abs(), libMesh::Elem::active(), libMesh::C_ONE, libMesh::C_ZERO, libMesh::DenseMatrix< T >::cholesky_solve(), libMesh::DofMap::constrain_p_dofs(), libMesh::FEType::default_quadrature_order(), libMesh::Elem::dim(), libMesh::DISCONTINUOUS, libMesh::DofMap::dof_indices(), libMesh::FEInterface::dofs_on_side(), libMesh::OrderWrapper::get_order(), libMesh::TensorTools::inner_product(), libMesh::DofObject::invalid_id, libMesh::FEInterface::inverse_map(), libMesh::DofMap::is_constrained_dof(), libMesh::Elem::level(), std::min(), libMesh::Elem::min_p_level_by_neighbor(), libMesh::Elem::n_neighbors(), libMesh::Elem::n_nodes(), libMesh::Elem::neighbor_ptr(), libMesh::FEType::order, libMesh::Elem::p_level(), libMesh::Real, libMesh::DenseVector< T >::resize(), libMesh::DenseMatrix< T >::resize(), libMesh::Elem::side_index_range(), libMesh::Threads::spin_mtx, libMesh::TOLERANCE, libMesh::DofMap::variable_type(), and libMesh::Elem::which_neighbor_am_i().

{
  libmesh_assert(elem);

  const unsigned int Dim = elem->dim();

  // Only constrain elements in 2,3D.
  if (Dim == 1)
    return;

  // Only constrain active elements with this method
  if (!elem->active())
    return;

  const FEType & base_fe_type = dof_map.variable_type(variable_number);

  // Construct FE objects for this element and its neighbors.
  std::unique_ptr<FEGenericBase<OutputShape>> my_fe
    (FEGenericBase<OutputShape>::build(Dim, base_fe_type));
  const FEContinuity cont = my_fe->get_continuity();

  // We don't need to constrain discontinuous elements
  if (cont == DISCONTINUOUS)
    return;
  libmesh_assert (cont == C_ZERO || cont == C_ONE);

  std::unique_ptr<FEGenericBase<OutputShape>> neigh_fe
    (FEGenericBase<OutputShape>::build(Dim, base_fe_type));

  QGauss my_qface(Dim-1, base_fe_type.default_quadrature_order());
  my_fe->attach_quadrature_rule (&my_qface);
  std::vector<Point> neigh_qface;

  const std::vector<Real> & JxW = my_fe->get_JxW();
  const std::vector<Point> & q_point = my_fe->get_xyz();
  const std::vector<std::vector<OutputShape>> & phi = my_fe->get_phi();
  const std::vector<std::vector<OutputShape>> & neigh_phi =
    neigh_fe->get_phi();
  const std::vector<Point> * face_normals = nullptr;
  const std::vector<std::vector<OutputGradient>> * dphi = nullptr;
  const std::vector<std::vector<OutputGradient>> * neigh_dphi = nullptr;

  std::vector<dof_id_type> my_dof_indices, neigh_dof_indices;
  std::vector<unsigned int> my_side_dofs, neigh_side_dofs;

  if (cont != C_ZERO)
    {
      const std::vector<Point> & ref_face_normals =
        my_fe->get_normals();
      face_normals = &ref_face_normals;
      const std::vector<std::vector<OutputGradient>> & ref_dphi =
        my_fe->get_dphi();
      dphi = &ref_dphi;
      const std::vector<std::vector<OutputGradient>> & ref_neigh_dphi =
        neigh_fe->get_dphi();
      neigh_dphi = &ref_neigh_dphi;
    }

  DenseMatrix<Real> Ke;
  DenseVector<Real> Fe;
  std::vector<DenseVector<Real>> Ue;

  // Look at the element faces.  Check to see if we need to
  // build constraints.
  for (auto s : elem->side_index_range())
    if (elem->neighbor_ptr(s) != nullptr)
      {
        // Get pointers to the element's neighbor.
        const Elem * neigh = elem->neighbor_ptr(s);

        // h refinement constraints:
        // constrain dofs shared between
        // this element and ones coarser
        // than this element.
        if (neigh->level() < elem->level())
          {
            unsigned int s_neigh = neigh->which_neighbor_am_i(elem);
            libmesh_assert_less (s_neigh, neigh->n_neighbors());

            // Find the minimum p level; we build the h constraint
            // matrix with this and then constrain away all higher p
            // DoFs.
            libmesh_assert(neigh->active());
            const unsigned int min_p_level =
              std::min(elem->p_level(), neigh->p_level());

            // we may need to make the FE objects reinit with the
            // minimum shared p_level
            const unsigned int old_elem_level = elem->p_level();
            if (elem->p_level() != min_p_level)
              my_fe->set_fe_order(my_fe->get_fe_type().order.get_order() - old_elem_level + min_p_level);
            const unsigned int old_neigh_level = neigh->p_level();
            if (old_neigh_level != min_p_level)
              neigh_fe->set_fe_order(neigh_fe->get_fe_type().order.get_order() - old_neigh_level + min_p_level);

            my_fe->reinit(elem, s);

            // This function gets called element-by-element, so there
            // will be a lot of memory allocation going on.  We can
            // at least minimize this for the case of the dof indices
            // by efficiently preallocating the requisite storage.
            // n_nodes is not necessarily n_dofs, but it is better
            // than nothing!
            my_dof_indices.reserve    (elem->n_nodes());
            neigh_dof_indices.reserve (neigh->n_nodes());

            dof_map.dof_indices (elem, my_dof_indices,
                                 variable_number,
                                 min_p_level);
            dof_map.dof_indices (neigh, neigh_dof_indices,
                                 variable_number,
                                 min_p_level);

            const unsigned int n_qp = my_qface.n_points();

            FEInterface::inverse_map (Dim, base_fe_type, neigh,
                                      q_point, neigh_qface);

            neigh_fe->reinit(neigh, &neigh_qface);

            // We're only concerned with DOFs whose values (and/or first
            // derivatives for C1 elements) are supported on side nodes
            FEType elem_fe_type = base_fe_type;
            if (old_elem_level != min_p_level)
              elem_fe_type.order = base_fe_type.order.get_order() - old_elem_level + min_p_level;
            FEType neigh_fe_type = base_fe_type;
            if (old_neigh_level != min_p_level)
              neigh_fe_type.order = base_fe_type.order.get_order() - old_neigh_level + min_p_level;
            FEInterface::dofs_on_side(elem,  Dim, elem_fe_type,  s,       my_side_dofs);
            FEInterface::dofs_on_side(neigh, Dim, neigh_fe_type, s_neigh, neigh_side_dofs);

            const unsigned int n_side_dofs =
              cast_int<unsigned int>(my_side_dofs.size());
            libmesh_assert_equal_to (n_side_dofs, neigh_side_dofs.size());

            Ke.resize (n_side_dofs, n_side_dofs);
            Ue.resize(n_side_dofs);

            // Form the projection matrix, (inner product of fine basis
            // functions against fine test functions)
            for (unsigned int is = 0; is != n_side_dofs; ++is)
              {
                const unsigned int i = my_side_dofs[is];
                for (unsigned int js = 0; js != n_side_dofs; ++js)
                  {
                    const unsigned int j = my_side_dofs[js];
                    for (unsigned int qp = 0; qp != n_qp; ++qp)
                      {
                        Ke(is,js) += JxW[qp] * TensorTools::inner_product(phi[i][qp], phi[j][qp]);
                        if (cont != C_ZERO)
                          Ke(is,js) += JxW[qp] *
                            TensorTools::inner_product((*dphi)[i][qp] *
                                                       (*face_normals)[qp],
                                                       (*dphi)[j][qp] *
                                                       (*face_normals)[qp]);
                      }
                  }
              }

            // Form the right hand sides, (inner product of coarse basis
            // functions against fine test functions)
            for (unsigned int is = 0; is != n_side_dofs; ++is)
              {
                const unsigned int i = neigh_side_dofs[is];
                Fe.resize (n_side_dofs);
                for (unsigned int js = 0; js != n_side_dofs; ++js)
                  {
                    const unsigned int j = my_side_dofs[js];
                    for (unsigned int qp = 0; qp != n_qp; ++qp)
                      {
                        Fe(js) += JxW[qp] *
                          TensorTools::inner_product(neigh_phi[i][qp],
                                                     phi[j][qp]);
                        if (cont != C_ZERO)
                          Fe(js) += JxW[qp] *
                            TensorTools::inner_product((*neigh_dphi)[i][qp] *
                                                       (*face_normals)[qp],
                                                       (*dphi)[j][qp] *
                                                       (*face_normals)[qp]);
                      }
                  }
                Ke.cholesky_solve(Fe, Ue[is]);
              }

            for (unsigned int js = 0; js != n_side_dofs; ++js)
              {
                const unsigned int j = my_side_dofs[js];
                const dof_id_type my_dof_g = my_dof_indices[j];
                libmesh_assert_not_equal_to (my_dof_g, DofObject::invalid_id);

                // Hunt for "constraining against myself" cases before
                // we bother creating a constraint row
                bool self_constraint = false;
                for (unsigned int is = 0; is != n_side_dofs; ++is)
                  {
                    const unsigned int i = neigh_side_dofs[is];
                    const dof_id_type their_dof_g = neigh_dof_indices[i];
                    libmesh_assert_not_equal_to (their_dof_g, DofObject::invalid_id);

                    if (their_dof_g == my_dof_g)
                      {
#ifndef NDEBUG
                        const Real their_dof_value = Ue[is](js);
                        libmesh_assert_less (std::abs(their_dof_value-1.),
                                             10*TOLERANCE);

                        for (unsigned int k = 0; k != n_side_dofs; ++k)
                          libmesh_assert(k == is ||
                                         std::abs(Ue[k](js)) <
                                         10*TOLERANCE);
#endif

                        self_constraint = true;
                        break;
                      }
                  }

                if (self_constraint)
                  continue;

                DofConstraintRow * constraint_row;

                // we may be running constraint methods concurrently
                // on multiple threads, so we need a lock to
                // ensure that this constraint is "ours"
                {
                  Threads::spin_mutex::scoped_lock lock(Threads::spin_mtx);

                  if (dof_map.is_constrained_dof(my_dof_g))
                    continue;

                  constraint_row = &(constraints[my_dof_g]);
                  libmesh_assert(constraint_row->empty());
                }

                for (unsigned int is = 0; is != n_side_dofs; ++is)
                  {
                    const unsigned int i = neigh_side_dofs[is];
                    const dof_id_type their_dof_g = neigh_dof_indices[i];
                    libmesh_assert_not_equal_to (their_dof_g, DofObject::invalid_id);
                    libmesh_assert_not_equal_to (their_dof_g, my_dof_g);

                    const Real their_dof_value = Ue[is](js);

                    if (std::abs(their_dof_value) < 10*TOLERANCE)
                      continue;

                    constraint_row->insert(std::make_pair(their_dof_g,
                                                          their_dof_value));
                  }
              }

            my_fe->set_fe_order(my_fe->get_fe_type().order.get_order() + old_elem_level - min_p_level);
            neigh_fe->set_fe_order(neigh_fe->get_fe_type().order.get_order() + old_neigh_level - min_p_level);
          }

        // p refinement constraints:
        // constrain dofs shared between
        // active elements and neighbors with
        // lower polynomial degrees
        const unsigned int min_p_level =
          neigh->min_p_level_by_neighbor(elem, elem->p_level());
        if (min_p_level < elem->p_level())
          {
            // Adaptive p refinement of non-hierarchic bases will
            // require more coding
            libmesh_assert(my_fe->is_hierarchic());
            dof_map.constrain_p_dofs(variable_number, elem,
                                     s, min_p_level);
          }
      }
}

compute_shape_functions()

void libMesh::FEGenericBase< FEOutputType< T >::type >::compute_shape_functions ( const Elem *  elem, const std::vector< Point > &  qp  )

protectedvirtualinherited

After having updated the jacobian and the transformation from local to global coordinates in FEAbstract::compute_map(), the first derivatives of the shape functions are transformed to global coordinates, giving dphi, dphidx, dphidy, and dphidz. This method should rarely be re-defined in derived classes, but still should be usable for children. Therefore, keep it protected.

Implements libMesh::FEAbstract.

Reimplemented in libMesh::FEXYZ< Dim >.

Definition at line 669 of file fe_base.C.

{
  //-------------------------------------------------------------------------
  // Compute the shape function values (and derivatives)
  // at the Quadrature points.  Note that the actual values
  // have already been computed via init_shape_functions

  // Start logging the shape function computation
  LOG_SCOPE("compute_shape_functions()", "FE");

  this->determine_calculations();

  if (calculate_phi)
    this->_fe_trans->map_phi(this->dim, elem, qp, (*this), this->phi);

  if (calculate_dphi)
    this->_fe_trans->map_dphi(this->dim, elem, qp, (*this), this->dphi,
                              this->dphidx, this->dphidy, this->dphidz);

#ifdef LIBMESH_ENABLE_SECOND_DERIVATIVES
  if (calculate_d2phi)
    this->_fe_trans->map_d2phi(this->dim, qp, (*this), this->d2phi,
                               this->d2phidx2, this->d2phidxdy, this->d2phidxdz,
                               this->d2phidy2, this->d2phidydz, this->d2phidz2);
#endif //LIBMESH_ENABLE_SECOND_DERIVATIVES

  // Only compute curl for vector-valued elements
  if (calculate_curl_phi && TypesEqual<OutputType,RealGradient>::value)
    this->_fe_trans->map_curl(this->dim, elem, qp, (*this), this->curl_phi);

  // Only compute div for vector-valued elements
  if (calculate_div_phi && TypesEqual<OutputType,RealGradient>::value)
    this->_fe_trans->map_div(this->dim, elem, qp, (*this), this->div_phi);
}

determine_calculations()

void libMesh::FEGenericBase< FEOutputType< T >::type >::determine_calculations ( )

protectedinherited

Determine which values are to be calculated, for both the FE itself and for the FEMap.

Definition at line 728 of file fe_base.C.

References libMesh::FEInterface::field_type(), and libMesh::TYPE_VECTOR.

{
  this->calculations_started = true;

  // If the user forgot to request anything, we'll be safe and
  // calculate everything:
#ifdef LIBMESH_ENABLE_SECOND_DERIVATIVES
  if (!this->calculate_phi && !this->calculate_dphi && !this->calculate_d2phi
      && !this->calculate_curl_phi && !this->calculate_div_phi)
    {
      this->calculate_phi = this->calculate_dphi = this->calculate_d2phi = this->calculate_dphiref = true;
      if (FEInterface::field_type(fe_type.family) == TYPE_VECTOR)
        {
          this->calculate_curl_phi = true;
          this->calculate_div_phi  = true;
        }
    }
#else
  if (!this->calculate_phi && !this->calculate_dphi && !this->calculate_curl_phi && !this->calculate_div_phi)
    {
      this->calculate_phi = this->calculate_dphi = this->calculate_dphiref = true;
      if (FEInterface::field_type(fe_type.family) == TYPE_VECTOR)
        {
          this->calculate_curl_phi = true;
          this->calculate_div_phi  = true;
        }
    }
#endif // LIBMESH_ENABLE_SECOND_DERIVATIVES

  // Request whichever terms are necessary from the FEMap
  if (this->calculate_phi)
    this->_fe_trans->init_map_phi(*this);

  if (this->calculate_dphiref)
    this->_fe_trans->init_map_dphi(*this);

#ifdef LIBMESH_ENABLE_SECOND_DERIVATIVES
  if (this->calculate_d2phi)
    this->_fe_trans->init_map_d2phi(*this);
#endif //LIBMESH_ENABLE_SECOND_DERIVATIVES
}

disable_print_counter_info()

void libMesh::ReferenceCounter::disable_print_counter_info ( )

staticinherited

Definition at line 106 of file reference_counter.C.

References libMesh::ReferenceCounter::_enable_print_counter.

Referenced by libMesh::LibMeshInit::LibMeshInit().

{
  _enable_print_counter = false;
  return;
}

dofs_on_edge() [1/2]

void libMesh::FE< Dim, T >::dofs_on_edge ( const Elem *const  elem, const Order  o, unsigned int  e, std::vector< unsigned int > &  di  )

staticinherited

Fills the vector di with the local degree of freedom indices associated with edge e of element elem

On a p-refined element, o should be the base order of the element.

Definition at line 103 of file fe.C.

{
  libmesh_assert(elem);
  libmesh_assert_less (e, elem->n_edges());

  di.clear();
  unsigned int nodenum = 0;
  const unsigned int n_nodes = elem->n_nodes();
  for (unsigned int n = 0; n != n_nodes; ++n)
    {
      const unsigned int n_dofs = n_dofs_at_node(elem->type(),
                                                 static_cast<Order>(o + elem->p_level()), n);
      if (elem->is_node_on_edge(n, e))
        for (unsigned int i = 0; i != n_dofs; ++i)
          di.push_back(nodenum++);
      else
        nodenum += n_dofs;
    }
}

dofs_on_edge() [2/2]

void libMesh::FE< 2, SUBDIVISION >::dofs_on_edge ( const Elem *  const, const Order  , unsigned  int, std::vector< unsigned int > &  di  )

inherited

Definition at line 907 of file fe_subdivision_2D.C.

{ di.resize(0); }

dofs_on_side() [1/2]

void libMesh::FE< Dim, T >::dofs_on_side ( const Elem *const  elem, const Order  o, unsigned int  s, std::vector< unsigned int > &  di  )

staticinherited

Fills the vector di with the local degree of freedom indices associated with side s of element elem

On a p-refined element, o should be the base order of the element.

Definition at line 77 of file fe.C.

{
  libmesh_assert(elem);
  libmesh_assert_less (s, elem->n_sides());

  di.clear();
  unsigned int nodenum = 0;
  const unsigned int n_nodes = elem->n_nodes();
  for (unsigned int n = 0; n != n_nodes; ++n)
    {
      const unsigned int n_dofs = n_dofs_at_node(elem->type(),
                                                 static_cast<Order>(o + elem->p_level()), n);
      if (elem->is_node_on_side(n, s))
        for (unsigned int i = 0; i != n_dofs; ++i)
          di.push_back(nodenum++);
      else
        nodenum += n_dofs;
    }
}

dofs_on_side() [2/2]

void libMesh::FE< 2, SUBDIVISION >::dofs_on_side ( const Elem *  const, const Order  , unsigned  int, std::vector< unsigned int > &  di  )

inherited

Definition at line 906 of file fe_subdivision_2D.C.

{ di.resize(0); }

edge_reinit() [1/2]

void libMesh::FE< Dim, T >::edge_reinit ( const Elem *  elem, const unsigned int  edge, const Real  tolerance = TOLERANCE, const std::vector< Point > *const  pts = nullptr, const std::vector< Real > *const  weights = nullptr  )

overridevirtualinherited

Reinitializes all the physical element-dependent data based on the edge. The tolerance parameter is passed to the involved call to inverse_map(). By default the shape functions and associated data are computed at the quadrature points specified by the quadrature rule qrule, but may be any points specified on the reference side element specified in the optional argument pts.

Implements libMesh::FEAbstract.

Definition at line 260 of file fe_boundary.C.

{
  libmesh_assert(elem);
  libmesh_assert (this->qrule != nullptr || pts != nullptr);
  // We don't do this for 1D elements!
  libmesh_assert_not_equal_to (Dim, 1);

  // We're (possibly re-) calculating now!  Time to determine what.
  // FIXME - we currently just assume that we're using JxW and calling
  // edge_map later.
  this->_fe_map->calculations_started = false;
  this->_fe_map->get_JxW();
  this->_fe_map->get_xyz();
  this->determine_calculations();

  // Build the side of interest
  const std::unique_ptr<const Elem> edge(elem->build_edge_ptr(e));

  // Initialize the shape functions at the user-specified
  // points
  if (pts != nullptr)
    {
      // The shape functions do not correspond to the qrule
      this->shapes_on_quadrature = false;

      // Initialize the edge shape functions
      this->_fe_map->template init_edge_shape_functions<Dim> (*pts, edge.get());

      // Compute the Jacobian*Weight on the face for integration
      if (weights != nullptr)
        {
          this->_fe_map->compute_edge_map (Dim, *weights, edge.get());
        }
      else
        {
          std::vector<Real> dummy_weights (pts->size(), 1.);
          this->_fe_map->compute_edge_map (Dim, dummy_weights, edge.get());
        }
    }
  // If there are no user specified points, we use the
  // quadrature rule
  else
    {
      // initialize quadrature rule
      this->qrule->init(edge->type(), elem->p_level());

      if (this->qrule->shapes_need_reinit())
        this->shapes_on_quadrature = false;

      // We might not need to reinitialize the shape functions
      if ((this->get_type() != elem->type())                   ||
          (edge->type() != static_cast<int>(last_edge))        || // Comparison between enum and unsigned, cast the unsigned to int
          this->shapes_need_reinit()                           ||
          !this->shapes_on_quadrature)
        {
          // Set the element type
          this->elem_type = elem->type();

          // Set the last_edge
          last_edge = edge->type();

          // Initialize the edge shape functions
          this->_fe_map->template init_edge_shape_functions<Dim> (this->qrule->get_points(), edge.get());
        }

      // Compute the Jacobian*Weight on the face for integration
      this->_fe_map->compute_edge_map (Dim, this->qrule->get_weights(), edge.get());

      // The shape functions correspond to the qrule
      this->shapes_on_quadrature = true;
    }

  // make a copy of the Jacobian for integration
  const std::vector<Real> JxW_int(this->_fe_map->get_JxW());

  // Find where the integration points are located on the
  // full element.
  std::vector<Point> qp;
  this->inverse_map (elem, this->_fe_map->get_xyz(), qp, tolerance);

  // compute the shape function and derivative values
  // at the points qp
  this->reinit  (elem, &qp);

  // copy back old data
  this->_fe_map->get_JxW() = JxW_int;
}

edge_reinit() [2/2]

void libMesh::FE< 2, SUBDIVISION >::edge_reinit ( Elem const *  , unsigned int  , Real  , const std::vector< Point > *  const, const std::vector< Real > *  const  )

virtualinherited

Reinitializes all the physical element-dependent data based on the edge of the element elem. The tolerance parameter is passed to the involved call to inverse_map(). By default the element data are computed at the quadrature points specified by the quadrature rule qrule, but any set of points on the reference edge element may be specified in the optional argument pts.

Implements libMesh::FEAbstract.

Definition at line 866 of file fe_subdivision_2D.C.

{
  libmesh_not_implemented();
}

enable_print_counter_info()

void libMesh::ReferenceCounter::enable_print_counter_info ( )

staticinherited

Methods to enable/disable the reference counter output from print_info()

Definition at line 100 of file reference_counter.C.

References libMesh::ReferenceCounter::_enable_print_counter.

{
  _enable_print_counter = true;
  return;
}

get_continuity() [1/54]

FEContinuity libMesh::FE< 0, SCALAR >::get_continuity ( ) const

virtualinherited

Returns

The continuity level of the finite element.

Implements libMesh::FEAbstract.

Definition at line 107 of file fe_scalar.C.

{ return DISCONTINUOUS; }

get_continuity() [2/54]

FEContinuity libMesh::FE< 1, SCALAR >::get_continuity ( ) const

virtualinherited

Returns

The continuity level of the finite element.

Implements libMesh::FEAbstract.

Definition at line 108 of file fe_scalar.C.

{ return DISCONTINUOUS; }

get_continuity() [3/54]

FEContinuity libMesh::FE< 2, SCALAR >::get_continuity ( ) const

virtualinherited

Returns

The continuity level of the finite element.

Implements libMesh::FEAbstract.

Definition at line 109 of file fe_scalar.C.

{ return DISCONTINUOUS; }

get_continuity() [4/54]

FEContinuity libMesh::FE< 3, SCALAR >::get_continuity ( ) const

virtualinherited

Returns

The continuity level of the finite element.

Implements libMesh::FEAbstract.

Definition at line 110 of file fe_scalar.C.

{ return DISCONTINUOUS; }

get_continuity() [5/54]

FEContinuity libMesh::FE< 0, L2_HIERARCHIC >::get_continuity ( ) const

virtualinherited

Returns

The continuity level of the finite element.

Implements libMesh::FEAbstract.

Definition at line 193 of file fe_l2_hierarchic.C.

{ return DISCONTINUOUS; }

get_continuity() [6/54]

FEContinuity libMesh::FE< 1, L2_HIERARCHIC >::get_continuity ( ) const

virtualinherited

Returns

The continuity level of the finite element.

Implements libMesh::FEAbstract.

Definition at line 194 of file fe_l2_hierarchic.C.

{ return DISCONTINUOUS; }

get_continuity() [7/54]

FEContinuity libMesh::FE< 2, L2_HIERARCHIC >::get_continuity ( ) const

virtualinherited

Returns

The continuity level of the finite element.

Implements libMesh::FEAbstract.

Definition at line 195 of file fe_l2_hierarchic.C.

{ return DISCONTINUOUS; }

get_continuity() [8/54]

FEContinuity libMesh::FE< 3, L2_HIERARCHIC >::get_continuity ( ) const

virtualinherited

Returns

The continuity level of the finite element.

Implements libMesh::FEAbstract.

Definition at line 196 of file fe_l2_hierarchic.C.

{ return DISCONTINUOUS; }

get_continuity() [9/54]

virtual FEContinuity libMesh::FE< Dim, T >::get_continuity ( ) const

overridevirtualinherited

Returns

The continuity level of the finite element.

Implements libMesh::FEAbstract.

get_continuity() [10/54]

FEContinuity libMesh::FE< 0, CLOUGH >::get_continuity ( ) const

virtualinherited

Returns

The continuity level of the finite element.

Implements libMesh::FEAbstract.

Definition at line 290 of file fe_clough.C.

{ return C_ONE; }

get_continuity() [11/54]

FEContinuity libMesh::FE< 1, CLOUGH >::get_continuity ( ) const

virtualinherited

Returns

The continuity level of the finite element.

Implements libMesh::FEAbstract.

Definition at line 291 of file fe_clough.C.

{ return C_ONE; }

get_continuity() [12/54]

FEContinuity libMesh::FE< 2, CLOUGH >::get_continuity ( ) const

virtualinherited

Returns

The continuity level of the finite element.

Implements libMesh::FEAbstract.

Definition at line 292 of file fe_clough.C.

{ return C_ONE; }

get_continuity() [13/54]

FEContinuity libMesh::FE< 3, CLOUGH >::get_continuity ( ) const

virtualinherited

Returns

The continuity level of the finite element.

Implements libMesh::FEAbstract.

Definition at line 293 of file fe_clough.C.

{ return C_ONE; }

get_continuity() [14/54]

FEContinuity libMesh::FE< 0, HERMITE >::get_continuity ( ) const

virtualinherited

Returns

The continuity level of the finite element.

Implements libMesh::FEAbstract.

Definition at line 340 of file fe_hermite.C.

{ return C_ONE; }

get_continuity() [15/54]

FEContinuity libMesh::FE< 1, HERMITE >::get_continuity ( ) const

virtualinherited

Returns

The continuity level of the finite element.

Implements libMesh::FEAbstract.

Definition at line 341 of file fe_hermite.C.

{ return C_ONE; }

get_continuity() [16/54]

FEContinuity libMesh::FE< 2, HERMITE >::get_continuity ( ) const

virtualinherited

Returns

The continuity level of the finite element.

Implements libMesh::FEAbstract.

Definition at line 342 of file fe_hermite.C.

{ return C_ONE; }

get_continuity() [17/54]

FEContinuity libMesh::FE< 3, HERMITE >::get_continuity ( ) const

virtualinherited

Returns

The continuity level of the finite element.

Implements libMesh::FEAbstract.

Definition at line 343 of file fe_hermite.C.

{ return C_ONE; }

get_continuity() [18/54]

FEContinuity libMesh::FE< 0, HIERARCHIC >::get_continuity ( ) const

virtualinherited

Returns

The continuity level of the finite element.

Implements libMesh::FEAbstract.

Definition at line 369 of file fe_hierarchic.C.

{ return C_ZERO; }

get_continuity() [19/54]

FEContinuity libMesh::FE< 1, HIERARCHIC >::get_continuity ( ) const

virtualinherited

Returns

The continuity level of the finite element.

Implements libMesh::FEAbstract.

Definition at line 370 of file fe_hierarchic.C.

{ return C_ZERO; }

get_continuity() [20/54]

FEContinuity libMesh::FE< 2, HIERARCHIC >::get_continuity ( ) const

virtualinherited

Returns

The continuity level of the finite element.

Implements libMesh::FEAbstract.

Definition at line 371 of file fe_hierarchic.C.

{ return C_ZERO; }

get_continuity() [21/54]

FEContinuity libMesh::FE< 3, HIERARCHIC >::get_continuity ( ) const

virtualinherited

Returns

The continuity level of the finite element.

Implements libMesh::FEAbstract.

Definition at line 372 of file fe_hierarchic.C.

{ return C_ZERO; }

get_continuity() [22/54]

FEContinuity libMesh::FE< 0, MONOMIAL >::get_continuity ( ) const

virtualinherited

Returns

The continuity level of the finite element.

Implements libMesh::FEAbstract.

Definition at line 402 of file fe_monomial.C.

{ return DISCONTINUOUS; }

get_continuity() [23/54]

FEContinuity libMesh::FE< 1, MONOMIAL >::get_continuity ( ) const

virtualinherited

Returns

The continuity level of the finite element.

Implements libMesh::FEAbstract.

Definition at line 403 of file fe_monomial.C.

{ return DISCONTINUOUS; }

get_continuity() [24/54]

FEContinuity libMesh::FE< 2, MONOMIAL >::get_continuity ( ) const

virtualinherited

Returns

The continuity level of the finite element.

Implements libMesh::FEAbstract.

Definition at line 404 of file fe_monomial.C.

{ return DISCONTINUOUS; }

get_continuity() [25/54]

FEContinuity libMesh::FE< 3, MONOMIAL >::get_continuity ( ) const

virtualinherited

Returns

The continuity level of the finite element.

Implements libMesh::FEAbstract.

Definition at line 405 of file fe_monomial.C.

{ return DISCONTINUOUS; }

get_continuity() [26/54]

FEContinuity libMesh::FE< 0, BERNSTEIN >::get_continuity ( ) const

virtualinherited

Returns

The continuity level of the finite element.

Implements libMesh::FEAbstract.

Definition at line 436 of file fe_bernstein.C.

{ return C_ZERO; }

get_continuity() [27/54]

FEContinuity libMesh::FE< 1, BERNSTEIN >::get_continuity ( ) const

virtualinherited

Returns

The continuity level of the finite element.

Implements libMesh::FEAbstract.

Definition at line 437 of file fe_bernstein.C.

{ return C_ZERO; }

get_continuity() [28/54]

FEContinuity libMesh::FE< 2, BERNSTEIN >::get_continuity ( ) const

virtualinherited

Returns

The continuity level of the finite element.

Implements libMesh::FEAbstract.

Definition at line 438 of file fe_bernstein.C.

{ return C_ZERO; }

get_continuity() [29/54]

FEContinuity libMesh::FE< 3, BERNSTEIN >::get_continuity ( ) const

virtualinherited

Returns

The continuity level of the finite element.

Implements libMesh::FEAbstract.

Definition at line 439 of file fe_bernstein.C.

{ return C_ZERO; }

get_continuity() [30/54]

FEContinuity libMesh::FE< 0, L2_LAGRANGE >::get_continuity ( ) const

virtualinherited

Returns

The continuity level of the finite element.

Implements libMesh::FEAbstract.

Definition at line 474 of file fe_l2_lagrange.C.

{ return DISCONTINUOUS; }

get_continuity() [31/54]

FEContinuity libMesh::FE< 1, L2_LAGRANGE >::get_continuity ( ) const

virtualinherited

Returns

The continuity level of the finite element.

Implements libMesh::FEAbstract.

Definition at line 475 of file fe_l2_lagrange.C.

{ return DISCONTINUOUS; }

get_continuity() [32/54]

FEContinuity libMesh::FE< 2, L2_LAGRANGE >::get_continuity ( ) const

virtualinherited

Returns

The continuity level of the finite element.

Implements libMesh::FEAbstract.

Definition at line 476 of file fe_l2_lagrange.C.

{ return DISCONTINUOUS; }

get_continuity() [33/54]

FEContinuity libMesh::FE< 3, L2_LAGRANGE >::get_continuity ( ) const

virtualinherited

Returns

The continuity level of the finite element.

Implements libMesh::FEAbstract.

Definition at line 477 of file fe_l2_lagrange.C.

{ return DISCONTINUOUS; }

get_continuity() [34/54]

FEContinuity libMesh::FE< 0, NEDELEC_ONE >::get_continuity ( ) const

virtualinherited

Returns

The continuity level of the finite element.

Implements libMesh::FEAbstract.

Definition at line 544 of file fe_nedelec_one.C.

{ NEDELEC_LOW_D_ERROR_MESSAGE }

get_continuity() [35/54]

FEContinuity libMesh::FE< 1, NEDELEC_ONE >::get_continuity ( ) const

virtualinherited

Returns

The continuity level of the finite element.

Implements libMesh::FEAbstract.

Definition at line 545 of file fe_nedelec_one.C.

{ NEDELEC_LOW_D_ERROR_MESSAGE }

get_continuity() [36/54]

FEContinuity libMesh::FE< 2, NEDELEC_ONE >::get_continuity ( ) const

virtualinherited

Returns

The continuity level of the finite element.

Implements libMesh::FEAbstract.

Definition at line 546 of file fe_nedelec_one.C.

{ return H_CURL; }

get_continuity() [37/54]

FEContinuity libMesh::FE< 3, NEDELEC_ONE >::get_continuity ( ) const

virtualinherited

Returns

The continuity level of the finite element.

Implements libMesh::FEAbstract.

Definition at line 547 of file fe_nedelec_one.C.

{ return H_CURL; }

get_continuity() [38/54]

FEContinuity libMesh::FE< 0, LAGRANGE >::get_continuity ( ) const

virtualinherited

Returns

The continuity level of the finite element.

Implements libMesh::FEAbstract.

Definition at line 877 of file fe_lagrange.C.

{ return C_ZERO; }

get_continuity() [39/54]

FEContinuity libMesh::FE< 1, LAGRANGE >::get_continuity ( ) const

virtualinherited

Returns

The continuity level of the finite element.

Implements libMesh::FEAbstract.

Definition at line 878 of file fe_lagrange.C.

{ return C_ZERO; }

get_continuity() [40/54]

FEContinuity libMesh::FE< 2, LAGRANGE >::get_continuity ( ) const

virtualinherited

Returns

The continuity level of the finite element.

Implements libMesh::FEAbstract.

Definition at line 879 of file fe_lagrange.C.

{ return C_ZERO; }

get_continuity() [41/54]

FEContinuity libMesh::FE< 3, LAGRANGE >::get_continuity ( ) const

virtualinherited

Returns

The continuity level of the finite element.

Implements libMesh::FEAbstract.

Definition at line 880 of file fe_lagrange.C.

{ return C_ZERO; }

get_continuity() [42/54]

FEContinuity libMesh::FE< 2, SUBDIVISION >::get_continuity ( ) const

virtualinherited

Returns

The continuity level of the finite element.

Implements libMesh::FEAbstract.

Definition at line 910 of file fe_subdivision_2D.C.

{ return C_ONE; }

get_continuity() [43/54]

FEContinuity libMesh::FE< 0, XYZ >::get_continuity ( ) const

virtualinherited

Returns

The continuity level of the finite element.

Implements libMesh::FEAbstract.

Definition at line 925 of file fe_xyz.C.

{ return DISCONTINUOUS; }

get_continuity() [44/54]

FEContinuity libMesh::FE< 1, XYZ >::get_continuity ( ) const

virtualinherited

Returns

The continuity level of the finite element.

Implements libMesh::FEAbstract.

Definition at line 926 of file fe_xyz.C.

{ return DISCONTINUOUS; }

get_continuity() [45/54]

FEContinuity libMesh::FE< 0, LAGRANGE_VEC >::get_continuity ( ) const

virtualinherited

Returns

The continuity level of the finite element.

Implements libMesh::FEAbstract.

Definition at line 927 of file fe_lagrange_vec.C.

{ return C_ZERO; }

get_continuity() [46/54]

FEContinuity libMesh::FE< 2, XYZ >::get_continuity ( ) const

virtualinherited

Returns

The continuity level of the finite element.

Implements libMesh::FEAbstract.

Definition at line 927 of file fe_xyz.C.

{ return DISCONTINUOUS; }

get_continuity() [47/54]

FEContinuity libMesh::FE< 1, LAGRANGE_VEC >::get_continuity ( ) const

virtualinherited

Returns

The continuity level of the finite element.

Implements libMesh::FEAbstract.

Definition at line 928 of file fe_lagrange_vec.C.

{ return C_ZERO; }

get_continuity() [48/54]

FEContinuity libMesh::FE< 3, XYZ >::get_continuity ( ) const

virtualinherited

Returns

The continuity level of the finite element.

Implements libMesh::FEAbstract.

Definition at line 928 of file fe_xyz.C.

{ return DISCONTINUOUS; }

get_continuity() [49/54]

FEContinuity libMesh::FE< 2, LAGRANGE_VEC >::get_continuity ( ) const

virtualinherited

Returns

The continuity level of the finite element.

Implements libMesh::FEAbstract.

Definition at line 929 of file fe_lagrange_vec.C.

{ return C_ZERO; }

get_continuity() [50/54]

FEContinuity libMesh::FE< 3, LAGRANGE_VEC >::get_continuity ( ) const

virtualinherited

Returns

The continuity level of the finite element.

Implements libMesh::FEAbstract.

Definition at line 930 of file fe_lagrange_vec.C.

{ return C_ZERO; }

get_continuity() [51/54]

FEContinuity libMesh::FE< 0, SZABAB >::get_continuity ( ) const

virtualinherited

Returns

The continuity level of the finite element.

Implements libMesh::FEAbstract.

Definition at line 1273 of file fe_szabab.C.

{ return C_ZERO; }

get_continuity() [52/54]

FEContinuity libMesh::FE< 1, SZABAB >::get_continuity ( ) const

virtualinherited

Returns

The continuity level of the finite element.

Implements libMesh::FEAbstract.

Definition at line 1274 of file fe_szabab.C.

{ return C_ZERO; }

get_continuity() [53/54]

FEContinuity libMesh::FE< 2, SZABAB >::get_continuity ( ) const

virtualinherited

Returns

The continuity level of the finite element.

Implements libMesh::FEAbstract.

Definition at line 1275 of file fe_szabab.C.

{ return C_ZERO; }

get_continuity() [54/54]

FEContinuity libMesh::FE< 3, SZABAB >::get_continuity ( ) const

virtualinherited

Returns

The continuity level of the finite element.

Implements libMesh::FEAbstract.

Definition at line 1276 of file fe_szabab.C.

{ return C_ZERO; }

get_curl_phi()

const std::vector<std::vector<OutputShape> >& libMesh::FEGenericBase< FEOutputType< T >::type >::get_curl_phi ( ) const

inlineinherited

Returns

The curl of the shape function at the quadrature points.

Definition at line 223 of file fe_base.h.

References libMesh::FEAbstract::calculate_curl_phi, libMesh::FEAbstract::calculate_dphiref, libMesh::FEAbstract::calculations_started, and libMesh::FEGenericBase< OutputType >::curl_phi.

  { libmesh_assert(!calculations_started || calculate_curl_phi);
    calculate_curl_phi = calculate_dphiref = true; return curl_phi; }

get_curvatures()

const std::vector<Real>& libMesh::FEAbstract::get_curvatures ( ) const

inlineinherited

Returns

The curvatures for use in face integration.

Definition at line 391 of file fe_abstract.h.

References libMesh::FEAbstract::_fe_map.

  { return this->_fe_map->get_curvatures();}

get_d2phi()

const std::vector<std::vector<OutputTensor> >& libMesh::FEGenericBase< FEOutputType< T >::type >::get_d2phi ( ) const

inlineinherited

Returns

The shape function second derivatives at the quadrature points.

Definition at line 289 of file fe_base.h.

References libMesh::FEAbstract::calculate_d2phi, libMesh::FEAbstract::calculate_dphiref, libMesh::FEAbstract::calculations_started, and libMesh::FEGenericBase< OutputType >::d2phi.

  { libmesh_assert(!calculations_started || calculate_d2phi);
    calculate_d2phi = calculate_dphiref = true; return d2phi; }

get_d2phideta2()

const std::vector<std::vector<OutputShape> >& libMesh::FEGenericBase< FEOutputType< T >::type >::get_d2phideta2 ( ) const

inlineinherited

Returns

The shape function second derivatives at the quadrature points, in reference coordinates

Definition at line 369 of file fe_base.h.

References libMesh::FEAbstract::calculate_d2phi, libMesh::FEAbstract::calculate_dphiref, libMesh::FEAbstract::calculations_started, and libMesh::FEGenericBase< OutputType >::d2phideta2.

  { libmesh_assert(!calculations_started || calculate_d2phi);
    calculate_d2phi = calculate_dphiref = true; return d2phideta2; }

get_d2phidetadzeta()

const std::vector<std::vector<OutputShape> >& libMesh::FEGenericBase< FEOutputType< T >::type >::get_d2phidetadzeta ( ) const

inlineinherited

Returns

The shape function second derivatives at the quadrature points, in reference coordinates

Definition at line 377 of file fe_base.h.

References libMesh::FEAbstract::calculate_d2phi, libMesh::FEAbstract::calculate_dphiref, libMesh::FEAbstract::calculations_started, and libMesh::FEGenericBase< OutputType >::d2phidetadzeta.

  { libmesh_assert(!calculations_started || calculate_d2phi);
    calculate_d2phi = calculate_dphiref = true; return d2phidetadzeta; }

get_d2phidx2()

const std::vector<std::vector<OutputShape> >& libMesh::FEGenericBase< FEOutputType< T >::type >::get_d2phidx2 ( ) const

inlineinherited

Returns

The shape function second derivatives at the quadrature points.

Definition at line 297 of file fe_base.h.

References libMesh::FEAbstract::calculate_d2phi, libMesh::FEAbstract::calculate_dphiref, libMesh::FEAbstract::calculations_started, and libMesh::FEGenericBase< OutputType >::d2phidx2.

  { libmesh_assert(!calculations_started || calculate_d2phi);
    calculate_d2phi = calculate_dphiref = true; return d2phidx2; }

get_d2phidxdy()

const std::vector<std::vector<OutputShape> >& libMesh::FEGenericBase< FEOutputType< T >::type >::get_d2phidxdy ( ) const

inlineinherited

Returns

The shape function second derivatives at the quadrature points.

Definition at line 305 of file fe_base.h.

References libMesh::FEAbstract::calculate_d2phi, libMesh::FEAbstract::calculate_dphiref, libMesh::FEAbstract::calculations_started, and libMesh::FEGenericBase< OutputType >::d2phidxdy.

  { libmesh_assert(!calculations_started || calculate_d2phi);
    calculate_d2phi = calculate_dphiref = true; return d2phidxdy; }

get_d2phidxdz()

const std::vector<std::vector<OutputShape> >& libMesh::FEGenericBase< FEOutputType< T >::type >::get_d2phidxdz ( ) const

inlineinherited

Returns

The shape function second derivatives at the quadrature points.

Definition at line 313 of file fe_base.h.

References libMesh::FEAbstract::calculate_d2phi, libMesh::FEAbstract::calculate_dphiref, libMesh::FEAbstract::calculations_started, and libMesh::FEGenericBase< OutputType >::d2phidxdz.

  { libmesh_assert(!calculations_started || calculate_d2phi);
    calculate_d2phi = calculate_dphiref = true; return d2phidxdz; }

get_d2phidxi2()

const std::vector<std::vector<OutputShape> >& libMesh::FEGenericBase< FEOutputType< T >::type >::get_d2phidxi2 ( ) const

inlineinherited

Returns

The shape function second derivatives at the quadrature points, in reference coordinates

Definition at line 345 of file fe_base.h.

References libMesh::FEAbstract::calculate_d2phi, libMesh::FEAbstract::calculate_dphiref, libMesh::FEAbstract::calculations_started, and libMesh::FEGenericBase< OutputType >::d2phidxi2.

  { libmesh_assert(!calculations_started || calculate_d2phi);
    calculate_d2phi = calculate_dphiref = true; return d2phidxi2; }

get_d2phidxideta()

const std::vector<std::vector<OutputShape> >& libMesh::FEGenericBase< FEOutputType< T >::type >::get_d2phidxideta ( ) const

inlineinherited

Returns

The shape function second derivatives at the quadrature points, in reference coordinates

Definition at line 353 of file fe_base.h.

References libMesh::FEAbstract::calculate_d2phi, libMesh::FEAbstract::calculate_dphiref, libMesh::FEAbstract::calculations_started, and libMesh::FEGenericBase< OutputType >::d2phidxideta.

  { libmesh_assert(!calculations_started || calculate_d2phi);
    calculate_d2phi = calculate_dphiref = true; return d2phidxideta; }

get_d2phidxidzeta()

const std::vector<std::vector<OutputShape> >& libMesh::FEGenericBase< FEOutputType< T >::type >::get_d2phidxidzeta ( ) const

inlineinherited

Returns

The shape function second derivatives at the quadrature points, in reference coordinates

Definition at line 361 of file fe_base.h.

References libMesh::FEAbstract::calculate_d2phi, libMesh::FEAbstract::calculate_dphiref, libMesh::FEAbstract::calculations_started, and libMesh::FEGenericBase< OutputType >::d2phidxidzeta.

  { libmesh_assert(!calculations_started || calculate_d2phi);
    calculate_d2phi = calculate_dphiref = true; return d2phidxidzeta; }

get_d2phidy2()

const std::vector<std::vector<OutputShape> >& libMesh::FEGenericBase< FEOutputType< T >::type >::get_d2phidy2 ( ) const

inlineinherited

Returns

The shape function second derivatives at the quadrature points.

Definition at line 321 of file fe_base.h.

References libMesh::FEAbstract::calculate_d2phi, libMesh::FEAbstract::calculate_dphiref, libMesh::FEAbstract::calculations_started, and libMesh::FEGenericBase< OutputType >::d2phidy2.

  { libmesh_assert(!calculations_started || calculate_d2phi);
    calculate_d2phi =  calculate_dphiref = true; return d2phidy2; }

get_d2phidydz()

const std::vector<std::vector<OutputShape> >& libMesh::FEGenericBase< FEOutputType< T >::type >::get_d2phidydz ( ) const

inlineinherited

Returns

The shape function second derivatives at the quadrature points.

Definition at line 329 of file fe_base.h.

References libMesh::FEAbstract::calculate_d2phi, libMesh::FEAbstract::calculate_dphiref, libMesh::FEAbstract::calculations_started, and libMesh::FEGenericBase< OutputType >::d2phidydz.

  { libmesh_assert(!calculations_started || calculate_d2phi);
    calculate_d2phi = calculate_dphiref = true; return d2phidydz; }

get_d2phidz2()

const std::vector<std::vector<OutputShape> >& libMesh::FEGenericBase< FEOutputType< T >::type >::get_d2phidz2 ( ) const

inlineinherited

Returns

The shape function second derivatives at the quadrature points.

Definition at line 337 of file fe_base.h.

References libMesh::FEAbstract::calculate_d2phi, libMesh::FEAbstract::calculate_dphiref, libMesh::FEAbstract::calculations_started, and libMesh::FEGenericBase< OutputType >::d2phidz2.

  { libmesh_assert(!calculations_started || calculate_d2phi);
    calculate_d2phi = calculate_dphiref = true; return d2phidz2; }

get_d2phidzeta2()

const std::vector<std::vector<OutputShape> >& libMesh::FEGenericBase< FEOutputType< T >::type >::get_d2phidzeta2 ( ) const

inlineinherited

Returns

The shape function second derivatives at the quadrature points, in reference coordinates

Definition at line 385 of file fe_base.h.

References libMesh::FEAbstract::calculate_d2phi, libMesh::FEAbstract::calculate_dphiref, libMesh::FEAbstract::calculations_started, and libMesh::FEGenericBase< OutputType >::d2phidzeta2.

  { libmesh_assert(!calculations_started || calculate_d2phi);
    calculate_d2phi = calculate_dphiref = true; return d2phidzeta2; }

get_d2xyzdeta2()

const std::vector<RealGradient>& libMesh::FEAbstract::get_d2xyzdeta2 ( ) const

inlineinherited

Returns

The second partial derivatives in eta.

Definition at line 278 of file fe_abstract.h.

References libMesh::FEAbstract::_fe_map.

  { return this->_fe_map->get_d2xyzdeta2(); }

get_d2xyzdetadzeta()

const std::vector<RealGradient>& libMesh::FEAbstract::get_d2xyzdetadzeta ( ) const

inlineinherited

Returns

The second partial derivatives in eta-zeta.

Definition at line 308 of file fe_abstract.h.

References libMesh::FEAbstract::_fe_map.

  { return this->_fe_map->get_d2xyzdetadzeta(); }

get_d2xyzdxi2()

const std::vector<RealGradient>& libMesh::FEAbstract::get_d2xyzdxi2 ( ) const

inlineinherited

Returns

The second partial derivatives in xi.

Definition at line 272 of file fe_abstract.h.

References libMesh::FEAbstract::_fe_map.

  { return this->_fe_map->get_d2xyzdxi2(); }

get_d2xyzdxideta()

const std::vector<RealGradient>& libMesh::FEAbstract::get_d2xyzdxideta ( ) const

inlineinherited

Returns

The second partial derivatives in xi-eta.

Definition at line 294 of file fe_abstract.h.

References libMesh::FEAbstract::_fe_map.

  { return this->_fe_map->get_d2xyzdxideta(); }

get_d2xyzdxidzeta()

const std::vector<RealGradient>& libMesh::FEAbstract::get_d2xyzdxidzeta ( ) const

inlineinherited

Returns

The second partial derivatives in xi-zeta.

Definition at line 302 of file fe_abstract.h.

References libMesh::FEAbstract::_fe_map.

  { return this->_fe_map->get_d2xyzdxidzeta(); }

get_d2xyzdzeta2()

const std::vector<RealGradient>& libMesh::FEAbstract::get_d2xyzdzeta2 ( ) const

inlineinherited

Returns

The second partial derivatives in zeta.

Definition at line 286 of file fe_abstract.h.

References libMesh::FEAbstract::_fe_map.

  { return this->_fe_map->get_d2xyzdzeta2(); }

get_detadx()

const std::vector<Real>& libMesh::FEAbstract::get_detadx ( ) const

inlineinherited

Returns

The deta/dx entry in the transformation matrix from physical to local coordinates.

Definition at line 338 of file fe_abstract.h.

References libMesh::FEAbstract::_fe_map.

  { return this->_fe_map->get_detadx(); }

get_detady()

const std::vector<Real>& libMesh::FEAbstract::get_detady ( ) const

inlineinherited

Returns

The deta/dy entry in the transformation matrix from physical to local coordinates.

Definition at line 345 of file fe_abstract.h.

References libMesh::FEAbstract::_fe_map.

  { return this->_fe_map->get_detady(); }

get_detadz()

const std::vector<Real>& libMesh::FEAbstract::get_detadz ( ) const

inlineinherited

Returns

The deta/dz entry in the transformation matrix from physical to local coordinates.

Definition at line 352 of file fe_abstract.h.

References libMesh::FEAbstract::_fe_map.

  { return this->_fe_map->get_detadz(); }

get_dim()

unsigned int libMesh::FEAbstract::get_dim ( ) const

inlineinherited

Returns

the dimension of this FE

Definition at line 231 of file fe_abstract.h.

References libMesh::FEAbstract::dim.

  { return dim; }

get_div_phi()

const std::vector<std::vector<OutputDivergence> >& libMesh::FEGenericBase< FEOutputType< T >::type >::get_div_phi ( ) const

inlineinherited

Returns

The divergence of the shape function at the quadrature points.

Definition at line 231 of file fe_base.h.

References libMesh::FEAbstract::calculate_div_phi, libMesh::FEAbstract::calculate_dphiref, libMesh::FEAbstract::calculations_started, and libMesh::FEGenericBase< OutputType >::div_phi.

  { libmesh_assert(!calculations_started || calculate_div_phi);
    calculate_div_phi = calculate_dphiref = true; return div_phi; }

get_dphase()

const std::vector<OutputGradient>& libMesh::FEGenericBase< FEOutputType< T >::type >::get_dphase ( ) const

inlineinherited

Returns

The global first derivative of the phase term which is used in infinite elements, evaluated at the quadrature points.

In case of the general finite element class FE this field is initialized to all zero, so that the variational formulation for an infinite element produces correct element matrices for a mesh using both finite and infinite elements.

Definition at line 403 of file fe_base.h.

References libMesh::FEGenericBase< OutputType >::dphase.

  { return dphase; }

get_dphi()

const std::vector<std::vector<OutputGradient> >& libMesh::FEGenericBase< FEOutputType< T >::type >::get_dphi ( ) const

inlineinherited

Returns

The shape function derivatives at the quadrature points.

Definition at line 215 of file fe_base.h.

References libMesh::FEAbstract::calculate_dphi, libMesh::FEAbstract::calculate_dphiref, libMesh::FEAbstract::calculations_started, and libMesh::FEGenericBase< OutputType >::dphi.

  { libmesh_assert(!calculations_started || calculate_dphi);
    calculate_dphi = calculate_dphiref = true; return dphi; }

get_dphideta()

const std::vector<std::vector<OutputShape> >& libMesh::FEGenericBase< FEOutputType< T >::type >::get_dphideta ( ) const

inlineinherited

Returns

The shape function eta-derivative at the quadrature points.

Definition at line 271 of file fe_base.h.

References libMesh::FEAbstract::calculate_dphiref, libMesh::FEAbstract::calculations_started, and libMesh::FEGenericBase< OutputType >::dphideta.

  { libmesh_assert(!calculations_started || calculate_dphiref);
    calculate_dphiref = true; return dphideta; }

get_dphidx()

const std::vector<std::vector<OutputShape> >& libMesh::FEGenericBase< FEOutputType< T >::type >::get_dphidx ( ) const

inlineinherited

Returns

The shape function x-derivative at the quadrature points.

Definition at line 239 of file fe_base.h.

References libMesh::FEAbstract::calculate_dphi, libMesh::FEAbstract::calculate_dphiref, libMesh::FEAbstract::calculations_started, and libMesh::FEGenericBase< OutputType >::dphidx.

  { libmesh_assert(!calculations_started || calculate_dphi);
    calculate_dphi = calculate_dphiref = true; return dphidx; }

get_dphidxi()

const std::vector<std::vector<OutputShape> >& libMesh::FEGenericBase< FEOutputType< T >::type >::get_dphidxi ( ) const

inlineinherited

Returns

The shape function xi-derivative at the quadrature points.

Definition at line 263 of file fe_base.h.

References libMesh::FEAbstract::calculate_dphiref, libMesh::FEAbstract::calculations_started, and libMesh::FEGenericBase< OutputType >::dphidxi.

  { libmesh_assert(!calculations_started || calculate_dphiref);
    calculate_dphiref = true; return dphidxi; }

get_dphidy()

const std::vector<std::vector<OutputShape> >& libMesh::FEGenericBase< FEOutputType< T >::type >::get_dphidy ( ) const

inlineinherited

Returns

The shape function y-derivative at the quadrature points.

Definition at line 247 of file fe_base.h.

References libMesh::FEAbstract::calculate_dphi, libMesh::FEAbstract::calculate_dphiref, libMesh::FEAbstract::calculations_started, and libMesh::FEGenericBase< OutputType >::dphidy.

  { libmesh_assert(!calculations_started || calculate_dphi);
    calculate_dphi = calculate_dphiref = true; return dphidy; }

get_dphidz()

const std::vector<std::vector<OutputShape> >& libMesh::FEGenericBase< FEOutputType< T >::type >::get_dphidz ( ) const

inlineinherited

Returns

The shape function z-derivative at the quadrature points.

Definition at line 255 of file fe_base.h.

References libMesh::FEAbstract::calculate_dphi, libMesh::FEAbstract::calculate_dphiref, libMesh::FEAbstract::calculations_started, and libMesh::FEGenericBase< OutputType >::dphidz.

  { libmesh_assert(!calculations_started || calculate_dphi);
    calculate_dphi = calculate_dphiref = true; return dphidz; }

get_dphidzeta()

const std::vector<std::vector<OutputShape> >& libMesh::FEGenericBase< FEOutputType< T >::type >::get_dphidzeta ( ) const

inlineinherited

Returns

The shape function zeta-derivative at the quadrature points.

Definition at line 279 of file fe_base.h.

References libMesh::FEAbstract::calculate_dphiref, libMesh::FEAbstract::calculations_started, and libMesh::FEGenericBase< OutputType >::dphidzeta.

  { libmesh_assert(!calculations_started || calculate_dphiref);
    calculate_dphiref = true; return dphidzeta; }

get_dxidx()

const std::vector<Real>& libMesh::FEAbstract::get_dxidx ( ) const

inlineinherited

Returns

The dxi/dx entry in the transformation matrix from physical to local coordinates.

Definition at line 317 of file fe_abstract.h.

References libMesh::FEAbstract::_fe_map.

  { return this->_fe_map->get_dxidx(); }

get_dxidy()

const std::vector<Real>& libMesh::FEAbstract::get_dxidy ( ) const

inlineinherited

Returns

The dxi/dy entry in the transformation matrix from physical to local coordinates.

Definition at line 324 of file fe_abstract.h.

References libMesh::FEAbstract::_fe_map.

  { return this->_fe_map->get_dxidy(); }

get_dxidz()

const std::vector<Real>& libMesh::FEAbstract::get_dxidz ( ) const

inlineinherited

Returns

The dxi/dz entry in the transformation matrix from physical to local coordinates.

Definition at line 331 of file fe_abstract.h.

References libMesh::FEAbstract::_fe_map.

  { return this->_fe_map->get_dxidz(); }

get_dxyzdeta()

const std::vector<RealGradient>& libMesh::FEAbstract::get_dxyzdeta ( ) const

inlineinherited

Returns

The element tangents in eta-direction at the quadrature points.

Definition at line 259 of file fe_abstract.h.

References libMesh::FEAbstract::_fe_map.

  { return this->_fe_map->get_dxyzdeta(); }

get_dxyzdxi()

const std::vector<RealGradient>& libMesh::FEAbstract::get_dxyzdxi ( ) const

inlineinherited

Returns

The element tangents in xi-direction at the quadrature points.

Definition at line 252 of file fe_abstract.h.

References libMesh::FEAbstract::_fe_map.

  { return this->_fe_map->get_dxyzdxi(); }

get_dxyzdzeta()

const std::vector<RealGradient>& libMesh::FEAbstract::get_dxyzdzeta ( ) const

inlineinherited

Returns

The element tangents in zeta-direction at the quadrature points.

Definition at line 266 of file fe_abstract.h.

References libMesh::FEAbstract::_fe_map.

  { return _fe_map->get_dxyzdzeta(); }

get_dzetadx()

const std::vector<Real>& libMesh::FEAbstract::get_dzetadx ( ) const

inlineinherited

Returns

The dzeta/dx entry in the transformation matrix from physical to local coordinates.

Definition at line 359 of file fe_abstract.h.

References libMesh::FEAbstract::_fe_map.

  { return this->_fe_map->get_dzetadx(); }

get_dzetady()

const std::vector<Real>& libMesh::FEAbstract::get_dzetady ( ) const

inlineinherited

Returns

The dzeta/dy entry in the transformation matrix from physical to local coordinates.

Definition at line 366 of file fe_abstract.h.

References libMesh::FEAbstract::_fe_map.

  { return this->_fe_map->get_dzetady(); }

get_dzetadz()

const std::vector<Real>& libMesh::FEAbstract::get_dzetadz ( ) const

inlineinherited

Returns

The dzeta/dz entry in the transformation matrix from physical to local coordinates.

Definition at line 373 of file fe_abstract.h.

References libMesh::FEAbstract::_fe_map.

  { return this->_fe_map->get_dzetadz(); }

get_family()

FEFamily libMesh::FEAbstract::get_family ( ) const

inlineinherited

Returns

The finite element family of this element.

Definition at line 455 of file fe_abstract.h.

References libMesh::FEType::family, and libMesh::FEAbstract::fe_type.

{ return fe_type.family; }

get_fe_map()

const FEMap& libMesh::FEAbstract::get_fe_map ( ) const

inlineinherited

Returns

The mapping object

Definition at line 460 of file fe_abstract.h.

References libMesh::FEAbstract::_fe_map.

Referenced by libMesh::HCurlFETransformation< OutputShape >::init_map_d2phi(), libMesh::H1FETransformation< OutputShape >::init_map_d2phi(), libMesh::HCurlFETransformation< OutputShape >::init_map_dphi(), libMesh::H1FETransformation< OutputShape >::init_map_dphi(), libMesh::HCurlFETransformation< OutputShape >::init_map_phi(), libMesh::HCurlFETransformation< OutputShape >::map_curl(), libMesh::H1FETransformation< OutputShape >::map_curl(), libMesh::H1FETransformation< OutputShape >::map_d2phi(), libMesh::H1FETransformation< OutputShape >::map_div(), libMesh::H1FETransformation< OutputShape >::map_dphi(), and libMesh::HCurlFETransformation< OutputShape >::map_phi().

{ return *_fe_map.get(); }

get_fe_type()

FEType libMesh::FEAbstract::get_fe_type ( ) const

inlineinherited

Returns

The FE Type (approximation order and family) of the finite element.

Definition at line 429 of file fe_abstract.h.

References libMesh::FEAbstract::fe_type.

Referenced by libMesh::FEMContext::build_new_fe(), libMesh::HCurlFETransformation< OutputShape >::map_phi(), libMesh::H1FETransformation< OutputShape >::map_phi(), libMesh::GenericProjector< FFunctor, GFunctor, FValue, ProjectionAction >::operator()(), and libMesh::JumpErrorEstimator::reinit_sides().

{ return fe_type; }

get_info()

std::string libMesh::ReferenceCounter::get_info ( )

staticinherited

Gets a string containing the reference information.

Definition at line 47 of file reference_counter.C.

References libMesh::ReferenceCounter::_counts, and libMesh::Quality::name().

Referenced by libMesh::ReferenceCounter::print_info().

{
#if defined(LIBMESH_ENABLE_REFERENCE_COUNTING) && defined(DEBUG)

  std::ostringstream oss;

  oss << '\n'
      << " ---------------------------------------------------------------------------- \n"
      << "| Reference count information                                                |\n"
      << " ---------------------------------------------------------------------------- \n";

  for (const auto & pr : _counts)
    {
      const std::string name(pr.first);
      const unsigned int creations    = pr.second.first;
      const unsigned int destructions = pr.second.second;

      oss << "| " << name << " reference count information:\n"
          << "|  Creations:    " << creations    << '\n'
          << "|  Destructions: " << destructions << '\n';
    }

  oss << " ---------------------------------------------------------------------------- \n";

  return oss.str();

#else

  return "";

#endif
}

get_JxW()

const std::vector<Real>& libMesh::FEAbstract::get_JxW ( ) const

inlineinherited

Returns

The element Jacobian times the quadrature weight for each quadrature point.

Definition at line 245 of file fe_abstract.h.

References libMesh::FEAbstract::_fe_map.

Referenced by libMesh::ExactSolution::_compute_error(), libMesh::DiscontinuityMeasure::boundary_side_integration(), libMesh::KellyErrorEstimator::boundary_side_integration(), libMesh::System::calculate_norm(), libMesh::ExactErrorEstimator::find_squared_element_error(), libMesh::FEMSystem::init_context(), libMesh::LaplacianErrorEstimator::internal_side_integration(), libMesh::DiscontinuityMeasure::internal_side_integration(), libMesh::KellyErrorEstimator::internal_side_integration(), and libMesh::GenericProjector< FFunctor, GFunctor, FValue, ProjectionAction >::operator()().

  { return this->_fe_map->get_JxW(); }

get_normals()

const std::vector<Point>& libMesh::FEAbstract::get_normals ( ) const

inlineinherited

Returns

The outward pointing normal vectors for face integration.

Definition at line 385 of file fe_abstract.h.

References libMesh::FEAbstract::_fe_map.

Referenced by libMesh::KellyErrorEstimator::boundary_side_integration(), libMesh::ParsedFEMFunction< T >::eval_args(), libMesh::ParsedFEMFunction< T >::init_context(), libMesh::KellyErrorEstimator::init_context(), and libMesh::KellyErrorEstimator::internal_side_integration().

  { return this->_fe_map->get_normals(); }

get_order()

Order libMesh::FEAbstract::get_order ( ) const

inlineinherited

Returns

The approximation order of the finite element.

Definition at line 434 of file fe_abstract.h.

References libMesh::FEAbstract::_p_level, libMesh::FEAbstract::fe_type, and libMesh::FEType::order.

{ return static_cast<Order>(fe_type.order + _p_level); }

get_p_level()

unsigned int libMesh::FEAbstract::get_p_level ( ) const

inlineinherited

Returns

The p refinement level that the current shape functions have been calculated for.

Definition at line 424 of file fe_abstract.h.

References libMesh::FEAbstract::_p_level.

{ return _p_level; }

get_phi()

const std::vector<std::vector<OutputShape> >& libMesh::FEGenericBase< FEOutputType< T >::type >::get_phi ( ) const

inlineinherited

Returns

The shape function values at the quadrature points on the element.

Definition at line 207 of file fe_base.h.

References libMesh::FEAbstract::calculate_phi, libMesh::FEAbstract::calculations_started, and libMesh::FEGenericBase< OutputType >::phi.

  { libmesh_assert(!calculations_started || calculate_phi);
    calculate_phi = true; return phi; }

get_refspace_nodes()

void libMesh::FEAbstract::get_refspace_nodes ( const ElemType  t, std::vector< Point > &  nodes  )

staticinherited

Returns

The reference space coordinates of nodes based on the element type.

Definition at line 283 of file fe_abstract.C.

References libMesh::EDGE2, libMesh::EDGE3, libMesh::HEX20, libMesh::HEX27, libMesh::HEX8, libMesh::PRISM15, libMesh::PRISM18, libMesh::PRISM6, libMesh::PYRAMID13, libMesh::PYRAMID14, libMesh::PYRAMID5, libMesh::QUAD4, libMesh::QUAD8, libMesh::QUAD9, libMesh::QUADSHELL4, libMesh::QUADSHELL8, libMesh::TET10, libMesh::TET4, libMesh::TRI3, libMesh::TRI6, and libMesh::TRISHELL3.

{
  switch(itemType)
    {
    case EDGE2:
      {
        nodes.resize(2);
        nodes[0] = Point (-1.,0.,0.);
        nodes[1] = Point (1.,0.,0.);
        return;
      }
    case EDGE3:
      {
        nodes.resize(3);
        nodes[0] = Point (-1.,0.,0.);
        nodes[1] = Point (1.,0.,0.);
        nodes[2] = Point (0.,0.,0.);
        return;
      }
    case TRI3:
    case TRISHELL3:
      {
        nodes.resize(3);
        nodes[0] = Point (0.,0.,0.);
        nodes[1] = Point (1.,0.,0.);
        nodes[2] = Point (0.,1.,0.);
        return;
      }
    case TRI6:
      {
        nodes.resize(6);
        nodes[0] = Point (0.,0.,0.);
        nodes[1] = Point (1.,0.,0.);
        nodes[2] = Point (0.,1.,0.);
        nodes[3] = Point (.5,0.,0.);
        nodes[4] = Point (.5,.5,0.);
        nodes[5] = Point (0.,.5,0.);
        return;
      }
    case QUAD4:
    case QUADSHELL4:
      {
        nodes.resize(4);
        nodes[0] = Point (-1.,-1.,0.);
        nodes[1] = Point (1.,-1.,0.);
        nodes[2] = Point (1.,1.,0.);
        nodes[3] = Point (-1.,1.,0.);
        return;
      }
    case QUAD8:
    case QUADSHELL8:
      {
        nodes.resize(8);
        nodes[0] = Point (-1.,-1.,0.);
        nodes[1] = Point (1.,-1.,0.);
        nodes[2] = Point (1.,1.,0.);
        nodes[3] = Point (-1.,1.,0.);
        nodes[4] = Point (0.,-1.,0.);
        nodes[5] = Point (1.,0.,0.);
        nodes[6] = Point (0.,1.,0.);
        nodes[7] = Point (-1.,0.,0.);
        return;
      }
    case QUAD9:
      {
        nodes.resize(9);
        nodes[0] = Point (-1.,-1.,0.);
        nodes[1] = Point (1.,-1.,0.);
        nodes[2] = Point (1.,1.,0.);
        nodes[3] = Point (-1.,1.,0.);
        nodes[4] = Point (0.,-1.,0.);
        nodes[5] = Point (1.,0.,0.);
        nodes[6] = Point (0.,1.,0.);
        nodes[7] = Point (-1.,0.,0.);
        nodes[8] = Point (0.,0.,0.);
        return;
      }
    case TET4:
      {
        nodes.resize(4);
        nodes[0] = Point (0.,0.,0.);
        nodes[1] = Point (1.,0.,0.);
        nodes[2] = Point (0.,1.,0.);
        nodes[3] = Point (0.,0.,1.);
        return;
      }
    case TET10:
      {
        nodes.resize(10);
        nodes[0] = Point (0.,0.,0.);
        nodes[1] = Point (1.,0.,0.);
        nodes[2] = Point (0.,1.,0.);
        nodes[3] = Point (0.,0.,1.);
        nodes[4] = Point (.5,0.,0.);
        nodes[5] = Point (.5,.5,0.);
        nodes[6] = Point (0.,.5,0.);
        nodes[7] = Point (0.,0.,.5);
        nodes[8] = Point (.5,0.,.5);
        nodes[9] = Point (0.,.5,.5);
        return;
      }
    case HEX8:
      {
        nodes.resize(8);
        nodes[0] = Point (-1.,-1.,-1.);
        nodes[1] = Point (1.,-1.,-1.);
        nodes[2] = Point (1.,1.,-1.);
        nodes[3] = Point (-1.,1.,-1.);
        nodes[4] = Point (-1.,-1.,1.);
        nodes[5] = Point (1.,-1.,1.);
        nodes[6] = Point (1.,1.,1.);
        nodes[7] = Point (-1.,1.,1.);
        return;
      }
    case HEX20:
      {
        nodes.resize(20);
        nodes[0] = Point (-1.,-1.,-1.);
        nodes[1] = Point (1.,-1.,-1.);
        nodes[2] = Point (1.,1.,-1.);
        nodes[3] = Point (-1.,1.,-1.);
        nodes[4] = Point (-1.,-1.,1.);
        nodes[5] = Point (1.,-1.,1.);
        nodes[6] = Point (1.,1.,1.);
        nodes[7] = Point (-1.,1.,1.);
        nodes[8] = Point (0.,-1.,-1.);
        nodes[9] = Point (1.,0.,-1.);
        nodes[10] = Point (0.,1.,-1.);
        nodes[11] = Point (-1.,0.,-1.);
        nodes[12] = Point (-1.,-1.,0.);
        nodes[13] = Point (1.,-1.,0.);
        nodes[14] = Point (1.,1.,0.);
        nodes[15] = Point (-1.,1.,0.);
        nodes[16] = Point (0.,-1.,1.);
        nodes[17] = Point (1.,0.,1.);
        nodes[18] = Point (0.,1.,1.);
        nodes[19] = Point (-1.,0.,1.);
        return;
      }
    case HEX27:
      {
        nodes.resize(27);
        nodes[0] = Point (-1.,-1.,-1.);
        nodes[1] = Point (1.,-1.,-1.);
        nodes[2] = Point (1.,1.,-1.);
        nodes[3] = Point (-1.,1.,-1.);
        nodes[4] = Point (-1.,-1.,1.);
        nodes[5] = Point (1.,-1.,1.);
        nodes[6] = Point (1.,1.,1.);
        nodes[7] = Point (-1.,1.,1.);
        nodes[8] = Point (0.,-1.,-1.);
        nodes[9] = Point (1.,0.,-1.);
        nodes[10] = Point (0.,1.,-1.);
        nodes[11] = Point (-1.,0.,-1.);
        nodes[12] = Point (-1.,-1.,0.);
        nodes[13] = Point (1.,-1.,0.);
        nodes[14] = Point (1.,1.,0.);
        nodes[15] = Point (-1.,1.,0.);
        nodes[16] = Point (0.,-1.,1.);
        nodes[17] = Point (1.,0.,1.);
        nodes[18] = Point (0.,1.,1.);
        nodes[19] = Point (-1.,0.,1.);
        nodes[20] = Point (0.,0.,-1.);
        nodes[21] = Point (0.,-1.,0.);
        nodes[22] = Point (1.,0.,0.);
        nodes[23] = Point (0.,1.,0.);
        nodes[24] = Point (-1.,0.,0.);
        nodes[25] = Point (0.,0.,1.);
        nodes[26] = Point (0.,0.,0.);
        return;
      }
    case PRISM6:
      {
        nodes.resize(6);
        nodes[0] = Point (0.,0.,-1.);
        nodes[1] = Point (1.,0.,-1.);
        nodes[2] = Point (0.,1.,-1.);
        nodes[3] = Point (0.,0.,1.);
        nodes[4] = Point (1.,0.,1.);
        nodes[5] = Point (0.,1.,1.);
        return;
      }
    case PRISM15:
      {
        nodes.resize(15);
        nodes[0] = Point (0.,0.,-1.);
        nodes[1] = Point (1.,0.,-1.);
        nodes[2] = Point (0.,1.,-1.);
        nodes[3] = Point (0.,0.,1.);
        nodes[4] = Point (1.,0.,1.);
        nodes[5] = Point (0.,1.,1.);
        nodes[6] = Point (.5,0.,-1.);
        nodes[7] = Point (.5,.5,-1.);
        nodes[8] = Point (0.,.5,-1.);
        nodes[9] = Point (0.,0.,0.);
        nodes[10] = Point (1.,0.,0.);
        nodes[11] = Point (0.,1.,0.);
        nodes[12] = Point (.5,0.,1.);
        nodes[13] = Point (.5,.5,1.);
        nodes[14] = Point (0.,.5,1.);
        return;
      }
    case PRISM18:
      {
        nodes.resize(18);
        nodes[0] = Point (0.,0.,-1.);
        nodes[1] = Point (1.,0.,-1.);
        nodes[2] = Point (0.,1.,-1.);
        nodes[3] = Point (0.,0.,1.);
        nodes[4] = Point (1.,0.,1.);
        nodes[5] = Point (0.,1.,1.);
        nodes[6] = Point (.5,0.,-1.);
        nodes[7] = Point (.5,.5,-1.);
        nodes[8] = Point (0.,.5,-1.);
        nodes[9] = Point (0.,0.,0.);
        nodes[10] = Point (1.,0.,0.);
        nodes[11] = Point (0.,1.,0.);
        nodes[12] = Point (.5,0.,1.);
        nodes[13] = Point (.5,.5,1.);
        nodes[14] = Point (0.,.5,1.);
        nodes[15] = Point (.5,0.,0.);
        nodes[16] = Point (.5,.5,0.);
        nodes[17] = Point (0.,.5,0.);
        return;
      }
    case PYRAMID5:
      {
        nodes.resize(5);
        nodes[0] = Point (-1.,-1.,0.);
        nodes[1] = Point (1.,-1.,0.);
        nodes[2] = Point (1.,1.,0.);
        nodes[3] = Point (-1.,1.,0.);
        nodes[4] = Point (0.,0.,1.);
        return;
      }
    case PYRAMID13:
      {
        nodes.resize(13);

        // base corners
        nodes[0] = Point (-1.,-1.,0.);
        nodes[1] = Point (1.,-1.,0.);
        nodes[2] = Point (1.,1.,0.);
        nodes[3] = Point (-1.,1.,0.);

        // apex
        nodes[4] = Point (0.,0.,1.);

        // base midedge
        nodes[5] = Point (0.,-1.,0.);
        nodes[6] = Point (1.,0.,0.);
        nodes[7] = Point (0.,1.,0.);
        nodes[8] = Point (-1,0.,0.);

        // lateral midedge
        nodes[9] = Point (-.5,-.5,.5);
        nodes[10] = Point (.5,-.5,.5);
        nodes[11] = Point (.5,.5,.5);
        nodes[12] = Point (-.5,.5,.5);

        return;
      }
    case PYRAMID14:
      {
        nodes.resize(14);

        // base corners
        nodes[0] = Point (-1.,-1.,0.);
        nodes[1] = Point (1.,-1.,0.);
        nodes[2] = Point (1.,1.,0.);
        nodes[3] = Point (-1.,1.,0.);

        // apex
        nodes[4] = Point (0.,0.,1.);

        // base midedge
        nodes[5] = Point (0.,-1.,0.);
        nodes[6] = Point (1.,0.,0.);
        nodes[7] = Point (0.,1.,0.);
        nodes[8] = Point (-1,0.,0.);

        // lateral midedge
        nodes[9] = Point (-.5,-.5,.5);
        nodes[10] = Point (.5,-.5,.5);
        nodes[11] = Point (.5,.5,.5);
        nodes[12] = Point (-.5,.5,.5);

        // base center
        nodes[13] = Point (0.,0.,0.);

        return;
      }

    default:
      libmesh_error_msg("ERROR: Unknown element type " << itemType);
    }
}

get_Sobolev_dweight()

const std::vector<RealGradient>& libMesh::FEGenericBase< FEOutputType< T >::type >::get_Sobolev_dweight ( ) const

inlineinherited

Returns

The first global derivative of the multiplicative weight at each quadrature point. See get_Sobolev_weight() for details. In case of FE initialized to all zero.

Definition at line 427 of file fe_base.h.

References libMesh::FEGenericBase< OutputType >::dweight.

  { return dweight; }

get_Sobolev_weight()

const std::vector<Real>& libMesh::FEGenericBase< FEOutputType< T >::type >::get_Sobolev_weight ( ) const

inlineinherited

Returns

The multiplicative weight at each quadrature point. This weight is used for certain infinite element weak formulations, so that weighted Sobolev spaces are used for the trial function space. This renders the variational form easily computable.

In case of the general finite element class FE this field is initialized to all ones, so that the variational formulation for an infinite element produces correct element matrices for a mesh using both finite and infinite elements.

Definition at line 419 of file fe_base.h.

References libMesh::FEGenericBase< OutputType >::weight.

  { return weight; }

get_tangents()

const std::vector<std::vector<Point> >& libMesh::FEAbstract::get_tangents ( ) const

inlineinherited

Returns

The tangent vectors for face integration.

Definition at line 379 of file fe_abstract.h.

References libMesh::FEAbstract::_fe_map.

  { return this->_fe_map->get_tangents(); }

get_type()

ElemType libMesh::FEAbstract::get_type ( ) const

inlineinherited

Returns

The element type that the current shape functions have been calculated for. Useful in determining when shape functions must be recomputed.

Definition at line 418 of file fe_abstract.h.

References libMesh::FEAbstract::elem_type.

{ return elem_type; }

get_xyz()

const std::vector<Point>& libMesh::FEAbstract::get_xyz ( ) const

inlineinherited

Returns

The xyz spatial locations of the quadrature points on the element.

Definition at line 238 of file fe_abstract.h.

References libMesh::FEAbstract::_fe_map.

Referenced by libMesh::ExactSolution::_compute_error(), libMesh::DiscontinuityMeasure::boundary_side_integration(), libMesh::KellyErrorEstimator::boundary_side_integration(), libMesh::JumpErrorEstimator::estimate_error(), libMesh::ParsedFEMFunction< T >::eval_args(), libMesh::ExactErrorEstimator::find_squared_element_error(), libMesh::ParsedFEMFunction< T >::init_context(), libMesh::DGFEMContext::neighbor_side_fe_reinit(), and libMesh::GenericProjector< FFunctor, GFunctor, FValue, ProjectionAction >::operator()().

  { return this->_fe_map->get_xyz(); }

increment_constructor_count()

void libMesh::ReferenceCounter::increment_constructor_count ( const std::string &  name )

inlineprotectedinherited

Increments the construction counter. Should be called in the constructor of any derived class that will be reference counted.

Definition at line 181 of file reference_counter.h.

References libMesh::ReferenceCounter::_counts, libMesh::Quality::name(), and libMesh::Threads::spin_mtx.

Referenced by libMesh::ReferenceCountedObject< RBParametrized >::ReferenceCountedObject().

{
  Threads::spin_mutex::scoped_lock lock(Threads::spin_mtx);
  std::pair<unsigned int, unsigned int> & p = _counts[name];

  p.first++;
}

increment_destructor_count()

void libMesh::ReferenceCounter::increment_destructor_count ( const std::string &  name )

inlineprotectedinherited

Increments the destruction counter. Should be called in the destructor of any derived class that will be reference counted.

Definition at line 194 of file reference_counter.h.

References libMesh::ReferenceCounter::_counts, libMesh::Quality::name(), and libMesh::Threads::spin_mtx.

Referenced by libMesh::ReferenceCountedObject< RBParametrized >::~ReferenceCountedObject().

{
  Threads::spin_mutex::scoped_lock lock(Threads::spin_mtx);
  std::pair<unsigned int, unsigned int> & p = _counts[name];

  p.second++;
}

init_base_shape_functions()

void libMesh::FE< Dim, T >::init_base_shape_functions ( const std::vector< Point > &  qp, const Elem *  e  )

overrideprotectedvirtualinherited

Initialize the data fields for the base of an an infinite element.

Implements libMesh::FEGenericBase< FEOutputType< T >::type >.

Definition at line 550 of file fe.C.

{
  this->elem_type = e->type();
  this->_fe_map->template init_reference_to_physical_map<Dim>(qp, e);
  init_shape_functions(qp, e);
}

init_shape_functions()

void libMesh::FESubdivision::init_shape_functions ( const std::vector< Point > &  qp, const Elem *  elem  )

overridevirtual

Update the various member data fields phi, dphidxi, dphideta, dphidzeta, etc. for the current element. These data will be computed at the points qp, which are generally (but need not be) the quadrature points.

Reimplemented from libMesh::FE< 2, SUBDIVISION >.

Definition at line 404 of file fe_subdivision_2D.C.

References libMesh::FEAbstract::_fe_map, A, libMesh::FEAbstract::calculations_started, libMesh::FEGenericBase< FEOutputType< T >::type >::d2phi, libMesh::FEGenericBase< FEOutputType< T >::type >::d2phideta2, libMesh::FEGenericBase< FEOutputType< T >::type >::d2phidxi2, libMesh::FEGenericBase< FEOutputType< T >::type >::d2phidxideta, libMesh::FEGenericBase< FEOutputType< T >::type >::dphi, libMesh::FEGenericBase< FEOutputType< T >::type >::dphideta, libMesh::FEGenericBase< FEOutputType< T >::type >::dphidxi, libMesh::FEAbstract::fe_type, init_subdivision_matrix(), std::max(), std::min(), libMesh::FEType::order, libMesh::FEGenericBase< FEOutputType< T >::type >::phi, std::pow(), libMesh::Real, libMesh::DenseMatrix< T >::right_multiply(), libMesh::FE< Dim, T >::shape(), libMesh::FE< Dim, T >::shape_deriv(), libMesh::FE< Dim, T >::shape_second_deriv(), libMesh::TRI3SUBDIVISION, and libMesh::Elem::type().

{
  libmesh_assert(elem);
  libmesh_assert_equal_to(elem->type(), TRI3SUBDIVISION);
  const Tri3Subdivision * sd_elem = static_cast<const Tri3Subdivision *>(elem);

  LOG_SCOPE("init_shape_functions()", "FESubdivision");

  calculations_started = true;

  const unsigned int valence = sd_elem->get_ordered_valence(0);
  const unsigned int n_qp = cast_int<unsigned int>(qp.size());
  const unsigned int n_approx_shape_functions = valence + 6;

  // resize the vectors to hold current data
  phi.resize         (n_approx_shape_functions);
  dphi.resize        (n_approx_shape_functions);
  dphidxi.resize     (n_approx_shape_functions);
  dphideta.resize    (n_approx_shape_functions);
#ifdef LIBMESH_ENABLE_SECOND_DERIVATIVES
  d2phi.resize       (n_approx_shape_functions);
  d2phidxi2.resize   (n_approx_shape_functions);
  d2phidxideta.resize(n_approx_shape_functions);
  d2phideta2.resize  (n_approx_shape_functions);
#endif

  for (unsigned int i = 0; i < n_approx_shape_functions; ++i)
    {
      phi[i].resize         (n_qp);
      dphi[i].resize        (n_qp);
      dphidxi[i].resize     (n_qp);
      dphideta[i].resize    (n_qp);
#ifdef LIBMESH_ENABLE_SECOND_DERIVATIVES
      d2phi[i].resize       (n_qp);
      d2phidxi2[i].resize   (n_qp);
      d2phidxideta[i].resize(n_qp);
      d2phideta2[i].resize  (n_qp);
#endif
    }

  // Renumbering of the shape functions
  static const unsigned int cvi[12] = {3,6,2,0,1,4,7,10,9,5,11,8};

  if (valence == 6) // This means that all vertices are regular, i.e. we have 12 shape functions
    {
      for (unsigned int i = 0; i < n_approx_shape_functions; ++i)
        {
          for (unsigned int p = 0; p < n_qp; ++p)
            {
              phi[i][p]          = FE<2,SUBDIVISION>::shape             (elem, fe_type.order, cvi[i],    qp[p]);
              dphidxi[i][p]      = FE<2,SUBDIVISION>::shape_deriv       (elem, fe_type.order, cvi[i], 0, qp[p]);
              dphideta[i][p]     = FE<2,SUBDIVISION>::shape_deriv       (elem, fe_type.order, cvi[i], 1, qp[p]);
              dphi[i][p](0)      = dphidxi[i][p];
              dphi[i][p](1)      = dphideta[i][p];
#ifdef LIBMESH_ENABLE_SECOND_DERIVATIVES
              d2phidxi2[i][p]    = FE<2,SUBDIVISION>::shape_second_deriv(elem, fe_type.order, cvi[i], 0, qp[p]);
              d2phidxideta[i][p] = FE<2,SUBDIVISION>::shape_second_deriv(elem, fe_type.order, cvi[i], 1, qp[p]);
              d2phideta2[i][p]   = FE<2,SUBDIVISION>::shape_second_deriv(elem, fe_type.order, cvi[i], 2, qp[p]);
              d2phi[i][p](0,0)   = d2phidxi2[i][p];
              d2phi[i][p](0,1)   = d2phi[i][p](1,0) = d2phidxideta[i][p];
              d2phi[i][p](1,1)   = d2phideta2[i][p];
#endif
            }
        }
    }
  else // vertex 0 is irregular by construction of the mesh
    {
      static const Real eps = 1e-10;

      // temporary values
      std::vector<Real> tphi(12);
      std::vector<Real> tdphidxi(12);
      std::vector<Real> tdphideta(12);
#ifdef LIBMESH_ENABLE_SECOND_DERIVATIVES
      std::vector<Real> td2phidxi2(12);
      std::vector<Real> td2phidxideta(12);
      std::vector<Real> td2phideta2(12);
#endif

      for (unsigned int p = 0; p < n_qp; ++p)
        {
          // evaluate the number of the required subdivisions
          Real v = qp[p](0);
          Real w = qp[p](1);
          Real u = 1 - v - w;
          Real min = 0, max = 0.5;
          int n = 0;
          while (!(u > min-eps && u < max+eps))
            {
              ++n;
              min = max;
              max += std::pow((Real)(2), -n-1);
            }

          // transform u, v and w according to the number of subdivisions required.
          const Real pow2 = std::pow((Real)(2), n);
          v *= pow2;
          w *= pow2;
          u = 1 - v - w;
          libmesh_assert_less(u, 0.5 + eps);
          libmesh_assert_greater(u, -eps);

          // find out in which subdivided patch we are and setup the "selection matrix" P and the transformation Jacobian
          // (see Int. J. Numer. Meth. Engng. 2000; 47:2039-2072, Appendix A.2.)
          const int k = n+1;
          Real jfac; // the additional factor per derivative order
          DenseMatrix<Real> P(12, valence+12);
          if (v > 0.5 - eps)
            {
              v = 2*v - 1;
              w = 2*w;
              jfac = std::pow((Real)(2), k);
              P( 0,2        ) = 1;
              P( 1,0        ) = 1;
              P( 2,valence+3) = 1;
              P( 3,1        ) = 1;
              P( 4,valence  ) = 1;
              P( 5,valence+8) = 1;
              P( 6,valence+2) = 1;
              P( 7,valence+1) = 1;
              P( 8,valence+4) = 1;
              P( 9,valence+7) = 1;
              P(10,valence+6) = 1;
              P(11,valence+9) = 1;
            }
          else if (w > 0.5 - eps)
            {
              v = 2*v;
              w = 2*w - 1;
              jfac = std::pow((Real)(2), k);
              P( 0,0         ) = 1;
              P( 1,valence- 1) = 1;
              P( 2,1         ) = 1;
              P( 3,valence   ) = 1;
              P( 4,valence+ 5) = 1;
              P( 5,valence+ 2) = 1;
              P( 6,valence+ 1) = 1;
              P( 7,valence+ 4) = 1;
              P( 8,valence+11) = 1;
              P( 9,valence+ 6) = 1;
              P(10,valence+ 9) = 1;
              P(11,valence+10) = 1;
            }
          else
            {
              v = 1 - 2*v;
              w = 1 - 2*w;
              jfac = std::pow((Real)(-2), k);
              P( 0,valence+9) = 1;
              P( 1,valence+6) = 1;
              P( 2,valence+4) = 1;
              P( 3,valence+1) = 1;
              P( 4,valence+2) = 1;
              P( 5,valence+5) = 1;
              P( 6,valence  ) = 1;
              P( 7,1        ) = 1;
              P( 8,valence+3) = 1;
              P( 9,valence-1) = 1;
              P(10,0        ) = 1;
              P(11,2        ) = 1;
            }

          u = 1 - v - w;
          if ((u > 1 + eps) || (u < -eps))
            libmesh_error_msg("SUBDIVISION irregular patch: u is outside valid range!");

          DenseMatrix<Real> A;
          init_subdivision_matrix(A, valence);

          // compute P*A^k
          if (k > 1)
            {
              DenseMatrix<Real> Acopy(A);
              for (int e = 1; e < k; ++e)
                A.right_multiply(Acopy);
            }
          P.right_multiply(A);

          const Point transformed_p(v,w);

          for (unsigned int i = 0; i < 12; ++i)
            {
              tphi[i]          = FE<2,SUBDIVISION>::shape             (elem, fe_type.order, i,    transformed_p);
              tdphidxi[i]      = FE<2,SUBDIVISION>::shape_deriv       (elem, fe_type.order, i, 0, transformed_p);
              tdphideta[i]     = FE<2,SUBDIVISION>::shape_deriv       (elem, fe_type.order, i, 1, transformed_p);

#ifdef LIBMESH_ENABLE_SECOND_DERIVATIVES
              td2phidxi2[i]    = FE<2,SUBDIVISION>::shape_second_deriv(elem, fe_type.order, i, 0, transformed_p);
              td2phidxideta[i] = FE<2,SUBDIVISION>::shape_second_deriv(elem, fe_type.order, i, 1, transformed_p);
              td2phideta2[i]   = FE<2,SUBDIVISION>::shape_second_deriv(elem, fe_type.order, i, 2, transformed_p);
#endif
            }

          // Finally, we can compute the irregular shape functions as the product of P
          // and the regular shape functions:
          Real sum1, sum2, sum3, sum4, sum5, sum6;
          for (unsigned int j = 0; j < n_approx_shape_functions; ++j)
            {
              sum1 = sum2 = sum3 = sum4 = sum5 = sum6 = 0;
              for (unsigned int i = 0; i < 12; ++i)
                {
                  sum1 += P(i,j) * tphi[i];
                  sum2 += P(i,j) * tdphidxi[i];
                  sum3 += P(i,j) * tdphideta[i];

#ifdef LIBMESH_ENABLE_SECOND_DERIVATIVES
                  sum4 += P(i,j) * td2phidxi2[i];
                  sum5 += P(i,j) * td2phidxideta[i];
                  sum6 += P(i,j) * td2phideta2[i];
#endif
                }
              phi[j][p]          = sum1;
              dphidxi[j][p]      = sum2 * jfac;
              dphideta[j][p]     = sum3 * jfac;
              dphi[j][p](0)      = dphidxi[j][p];
              dphi[j][p](1)      = dphideta[j][p];

#ifdef LIBMESH_ENABLE_SECOND_DERIVATIVES
              d2phidxi2[j][p]    = sum4 * jfac * jfac;
              d2phidxideta[j][p] = sum5 * jfac * jfac;
              d2phideta2[j][p]   = sum6 * jfac * jfac;
              d2phi[j][p](0,0)   = d2phidxi2[j][p];
              d2phi[j][p](0,1)   = d2phi[j][p](1,0) = d2phidxideta[j][p];
              d2phi[j][p](1,1)   = d2phideta2[j][p];
#endif
            }
        } // end quadrature loop
    } // end irregular vertex

  // Let the FEMap use the same initialized shape functions
  this->_fe_map->get_phi_map()          = phi;
  this->_fe_map->get_dphidxi_map()      = dphidxi;
  this->_fe_map->get_dphideta_map()     = dphideta;

#ifdef LIBMESH_ENABLE_SECOND_DERIVATIVES
  this->_fe_map->get_d2phidxi2_map()    = d2phidxi2;
  this->_fe_map->get_d2phideta2_map()   = d2phideta2;
  this->_fe_map->get_d2phidxideta_map() = d2phidxideta;
#endif
}

init_subdivision_matrix()

void libMesh::FESubdivision::init_subdivision_matrix ( DenseMatrix< Real > &  A, unsigned int  valence  )

static

Builds the subdivision matrix A for the Loop scheme. The size depends on the element's valence.

Definition at line 42 of file fe_subdivision_2D.C.

References A, and loop_subdivision_mask().

Referenced by init_shape_functions().

{
  A.resize(valence + 12, valence + 12);

  // A = (S11 0; S21 S22), see Cirak et al.,
  // Int. J. Numer. Meth. Engng. 2000; 47:2039-2072, Appendix A.2.

  // First, set the static S21 part
  A(valence+ 1,0        ) = 0.125;
  A(valence+ 1,1        ) = 0.375;
  A(valence+ 1,valence  ) = 0.375;
  A(valence+ 2,0        ) = 0.0625;
  A(valence+ 2,1        ) = 0.625;
  A(valence+ 2,2        ) = 0.0625;
  A(valence+ 2,valence  ) = 0.0625;
  A(valence+ 3,0        ) = 0.125;
  A(valence+ 3,1        ) = 0.375;
  A(valence+ 3,2        ) = 0.375;
  A(valence+ 4,0        ) = 0.0625;
  A(valence+ 4,1        ) = 0.0625;
  A(valence+ 4,valence-1) = 0.0625;
  A(valence+ 4,valence  ) = 0.625;
  A(valence+ 5,0        ) = 0.125;
  A(valence+ 5,valence-1) = 0.375;
  A(valence+ 5,valence  ) = 0.375;
  A(valence+ 6,1        ) = 0.375;
  A(valence+ 6,valence  ) = 0.125;
  A(valence+ 7,1        ) = 0.375;
  A(valence+ 8,1        ) = 0.375;
  A(valence+ 8,2        ) = 0.125;
  A(valence+ 9,1        ) = 0.125;
  A(valence+ 9,valence  ) = 0.375;
  A(valence+10,valence  ) = 0.375;
  A(valence+11,valence-1) = 0.125;
  A(valence+11,valence  ) = 0.375;

  // Next, set the static S22 part
  A(valence+ 1,valence+1) = 0.125;
  A(valence+ 2,valence+1) = 0.0625;
  A(valence+ 2,valence+2) = 0.0625;
  A(valence+ 2,valence+3) = 0.0625;
  A(valence+ 3,valence+3) = 0.125;
  A(valence+ 4,valence+1) = 0.0625;
  A(valence+ 4,valence+4) = 0.0625;
  A(valence+ 4,valence+5) = 0.0625;
  A(valence+ 5,valence+5) = 0.125;
  A(valence+ 6,valence+1) = 0.375;
  A(valence+ 6,valence+2) = 0.125;
  A(valence+ 7,valence+1) = 0.125;
  A(valence+ 7,valence+2) = 0.375;
  A(valence+ 7,valence+3) = 0.125;
  A(valence+ 8,valence+2) = 0.125;
  A(valence+ 8,valence+3) = 0.375;
  A(valence+ 9,valence+1) = 0.375;
  A(valence+ 9,valence+4) = 0.125;
  A(valence+10,valence+1) = 0.125;
  A(valence+10,valence+4) = 0.375;
  A(valence+10,valence+5) = 0.125;
  A(valence+11,valence+4) = 0.125;
  A(valence+11,valence+5) = 0.375;

  // Last, set the S11 part: first row
  std::vector<Real> weights;
  loop_subdivision_mask(weights, valence);
  for (unsigned int i = 0; i <= valence; ++i)
    A(0,i) = weights[i];

  // second row
  A(1,0) = 0.375;
  A(1,1) = 0.375;
  A(1,2) = 0.125;
  A(1,valence) = 0.125;

  // third to second-to-last rows
  for (unsigned int i = 2; i < valence; ++i)
    {
      A(i,0  ) = 0.375;
      A(i,i-1) = 0.125;
      A(i,i  ) = 0.375;
      A(i,i+1) = 0.125;
    }

  // last row
  A(valence,0) = 0.375;
  A(valence,1) = 0.125;
  A(valence,valence-1) = 0.125;
  A(valence,valence  ) = 0.375;
}

inverse_map() [1/4]

Point libMesh::FE< Dim, T >::inverse_map ( const Elem *  elem, const Point &  p, const Real  tolerance = TOLERANCE, const bool  secure = true  )

staticinherited

Returns

The location (on the reference element) of the point p located in physical space. This function requires inverting the (possibly nonlinear) transformation map, so it is not trivial. The optional parameter tolerance defines how close is "good enough." The map inversion iteration computes the sequence , and the iteration is terminated when

Definition at line 1576 of file fe_map.C.

{
  libmesh_assert(elem);
  libmesh_assert_greater_equal (tolerance, 0.);


  // Start logging the map inversion.
  LOG_SCOPE("inverse_map()", "FE");

  // How much did the point on the reference
  // element change by in this Newton step?
  Real inverse_map_error = 0.;

  //  The point on the reference element.  This is
  //  the "initial guess" for Newton's method.  The
  //  centroid seems like a good idea, but computing
  //  it is a little more intensive than, say taking
  //  the zero point.
  //
  //  Convergence should be insensitive of this choice
  //  for "good" elements.
  Point p; // the zero point.  No computation required

  //  The number of iterations in the map inversion process.
  unsigned int cnt = 0;

  //  The number of iterations after which we give up and declare
  //  divergence
  const unsigned int max_cnt = 10;

  //  The distance (in master element space) beyond which we give up
  //  and declare divergence.  This is no longer used...
  // Real max_step_length = 4.;



  //  Newton iteration loop.
  do
    {
      //  Where our current iterate \p p maps to.
      const Point physical_guess = FE<Dim,T>::map (elem, p);

      //  How far our current iterate is from the actual point.
      const Point delta = physical_point - physical_guess;

      //  Increment in current iterate \p p, will be computed.
      Point dp;


      //  The form of the map and how we invert it depends
      //  on the dimension that we are in.
      switch (Dim)
        {
          // ------------------------------------------------------------------
          //  0D map inversion is trivial
        case 0:
          {
            break;
          }

          // ------------------------------------------------------------------
          //  1D map inversion
          //
          //  Here we find the point on a 1D reference element that maps to
          //  the point \p physical_point in the domain.  This is a bit tricky
          //  since we do not want to assume that the point \p physical_point
          //  is also in a 1D domain.  In particular, this method might get
          //  called on the edge of a 3D element, in which case
          //  \p physical_point actually lives in 3D.
        case 1:
          {
            const Point dxi = FE<Dim,T>::map_xi (elem, p);

            //  Newton's method in this case looks like
            //
            //  {X} - {X_n} = [J]*dp
            //
            //  Where {X}, {X_n} are 3x1 vectors, [J] is a 3x1 matrix
            //  d(x,y,z)/dxi, and we seek dp, a scalar.  Since the above
            //  system is either overdetermined or rank-deficient, we will
            //  solve the normal equations for this system
            //
            //  [J]^T ({X} - {X_n}) = [J]^T [J] {dp}
            //
            //  which involves the trivial inversion of the scalar
            //  G = [J]^T [J]
            const Real G = dxi*dxi;

            if (secure)
              libmesh_assert_greater (G, 0.);

            const Real Ginv = 1./G;

            const Real  dxidelta = dxi*delta;

            dp(0) = Ginv*dxidelta;

            // No master elements have radius > 4, but sometimes we
            // can take a step that big while still converging
            // if (secure)
            // libmesh_assert_less (dp.size(), max_step_length);

            break;
          }



          // ------------------------------------------------------------------
          //  2D map inversion
          //
          //  Here we find the point on a 2D reference element that maps to
          //  the point \p physical_point in the domain.  This is a bit tricky
          //  since we do not want to assume that the point \p physical_point
          //  is also in a 2D domain.  In particular, this method might get
          //  called on the face of a 3D element, in which case
          //  \p physical_point actually lives in 3D.
        case 2:
          {
            const Point dxi  = FE<Dim,T>::map_xi  (elem, p);
            const Point deta = FE<Dim,T>::map_eta (elem, p);

            //  Newton's method in this case looks like
            //
            //  {X} - {X_n} = [J]*{dp}
            //
            //  Where {X}, {X_n} are 3x1 vectors, [J] is a 3x2 matrix
            //  d(x,y,z)/d(xi,eta), and we seek {dp}, a 2x1 vector.  Since
            //  the above system is either over-determined or rank-deficient,
            //  we will solve the normal equations for this system
            //
            //  [J]^T ({X} - {X_n}) = [J]^T [J] {dp}
            //
            //  which involves the inversion of the 2x2 matrix
            //  [G] = [J]^T [J]
            const Real
              G11 = dxi*dxi,  G12 = dxi*deta,
              G21 = dxi*deta, G22 = deta*deta;


            const Real det = (G11*G22 - G12*G21);

            if (secure)
              libmesh_assert_not_equal_to (det, 0.);

            const Real inv_det = 1./det;

            const Real
              Ginv11 =  G22*inv_det,
              Ginv12 = -G12*inv_det,

              Ginv21 = -G21*inv_det,
              Ginv22 =  G11*inv_det;


            const Real  dxidelta  = dxi*delta;
            const Real  detadelta = deta*delta;

            dp(0) = (Ginv11*dxidelta + Ginv12*detadelta);
            dp(1) = (Ginv21*dxidelta + Ginv22*detadelta);

            // No master elements have radius > 4, but sometimes we
            // can take a step that big while still converging
            // if (secure)
            // libmesh_assert_less (dp.size(), max_step_length);

            break;
          }



          // ------------------------------------------------------------------
          //  3D map inversion
          //
          //  Here we find the point in a 3D reference element that maps to
          //  the point \p physical_point in a 3D domain. Nothing special
          //  has to happen here, since (unless the map is singular because
          //  you have a BAD element) the map will be invertible and we can
          //  apply Newton's method directly.
        case 3:
          {
            const Point dxi   = FE<Dim,T>::map_xi   (elem, p);
            const Point deta  = FE<Dim,T>::map_eta  (elem, p);
            const Point dzeta = FE<Dim,T>::map_zeta (elem, p);

            //  Newton's method in this case looks like
            //
            //  {X} = {X_n} + [J]*{dp}
            //
            //  Where {X}, {X_n} are 3x1 vectors, [J] is a 3x3 matrix
            //  d(x,y,z)/d(xi,eta,zeta), and we seek {dp}, a 3x1 vector.
            //  Since the above system is nonsingular for invertible maps
            //  we will solve
            //
            //  {dp} = [J]^-1 ({X} - {X_n})
            //
            //  which involves the inversion of the 3x3 matrix [J]
            libmesh_try
              {
                RealTensorValue(dxi(0), deta(0), dzeta(0),
                                dxi(1), deta(1), dzeta(1),
                                dxi(2), deta(2), dzeta(2)).solve(delta, dp);
              }
            libmesh_catch (ConvergenceFailure &)
              {
                // We encountered a singular Jacobian.  The value of
                // dp is zero, since it was never changed during the
                // call to RealTensorValue::solve().  We don't want to
                // continue iterating until max_cnt since there is no
                // update to the Newton iterate, and we don't want to
                // print the inverse_map_error value since it will
                // confusingly be 0.  Therefore, in the secure case we
                // need to throw an error message while in the !secure
                // case we can just return a far away point.
                if (secure)
                  {
                    libMesh::err << "ERROR: Newton scheme encountered a singular Jacobian in element: "
                                 << elem->id()
                                 << std::endl;

                    elem->print_info(libMesh::err);

                    libmesh_error_msg("Exiting...");
                  }
                else
                  {
                    for (unsigned int i=0; i != Dim; ++i)
                      p(i) = 1e6;
                    return p;
                  }
              }

            // No master elements have radius > 4, but sometimes we
            // can take a step that big while still converging
            // if (secure)
            // libmesh_assert_less (dp.size(), max_step_length);

            break;
          }


          //  Some other dimension?
        default:
          libmesh_error_msg("Invalid Dim = " << Dim);
        } // end switch(Dim), dp now computed



      //  ||P_n+1 - P_n||
      inverse_map_error = dp.norm();

      //  P_n+1 = P_n + dp
      p.add (dp);

      //  Increment the iteration count.
      cnt++;

      //  Watch for divergence of Newton's
      //  method.  Here's how it goes:
      //  (1) For good elements, we expect convergence in 10
      //      iterations, with no too-large steps.
      //      - If called with (secure == true) and we have not yet converged
      //        print out a warning message.
      //      - If called with (secure == true) and we have not converged in
      //        20 iterations abort
      //  (2) This method may be called in cases when the target point is not
      //      inside the element and we have no business expecting convergence.
      //      For these cases if we have not converged in 10 iterations forget
      //      about it.
      if (cnt > max_cnt)
        {
          //  Warn about divergence when secure is true - this
          //  shouldn't happen
          if (secure)
            {
              // Print every time in devel/dbg modes
#ifndef NDEBUG
              libmesh_here();
              libMesh::err << "WARNING: Newton scheme has not converged in "
                           << cnt << " iterations:" << std::endl
                           << "   physical_point="
                           << physical_point
                           << "   physical_guess="
                           << physical_guess
                           << "   dp="
                           << dp
                           << "   p="
                           << p
                           << "   error=" << inverse_map_error
                           << "   in element " << elem->id()
                           << std::endl;

              elem->print_info(libMesh::err);
#else
              // In optimized mode, just print once that an inverse_map() call
              // had trouble converging its Newton iteration.
              libmesh_do_once(libMesh::err << "WARNING: At least one element took more than "
                              << max_cnt
                              << " iterations to converge in inverse_map()...\n"
                              << "Rerun in devel/dbg mode for more details."
                              << std::endl;);

#endif // NDEBUG

              if (cnt > 2*max_cnt)
                {
                  libMesh::err << "ERROR: Newton scheme FAILED to converge in "
                               << cnt
                               << " iterations in element "
                               << elem->id()
                               << " for physical point = "
                               << physical_point
                               << std::endl;

                  elem->print_info(libMesh::err);

                  libmesh_error_msg("Exiting...");
                }
            }
          //  Return a far off point when secure is false - this
          //  should only happen when we're trying to map a point
          //  that's outside the element
          else
            {
              for (unsigned int i=0; i != Dim; ++i)
                p(i) = 1e6;

              return p;
            }
        }
    }
  while (inverse_map_error > tolerance);



  //  If we are in debug mode do two sanity checks.
#ifdef DEBUG

  if (secure)
    {
      // Make sure the point \p p on the reference element actually
      // does map to the point \p physical_point within a tolerance.

      const Point check = FE<Dim,T>::map (elem, p);
      const Point diff  = physical_point - check;

      if (diff.norm() > tolerance)
        {
          libmesh_here();
          libMesh::err << "WARNING:  diff is "
                       << diff.norm()
                       << std::endl
                       << " point="
                       << physical_point;
          libMesh::err << " local=" << check;
          libMesh::err << " lref= " << p;

          elem->print_info(libMesh::err);
        }

      // Make sure the point \p p on the reference element actually
      // is

      if (!FEAbstract::on_reference_element(p, elem->type(), 2*tolerance))
        {
          libmesh_here();
          libMesh::err << "WARNING:  inverse_map of physical point "
                       << physical_point
                       << " is not on element." << '\n';
          elem->print_info(libMesh::err);
        }
    }

#endif

  return p;
}

inverse_map() [2/4]

void libMesh::FE< Dim, T >::inverse_map ( const Elem *  elem, const std::vector< Point > &  physical_points, std::vector< Point > &  reference_points, const Real  tolerance = TOLERANCE, const bool  secure = true  )

staticinherited

Takes a number points in physical space (in the physical_points vector) and finds their location on the reference element for the input element elem. The values on the reference element are returned in the vector reference_points. The optional parameter tolerance defines how close is "good enough." The map inversion iteration computes the sequence , and the iteration is terminated when

Definition at line 1960 of file fe_map.C.

{
  // The number of points to find the
  // inverse map of
  const std::size_t n_points = physical_points.size();

  // Resize the vector to hold the points
  // on the reference element
  reference_points.resize(n_points);

  // Find the coordinates on the reference
  // element of each point in physical space
  for (std::size_t p=0; p<n_points; p++)
    reference_points[p] =
      FE<Dim,T>::inverse_map (elem, physical_points[p], tolerance, secure);
}

inverse_map() [3/4]

Point libMesh::FE< 2, SUBDIVISION >::inverse_map ( const Elem *  , const Point &  , const Real  , const bool    )

inherited

Definition at line 876 of file fe_subdivision_2D.C.

{
  libmesh_not_implemented();
}

inverse_map() [4/4]

void libMesh::FE< 2, SUBDIVISION >::inverse_map ( const Elem *  , const std::vector< Point > &  , std::vector< Point > &  , Real  , bool    )

inherited

Definition at line 885 of file fe_subdivision_2D.C.

{
  libmesh_not_implemented();
}

is_hierarchic() [1/54]

bool libMesh::FE< 0, SCALAR >::is_hierarchic ( ) const

virtualinherited

Returns

true if the finite element's higher order shape functions are hierarchic

Implements libMesh::FEAbstract.

Definition at line 113 of file fe_scalar.C.

{ return false; }

is_hierarchic() [2/54]

bool libMesh::FE< 1, SCALAR >::is_hierarchic ( ) const

virtualinherited

Returns

true if the finite element's higher order shape functions are hierarchic

Implements libMesh::FEAbstract.

Definition at line 114 of file fe_scalar.C.

{ return false; }

is_hierarchic() [3/54]

bool libMesh::FE< 2, SCALAR >::is_hierarchic ( ) const

virtualinherited

Returns

true if the finite element's higher order shape functions are hierarchic

Implements libMesh::FEAbstract.

Definition at line 115 of file fe_scalar.C.

{ return false; }

is_hierarchic() [4/54]

bool libMesh::FE< 3, SCALAR >::is_hierarchic ( ) const

virtualinherited

Returns

true if the finite element's higher order shape functions are hierarchic

Implements libMesh::FEAbstract.

Definition at line 116 of file fe_scalar.C.

{ return false; }

is_hierarchic() [5/54]

bool libMesh::FE< 0, L2_HIERARCHIC >::is_hierarchic ( ) const

virtualinherited

Returns

true if the finite element's higher order shape functions are hierarchic

Implements libMesh::FEAbstract.

Definition at line 199 of file fe_l2_hierarchic.C.

{ return true; }

is_hierarchic() [6/54]

bool libMesh::FE< 1, L2_HIERARCHIC >::is_hierarchic ( ) const

virtualinherited

Returns

true if the finite element's higher order shape functions are hierarchic

Implements libMesh::FEAbstract.

Definition at line 200 of file fe_l2_hierarchic.C.

{ return true; }

is_hierarchic() [7/54]

bool libMesh::FE< 2, L2_HIERARCHIC >::is_hierarchic ( ) const

virtualinherited

Returns

true if the finite element's higher order shape functions are hierarchic

Implements libMesh::FEAbstract.

Definition at line 201 of file fe_l2_hierarchic.C.

{ return true; }

is_hierarchic() [8/54]

bool libMesh::FE< 3, L2_HIERARCHIC >::is_hierarchic ( ) const

virtualinherited

Returns

true if the finite element's higher order shape functions are hierarchic

Implements libMesh::FEAbstract.

Definition at line 202 of file fe_l2_hierarchic.C.

{ return true; }

is_hierarchic() [9/54]

virtual bool libMesh::FE< Dim, T >::is_hierarchic ( ) const

overridevirtualinherited

Returns

true if the finite element's higher order shape functions are hierarchic

Implements libMesh::FEAbstract.

is_hierarchic() [10/54]

bool libMesh::FE< 0, CLOUGH >::is_hierarchic ( ) const

virtualinherited

Returns

true if the finite element's higher order shape functions are hierarchic

Implements libMesh::FEAbstract.

Definition at line 296 of file fe_clough.C.

{ return false; } // FIXME - this will be changed

is_hierarchic() [11/54]

bool libMesh::FE< 1, CLOUGH >::is_hierarchic ( ) const

virtualinherited

Returns

true if the finite element's higher order shape functions are hierarchic

Implements libMesh::FEAbstract.

Definition at line 297 of file fe_clough.C.

{ return false; } // FIXME - this will be changed

is_hierarchic() [12/54]

bool libMesh::FE< 2, CLOUGH >::is_hierarchic ( ) const

virtualinherited

Returns

true if the finite element's higher order shape functions are hierarchic

Implements libMesh::FEAbstract.

Definition at line 298 of file fe_clough.C.

{ return false; } // FIXME - this will be changed

is_hierarchic() [13/54]

bool libMesh::FE< 3, CLOUGH >::is_hierarchic ( ) const

virtualinherited

Returns

true if the finite element's higher order shape functions are hierarchic

Implements libMesh::FEAbstract.

Definition at line 299 of file fe_clough.C.

{ return false; } // FIXME - this will be changed

is_hierarchic() [14/54]

bool libMesh::FE< 0, HERMITE >::is_hierarchic ( ) const

virtualinherited

Returns

true if the finite element's higher order shape functions are hierarchic

Implements libMesh::FEAbstract.

Definition at line 346 of file fe_hermite.C.

{ return true; }

is_hierarchic() [15/54]

bool libMesh::FE< 1, HERMITE >::is_hierarchic ( ) const

virtualinherited

Returns

true if the finite element's higher order shape functions are hierarchic

Implements libMesh::FEAbstract.

Definition at line 347 of file fe_hermite.C.

{ return true; }

is_hierarchic() [16/54]

bool libMesh::FE< 2, HERMITE >::is_hierarchic ( ) const

virtualinherited

Returns

true if the finite element's higher order shape functions are hierarchic

Implements libMesh::FEAbstract.

Definition at line 348 of file fe_hermite.C.

{ return true; }

is_hierarchic() [17/54]

bool libMesh::FE< 3, HERMITE >::is_hierarchic ( ) const

virtualinherited

Returns

true if the finite element's higher order shape functions are hierarchic

Implements libMesh::FEAbstract.

Definition at line 349 of file fe_hermite.C.

{ return true; }

is_hierarchic() [18/54]

bool libMesh::FE< 0, HIERARCHIC >::is_hierarchic ( ) const

virtualinherited

Returns

true if the finite element's higher order shape functions are hierarchic

Implements libMesh::FEAbstract.

Definition at line 375 of file fe_hierarchic.C.

{ return true; }

is_hierarchic() [19/54]

bool libMesh::FE< 1, HIERARCHIC >::is_hierarchic ( ) const

virtualinherited

Returns

true if the finite element's higher order shape functions are hierarchic

Implements libMesh::FEAbstract.

Definition at line 376 of file fe_hierarchic.C.

{ return true; }

is_hierarchic() [20/54]

bool libMesh::FE< 2, HIERARCHIC >::is_hierarchic ( ) const

virtualinherited

Returns

true if the finite element's higher order shape functions are hierarchic

Implements libMesh::FEAbstract.

Definition at line 377 of file fe_hierarchic.C.

{ return true; }

is_hierarchic() [21/54]

bool libMesh::FE< 3, HIERARCHIC >::is_hierarchic ( ) const

virtualinherited

Returns

true if the finite element's higher order shape functions are hierarchic

Implements libMesh::FEAbstract.

Definition at line 378 of file fe_hierarchic.C.

{ return true; }

is_hierarchic() [22/54]

bool libMesh::FE< 0, MONOMIAL >::is_hierarchic ( ) const

virtualinherited

Returns

true if the finite element's higher order shape functions are hierarchic

Implements libMesh::FEAbstract.

Definition at line 409 of file fe_monomial.C.

{ return true; }

is_hierarchic() [23/54]

bool libMesh::FE< 1, MONOMIAL >::is_hierarchic ( ) const

virtualinherited

Returns

true if the finite element's higher order shape functions are hierarchic

Implements libMesh::FEAbstract.

Definition at line 410 of file fe_monomial.C.

{ return true; }

is_hierarchic() [24/54]

bool libMesh::FE< 2, MONOMIAL >::is_hierarchic ( ) const

virtualinherited

Returns

true if the finite element's higher order shape functions are hierarchic

Implements libMesh::FEAbstract.

Definition at line 411 of file fe_monomial.C.

{ return true; }

is_hierarchic() [25/54]

bool libMesh::FE< 3, MONOMIAL >::is_hierarchic ( ) const

virtualinherited

Returns

true if the finite element's higher order shape functions are hierarchic

Implements libMesh::FEAbstract.

Definition at line 412 of file fe_monomial.C.

{ return true; }

is_hierarchic() [26/54]

bool libMesh::FE< 0, BERNSTEIN >::is_hierarchic ( ) const

virtualinherited

Returns

true if the finite element's higher order shape functions are hierarchic

Implements libMesh::FEAbstract.

Definition at line 442 of file fe_bernstein.C.

{ return false; }

is_hierarchic() [27/54]

bool libMesh::FE< 1, BERNSTEIN >::is_hierarchic ( ) const

virtualinherited

Returns

true if the finite element's higher order shape functions are hierarchic

Implements libMesh::FEAbstract.

Definition at line 443 of file fe_bernstein.C.

{ return false; }

is_hierarchic() [28/54]

bool libMesh::FE< 2, BERNSTEIN >::is_hierarchic ( ) const

virtualinherited

Returns

true if the finite element's higher order shape functions are hierarchic

Implements libMesh::FEAbstract.

Definition at line 444 of file fe_bernstein.C.

{ return false; }

is_hierarchic() [29/54]

bool libMesh::FE< 3, BERNSTEIN >::is_hierarchic ( ) const

virtualinherited

Returns

true if the finite element's higher order shape functions are hierarchic

Implements libMesh::FEAbstract.

Definition at line 445 of file fe_bernstein.C.

{ return false; }

is_hierarchic() [30/54]

bool libMesh::FE< 0, L2_LAGRANGE >::is_hierarchic ( ) const

virtualinherited

Returns

true if the finite element's higher order shape functions are hierarchic

Implements libMesh::FEAbstract.

Definition at line 480 of file fe_l2_lagrange.C.

{ return false; }

is_hierarchic() [31/54]

bool libMesh::FE< 1, L2_LAGRANGE >::is_hierarchic ( ) const

virtualinherited

Returns

true if the finite element's higher order shape functions are hierarchic

Implements libMesh::FEAbstract.

Definition at line 481 of file fe_l2_lagrange.C.

{ return false; }

is_hierarchic() [32/54]

bool libMesh::FE< 2, L2_LAGRANGE >::is_hierarchic ( ) const

virtualinherited

Returns

true if the finite element's higher order shape functions are hierarchic

Implements libMesh::FEAbstract.

Definition at line 482 of file fe_l2_lagrange.C.

{ return false; }

is_hierarchic() [33/54]

bool libMesh::FE< 3, L2_LAGRANGE >::is_hierarchic ( ) const

virtualinherited

Returns

true if the finite element's higher order shape functions are hierarchic

Implements libMesh::FEAbstract.

Definition at line 483 of file fe_l2_lagrange.C.

{ return false; }

is_hierarchic() [34/54]

bool libMesh::FE< 0, NEDELEC_ONE >::is_hierarchic ( ) const

virtualinherited

Returns

true if the finite element's higher order shape functions are hierarchic

Implements libMesh::FEAbstract.

Definition at line 550 of file fe_nedelec_one.C.

{ NEDELEC_LOW_D_ERROR_MESSAGE }

is_hierarchic() [35/54]

bool libMesh::FE< 1, NEDELEC_ONE >::is_hierarchic ( ) const

virtualinherited

Returns

true if the finite element's higher order shape functions are hierarchic

Implements libMesh::FEAbstract.

Definition at line 551 of file fe_nedelec_one.C.

{ NEDELEC_LOW_D_ERROR_MESSAGE }

is_hierarchic() [36/54]

bool libMesh::FE< 2, NEDELEC_ONE >::is_hierarchic ( ) const

virtualinherited

Returns

true if the finite element's higher order shape functions are hierarchic

Implements libMesh::FEAbstract.

Definition at line 552 of file fe_nedelec_one.C.

{ return false; }

is_hierarchic() [37/54]

bool libMesh::FE< 3, NEDELEC_ONE >::is_hierarchic ( ) const

virtualinherited

Returns

true if the finite element's higher order shape functions are hierarchic

Implements libMesh::FEAbstract.

Definition at line 553 of file fe_nedelec_one.C.

{ return false; }

is_hierarchic() [38/54]

bool libMesh::FE< 0, LAGRANGE >::is_hierarchic ( ) const

virtualinherited

Returns

true if the finite element's higher order shape functions are hierarchic

Implements libMesh::FEAbstract.

Definition at line 883 of file fe_lagrange.C.

{ return false; }

is_hierarchic() [39/54]

bool libMesh::FE< 1, LAGRANGE >::is_hierarchic ( ) const

virtualinherited

Returns

true if the finite element's higher order shape functions are hierarchic

Implements libMesh::FEAbstract.

Definition at line 884 of file fe_lagrange.C.

{ return false; }

is_hierarchic() [40/54]

bool libMesh::FE< 2, LAGRANGE >::is_hierarchic ( ) const

virtualinherited

Returns

true if the finite element's higher order shape functions are hierarchic

Implements libMesh::FEAbstract.

Definition at line 885 of file fe_lagrange.C.

{ return false; }

is_hierarchic() [41/54]

bool libMesh::FE< 3, LAGRANGE >::is_hierarchic ( ) const

virtualinherited

Returns

true if the finite element's higher order shape functions are hierarchic

Implements libMesh::FEAbstract.

Definition at line 886 of file fe_lagrange.C.

{ return false; }

is_hierarchic() [42/54]

bool libMesh::FE< 2, SUBDIVISION >::is_hierarchic ( ) const

virtualinherited

Returns

true if the finite element's higher order shape functions are hierarchic

Implements libMesh::FEAbstract.

Definition at line 913 of file fe_subdivision_2D.C.

{ return false; }

is_hierarchic() [43/54]

bool libMesh::FE< 0, XYZ >::is_hierarchic ( ) const

virtualinherited

Returns

true if the finite element's higher order shape functions are hierarchic

Implements libMesh::FEAbstract.

Definition at line 932 of file fe_xyz.C.

{ return true; }

is_hierarchic() [44/54]

bool libMesh::FE< 0, LAGRANGE_VEC >::is_hierarchic ( ) const

virtualinherited

Returns

true if the finite element's higher order shape functions are hierarchic

Implements libMesh::FEAbstract.

Definition at line 933 of file fe_lagrange_vec.C.

{ return false; }

is_hierarchic() [45/54]

bool libMesh::FE< 1, XYZ >::is_hierarchic ( ) const

virtualinherited

Returns

true if the finite element's higher order shape functions are hierarchic

Implements libMesh::FEAbstract.

Definition at line 933 of file fe_xyz.C.

{ return true; }

is_hierarchic() [46/54]

bool libMesh::FE< 2, XYZ >::is_hierarchic ( ) const

virtualinherited

Returns

true if the finite element's higher order shape functions are hierarchic

Implements libMesh::FEAbstract.

Definition at line 934 of file fe_xyz.C.

{ return true; }

is_hierarchic() [47/54]

bool libMesh::FE< 1, LAGRANGE_VEC >::is_hierarchic ( ) const

virtualinherited

Returns

true if the finite element's higher order shape functions are hierarchic

Implements libMesh::FEAbstract.

Definition at line 934 of file fe_lagrange_vec.C.

{ return false; }

is_hierarchic() [48/54]

bool libMesh::FE< 2, LAGRANGE_VEC >::is_hierarchic ( ) const

virtualinherited

Returns

true if the finite element's higher order shape functions are hierarchic

Implements libMesh::FEAbstract.

Definition at line 935 of file fe_lagrange_vec.C.

{ return false; }

is_hierarchic() [49/54]

bool libMesh::FE< 3, XYZ >::is_hierarchic ( ) const

virtualinherited

Returns

true if the finite element's higher order shape functions are hierarchic

Implements libMesh::FEAbstract.

Definition at line 935 of file fe_xyz.C.

{ return true; }

is_hierarchic() [50/54]

bool libMesh::FE< 3, LAGRANGE_VEC >::is_hierarchic ( ) const

virtualinherited

Returns

true if the finite element's higher order shape functions are hierarchic

Implements libMesh::FEAbstract.

Definition at line 936 of file fe_lagrange_vec.C.

{ return false; }

is_hierarchic() [51/54]

bool libMesh::FE< 0, SZABAB >::is_hierarchic ( ) const

virtualinherited

Returns

true if the finite element's higher order shape functions are hierarchic

Implements libMesh::FEAbstract.

Definition at line 1279 of file fe_szabab.C.

{ return true; }

is_hierarchic() [52/54]

bool libMesh::FE< 1, SZABAB >::is_hierarchic ( ) const

virtualinherited

Returns

true if the finite element's higher order shape functions are hierarchic

Implements libMesh::FEAbstract.

Definition at line 1280 of file fe_szabab.C.

{ return true; }

is_hierarchic() [53/54]

bool libMesh::FE< 2, SZABAB >::is_hierarchic ( ) const

virtualinherited

Returns

true if the finite element's higher order shape functions are hierarchic

Implements libMesh::FEAbstract.

Definition at line 1281 of file fe_szabab.C.

{ return true; }

is_hierarchic() [54/54]

bool libMesh::FE< 3, SZABAB >::is_hierarchic ( ) const

virtualinherited

Returns

true if the finite element's higher order shape functions are hierarchic

Implements libMesh::FEAbstract.

Definition at line 1282 of file fe_szabab.C.

{ return true; }

loop_subdivision_mask()

void libMesh::FESubdivision::loop_subdivision_mask ( std::vector< Real > &  weights, const unsigned int  valence  )

static

Fills the vector weights with the weight coefficients of the Loop subdivision mask for evaluating the limit surface at a node explicitly. The size of weights will be 1 + valence, where valence is the number of neighbor nodes of the node where the limit surface is to be evaluated. The weight for the node itself is the first element of weights.

Definition at line 392 of file fe_subdivision_2D.C.

References libMesh::pi, and libMesh::Real.

Referenced by init_subdivision_matrix().

{
  libmesh_assert_greater(valence, 0);
  const Real cs = std::cos(2 * libMesh::pi / valence);
  const Real nb_weight = (0.625 - Utility::pow<2>(0.375 + 0.25 * cs)) / valence;
  weights.resize(1 + valence, nb_weight);
  weights[0] = 1.0 - valence * nb_weight;
}

map()

Point libMesh::FE< Dim, T >::map ( const Elem *  elem, const Point &  reference_point  )

staticinherited

Returns

The location (in physical space) of the point p located on the reference element.

Definition at line 1985 of file fe_map.C.

{
  libmesh_assert(elem);

  Point p;

  const ElemType type     = elem->type();
  const Order order       = elem->default_order();
  const unsigned int n_sf = FE<Dim,LAGRANGE>::n_shape_functions(type, order);

  // Lagrange basis functions are used for mapping
  for (unsigned int i=0; i<n_sf; i++)
    p.add_scaled (elem->point(i),
                  FE<Dim,LAGRANGE>::shape(type,
                                          order,
                                          i,
                                          reference_point)
                  );

  return p;
}

map_eta()

Point libMesh::FE< Dim, T >::map_eta ( const Elem *  elem, const Point &  reference_point  )

staticinherited

Returns

d(xyz)/deta (in physical space) of the point p located on the reference element.

Definition at line 2040 of file fe_map.C.

{
  libmesh_assert(elem);

  Point p;

  const ElemType type     = elem->type();
  const Order order       = elem->default_order();
  const unsigned int n_sf = FE<Dim,LAGRANGE>::n_shape_functions(type, order);

  // Lagrange basis functions are used for mapping
  for (unsigned int i=0; i<n_sf; i++)
    p.add_scaled (elem->point(i),
                  FE<Dim,LAGRANGE>::shape_deriv(type,
                                                order,
                                                i,
                                                1,
                                                reference_point)
                  );

  return p;
}

map_xi()

Point libMesh::FE< Dim, T >::map_xi ( const Elem *  elem, const Point &  reference_point  )

staticinherited

Returns

d(xyz)/dxi (in physical space) of the point p located on the reference element.

Definition at line 2012 of file fe_map.C.

{
  libmesh_assert(elem);

  Point p;

  const ElemType type     = elem->type();
  const Order order       = elem->default_order();
  const unsigned int n_sf = FE<Dim,LAGRANGE>::n_shape_functions(type, order);

  // Lagrange basis functions are used for mapping
  for (unsigned int i=0; i<n_sf; i++)
    p.add_scaled (elem->point(i),
                  FE<Dim,LAGRANGE>::shape_deriv(type,
                                                order,
                                                i,
                                                0,
                                                reference_point)
                  );

  return p;
}

map_zeta()

Point libMesh::FE< Dim, T >::map_zeta ( const Elem *  elem, const Point &  reference_point  )

staticinherited

Returns

d(xyz)/dzeta (in physical space) of the point p located on the reference element.

Definition at line 2068 of file fe_map.C.

{
  libmesh_assert(elem);

  Point p;

  const ElemType type     = elem->type();
  const Order order       = elem->default_order();
  const unsigned int n_sf = FE<Dim,LAGRANGE>::n_shape_functions(type, order);

  // Lagrange basis functions are used for mapping
  for (unsigned int i=0; i<n_sf; i++)
    p.add_scaled (elem->point(i),
                  FE<Dim,LAGRANGE>::shape_deriv(type,
                                                order,
                                                i,
                                                2,
                                                reference_point)
                  );

  return p;
}

n_dofs() [1/54]

unsigned int libMesh::FE< 0, SCALAR >::n_dofs ( const ElemType  , const Order  o  )

inherited

Definition at line 87 of file fe_scalar.C.

{ return o; }

n_dofs() [2/54]

unsigned int libMesh::FE< 1, SCALAR >::n_dofs ( const ElemType  , const Order  o  )

inherited

Definition at line 88 of file fe_scalar.C.

{ return o; }

n_dofs() [3/54]

unsigned int libMesh::FE< 2, SCALAR >::n_dofs ( const ElemType  , const Order  o  )

inherited

Definition at line 89 of file fe_scalar.C.

{ return o; }

n_dofs() [4/54]

unsigned int libMesh::FE< 3, SCALAR >::n_dofs ( const ElemType  , const Order  o  )

inherited

Definition at line 90 of file fe_scalar.C.

{ return o; }

n_dofs() [5/54]

unsigned int libMesh::FE< 0, L2_HIERARCHIC >::n_dofs ( const ElemType  t, const Order  o  )

inherited

Definition at line 174 of file fe_l2_hierarchic.C.

{ return l2_hierarchic_n_dofs(t, o); }

n_dofs() [6/54]

unsigned int libMesh::FE< 1, L2_HIERARCHIC >::n_dofs ( const ElemType  t, const Order  o  )

inherited

Definition at line 175 of file fe_l2_hierarchic.C.

{ return l2_hierarchic_n_dofs(t, o); }

n_dofs() [7/54]

unsigned int libMesh::FE< 2, L2_HIERARCHIC >::n_dofs ( const ElemType  t, const Order  o  )

inherited

Definition at line 176 of file fe_l2_hierarchic.C.

{ return l2_hierarchic_n_dofs(t, o); }

n_dofs() [8/54]

unsigned int libMesh::FE< 3, L2_HIERARCHIC >::n_dofs ( const ElemType  t, const Order  o  )

inherited

Definition at line 177 of file fe_l2_hierarchic.C.

{ return l2_hierarchic_n_dofs(t, o); }

n_dofs() [9/54]

static unsigned int libMesh::FE< Dim, T >::n_dofs ( const ElemType  t, const Order  o  )

staticinherited

Returns

The number of shape functions associated with this finite element.

On a p-refined element, o should be the total order of the element.

n_dofs() [10/54]

unsigned int libMesh::FE< 0, CLOUGH >::n_dofs ( const ElemType  t, const Order  o  )

inherited

Definition at line 271 of file fe_clough.C.

{ return clough_n_dofs(t, o); }

n_dofs() [11/54]

unsigned int libMesh::FE< 1, CLOUGH >::n_dofs ( const ElemType  t, const Order  o  )

inherited

Definition at line 272 of file fe_clough.C.

{ return clough_n_dofs(t, o); }

n_dofs() [12/54]

unsigned int libMesh::FE< 2, CLOUGH >::n_dofs ( const ElemType  t, const Order  o  )

inherited

Definition at line 273 of file fe_clough.C.

{ return clough_n_dofs(t, o); }

n_dofs() [13/54]

unsigned int libMesh::FE< 3, CLOUGH >::n_dofs ( const ElemType  t, const Order  o  )

inherited

Definition at line 274 of file fe_clough.C.

{ return clough_n_dofs(t, o); }

n_dofs() [14/54]

unsigned int libMesh::FE< 0, HERMITE >::n_dofs ( const ElemType  t, const Order  o  )

inherited

Definition at line 321 of file fe_hermite.C.

{ return hermite_n_dofs(t, o); }

n_dofs() [15/54]

unsigned int libMesh::FE< 1, HERMITE >::n_dofs ( const ElemType  t, const Order  o  )

inherited

Definition at line 322 of file fe_hermite.C.

{ return hermite_n_dofs(t, o); }

n_dofs() [16/54]

unsigned int libMesh::FE< 2, HERMITE >::n_dofs ( const ElemType  t, const Order  o  )

inherited

Definition at line 323 of file fe_hermite.C.

{ return hermite_n_dofs(t, o); }

n_dofs() [17/54]

unsigned int libMesh::FE< 3, HERMITE >::n_dofs ( const ElemType  t, const Order  o  )

inherited

Definition at line 324 of file fe_hermite.C.

{ return hermite_n_dofs(t, o); }

n_dofs() [18/54]

unsigned int libMesh::FE< 0, HIERARCHIC >::n_dofs ( const ElemType  t, const Order  o  )

inherited

Definition at line 351 of file fe_hierarchic.C.

{ return hierarchic_n_dofs(t, o); }

n_dofs() [19/54]

unsigned int libMesh::FE< 1, HIERARCHIC >::n_dofs ( const ElemType  t, const Order  o  )

inherited

Definition at line 352 of file fe_hierarchic.C.

{ return hierarchic_n_dofs(t, o); }

n_dofs() [20/54]

unsigned int libMesh::FE< 2, HIERARCHIC >::n_dofs ( const ElemType  t, const Order  o  )

inherited

Definition at line 353 of file fe_hierarchic.C.

{ return hierarchic_n_dofs(t, o); }

n_dofs() [21/54]

unsigned int libMesh::FE< 3, HIERARCHIC >::n_dofs ( const ElemType  t, const Order  o  )

inherited

Definition at line 354 of file fe_hierarchic.C.

{ return hierarchic_n_dofs(t, o); }

n_dofs() [22/54]

unsigned int libMesh::FE< 0, MONOMIAL >::n_dofs ( const ElemType  t, const Order  o  )

inherited

Definition at line 382 of file fe_monomial.C.

{ return monomial_n_dofs(t, o); }

n_dofs() [23/54]

unsigned int libMesh::FE< 1, MONOMIAL >::n_dofs ( const ElemType  t, const Order  o  )

inherited

Definition at line 383 of file fe_monomial.C.

{ return monomial_n_dofs(t, o); }

n_dofs() [24/54]

unsigned int libMesh::FE< 2, MONOMIAL >::n_dofs ( const ElemType  t, const Order  o  )

inherited

Definition at line 384 of file fe_monomial.C.

{ return monomial_n_dofs(t, o); }

n_dofs() [25/54]

unsigned int libMesh::FE< 3, MONOMIAL >::n_dofs ( const ElemType  t, const Order  o  )

inherited

Definition at line 385 of file fe_monomial.C.

{ return monomial_n_dofs(t, o); }

n_dofs() [26/54]

unsigned int libMesh::FE< 0, BERNSTEIN >::n_dofs ( const ElemType  t, const Order  o  )

inherited

Definition at line 418 of file fe_bernstein.C.

{ return bernstein_n_dofs(t, o); }

n_dofs() [27/54]

unsigned int libMesh::FE< 1, BERNSTEIN >::n_dofs ( const ElemType  t, const Order  o  )

inherited

Definition at line 419 of file fe_bernstein.C.

{ return bernstein_n_dofs(t, o); }

n_dofs() [28/54]

unsigned int libMesh::FE< 2, BERNSTEIN >::n_dofs ( const ElemType  t, const Order  o  )

inherited

Definition at line 420 of file fe_bernstein.C.

{ return bernstein_n_dofs(t, o); }

n_dofs() [29/54]

unsigned int libMesh::FE< 3, BERNSTEIN >::n_dofs ( const ElemType  t, const Order  o  )

inherited

Definition at line 421 of file fe_bernstein.C.

{ return bernstein_n_dofs(t, o); }

n_dofs() [30/54]

unsigned int libMesh::FE< 0, L2_LAGRANGE >::n_dofs ( const ElemType  t, const Order  o  )

inherited

Definition at line 454 of file fe_l2_lagrange.C.

{ return l2_lagrange_n_dofs(t, o); }

n_dofs() [31/54]

unsigned int libMesh::FE< 1, L2_LAGRANGE >::n_dofs ( const ElemType  t, const Order  o  )

inherited

Definition at line 455 of file fe_l2_lagrange.C.

{ return l2_lagrange_n_dofs(t, o); }

n_dofs() [32/54]

unsigned int libMesh::FE< 2, L2_LAGRANGE >::n_dofs ( const ElemType  t, const Order  o  )

inherited

Definition at line 456 of file fe_l2_lagrange.C.

{ return l2_lagrange_n_dofs(t, o); }

n_dofs() [33/54]

unsigned int libMesh::FE< 3, L2_LAGRANGE >::n_dofs ( const ElemType  t, const Order  o  )

inherited

Definition at line 457 of file fe_l2_lagrange.C.

{ return l2_lagrange_n_dofs(t, o); }

n_dofs() [34/54]

unsigned int libMesh::FE< 0, NEDELEC_ONE >::n_dofs ( const ElemType  , const Order    )

inherited

Definition at line 522 of file fe_nedelec_one.C.

{ NEDELEC_LOW_D_ERROR_MESSAGE }

n_dofs() [35/54]

unsigned int libMesh::FE< 1, NEDELEC_ONE >::n_dofs ( const ElemType  , const Order    )

inherited

Definition at line 523 of file fe_nedelec_one.C.

{ NEDELEC_LOW_D_ERROR_MESSAGE }

n_dofs() [36/54]

unsigned int libMesh::FE< 2, NEDELEC_ONE >::n_dofs ( const ElemType  t, const Order  o  )

inherited

Definition at line 524 of file fe_nedelec_one.C.

{ return nedelec_one_n_dofs(t, o); }

n_dofs() [37/54]

unsigned int libMesh::FE< 3, NEDELEC_ONE >::n_dofs ( const ElemType  t, const Order  o  )

inherited

Definition at line 525 of file fe_nedelec_one.C.

{ return nedelec_one_n_dofs(t, o); }

n_dofs() [38/54]

unsigned int libMesh::FE< 0, LAGRANGE >::n_dofs ( const ElemType  t, const Order  o  )

inherited

Definition at line 855 of file fe_lagrange.C.

{ return lagrange_n_dofs(t, o); }

n_dofs() [39/54]

unsigned int libMesh::FE< 1, LAGRANGE >::n_dofs ( const ElemType  t, const Order  o  )

inherited

Definition at line 856 of file fe_lagrange.C.

{ return lagrange_n_dofs(t, o); }

n_dofs() [40/54]

unsigned int libMesh::FE< 2, LAGRANGE >::n_dofs ( const ElemType  t, const Order  o  )

inherited

Definition at line 857 of file fe_lagrange.C.

{ return lagrange_n_dofs(t, o); }

n_dofs() [41/54]

unsigned int libMesh::FE< 3, LAGRANGE >::n_dofs ( const ElemType  t, const Order  o  )

inherited

Definition at line 858 of file fe_lagrange.C.

{ return lagrange_n_dofs(t, o); }

n_dofs() [42/54]

unsigned int libMesh::FE< 2, SUBDIVISION >::n_dofs ( const ElemType  , const Order    )

inherited

Definition at line 897 of file fe_subdivision_2D.C.

{ libmesh_not_implemented(); return 0; }

n_dofs() [43/54]

unsigned int libMesh::FE< 0, LAGRANGE_VEC >::n_dofs ( const ElemType  t, const Order  o  )

inherited

Definition at line 905 of file fe_lagrange_vec.C.

{ return FE<0,LAGRANGE>::n_dofs(t,o); }

n_dofs() [44/54]

unsigned int libMesh::FE< 1, LAGRANGE_VEC >::n_dofs ( const ElemType  t, const Order  o  )

inherited

Definition at line 906 of file fe_lagrange_vec.C.

{ return FE<1,LAGRANGE>::n_dofs(t,o); }

n_dofs() [45/54]

unsigned int libMesh::FE< 0, XYZ >::n_dofs ( const ElemType  t, const Order  o  )

inherited

Definition at line 906 of file fe_xyz.C.

{ return xyz_n_dofs(t, o); }

n_dofs() [46/54]

unsigned int libMesh::FE< 2, LAGRANGE_VEC >::n_dofs ( const ElemType  t, const Order  o  )

inherited

Definition at line 907 of file fe_lagrange_vec.C.

{ return 2*FE<2,LAGRANGE>::n_dofs(t,o); }

n_dofs() [47/54]

unsigned int libMesh::FE< 1, XYZ >::n_dofs ( const ElemType  t, const Order  o  )

inherited

Definition at line 907 of file fe_xyz.C.

{ return xyz_n_dofs(t, o); }

n_dofs() [48/54]

unsigned int libMesh::FE< 3, LAGRANGE_VEC >::n_dofs ( const ElemType  t, const Order  o  )

inherited

Definition at line 908 of file fe_lagrange_vec.C.

{ return 3*FE<3,LAGRANGE>::n_dofs(t,o); }

n_dofs() [49/54]

unsigned int libMesh::FE< 2, XYZ >::n_dofs ( const ElemType  t, const Order  o  )

inherited

Definition at line 908 of file fe_xyz.C.

{ return xyz_n_dofs(t, o); }

n_dofs() [50/54]

unsigned int libMesh::FE< 3, XYZ >::n_dofs ( const ElemType  t, const Order  o  )

inherited

Definition at line 909 of file fe_xyz.C.

{ return xyz_n_dofs(t, o); }

n_dofs() [51/54]

unsigned int libMesh::FE< 0, SZABAB >::n_dofs ( const ElemType  t, const Order  o  )

inherited

Definition at line 1255 of file fe_szabab.C.

{ return szabab_n_dofs(t, o); }

n_dofs() [52/54]

unsigned int libMesh::FE< 1, SZABAB >::n_dofs ( const ElemType  t, const Order  o  )

inherited

Definition at line 1256 of file fe_szabab.C.

{ return szabab_n_dofs(t, o); }

n_dofs() [53/54]

unsigned int libMesh::FE< 2, SZABAB >::n_dofs ( const ElemType  t, const Order  o  )

inherited

Definition at line 1257 of file fe_szabab.C.

{ return szabab_n_dofs(t, o); }

n_dofs() [54/54]

unsigned int libMesh::FE< 3, SZABAB >::n_dofs ( const ElemType  t, const Order  o  )

inherited

Definition at line 1258 of file fe_szabab.C.

{ return szabab_n_dofs(t, o); }

n_dofs_at_node() [1/54]

unsigned int libMesh::FE< 0, SCALAR >::n_dofs_at_node ( const ElemType  , const Order  , const unsigned  int  )

inherited

Definition at line 94 of file fe_scalar.C.

{ return 0; }

n_dofs_at_node() [2/54]

unsigned int libMesh::FE< 1, SCALAR >::n_dofs_at_node ( const ElemType  , const Order  , const unsigned  int  )

inherited

Definition at line 95 of file fe_scalar.C.

{ return 0; }

n_dofs_at_node() [3/54]

unsigned int libMesh::FE< 2, SCALAR >::n_dofs_at_node ( const ElemType  , const Order  , const unsigned  int  )

inherited

Definition at line 96 of file fe_scalar.C.

{ return 0; }

n_dofs_at_node() [4/54]

unsigned int libMesh::FE< 3, SCALAR >::n_dofs_at_node ( const ElemType  , const Order  , const unsigned  int  )

inherited

Definition at line 97 of file fe_scalar.C.

{ return 0; }

n_dofs_at_node() [5/54]

unsigned int libMesh::FE< 0, L2_HIERARCHIC >::n_dofs_at_node ( const ElemType  , const Order  , const unsigned  int  )

inherited

Definition at line 181 of file fe_l2_hierarchic.C.

{ return 0; }

n_dofs_at_node() [6/54]

unsigned int libMesh::FE< 1, L2_HIERARCHIC >::n_dofs_at_node ( const ElemType  , const Order  , const unsigned  int  )

inherited

Definition at line 182 of file fe_l2_hierarchic.C.

{ return 0; }

n_dofs_at_node() [7/54]

unsigned int libMesh::FE< 2, L2_HIERARCHIC >::n_dofs_at_node ( const ElemType  , const Order  , const unsigned  int  )

inherited

Definition at line 183 of file fe_l2_hierarchic.C.

{ return 0; }

n_dofs_at_node() [8/54]

unsigned int libMesh::FE< 3, L2_HIERARCHIC >::n_dofs_at_node ( const ElemType  , const Order  , const unsigned  int  )

inherited

Definition at line 184 of file fe_l2_hierarchic.C.

{ return 0; }

n_dofs_at_node() [9/54]

static unsigned int libMesh::FE< Dim, T >::n_dofs_at_node ( const ElemType  t, const Order  o, const unsigned int  n  )

staticinherited

Returns

The number of dofs at node n for a finite element of type t and order o.

On a p-refined element, o should be the total order of the element.

n_dofs_at_node() [10/54]

unsigned int libMesh::FE< 0, CLOUGH >::n_dofs_at_node ( const ElemType  t, const Order  o, const unsigned int  n  )

inherited

Definition at line 278 of file fe_clough.C.

{ return clough_n_dofs_at_node(t, o, n); }

n_dofs_at_node() [11/54]

unsigned int libMesh::FE< 1, CLOUGH >::n_dofs_at_node ( const ElemType  t, const Order  o, const unsigned int  n  )

inherited

Definition at line 279 of file fe_clough.C.

{ return clough_n_dofs_at_node(t, o, n); }

n_dofs_at_node() [12/54]

unsigned int libMesh::FE< 2, CLOUGH >::n_dofs_at_node ( const ElemType  t, const Order  o, const unsigned int  n  )

inherited

Definition at line 280 of file fe_clough.C.

{ return clough_n_dofs_at_node(t, o, n); }

n_dofs_at_node() [13/54]

unsigned int libMesh::FE< 3, CLOUGH >::n_dofs_at_node ( const ElemType  t, const Order  o, const unsigned int  n  )

inherited

Definition at line 281 of file fe_clough.C.

{ return clough_n_dofs_at_node(t, o, n); }

n_dofs_at_node() [14/54]

unsigned int libMesh::FE< 0, HERMITE >::n_dofs_at_node ( const ElemType  t, const Order  o, const unsigned int  n  )

inherited

Definition at line 328 of file fe_hermite.C.

{ return hermite_n_dofs_at_node(t, o, n); }

n_dofs_at_node() [15/54]

unsigned int libMesh::FE< 1, HERMITE >::n_dofs_at_node ( const ElemType  t, const Order  o, const unsigned int  n  )

inherited

Definition at line 329 of file fe_hermite.C.

{ return hermite_n_dofs_at_node(t, o, n); }

n_dofs_at_node() [16/54]

unsigned int libMesh::FE< 2, HERMITE >::n_dofs_at_node ( const ElemType  t, const Order  o, const unsigned int  n  )

inherited

Definition at line 330 of file fe_hermite.C.

{ return hermite_n_dofs_at_node(t, o, n); }

n_dofs_at_node() [17/54]

unsigned int libMesh::FE< 3, HERMITE >::n_dofs_at_node ( const ElemType  t, const Order  o, const unsigned int  n  )

inherited

Definition at line 331 of file fe_hermite.C.

{ return hermite_n_dofs_at_node(t, o, n); }

n_dofs_at_node() [18/54]

unsigned int libMesh::FE< 0, HIERARCHIC >::n_dofs_at_node ( const ElemType  t, const Order  o, const unsigned int  n  )

inherited

Definition at line 357 of file fe_hierarchic.C.

{ return hierarchic_n_dofs_at_node(t, o, n); }

n_dofs_at_node() [19/54]

unsigned int libMesh::FE< 1, HIERARCHIC >::n_dofs_at_node ( const ElemType  t, const Order  o, const unsigned int  n  )

inherited

Definition at line 358 of file fe_hierarchic.C.

{ return hierarchic_n_dofs_at_node(t, o, n); }

n_dofs_at_node() [20/54]

unsigned int libMesh::FE< 2, HIERARCHIC >::n_dofs_at_node ( const ElemType  t, const Order  o, const unsigned int  n  )

inherited

Definition at line 359 of file fe_hierarchic.C.

{ return hierarchic_n_dofs_at_node(t, o, n); }

n_dofs_at_node() [21/54]

unsigned int libMesh::FE< 3, HIERARCHIC >::n_dofs_at_node ( const ElemType  t, const Order  o, const unsigned int  n  )

inherited

Definition at line 360 of file fe_hierarchic.C.

{ return hierarchic_n_dofs_at_node(t, o, n); }

n_dofs_at_node() [22/54]

unsigned int libMesh::FE< 0, MONOMIAL >::n_dofs_at_node ( const ElemType  , const Order  , const unsigned  int  )

inherited

Definition at line 389 of file fe_monomial.C.

{ return 0; }

n_dofs_at_node() [23/54]

unsigned int libMesh::FE< 1, MONOMIAL >::n_dofs_at_node ( const ElemType  , const Order  , const unsigned  int  )

inherited

Definition at line 390 of file fe_monomial.C.

{ return 0; }

n_dofs_at_node() [24/54]

unsigned int libMesh::FE< 2, MONOMIAL >::n_dofs_at_node ( const ElemType  , const Order  , const unsigned  int  )

inherited

Definition at line 391 of file fe_monomial.C.

{ return 0; }

n_dofs_at_node() [25/54]

unsigned int libMesh::FE< 3, MONOMIAL >::n_dofs_at_node ( const ElemType  , const Order  , const unsigned  int  )

inherited

Definition at line 392 of file fe_monomial.C.

{ return 0; }

n_dofs_at_node() [26/54]

unsigned int libMesh::FE< 0, BERNSTEIN >::n_dofs_at_node ( const ElemType  t, const Order  o, const unsigned int  n  )

inherited

Definition at line 424 of file fe_bernstein.C.

{ return bernstein_n_dofs_at_node(t, o, n); }

n_dofs_at_node() [27/54]

unsigned int libMesh::FE< 1, BERNSTEIN >::n_dofs_at_node ( const ElemType  t, const Order  o, const unsigned int  n  )

inherited

Definition at line 425 of file fe_bernstein.C.

{ return bernstein_n_dofs_at_node(t, o, n); }

n_dofs_at_node() [28/54]

unsigned int libMesh::FE< 2, BERNSTEIN >::n_dofs_at_node ( const ElemType  t, const Order  o, const unsigned int  n  )

inherited

Definition at line 426 of file fe_bernstein.C.

{ return bernstein_n_dofs_at_node(t, o, n); }

n_dofs_at_node() [29/54]

unsigned int libMesh::FE< 3, BERNSTEIN >::n_dofs_at_node ( const ElemType  t, const Order  o, const unsigned int  n  )

inherited

Definition at line 427 of file fe_bernstein.C.

{ return bernstein_n_dofs_at_node(t, o, n); }

n_dofs_at_node() [30/54]

unsigned int libMesh::FE< 0, L2_LAGRANGE >::n_dofs_at_node ( const ElemType  , const Order  , const unsigned  int  )

inherited

Definition at line 461 of file fe_l2_lagrange.C.

{ return 0; }

n_dofs_at_node() [31/54]

unsigned int libMesh::FE< 1, L2_LAGRANGE >::n_dofs_at_node ( const ElemType  , const Order  , const unsigned  int  )

inherited

Definition at line 462 of file fe_l2_lagrange.C.

{ return 0; }

n_dofs_at_node() [32/54]

unsigned int libMesh::FE< 2, L2_LAGRANGE >::n_dofs_at_node ( const ElemType  , const Order  , const unsigned  int  )

inherited

Definition at line 463 of file fe_l2_lagrange.C.

{ return 0; }

n_dofs_at_node() [33/54]

unsigned int libMesh::FE< 3, L2_LAGRANGE >::n_dofs_at_node ( const ElemType  , const Order  , const unsigned  int  )

inherited

Definition at line 464 of file fe_l2_lagrange.C.

{ return 0; }

n_dofs_at_node() [34/54]

unsigned int libMesh::FE< 0, NEDELEC_ONE >::n_dofs_at_node ( const ElemType  , const Order  , const unsigned  int  )

inherited

Definition at line 530 of file fe_nedelec_one.C.

{ NEDELEC_LOW_D_ERROR_MESSAGE }

n_dofs_at_node() [35/54]

unsigned int libMesh::FE< 1, NEDELEC_ONE >::n_dofs_at_node ( const ElemType  , const Order  , const unsigned  int  )

inherited

Definition at line 531 of file fe_nedelec_one.C.

{ NEDELEC_LOW_D_ERROR_MESSAGE }

n_dofs_at_node() [36/54]

unsigned int libMesh::FE< 2, NEDELEC_ONE >::n_dofs_at_node ( const ElemType  t, const Order  o, const unsigned int  n  )

inherited

Definition at line 532 of file fe_nedelec_one.C.

{ return nedelec_one_n_dofs_at_node(t, o, n); }

n_dofs_at_node() [37/54]

unsigned int libMesh::FE< 3, NEDELEC_ONE >::n_dofs_at_node ( const ElemType  t, const Order  o, const unsigned int  n  )

inherited

Definition at line 533 of file fe_nedelec_one.C.

{ return nedelec_one_n_dofs_at_node(t, o, n); }

n_dofs_at_node() [38/54]

unsigned int libMesh::FE< 0, LAGRANGE >::n_dofs_at_node ( const ElemType  t, const Order  o, const unsigned int  n  )

inherited

Definition at line 863 of file fe_lagrange.C.

{ return lagrange_n_dofs_at_node(t, o, n); }

n_dofs_at_node() [39/54]

unsigned int libMesh::FE< 1, LAGRANGE >::n_dofs_at_node ( const ElemType  t, const Order  o, const unsigned int  n  )

inherited

Definition at line 864 of file fe_lagrange.C.

{ return lagrange_n_dofs_at_node(t, o, n); }

n_dofs_at_node() [40/54]

unsigned int libMesh::FE< 2, LAGRANGE >::n_dofs_at_node ( const ElemType  t, const Order  o, const unsigned int  n  )

inherited

Definition at line 865 of file fe_lagrange.C.

{ return lagrange_n_dofs_at_node(t, o, n); }

n_dofs_at_node() [41/54]

unsigned int libMesh::FE< 3, LAGRANGE >::n_dofs_at_node ( const ElemType  t, const Order  o, const unsigned int  n  )

inherited

Definition at line 866 of file fe_lagrange.C.

{ return lagrange_n_dofs_at_node(t, o, n); }

n_dofs_at_node() [42/54]

unsigned int libMesh::FE< 2, SUBDIVISION >::n_dofs_at_node ( const ElemType  , const Order  , const unsigned  int  )

inherited

Definition at line 900 of file fe_subdivision_2D.C.

{ return 1; }

n_dofs_at_node() [43/54]

unsigned int libMesh::FE< 0, LAGRANGE_VEC >::n_dofs_at_node ( const ElemType  t, const Order  o, const unsigned int  n  )

inherited

Definition at line 913 of file fe_lagrange_vec.C.

{ return FE<0,LAGRANGE>::n_dofs_at_node(t,o,n); }

n_dofs_at_node() [44/54]

unsigned int libMesh::FE< 0, XYZ >::n_dofs_at_node ( const ElemType  , const Order  , const unsigned  int  )

inherited

Definition at line 913 of file fe_xyz.C.

{ return 0; }

n_dofs_at_node() [45/54]

unsigned int libMesh::FE< 1, LAGRANGE_VEC >::n_dofs_at_node ( const ElemType  t, const Order  o, const unsigned int  n  )

inherited

Definition at line 914 of file fe_lagrange_vec.C.

{ return FE<1,LAGRANGE>::n_dofs_at_node(t,o,n); }

n_dofs_at_node() [46/54]

unsigned int libMesh::FE< 1, XYZ >::n_dofs_at_node ( const ElemType  , const Order  , const unsigned  int  )

inherited

Definition at line 914 of file fe_xyz.C.

{ return 0; }

n_dofs_at_node() [47/54]

unsigned int libMesh::FE< 2, LAGRANGE_VEC >::n_dofs_at_node ( const ElemType  t, const Order  o, const unsigned int  n  )

inherited

Definition at line 915 of file fe_lagrange_vec.C.

{ return 2*FE<2,LAGRANGE>::n_dofs_at_node(t,o,n); }

n_dofs_at_node() [48/54]

unsigned int libMesh::FE< 2, XYZ >::n_dofs_at_node ( const ElemType  , const Order  , const unsigned  int  )

inherited

Definition at line 915 of file fe_xyz.C.

{ return 0; }

n_dofs_at_node() [49/54]

unsigned int libMesh::FE< 3, LAGRANGE_VEC >::n_dofs_at_node ( const ElemType  t, const Order  o, const unsigned int  n  )

inherited

Definition at line 916 of file fe_lagrange_vec.C.

{ return 3*FE<2,LAGRANGE>::n_dofs_at_node(t,o,n); }

n_dofs_at_node() [50/54]

unsigned int libMesh::FE< 3, XYZ >::n_dofs_at_node ( const ElemType  , const Order  , const unsigned  int  )

inherited

Definition at line 916 of file fe_xyz.C.

{ return 0; }

n_dofs_at_node() [51/54]

unsigned int libMesh::FE< 0, SZABAB >::n_dofs_at_node ( const ElemType  t, const Order  o, const unsigned int  n  )

inherited

Definition at line 1261 of file fe_szabab.C.

{ return szabab_n_dofs_at_node(t, o, n); }

n_dofs_at_node() [52/54]

unsigned int libMesh::FE< 1, SZABAB >::n_dofs_at_node ( const ElemType  t, const Order  o, const unsigned int  n  )

inherited

Definition at line 1262 of file fe_szabab.C.

{ return szabab_n_dofs_at_node(t, o, n); }

n_dofs_at_node() [53/54]

unsigned int libMesh::FE< 2, SZABAB >::n_dofs_at_node ( const ElemType  t, const Order  o, const unsigned int  n  )

inherited

Definition at line 1263 of file fe_szabab.C.

{ return szabab_n_dofs_at_node(t, o, n); }

n_dofs_at_node() [54/54]

unsigned int libMesh::FE< 3, SZABAB >::n_dofs_at_node ( const ElemType  t, const Order  o, const unsigned int  n  )

inherited

Definition at line 1264 of file fe_szabab.C.

{ return szabab_n_dofs_at_node(t, o, n); }

n_dofs_per_elem() [1/54]

unsigned int libMesh::FE< 0, SCALAR >::n_dofs_per_elem ( const ElemType  , const Order    )

inherited

Definition at line 101 of file fe_scalar.C.

{ return 0; }

n_dofs_per_elem() [2/54]

unsigned int libMesh::FE< 1, SCALAR >::n_dofs_per_elem ( const ElemType  , const Order    )

inherited

Definition at line 102 of file fe_scalar.C.

{ return 0; }

n_dofs_per_elem() [3/54]

unsigned int libMesh::FE< 2, SCALAR >::n_dofs_per_elem ( const ElemType  , const Order    )

inherited

Definition at line 103 of file fe_scalar.C.

{ return 0; }

n_dofs_per_elem() [4/54]

unsigned int libMesh::FE< 3, SCALAR >::n_dofs_per_elem ( const ElemType  , const Order    )

inherited

Definition at line 104 of file fe_scalar.C.

{ return 0; }

n_dofs_per_elem() [5/54]

unsigned int libMesh::FE< 0, L2_HIERARCHIC >::n_dofs_per_elem ( const ElemType  t, const Order  o  )

inherited

Definition at line 187 of file fe_l2_hierarchic.C.

{ return l2_hierarchic_n_dofs(t, o); }

n_dofs_per_elem() [6/54]

unsigned int libMesh::FE< 1, L2_HIERARCHIC >::n_dofs_per_elem ( const ElemType  t, const Order  o  )

inherited

Definition at line 188 of file fe_l2_hierarchic.C.

{ return l2_hierarchic_n_dofs(t, o); }

n_dofs_per_elem() [7/54]

unsigned int libMesh::FE< 2, L2_HIERARCHIC >::n_dofs_per_elem ( const ElemType  t, const Order  o  )

inherited

Definition at line 189 of file fe_l2_hierarchic.C.

{ return l2_hierarchic_n_dofs(t, o); }

n_dofs_per_elem() [8/54]

unsigned int libMesh::FE< 3, L2_HIERARCHIC >::n_dofs_per_elem ( const ElemType  t, const Order  o  )

inherited

Definition at line 190 of file fe_l2_hierarchic.C.

{ return l2_hierarchic_n_dofs(t, o); }

n_dofs_per_elem() [9/54]

static unsigned int libMesh::FE< Dim, T >::n_dofs_per_elem ( const ElemType  t, const Order  o  )

staticinherited

Returns

The number of dofs interior to the element, not associated with any interior nodes.

On a p-refined element, o should be the total order of the element.

n_dofs_per_elem() [10/54]

unsigned int libMesh::FE< 0, CLOUGH >::n_dofs_per_elem ( const ElemType  t, const Order  o  )

inherited

Definition at line 284 of file fe_clough.C.

{ return clough_n_dofs_per_elem(t, o); }

n_dofs_per_elem() [11/54]

unsigned int libMesh::FE< 1, CLOUGH >::n_dofs_per_elem ( const ElemType  t, const Order  o  )

inherited

Definition at line 285 of file fe_clough.C.

{ return clough_n_dofs_per_elem(t, o); }

n_dofs_per_elem() [12/54]

unsigned int libMesh::FE< 2, CLOUGH >::n_dofs_per_elem ( const ElemType  t, const Order  o  )

inherited

Definition at line 286 of file fe_clough.C.

{ return clough_n_dofs_per_elem(t, o); }

n_dofs_per_elem() [13/54]

unsigned int libMesh::FE< 3, CLOUGH >::n_dofs_per_elem ( const ElemType  t, const Order  o  )

inherited

Definition at line 287 of file fe_clough.C.

{ return clough_n_dofs_per_elem(t, o); }

n_dofs_per_elem() [14/54]

unsigned int libMesh::FE< 0, HERMITE >::n_dofs_per_elem ( const ElemType  t, const Order  o  )

inherited

Definition at line 334 of file fe_hermite.C.

{ return hermite_n_dofs_per_elem(t, o); }

n_dofs_per_elem() [15/54]

unsigned int libMesh::FE< 1, HERMITE >::n_dofs_per_elem ( const ElemType  t, const Order  o  )

inherited

Definition at line 335 of file fe_hermite.C.

{ return hermite_n_dofs_per_elem(t, o); }

n_dofs_per_elem() [16/54]

unsigned int libMesh::FE< 2, HERMITE >::n_dofs_per_elem ( const ElemType  t, const Order  o  )

inherited

Definition at line 336 of file fe_hermite.C.

{ return hermite_n_dofs_per_elem(t, o); }

n_dofs_per_elem() [17/54]

unsigned int libMesh::FE< 3, HERMITE >::n_dofs_per_elem ( const ElemType  t, const Order  o  )

inherited

Definition at line 337 of file fe_hermite.C.

{ return hermite_n_dofs_per_elem(t, o); }

n_dofs_per_elem() [18/54]

unsigned int libMesh::FE< 0, HIERARCHIC >::n_dofs_per_elem ( const ElemType  t, const Order  o  )

inherited

Definition at line 363 of file fe_hierarchic.C.

{ return hierarchic_n_dofs_per_elem(t, o); }

n_dofs_per_elem() [19/54]

unsigned int libMesh::FE< 1, HIERARCHIC >::n_dofs_per_elem ( const ElemType  t, const Order  o  )

inherited

Definition at line 364 of file fe_hierarchic.C.

{ return hierarchic_n_dofs_per_elem(t, o); }

n_dofs_per_elem() [20/54]

unsigned int libMesh::FE< 2, HIERARCHIC >::n_dofs_per_elem ( const ElemType  t, const Order  o  )

inherited

Definition at line 365 of file fe_hierarchic.C.

{ return hierarchic_n_dofs_per_elem(t, o); }

n_dofs_per_elem() [21/54]

unsigned int libMesh::FE< 3, HIERARCHIC >::n_dofs_per_elem ( const ElemType  t, const Order  o  )

inherited

Definition at line 366 of file fe_hierarchic.C.

{ return hierarchic_n_dofs_per_elem(t, o); }

n_dofs_per_elem() [22/54]

unsigned int libMesh::FE< 0, MONOMIAL >::n_dofs_per_elem ( const ElemType  t, const Order  o  )

inherited

Definition at line 395 of file fe_monomial.C.

{ return monomial_n_dofs(t, o); }

n_dofs_per_elem() [23/54]

unsigned int libMesh::FE< 1, MONOMIAL >::n_dofs_per_elem ( const ElemType  t, const Order  o  )

inherited

Definition at line 396 of file fe_monomial.C.

{ return monomial_n_dofs(t, o); }

n_dofs_per_elem() [24/54]

unsigned int libMesh::FE< 2, MONOMIAL >::n_dofs_per_elem ( const ElemType  t, const Order  o  )

inherited

Definition at line 397 of file fe_monomial.C.

{ return monomial_n_dofs(t, o); }

n_dofs_per_elem() [25/54]

unsigned int libMesh::FE< 3, MONOMIAL >::n_dofs_per_elem ( const ElemType  t, const Order  o  )

inherited

Definition at line 398 of file fe_monomial.C.

{ return monomial_n_dofs(t, o); }

n_dofs_per_elem() [26/54]

unsigned int libMesh::FE< 0, BERNSTEIN >::n_dofs_per_elem ( const ElemType  t, const Order  o  )

inherited

Definition at line 430 of file fe_bernstein.C.

{ return bernstein_n_dofs_per_elem(t, o); }

n_dofs_per_elem() [27/54]

unsigned int libMesh::FE< 1, BERNSTEIN >::n_dofs_per_elem ( const ElemType  t, const Order  o  )

inherited

Definition at line 431 of file fe_bernstein.C.

{ return bernstein_n_dofs_per_elem(t, o); }

n_dofs_per_elem() [28/54]

unsigned int libMesh::FE< 2, BERNSTEIN >::n_dofs_per_elem ( const ElemType  t, const Order  o  )

inherited

Definition at line 432 of file fe_bernstein.C.

{ return bernstein_n_dofs_per_elem(t, o); }

n_dofs_per_elem() [29/54]

unsigned int libMesh::FE< 3, BERNSTEIN >::n_dofs_per_elem ( const ElemType  t, const Order  o  )

inherited

Definition at line 433 of file fe_bernstein.C.

{ return bernstein_n_dofs_per_elem(t, o); }

n_dofs_per_elem() [30/54]

unsigned int libMesh::FE< 0, L2_LAGRANGE >::n_dofs_per_elem ( const ElemType  t, const Order  o  )

inherited

Definition at line 468 of file fe_l2_lagrange.C.

{ return l2_lagrange_n_dofs(t, o); }

n_dofs_per_elem() [31/54]

unsigned int libMesh::FE< 1, L2_LAGRANGE >::n_dofs_per_elem ( const ElemType  t, const Order  o  )

inherited

Definition at line 469 of file fe_l2_lagrange.C.

{ return l2_lagrange_n_dofs(t, o); }

n_dofs_per_elem() [32/54]

unsigned int libMesh::FE< 2, L2_LAGRANGE >::n_dofs_per_elem ( const ElemType  t, const Order  o  )

inherited

Definition at line 470 of file fe_l2_lagrange.C.

{ return l2_lagrange_n_dofs(t, o); }

n_dofs_per_elem() [33/54]

unsigned int libMesh::FE< 3, L2_LAGRANGE >::n_dofs_per_elem ( const ElemType  t, const Order  o  )

inherited

Definition at line 471 of file fe_l2_lagrange.C.

{ return l2_lagrange_n_dofs(t, o); }

n_dofs_per_elem() [34/54]

unsigned int libMesh::FE< 0, NEDELEC_ONE >::n_dofs_per_elem ( const ElemType  , const Order    )

inherited

Definition at line 538 of file fe_nedelec_one.C.

{ NEDELEC_LOW_D_ERROR_MESSAGE }

n_dofs_per_elem() [35/54]

unsigned int libMesh::FE< 1, NEDELEC_ONE >::n_dofs_per_elem ( const ElemType  , const Order    )

inherited

Definition at line 539 of file fe_nedelec_one.C.

{ NEDELEC_LOW_D_ERROR_MESSAGE }

n_dofs_per_elem() [36/54]

unsigned int libMesh::FE< 2, NEDELEC_ONE >::n_dofs_per_elem ( const ElemType  , const Order    )

inherited

Definition at line 540 of file fe_nedelec_one.C.

{ return 0; }

n_dofs_per_elem() [37/54]

unsigned int libMesh::FE< 3, NEDELEC_ONE >::n_dofs_per_elem ( const ElemType  , const Order    )

inherited

Definition at line 541 of file fe_nedelec_one.C.

{ return 0; }

n_dofs_per_elem() [38/54]

unsigned int libMesh::FE< 0, LAGRANGE >::n_dofs_per_elem ( const ElemType  , const Order    )

inherited

Definition at line 871 of file fe_lagrange.C.

{ return 0; }

n_dofs_per_elem() [39/54]

unsigned int libMesh::FE< 1, LAGRANGE >::n_dofs_per_elem ( const ElemType  , const Order    )

inherited

Definition at line 872 of file fe_lagrange.C.

{ return 0; }

n_dofs_per_elem() [40/54]

unsigned int libMesh::FE< 2, LAGRANGE >::n_dofs_per_elem ( const ElemType  , const Order    )

inherited

Definition at line 873 of file fe_lagrange.C.

{ return 0; }

n_dofs_per_elem() [41/54]

unsigned int libMesh::FE< 3, LAGRANGE >::n_dofs_per_elem ( const ElemType  , const Order    )

inherited

Definition at line 874 of file fe_lagrange.C.

{ return 0; }

n_dofs_per_elem() [42/54]

unsigned int libMesh::FE< 2, SUBDIVISION >::n_dofs_per_elem ( const ElemType  , const Order    )

inherited

Definition at line 903 of file fe_subdivision_2D.C.

{ return 0; }

n_dofs_per_elem() [43/54]

unsigned int libMesh::FE< 0, XYZ >::n_dofs_per_elem ( const ElemType  t, const Order  o  )

inherited

Definition at line 919 of file fe_xyz.C.

{ return xyz_n_dofs_per_elem(t, o); }

n_dofs_per_elem() [44/54]

unsigned int libMesh::FE< 1, XYZ >::n_dofs_per_elem ( const ElemType  t, const Order  o  )

inherited

Definition at line 920 of file fe_xyz.C.

{ return xyz_n_dofs_per_elem(t, o); }

n_dofs_per_elem() [45/54]

unsigned int libMesh::FE< 0, LAGRANGE_VEC >::n_dofs_per_elem ( const ElemType  , const Order    )

inherited

Definition at line 921 of file fe_lagrange_vec.C.

{ return 0; }

n_dofs_per_elem() [46/54]

unsigned int libMesh::FE< 2, XYZ >::n_dofs_per_elem ( const ElemType  t, const Order  o  )

inherited

Definition at line 921 of file fe_xyz.C.

{ return xyz_n_dofs_per_elem(t, o); }

n_dofs_per_elem() [47/54]

unsigned int libMesh::FE< 1, LAGRANGE_VEC >::n_dofs_per_elem ( const ElemType  , const Order    )

inherited

Definition at line 922 of file fe_lagrange_vec.C.

{ return 0; }

n_dofs_per_elem() [48/54]

unsigned int libMesh::FE< 3, XYZ >::n_dofs_per_elem ( const ElemType  t, const Order  o  )

inherited

Definition at line 922 of file fe_xyz.C.

{ return xyz_n_dofs_per_elem(t, o); }

n_dofs_per_elem() [49/54]

unsigned int libMesh::FE< 2, LAGRANGE_VEC >::n_dofs_per_elem ( const ElemType  , const Order    )

inherited

Definition at line 923 of file fe_lagrange_vec.C.

{ return 0; }

n_dofs_per_elem() [50/54]

unsigned int libMesh::FE< 3, LAGRANGE_VEC >::n_dofs_per_elem ( const ElemType  , const Order    )

inherited

Definition at line 924 of file fe_lagrange_vec.C.

{ return 0; }

n_dofs_per_elem() [51/54]

unsigned int libMesh::FE< 0, SZABAB >::n_dofs_per_elem ( const ElemType  t, const Order  o  )

inherited

Definition at line 1267 of file fe_szabab.C.

{ return szabab_n_dofs_per_elem(t, o); }

n_dofs_per_elem() [52/54]

unsigned int libMesh::FE< 1, SZABAB >::n_dofs_per_elem ( const ElemType  t, const Order  o  )

inherited

Definition at line 1268 of file fe_szabab.C.

{ return szabab_n_dofs_per_elem(t, o); }

n_dofs_per_elem() [53/54]

unsigned int libMesh::FE< 2, SZABAB >::n_dofs_per_elem ( const ElemType  t, const Order  o  )

inherited

Definition at line 1269 of file fe_szabab.C.

{ return szabab_n_dofs_per_elem(t, o); }

n_dofs_per_elem() [54/54]

unsigned int libMesh::FE< 3, SZABAB >::n_dofs_per_elem ( const ElemType  t, const Order  o  )

inherited

Definition at line 1270 of file fe_szabab.C.

{ return szabab_n_dofs_per_elem(t, o); }

n_objects()

static unsigned int libMesh::ReferenceCounter::n_objects ( )

inlinestaticinherited

Prints the number of outstanding (created, but not yet destroyed) objects.

Definition at line 83 of file reference_counter.h.

References libMesh::ReferenceCounter::_n_objects.

  { return _n_objects; }

n_quadrature_points()

unsigned int libMesh::FE< Dim, T >::n_quadrature_points ( ) const

overridevirtualinherited

Returns

The total number of quadrature points. Call this to get an upper bound for the for loop in your simulation for matrix assembly of the current element.

Implements libMesh::FEAbstract.

Definition at line 69 of file fe.C.

{
  libmesh_assert(this->qrule);
  return this->qrule->n_points();
}

n_shape_functions() [1/2]

unsigned int libMesh::FE< Dim, T >::n_shape_functions ( ) const

overridevirtualinherited

Returns

The number of shape functions associated with this finite element.

Implements libMesh::FEAbstract.

Definition at line 50 of file fe.C.

{
  return FE<Dim,T>::n_dofs (this->elem_type,
                            static_cast<Order>(this->fe_type.order + this->_p_level));
}

n_shape_functions() [2/2]

static unsigned int libMesh::FE< Dim, T >::n_shape_functions ( const ElemType  t, const Order  o  )

inlinestaticinherited

Returns

The number of shape functions associated with a finite element of type t and approximation order o.

On a p-refined element, o should be the total order of the element.

Definition at line 228 of file fe.h.

  { return FE<Dim,T>::n_dofs (t,o); }

nodal_soln() [1/54]

void libMesh::FE< 0, SCALAR >::nodal_soln ( const Elem *  elem, const Order  order, const std::vector< Number > &  elem_soln, std::vector< Number > &  nodal_soln  )

inherited

Definition at line 58 of file fe_scalar.C.

{ scalar_nodal_soln(elem, order, elem_soln, nodal_soln); }

nodal_soln() [2/54]

void libMesh::FE< 1, SCALAR >::nodal_soln ( const Elem *  elem, const Order  order, const std::vector< Number > &  elem_soln, std::vector< Number > &  nodal_soln  )

inherited

Definition at line 65 of file fe_scalar.C.

{ scalar_nodal_soln(elem, order, elem_soln, nodal_soln); }

nodal_soln() [3/54]

void libMesh::FE< 2, SCALAR >::nodal_soln ( const Elem *  elem, const Order  order, const std::vector< Number > &  elem_soln, std::vector< Number > &  nodal_soln  )

inherited

Definition at line 72 of file fe_scalar.C.

{ scalar_nodal_soln(elem, order, elem_soln, nodal_soln); }

nodal_soln() [4/54]

void libMesh::FE< 3, SCALAR >::nodal_soln ( const Elem *  elem, const Order  order, const std::vector< Number > &  elem_soln, std::vector< Number > &  nodal_soln  )

inherited

Definition at line 79 of file fe_scalar.C.

{ scalar_nodal_soln(elem, order, elem_soln, nodal_soln); }

nodal_soln() [5/54]

void libMesh::FE< 0, L2_HIERARCHIC >::nodal_soln ( const Elem *  elem, const Order  order, const std::vector< Number > &  elem_soln, std::vector< Number > &  nodal_soln  )

inherited

Definition at line 146 of file fe_l2_hierarchic.C.

{ l2_hierarchic_nodal_soln(elem, order, elem_soln, nodal_soln, /*Dim=*/0); }

nodal_soln() [6/54]

void libMesh::FE< 1, L2_HIERARCHIC >::nodal_soln ( const Elem *  elem, const Order  order, const std::vector< Number > &  elem_soln, std::vector< Number > &  nodal_soln  )

inherited

Definition at line 153 of file fe_l2_hierarchic.C.

{ l2_hierarchic_nodal_soln(elem, order, elem_soln, nodal_soln, /*Dim=*/1); }

nodal_soln() [7/54]

void libMesh::FE< 2, L2_HIERARCHIC >::nodal_soln ( const Elem *  elem, const Order  order, const std::vector< Number > &  elem_soln, std::vector< Number > &  nodal_soln  )

inherited

Definition at line 160 of file fe_l2_hierarchic.C.

{ l2_hierarchic_nodal_soln(elem, order, elem_soln, nodal_soln, /*Dim=*/2); }

nodal_soln() [8/54]

void libMesh::FE< 3, L2_HIERARCHIC >::nodal_soln ( const Elem *  elem, const Order  order, const std::vector< Number > &  elem_soln, std::vector< Number > &  nodal_soln  )

inherited

Definition at line 167 of file fe_l2_hierarchic.C.

{ l2_hierarchic_nodal_soln(elem, order, elem_soln, nodal_soln, /*Dim=*/3); }

nodal_soln() [9/54]

static void libMesh::FE< Dim, T >::nodal_soln ( const Elem *  elem, const Order  o, const std::vector< Number > &  elem_soln, std::vector< Number > &  nodal_soln  )

staticinherited

Build the nodal soln from the element soln. This is the solution that will be plotted.

On a p-refined element, o should be the base order of the element.

nodal_soln() [10/54]

void libMesh::FE< 0, CLOUGH >::nodal_soln ( const Elem *  elem, const Order  order, const std::vector< Number > &  elem_soln, std::vector< Number > &  nodal_soln  )

inherited

Definition at line 242 of file fe_clough.C.

{ clough_nodal_soln(elem, order, elem_soln, nodal_soln, /*Dim=*/0); }

nodal_soln() [11/54]

void libMesh::FE< 1, CLOUGH >::nodal_soln ( const Elem *  elem, const Order  order, const std::vector< Number > &  elem_soln, std::vector< Number > &  nodal_soln  )

inherited

Definition at line 249 of file fe_clough.C.

{ clough_nodal_soln(elem, order, elem_soln, nodal_soln, /*Dim=*/1); }

nodal_soln() [12/54]

void libMesh::FE< 2, CLOUGH >::nodal_soln ( const Elem *  elem, const Order  order, const std::vector< Number > &  elem_soln, std::vector< Number > &  nodal_soln  )

inherited

Definition at line 256 of file fe_clough.C.

{ clough_nodal_soln(elem, order, elem_soln, nodal_soln, /*Dim=*/2); }

nodal_soln() [13/54]

void libMesh::FE< 3, CLOUGH >::nodal_soln ( const Elem *  elem, const Order  order, const std::vector< Number > &  elem_soln, std::vector< Number > &  nodal_soln  )

inherited

Definition at line 263 of file fe_clough.C.

{ clough_nodal_soln(elem, order, elem_soln, nodal_soln, /*Dim=*/3); }

nodal_soln() [14/54]

void libMesh::FE< 0, HERMITE >::nodal_soln ( const Elem *  elem, const Order  order, const std::vector< Number > &  elem_soln, std::vector< Number > &  nodal_soln  )

inherited

Definition at line 288 of file fe_hermite.C.

{ hermite_nodal_soln(elem, order, elem_soln, nodal_soln, /*Dim=*/0); }

nodal_soln() [15/54]

void libMesh::FE< 1, HERMITE >::nodal_soln ( const Elem *  elem, const Order  order, const std::vector< Number > &  elem_soln, std::vector< Number > &  nodal_soln  )

inherited

Definition at line 295 of file fe_hermite.C.

{ hermite_nodal_soln(elem, order, elem_soln, nodal_soln, /*Dim=*/1); }

nodal_soln() [16/54]

void libMesh::FE< 2, HERMITE >::nodal_soln ( const Elem *  elem, const Order  order, const std::vector< Number > &  elem_soln, std::vector< Number > &  nodal_soln  )

inherited

Definition at line 302 of file fe_hermite.C.

{ hermite_nodal_soln(elem, order, elem_soln, nodal_soln, /*Dim=*/2); }

nodal_soln() [17/54]

void libMesh::FE< 3, HERMITE >::nodal_soln ( const Elem *  elem, const Order  order, const std::vector< Number > &  elem_soln, std::vector< Number > &  nodal_soln  )

inherited

Definition at line 309 of file fe_hermite.C.

{ hermite_nodal_soln(elem, order, elem_soln, nodal_soln, /*Dim=*/3); }

nodal_soln() [18/54]

void libMesh::FE< 0, HIERARCHIC >::nodal_soln ( const Elem *  elem, const Order  order, const std::vector< Number > &  elem_soln, std::vector< Number > &  nodal_soln  )

inherited

Definition at line 322 of file fe_hierarchic.C.

{ hierarchic_nodal_soln(elem, order, elem_soln, nodal_soln, /*Dim=*/0); }

nodal_soln() [19/54]

void libMesh::FE< 1, HIERARCHIC >::nodal_soln ( const Elem *  elem, const Order  order, const std::vector< Number > &  elem_soln, std::vector< Number > &  nodal_soln  )

inherited

Definition at line 329 of file fe_hierarchic.C.

{ hierarchic_nodal_soln(elem, order, elem_soln, nodal_soln, /*Dim=*/1); }

nodal_soln() [20/54]

void libMesh::FE< 2, HIERARCHIC >::nodal_soln ( const Elem *  elem, const Order  order, const std::vector< Number > &  elem_soln, std::vector< Number > &  nodal_soln  )

inherited

Definition at line 336 of file fe_hierarchic.C.

{ hierarchic_nodal_soln(elem, order, elem_soln, nodal_soln, /*Dim=*/2); }

nodal_soln() [21/54]

void libMesh::FE< 3, HIERARCHIC >::nodal_soln ( const Elem *  elem, const Order  order, const std::vector< Number > &  elem_soln, std::vector< Number > &  nodal_soln  )

inherited

Definition at line 343 of file fe_hierarchic.C.

{ hierarchic_nodal_soln(elem, order, elem_soln, nodal_soln, /*Dim=*/3); }

nodal_soln() [22/54]

void libMesh::FE< 0, MONOMIAL >::nodal_soln ( const Elem *  elem, const Order  order, const std::vector< Number > &  elem_soln, std::vector< Number > &  nodal_soln  )

inherited

Definition at line 353 of file fe_monomial.C.

{ monomial_nodal_soln(elem, order, elem_soln, nodal_soln, /*Dim=*/0); }

nodal_soln() [23/54]

void libMesh::FE< 1, MONOMIAL >::nodal_soln ( const Elem *  elem, const Order  order, const std::vector< Number > &  elem_soln, std::vector< Number > &  nodal_soln  )

inherited

Definition at line 360 of file fe_monomial.C.

{ monomial_nodal_soln(elem, order, elem_soln, nodal_soln, /*Dim=*/1); }

nodal_soln() [24/54]

void libMesh::FE< 2, MONOMIAL >::nodal_soln ( const Elem *  elem, const Order  order, const std::vector< Number > &  elem_soln, std::vector< Number > &  nodal_soln  )

inherited

Definition at line 367 of file fe_monomial.C.

{ monomial_nodal_soln(elem, order, elem_soln, nodal_soln, /*Dim=*/2); }

nodal_soln() [25/54]

void libMesh::FE< 3, MONOMIAL >::nodal_soln ( const Elem *  elem, const Order  order, const std::vector< Number > &  elem_soln, std::vector< Number > &  nodal_soln  )

inherited

Definition at line 374 of file fe_monomial.C.

{ monomial_nodal_soln(elem, order, elem_soln, nodal_soln, /*Dim=*/3); }

nodal_soln() [26/54]

void libMesh::FE< 0, BERNSTEIN >::nodal_soln ( const Elem *  elem, const Order  order, const std::vector< Number > &  elem_soln, std::vector< Number > &  nodal_soln  )

inherited

Definition at line 389 of file fe_bernstein.C.

{ bernstein_nodal_soln(elem, order, elem_soln, nodal_soln, /*Dim=*/0); }

nodal_soln() [27/54]

void libMesh::FE< 1, BERNSTEIN >::nodal_soln ( const Elem *  elem, const Order  order, const std::vector< Number > &  elem_soln, std::vector< Number > &  nodal_soln  )

inherited

Definition at line 396 of file fe_bernstein.C.

{ bernstein_nodal_soln(elem, order, elem_soln, nodal_soln, /*Dim=*/1); }

nodal_soln() [28/54]

void libMesh::FE< 2, BERNSTEIN >::nodal_soln ( const Elem *  elem, const Order  order, const std::vector< Number > &  elem_soln, std::vector< Number > &  nodal_soln  )

inherited

Definition at line 403 of file fe_bernstein.C.

{ bernstein_nodal_soln(elem, order, elem_soln, nodal_soln, /*Dim=*/2); }

nodal_soln() [29/54]

void libMesh::FE< 3, BERNSTEIN >::nodal_soln ( const Elem *  elem, const Order  order, const std::vector< Number > &  elem_soln, std::vector< Number > &  nodal_soln  )

inherited

Definition at line 410 of file fe_bernstein.C.

{ bernstein_nodal_soln(elem, order, elem_soln, nodal_soln, /*Dim=*/3); }

nodal_soln() [30/54]

void libMesh::FE< 0, L2_LAGRANGE >::nodal_soln ( const Elem *  elem, const Order  order, const std::vector< Number > &  elem_soln, std::vector< Number > &  nodal_soln  )

inherited

Definition at line 423 of file fe_l2_lagrange.C.

{ l2_lagrange_nodal_soln(elem, order, elem_soln, nodal_soln); }

nodal_soln() [31/54]

void libMesh::FE< 1, L2_LAGRANGE >::nodal_soln ( const Elem *  elem, const Order  order, const std::vector< Number > &  elem_soln, std::vector< Number > &  nodal_soln  )

inherited

Definition at line 430 of file fe_l2_lagrange.C.

{ l2_lagrange_nodal_soln(elem, order, elem_soln, nodal_soln); }

nodal_soln() [32/54]

void libMesh::FE< 2, L2_LAGRANGE >::nodal_soln ( const Elem *  elem, const Order  order, const std::vector< Number > &  elem_soln, std::vector< Number > &  nodal_soln  )

inherited

Definition at line 437 of file fe_l2_lagrange.C.

{ l2_lagrange_nodal_soln(elem, order, elem_soln, nodal_soln); }

nodal_soln() [33/54]

void libMesh::FE< 3, L2_LAGRANGE >::nodal_soln ( const Elem *  elem, const Order  order, const std::vector< Number > &  elem_soln, std::vector< Number > &  nodal_soln  )

inherited

Definition at line 444 of file fe_l2_lagrange.C.

{ l2_lagrange_nodal_soln(elem, order, elem_soln, nodal_soln); }

nodal_soln() [34/54]

void libMesh::FE< 0, NEDELEC_ONE >::nodal_soln ( const Elem *  , const Order  , const std::vector< Number > &  , std::vector< Number > &    )

inherited

Definition at line 491 of file fe_nedelec_one.C.

{ NEDELEC_LOW_D_ERROR_MESSAGE }

nodal_soln() [35/54]

void libMesh::FE< 1, NEDELEC_ONE >::nodal_soln ( const Elem *  , const Order  , const std::vector< Number > &  , std::vector< Number > &    )

inherited

Definition at line 498 of file fe_nedelec_one.C.

{ NEDELEC_LOW_D_ERROR_MESSAGE }

nodal_soln() [36/54]

void libMesh::FE< 2, NEDELEC_ONE >::nodal_soln ( const Elem *  elem, const Order  order, const std::vector< Number > &  elem_soln, std::vector< Number > &  nodal_soln  )

inherited

Definition at line 505 of file fe_nedelec_one.C.

{ nedelec_one_nodal_soln(elem, order, elem_soln, 2 /*dim*/, nodal_soln); }

nodal_soln() [37/54]

void libMesh::FE< 0, LAGRANGE_VEC >::nodal_soln ( const Elem *  elem, const Order  order, const std::vector< Number > &  elem_soln, std::vector< Number > &  nodal_soln  )

inherited

Definition at line 507 of file fe_lagrange_vec.C.

{ FE<0,LAGRANGE>::nodal_soln(elem, order, elem_soln, nodal_soln); }

nodal_soln() [38/54]

void libMesh::FE< 3, NEDELEC_ONE >::nodal_soln ( const Elem *  elem, const Order  order, const std::vector< Number > &  elem_soln, std::vector< Number > &  nodal_soln  )

inherited

Definition at line 512 of file fe_nedelec_one.C.

{ nedelec_one_nodal_soln(elem, order, elem_soln, 3 /*dim*/, nodal_soln); }

nodal_soln() [39/54]

void libMesh::FE< 1, LAGRANGE_VEC >::nodal_soln ( const Elem *  elem, const Order  order, const std::vector< Number > &  elem_soln, std::vector< Number > &  nodal_soln  )

inherited

Definition at line 514 of file fe_lagrange_vec.C.

{ FE<1,LAGRANGE>::nodal_soln(elem, order, elem_soln, nodal_soln); }

nodal_soln() [40/54]

void libMesh::FE< 2, LAGRANGE_VEC >::nodal_soln ( const Elem *  elem, const Order  order, const std::vector< Number > &  elem_soln, std::vector< Number > &  nodal_soln  )

inherited

Definition at line 521 of file fe_lagrange_vec.C.

{ lagrange_vec_nodal_soln(elem, order, elem_soln, 2 /*dimension*/, nodal_soln); }

nodal_soln() [41/54]

void libMesh::FE< 3, LAGRANGE_VEC >::nodal_soln ( const Elem *  elem, const Order  order, const std::vector< Number > &  elem_soln, std::vector< Number > &  nodal_soln  )

inherited

Definition at line 528 of file fe_lagrange_vec.C.

{ lagrange_vec_nodal_soln(elem, order, elem_soln, 3 /*dimension*/, nodal_soln); }

nodal_soln() [42/54]

void libMesh::FE< 2, SUBDIVISION >::nodal_soln ( const Elem *  elem, const Order  , const std::vector< Number > &  elem_soln, std::vector< Number > &  nodal_soln  )

inherited

Definition at line 818 of file fe_subdivision_2D.C.

{
  libmesh_assert(elem);
  libmesh_assert_equal_to(elem->type(), TRI3SUBDIVISION);
  const Tri3Subdivision * sd_elem = static_cast<const Tri3Subdivision *>(elem);

  nodal_soln.resize(3); // three nodes per element

  // Ghost nodes are auxiliary.
  if (sd_elem->is_ghost())
    {
      nodal_soln[0] = 0;
      nodal_soln[1] = 0;
      nodal_soln[2] = 0;
      return;
    }

  // First node (node 0 in the element patch):
  unsigned int j = sd_elem->local_node_number(sd_elem->get_ordered_node(0)->id());
  nodal_soln[j] = elem_soln[0];

  // Second node (node 1 in the element patch):
  j = sd_elem->local_node_number(sd_elem->get_ordered_node(1)->id());
  nodal_soln[j] = elem_soln[1];

  // Third node (node 'valence' in the element patch):
  j = sd_elem->local_node_number(sd_elem->get_ordered_node(2)->id());
  nodal_soln[j] = elem_soln[sd_elem->get_ordered_valence(0)];
}

nodal_soln() [43/54]

void libMesh::FE< 0, LAGRANGE >::nodal_soln ( const Elem *  elem, const Order  order, const std::vector< Number > &  elem_soln, std::vector< Number > &  nodal_soln  )

inherited

Definition at line 824 of file fe_lagrange.C.

{ lagrange_nodal_soln(elem, order, elem_soln, nodal_soln); }

nodal_soln() [44/54]

void libMesh::FE< 1, LAGRANGE >::nodal_soln ( const Elem *  elem, const Order  order, const std::vector< Number > &  elem_soln, std::vector< Number > &  nodal_soln  )

inherited

Definition at line 831 of file fe_lagrange.C.

{ lagrange_nodal_soln(elem, order, elem_soln, nodal_soln); }

nodal_soln() [45/54]

void libMesh::FE< 2, LAGRANGE >::nodal_soln ( const Elem *  elem, const Order  order, const std::vector< Number > &  elem_soln, std::vector< Number > &  nodal_soln  )

inherited

Definition at line 838 of file fe_lagrange.C.

{ lagrange_nodal_soln(elem, order, elem_soln, nodal_soln); }

nodal_soln() [46/54]

void libMesh::FE< 3, LAGRANGE >::nodal_soln ( const Elem *  elem, const Order  order, const std::vector< Number > &  elem_soln, std::vector< Number > &  nodal_soln  )

inherited

Definition at line 845 of file fe_lagrange.C.

{ lagrange_nodal_soln(elem, order, elem_soln, nodal_soln); }

nodal_soln() [47/54]

void libMesh::FE< 0, XYZ >::nodal_soln ( const Elem *  elem, const Order  order, const std::vector< Number > &  elem_soln, std::vector< Number > &  nodal_soln  )

inherited

Definition at line 876 of file fe_xyz.C.

{ xyz_nodal_soln(elem, order, elem_soln, nodal_soln, /*Dim=*/0); }

nodal_soln() [48/54]

void libMesh::FE< 1, XYZ >::nodal_soln ( const Elem *  elem, const Order  order, const std::vector< Number > &  elem_soln, std::vector< Number > &  nodal_soln  )

inherited

Definition at line 883 of file fe_xyz.C.

{ xyz_nodal_soln(elem, order, elem_soln, nodal_soln, /*Dim=*/1); }

nodal_soln() [49/54]

void libMesh::FE< 2, XYZ >::nodal_soln ( const Elem *  elem, const Order  order, const std::vector< Number > &  elem_soln, std::vector< Number > &  nodal_soln  )

inherited

Definition at line 890 of file fe_xyz.C.

{ xyz_nodal_soln(elem, order, elem_soln, nodal_soln, /*Dim=*/2); }

nodal_soln() [50/54]

void libMesh::FE< 3, XYZ >::nodal_soln ( const Elem *  elem, const Order  order, const std::vector< Number > &  elem_soln, std::vector< Number > &  nodal_soln  )

inherited

Definition at line 897 of file fe_xyz.C.

{ xyz_nodal_soln(elem, order, elem_soln, nodal_soln, /*Dim=*/3); }

nodal_soln() [51/54]

void libMesh::FE< 0, SZABAB >::nodal_soln ( const Elem *  elem, const Order  order, const std::vector< Number > &  elem_soln, std::vector< Number > &  nodal_soln  )

inherited

Definition at line 1226 of file fe_szabab.C.

{ szabab_nodal_soln(elem, order, elem_soln, nodal_soln, /*Dim=*/0); }

nodal_soln() [52/54]

void libMesh::FE< 1, SZABAB >::nodal_soln ( const Elem *  elem, const Order  order, const std::vector< Number > &  elem_soln, std::vector< Number > &  nodal_soln  )

inherited

Definition at line 1233 of file fe_szabab.C.

{ szabab_nodal_soln(elem, order, elem_soln, nodal_soln, /*Dim=*/1); }

nodal_soln() [53/54]

void libMesh::FE< 2, SZABAB >::nodal_soln ( const Elem *  elem, const Order  order, const std::vector< Number > &  elem_soln, std::vector< Number > &  nodal_soln  )

inherited

Definition at line 1240 of file fe_szabab.C.

{ szabab_nodal_soln(elem, order, elem_soln, nodal_soln, /*Dim=*/2); }

nodal_soln() [54/54]

void libMesh::FE< 3, SZABAB >::nodal_soln ( const Elem *  elem, const Order  order, const std::vector< Number > &  elem_soln, std::vector< Number > &  nodal_soln  )

inherited

Definition at line 1247 of file fe_szabab.C.

{ szabab_nodal_soln(elem, order, elem_soln, nodal_soln, /*Dim=*/3); }

on_reference_element()

bool libMesh::FEAbstract::on_reference_element ( const Point &  p, const ElemType  t, const Real  eps = TOLERANCE  )

staticinherited

Returns

true if the point p is located on the reference element for element type t, false otherwise. Since we are doing floating point comparisons here the parameter eps can be specified to indicate a tolerance. For example, becomes .

Definition at line 581 of file fe_abstract.C.

References libMesh::EDGE2, libMesh::EDGE3, libMesh::EDGE4, libMesh::HEX20, libMesh::HEX27, libMesh::HEX8, libMesh::INFHEX16, libMesh::INFHEX18, libMesh::INFHEX8, libMesh::INFPRISM12, libMesh::INFPRISM6, libMesh::NODEELEM, libMesh::PRISM15, libMesh::PRISM18, libMesh::PRISM6, libMesh::PYRAMID13, libMesh::PYRAMID14, libMesh::PYRAMID5, libMesh::QUAD4, libMesh::QUAD8, libMesh::QUAD9, libMesh::QUADSHELL4, libMesh::QUADSHELL8, libMesh::Real, libMesh::TET10, libMesh::TET4, libMesh::TRI3, libMesh::TRI6, and libMesh::TRISHELL3.

Referenced by libMesh::FEInterface::ifem_on_reference_element(), libMesh::FE< Dim, LAGRANGE_VEC >::inverse_map(), and libMesh::FEInterface::on_reference_element().

{
  libmesh_assert_greater_equal (eps, 0.);

  const Real xi   = p(0);
#if LIBMESH_DIM > 1
  const Real eta  = p(1);
#else
  const Real eta  = 0.;
#endif
#if LIBMESH_DIM > 2
  const Real zeta = p(2);
#else
  const Real zeta  = 0.;
#endif

  switch (t)
    {
    case NODEELEM:
      {
        return (!xi && !eta && !zeta);
      }
    case EDGE2:
    case EDGE3:
    case EDGE4:
      {
        // The reference 1D element is [-1,1].
        if ((xi >= -1.-eps) &&
            (xi <=  1.+eps))
          return true;

        return false;
      }


    case TRI3:
    case TRISHELL3:
    case TRI6:
      {
        // The reference triangle is isosceles
        // and is bound by xi=0, eta=0, and xi+eta=1.
        if ((xi  >= 0.-eps) &&
            (eta >= 0.-eps) &&
            ((xi + eta) <= 1.+eps))
          return true;

        return false;
      }


    case QUAD4:
    case QUADSHELL4:
    case QUAD8:
    case QUADSHELL8:
    case QUAD9:
      {
        // The reference quadrilateral element is [-1,1]^2.
        if ((xi  >= -1.-eps) &&
            (xi  <=  1.+eps) &&
            (eta >= -1.-eps) &&
            (eta <=  1.+eps))
          return true;

        return false;
      }


    case TET4:
    case TET10:
      {
        // The reference tetrahedral is isosceles
        // and is bound by xi=0, eta=0, zeta=0,
        // and xi+eta+zeta=1.
        if ((xi   >= 0.-eps) &&
            (eta  >= 0.-eps) &&
            (zeta >= 0.-eps) &&
            ((xi + eta + zeta) <= 1.+eps))
          return true;

        return false;
      }


    case HEX8:
    case HEX20:
    case HEX27:
      {
        /*
          if ((xi   >= -1.) &&
          (xi   <=  1.) &&
          (eta  >= -1.) &&
          (eta  <=  1.) &&
          (zeta >= -1.) &&
          (zeta <=  1.))
          return true;
        */

        // The reference hexahedral element is [-1,1]^3.
        if ((xi   >= -1.-eps) &&
            (xi   <=  1.+eps) &&
            (eta  >= -1.-eps) &&
            (eta  <=  1.+eps) &&
            (zeta >= -1.-eps) &&
            (zeta <=  1.+eps))
          {
            //    libMesh::out << "Strange Point:\n";
            //    p.print();
            return true;
          }

        return false;
      }

    case PRISM6:
    case PRISM15:
    case PRISM18:
      {
        // Figure this one out...
        // inside the reference triangle with zeta in [-1,1]
        if ((xi   >=  0.-eps) &&
            (eta  >=  0.-eps) &&
            (zeta >= -1.-eps) &&
            (zeta <=  1.+eps) &&
            ((xi + eta) <= 1.+eps))
          return true;

        return false;
      }


    case PYRAMID5:
    case PYRAMID13:
    case PYRAMID14:
      {
        // Check that the point is on the same side of all the faces
        // by testing whether:
        //
        // n_i.(x - x_i) <= 0
        //
        // for each i, where:
        //   n_i is the outward normal of face i,
        //   x_i is a point on face i.
        if ((-eta - 1. + zeta <= 0.+eps) &&
            (  xi - 1. + zeta <= 0.+eps) &&
            ( eta - 1. + zeta <= 0.+eps) &&
            ( -xi - 1. + zeta <= 0.+eps) &&
            (            zeta >= 0.-eps))
          return true;

        return false;
      }

#ifdef LIBMESH_ENABLE_INFINITE_ELEMENTS
    case INFHEX8:
    case INFHEX16:
    case INFHEX18:
      {
        // The reference infhex8 is a [-1,1]^3.
        if ((xi   >= -1.-eps) &&
            (xi   <=  1.+eps) &&
            (eta  >= -1.-eps) &&
            (eta  <=  1.+eps) &&
            (zeta >= -1.-eps) &&
            (zeta <=  1.+eps))
          {
            return true;
          }
        return false;
      }

    case INFPRISM6:
    case INFPRISM12:
      {
        // inside the reference triangle with zeta in [-1,1]
        if ((xi   >=  0.-eps) &&
            (eta  >=  0.-eps) &&
            (zeta >= -1.-eps) &&
            (zeta <=  1.+eps) &&
            ((xi + eta) <= 1.+eps))
          {
            return true;
          }

        return false;
      }
#endif

    default:
      libmesh_error_msg("ERROR: Unknown element type " << t);
    }

  // If we get here then the point is _not_ in the
  // reference element.   Better return false.

  return false;
}

print_d2phi()

void libMesh::FEGenericBase< FEOutputType< T >::type >::print_d2phi ( std::ostream &  os ) const

virtualinherited

Prints the value of each shape function's second derivatives at each quadrature point.

Implements libMesh::FEAbstract.

Definition at line 776 of file fe_base.C.

References libMesh::index_range().

{
  for (auto i : index_range(dphi))
    for (auto j : index_range(dphi[i]))
      os << " d2phi[" << i << "][" << j << "]=" << d2phi[i][j];
}

print_dphi()

void libMesh::FEGenericBase< FEOutputType< T >::type >::print_dphi ( std::ostream &  os ) const

virtualinherited

Prints the value of each shape function's derivative at each quadrature point.

Implements libMesh::FEAbstract.

Definition at line 718 of file fe_base.C.

References libMesh::index_range().

{
  for (auto i : index_range(dphi))
    for (auto j : index_range(dphi[i]))
      os << " dphi[" << i << "][" << j << "]=" << dphi[i][j];
}

print_info() [1/2]

void libMesh::ReferenceCounter::print_info ( std::ostream &  out = libMesh::out )

staticinherited

Prints the reference information, by default to libMesh::out.

Definition at line 87 of file reference_counter.C.

References libMesh::ReferenceCounter::_enable_print_counter, and libMesh::ReferenceCounter::get_info().

{
  if (_enable_print_counter)
    out_stream << ReferenceCounter::get_info();
}

print_info() [2/2]

void libMesh::FEAbstract::print_info ( std::ostream &  os ) const

inherited

Prints all the relevant information about the current element.

Definition at line 793 of file fe_abstract.C.

References libMesh::FEAbstract::print_dphi(), libMesh::FEAbstract::print_JxW(), libMesh::FEAbstract::print_phi(), and libMesh::FEAbstract::print_xyz().

Referenced by libMesh::operator<<().

{
  os << "phi[i][j]: Shape function i at quadrature pt. j" << std::endl;
  this->print_phi(os);

  os << "dphi[i][j]: Shape function i's gradient at quadrature pt. j" << std::endl;
  this->print_dphi(os);

  os << "XYZ locations of the quadrature pts." << std::endl;
  this->print_xyz(os);

  os << "Values of JxW at the quadrature pts." << std::endl;
  this->print_JxW(os);
}

print_JxW()

void libMesh::FEAbstract::print_JxW ( std::ostream &  os ) const

inherited

Prints the Jacobian times the weight for each quadrature point.

Definition at line 780 of file fe_abstract.C.

References libMesh::FEAbstract::_fe_map.

Referenced by libMesh::FEAbstract::print_info().

{
  this->_fe_map->print_JxW(os);
}

print_phi()

void libMesh::FEGenericBase< FEOutputType< T >::type >::print_phi ( std::ostream &  os ) const

virtualinherited

Prints the value of each shape function at each quadrature point.

Implements libMesh::FEAbstract.

Definition at line 707 of file fe_base.C.

References libMesh::index_range().

{
  for (auto i : index_range(phi))
    for (auto j : index_range(phi[i]))
      os << " phi[" << i << "][" << j << "]=" << phi[i][j] << std::endl;
}

print_xyz()

void libMesh::FEAbstract::print_xyz ( std::ostream &  os ) const

inherited

Prints the spatial location of each quadrature point (on the physical element).

Definition at line 787 of file fe_abstract.C.

References libMesh::FEAbstract::_fe_map.

Referenced by libMesh::FEAbstract::print_info().

{
  this->_fe_map->print_xyz(os);
}

regular_shape()

Real libMesh::FESubdivision::regular_shape ( const unsigned int  i, const Real  v, const Real  w  )

static

Returns

The value of the of the 12 quartic box splines interpolating a regular Loop subdivision element, evaluated at the barycentric coordinates v, w.

Definition at line 134 of file fe_subdivision_2D.C.

References libMesh::Utility::pow(), and libMesh::Real.

{
  // These are the 12 quartic box splines, see Cirak et al.,
  // Int. J. Numer. Meth. Engng. 2000; 47:2039-2072, Appendix A.1.

  const Real u = 1 - v - w;
  libmesh_assert_less_equal(0, v);
  libmesh_assert_less_equal(0, w);
  libmesh_assert_less_equal(0, u);

  using Utility::pow;
  const Real factor = 1. / 12;

  switch (i)
    {
    case 0:
      return factor*(pow<4>(u) + 2*u*u*u*v);
    case 1:
      return factor*(pow<4>(u) + 2*u*u*u*w);
    case 2:
      return factor*(pow<4>(u) + 2*u*u*u*w + 6*u*u*u*v + 6*u*u*v*w + 12*u*u*v*v + 6*u*v*v*w + 6*u*v*v*v +
                     2*v*v*v*w + pow<4>(v));
    case 3:
      return factor*(6*pow<4>(u) + 24*u*u*u*w + 24*u*u*w*w + 8*u*w*w*w + pow<4>(w) + 24*u*u*u*v +
                     60*u*u*v*w + 36*u*v*w*w + 6*v*w*w*w + 24*u*u*v*v + 36*u*v*v*w + 12*v*v*w*w + 8*u*v*v*v +
                     6*v*v*v*w + pow<4>(v));
    case 4:
      return factor*(pow<4>(u) + 6*u*u*u*w + 12*u*u*w*w + 6*u*w*w*w + pow<4>(w) + 2*u*u*u*v + 6*u*u*v*w +
                     6*u*v*w*w + 2*v*w*w*w);
    case 5:
      return factor*(2*u*v*v*v + pow<4>(v));
    case 6:
      return factor*(pow<4>(u) + 6*u*u*u*w + 12*u*u*w*w + 6*u*w*w*w + pow<4>(w) + 8*u*u*u*v + 36*u*u*v*w +
                     36*u*v*w*w + 8*v*w*w*w + 24*u*u*v*v + 60*u*v*v*w + 24*v*v*w*w + 24*u*v*v*v + 24*v*v*v*w + 6*pow<4>(v));
    case 7:
      return factor*(pow<4>(u) + 8*u*u*u*w + 24*u*u*w*w + 24*u*w*w*w + 6*pow<4>(w) + 6*u*u*u*v + 36*u*u*v*w +
                     60*u*v*w*w + 24*v*w*w*w + 12*u*u*v*v + 36*u*v*v*w + 24*v*v*w*w + 6*u*v*v*v + 8*v*v*v*w + pow<4>(v));
    case 8:
      return factor*(2*u*w*w*w + pow<4>(w));
    case 9:
      return factor*(2*v*v*v*w + pow<4>(v));
    case 10:
      return factor*(2*u*w*w*w + pow<4>(w) + 6*u*v*w*w + 6*v*w*w*w + 6*u*v*v*w + 12*v*v*w*w + 2*u*v*v*v +
                     6*v*v*v*w + pow<4>(v));
    case 11:
      return factor*(pow<4>(w) + 2*v*w*w*w);

    default:
      libmesh_error_msg("Invalid i = " << i);
    }
}

regular_shape_deriv()

Real libMesh::FESubdivision::regular_shape_deriv ( const unsigned int  i, const unsigned int  j, const Real  v, const Real  w  )

static

Returns

The derivative of the of the 12 quartic box splines interpolating a regular Loop subdivision element, evaluated at the barycentric coordinates v, w.

Definition at line 190 of file fe_subdivision_2D.C.

References libMesh::Real.

{
  const Real u = 1 - v - w;
  const Real factor = 1. / 12;

  switch (j) // j=0: xi-directional derivative, j=1: eta-directional derivative
    {
    case 0: // xi derivatives
      {
        switch (i) // shape function number
          {
          case 0:
            return factor*(-6*v*u*u - 2*u*u*u);
          case 1:
            return factor*(-4*u*u*u - 6*u*u*w);
          case 2:
            return factor*(-2*v*v*v - 6*v*v*u + 6*v*u*u + 2*u*u*u);
          case 3:
            return factor*(-4*v*v*v - 24*v*v*u - 24*v*u*u - 18*v*v*w - 48*v*u*w - 12*u*u*w -
                           12*v*w*w - 12*u*w*w - 2*w*w*w);
          case 4:
            return factor*(-6*v*u*u - 2*u*u*u - 12*v*u*w-12*u*u*w - 6*v*w*w - 18*u*w*w - 4*w*w*w);
          case 5:
            return factor*(2*v*v*v + 6*v*v*u);
          case 6:
            return factor*(24*v*v*u + 24*v*u*u + 4*u*u*u + 12*v*v*w + 48*v*u*w + 18*u*u*w +
                           12*v*w*w + 12*u*w*w + 2*w*w*w);
          case 7:
            return factor*(-2*v*v*v - 6*v*v*u + 6*v*u*u + 2*u*u*u - 12*v*v*w + 12*u*u*w -
                           12*v*w*w + 12*u*w*w);
          case 8:
            return -w*w*w/6;
          case 9:
            return factor*(4*v*v*v + 6*v*v*w);
          case 10:
            return factor*(2*v*v*v + 6*v*v*u + 12*v*v*w + 12*v*u*w + 18*v*w*w + 6*u*w*w + 4*w*w*w);
          case 11:
            return w*w*w/6;
          default:
            libmesh_error_msg("Invalid i = " << i);
          }
      }
    case 1: // eta derivatives
      {
        switch (i) // shape function number
          {
          case 0:
            return factor*(-6*v*u*u - 4*u*u*u);
          case 1:
            return factor*(-2*u*u*u - 6*u*u*w);
          case 2:
            return factor*(-4*v*v*v - 18*v*v*u - 12*v*u*u - 2*u*u*u - 6*v*v*w - 12*v*u*w -
                           6*u*u*w);
          case 3:
            return factor*(-2*v*v*v-12*v*v*u - 12*v*u*u - 12*v*v*w - 48*v*u*w - 24*u*u*w -
                           18*v*w*w - 24*u*w*w - 4*w*w*w);
          case 4:
            return factor*(2*u*u*u + 6*u*u*w - 6*u*w*w - 2*w*w*w);
          case 5:
            return -v*v*v/6;
          case 6:
            return factor*(12*v*v*u + 12*v*u*u + 2*u*u*u - 12*v*v*w + 6*u*u*w - 12*v*w*w -
                           6*u*w*w - 2*w*w*w);
          case 7:
            return factor*(2*v*v*v + 12*v*v*u + 18*v*u*u + 4*u*u*u + 12*v*v*w + 48*v*u*w +
                           24*u*u*w + 12*v*w*w + 24*u*w*w);
          case 8:
            return factor*(6*u*w*w + 2*w*w*w);
          case 9:
            return v*v*v/6;
          case 10:
            return factor*(4*v*v*v + 6*v*v*u + 18*v*v*w + 12*v*u*w + 12*v*w*w + 6*u*w*w +
                           2*w*w*w);
          case 11:
            return factor*(6*v*w*w + 4*w*w*w);
          default:
            libmesh_error_msg("Invalid i = " << i);
          }
      }
    default:
      libmesh_error_msg("Invalid j = " << j);
    }
}

regular_shape_second_deriv()

Real libMesh::FESubdivision::regular_shape_second_deriv ( const unsigned int  i, const unsigned int  j, const Real  v, const Real  w  )

static

Returns

The second derivative of the of the 12 quartic box splines interpolating a regular Loop subdivision element, evaluated at the barycentric coordinates v, w.

Definition at line 279 of file fe_subdivision_2D.C.

References libMesh::Real.

{
  const Real u = 1 - v - w;
  const Real factor = 1. / 12;

  switch (j)
    {
    case 0: // xi-xi derivative
      {
        switch (i) // shape function number
          {
          case 0:
            return v*u;
          case 1:
            return u*u + u*w;
          case 2:
            return -2*v*u;
          case 3:
            return v*v - 2*u*u + v*w - 2*u*w;
          case 4:
            return v*u + v*w + u*w + w*w;
          case 5:
            return v*u;
          case 6:
            return factor*(-24*v*v + 12*u*u - 24*v*w + 12*u*w);
          case 7:
            return -2*v*u - 2*v*w - 2*u*w - 2*w*w;
          case 8:
            return 0.;
          case 9:
            return v*v + v*w;
          case 10:
            return v*u + v*w + u*w + w*w;
          case 11:
            return 0.;
          default:
            libmesh_error_msg("Invalid i = " << i);
          }
      }
    case 1: //eta-xi derivative
      {
        switch (i)
          {
          case 0:
            return factor*(12*v*u + 6*u*u);
          case 1:
            return factor*(6*u*u + 12*u*w);
          case 2:
            return factor*(6*v*v - 12*v*u - 6*u*u);
          case 3:
            return factor*(6*v*v - 12*u*u + 24*v*w + 6*w*w);
          case 4:
            return factor*(-6*u*u - 12*u*w + 6*w*w);
          case 5:
            return -v*v/2.;
          case 6:
            return factor*(-12*v*v + 6*u*u - 24*v*w - 12*u*w - 6*w*w);
          case 7:
            return factor*(-6*v*v - 12*v*u + 6*u*u - 24*v*w - 12*w*w);
          case 8:
            return -w*w/2.;
          case 9:
            return v*v/2.;
          case 10:
            return factor*(6*v*v + 12*v*u + 24*v*w + 12*u*w + 6*w*w);
          case 11:
            return w*w/2.;
          default:
            libmesh_error_msg("Invalid i = " << i);
          }
      }
    case 2: // eta-eta derivative
      {
        switch (i)
          {
          case 0:
            return v*u + u*u;
          case 1:
            return u*w;
          case 2:
            return v*v + v*u + v*w + u*w;
          case 3:
            return -2*v*u - 2*u*u + v*w + w*w;
          case 4:
            return -2*u*w;
          case 5:
            return 0.;
          case 6:
            return -2*v*v - 2*v*u - 2*v*w - 2*u*w;
          case 7:
            return v*u + u*u - 2*v*w - 2*w*w;
          case 8:
            return u*w;
          case 9:
            return 0.;
          case 10:
            return v*v + v*u + v*w + u*w;
          case 11:
            return v*w + w*w;
          default:
            libmesh_error_msg("Invalid i = " << i);
          }
      }
    default:
      libmesh_error_msg("Invalid j = " << j);
    }
}

reinit() [1/2]

virtual void libMesh::FESubdivision::reinit ( const Elem *  elem, const std::vector< Point > *const  pts = nullptr, const std::vector< Real > *const  weights = nullptr  )

overridevirtual

This is at the core of this class. Use this for each new non-ghosted element in the mesh. Reinitializes all the physical element-dependent data based on the current element elem. By default the shape functions and associated data are computed at the quadrature points specified by the quadrature rule qrule, but may be any points specified on the reference element specified in the optional argument pts.

Reimplemented from libMesh::FE< 2, SUBDIVISION >.

reinit() [2/2]

virtual void libMesh::FESubdivision::reinit ( const Elem *  , const unsigned int  , const Real  = TOLERANCE, const std::vector< Point > *  const = nullptr, const std::vector< Real > *  const = nullptr  )

inlineoverridevirtual

This prevents some compilers being confused by partially overriding this virtual function.

Reimplemented from libMesh::FE< 2, SUBDIVISION >.

Definition at line 601 of file fe.h.

  { libmesh_not_implemented(); }

set_fe_order()

void libMesh::FEAbstract::set_fe_order ( int  new_order )

inlineinherited

Sets the base FE order of the finite element.

Definition at line 439 of file fe_abstract.h.

References libMesh::FEAbstract::fe_type, and libMesh::FEType::order.

{ fe_type.order = new_order; }

shape() [1/109]

Real libMesh::FE< 3, SCALAR >::shape ( const ElemType  , const Order  , const unsigned  int, const Point &    )

inherited

Definition at line 29 of file fe_scalar_shape_3D.C.

{
  return 1.;
}

shape() [2/109]

Real libMesh::FE< 2, SCALAR >::shape ( const ElemType  , const Order  , const unsigned  int, const Point &    )

inherited

Definition at line 29 of file fe_scalar_shape_2D.C.

{
  return 1.;
}

shape() [3/109]

RealGradient libMesh::FE< 2, NEDELEC_ONE >::shape ( const ElemType  , const Order  , const unsigned  int, const Point &    )

inherited

Definition at line 29 of file fe_nedelec_one_shape_2D.C.

{
  libmesh_error_msg("Nedelec elements require the element type \nbecause edge orientation is needed.");
  return RealGradient();
}

shape() [4/109]

RealGradient libMesh::FE< 3, NEDELEC_ONE >::shape ( const ElemType  , const Order  , const unsigned  int, const Point &    )

inherited

Definition at line 29 of file fe_nedelec_one_shape_3D.C.

{
  libmesh_error_msg("Nedelec elements require the element type \nbecause edge orientation is needed.");
  return RealGradient();
}

shape() [5/109]

Real libMesh::FE< 0, SCALAR >::shape ( const ElemType  , const Order  , const unsigned  int, const Point &    )

inherited

Definition at line 29 of file fe_scalar_shape_0D.C.

{
  return 1.;
}

shape() [6/109]

Real libMesh::FE< 1, SCALAR >::shape ( const ElemType  , const Order  , const unsigned  int, const Point &    )

inherited

Definition at line 29 of file fe_scalar_shape_1D.C.

{
  return 1.;
}

shape() [7/109]

Real libMesh::FE< 2, LAGRANGE >::shape ( const ElemType  type, const Order  order, const unsigned int  i, const Point &  p  )

inherited

Definition at line 32 of file fe_lagrange_shape_2D.C.

{
#if LIBMESH_DIM > 1

  switch (order)
    {
      // linear Lagrange shape functions
    case FIRST:
      {
        switch (type)
          {
          case QUAD4:
          case QUADSHELL4:
          case QUAD8:
          case QUADSHELL8:
          case QUAD9:
            {
              // Compute quad shape functions as a tensor-product
              const Real xi  = p(0);
              const Real eta = p(1);

              libmesh_assert_less (i, 4);

              //                                0  1  2  3
              static const unsigned int i0[] = {0, 1, 1, 0};
              static const unsigned int i1[] = {0, 0, 1, 1};

              return (FE<1,LAGRANGE>::shape(EDGE2, FIRST, i0[i], xi)*
                      FE<1,LAGRANGE>::shape(EDGE2, FIRST, i1[i], eta));
            }

          case TRI3:
          case TRISHELL3:
          case TRI6:
            {
              const Real zeta1 = p(0);
              const Real zeta2 = p(1);
              const Real zeta0 = 1. - zeta1 - zeta2;

              libmesh_assert_less (i, 3);

              switch(i)
                {
                case 0:
                  return zeta0;

                case 1:
                  return zeta1;

                case 2:
                  return zeta2;

                default:
                  libmesh_error_msg("Invalid shape function index i = " << i);
                }
            }

          default:
            libmesh_error_msg("ERROR: Unsupported 2D element type: " << type);
          }
      }


      // quadratic Lagrange shape functions
    case SECOND:
      {
        switch (type)
          {
          case QUAD8:
          case QUADSHELL8:
            {
              const Real xi  = p(0);
              const Real eta = p(1);

              libmesh_assert_less (i, 8);

              switch (i)
                {
                case 0:
                  return .25*(1. - xi)*(1. - eta)*(-1. - xi - eta);

                case 1:
                  return .25*(1. + xi)*(1. - eta)*(-1. + xi - eta);

                case 2:
                  return .25*(1. + xi)*(1. + eta)*(-1. + xi + eta);

                case 3:
                  return .25*(1. - xi)*(1. + eta)*(-1. - xi + eta);

                case 4:
                  return .5*(1. - xi*xi)*(1. - eta);

                case 5:
                  return .5*(1. + xi)*(1. - eta*eta);

                case 6:
                  return .5*(1. - xi*xi)*(1. + eta);

                case 7:
                  return .5*(1. - xi)*(1. - eta*eta);

                default:
                  libmesh_error_msg("Invalid shape function index i = " << i);
                }
            }

          case QUAD9:
            {
              // Compute quad shape functions as a tensor-product
              const Real xi  = p(0);
              const Real eta = p(1);

              libmesh_assert_less (i, 9);

              //                                0  1  2  3  4  5  6  7  8
              static const unsigned int i0[] = {0, 1, 1, 0, 2, 1, 2, 0, 2};
              static const unsigned int i1[] = {0, 0, 1, 1, 0, 2, 1, 2, 2};

              return (FE<1,LAGRANGE>::shape(EDGE3, SECOND, i0[i], xi)*
                      FE<1,LAGRANGE>::shape(EDGE3, SECOND, i1[i], eta));
            }

          case TRI6:
            {
              const Real zeta1 = p(0);
              const Real zeta2 = p(1);
              const Real zeta0 = 1. - zeta1 - zeta2;

              libmesh_assert_less (i, 6);

              switch(i)
                {
                case 0:
                  return 2.*zeta0*(zeta0-0.5);

                case 1:
                  return 2.*zeta1*(zeta1-0.5);

                case 2:
                  return 2.*zeta2*(zeta2-0.5);

                case 3:
                  return 4.*zeta0*zeta1;

                case 4:
                  return 4.*zeta1*zeta2;

                case 5:
                  return 4.*zeta2*zeta0;

                default:
                  libmesh_error_msg("Invalid shape function index i = " << i);
                }
            }

          default:
            libmesh_error_msg("ERROR: Unsupported 2D element type: " << type);
          }
      }



      // unsupported order
    default:
      libmesh_error_msg("ERROR: Unsupported 2D FE order: " << order);
    }
#else // LIBMESH_DIM > 1
  return 0.;
#endif
}

shape() [8/109]

Real libMesh::FE< 1, LAGRANGE >::shape ( const ElemType  , const Order  order, const unsigned int  i, const Point &  p  )

inherited

Definition at line 32 of file fe_lagrange_shape_1D.C.

{
  const Real xi = p(0);


  switch (order)
    {
      // Lagrange linears
    case FIRST:
      {
        libmesh_assert_less (i, 2);

        switch (i)
          {
          case 0:
            return .5*(1. - xi);

          case 1:
            return .5*(1. + xi);

          default:
            libmesh_error_msg("Invalid shape function index i = " << i);
          }
      }



      // Lagrange quadratics
    case SECOND:
      {
        libmesh_assert_less (i, 3);

        switch (i)
          {
          case 0:
            return .5*xi*(xi - 1.);

          case 1:
            return .5*xi*(xi + 1);

          case 2:
            return (1. - xi*xi);

          default:
            libmesh_error_msg("Invalid shape function index i = " << i);
          }
      }



      // Lagrange cubics
    case THIRD:
      {
        libmesh_assert_less (i, 4);

        switch (i)
          {
          case 0:
            return 9./16.*(1./9.-xi*xi)*(xi-1.);

          case 1:
            return -9./16.*(1./9.-xi*xi)*(xi+1.);

          case 2:
            return 27./16.*(1.-xi*xi)*(1./3.-xi);

          case 3:
            return 27./16.*(1.-xi*xi)*(1./3.+xi);

          default:
            libmesh_error_msg("Invalid shape function index i = " << i);
          }
      }

    default:
      libmesh_error_msg("ERROR: Unsupported polynomial order = " << order);
    }
}

shape() [9/109]

Real libMesh::FE< 2, L2_HIERARCHIC >::shape ( const ElemType  , const Order  , const unsigned  int, const Point &    )

inherited

Definition at line 32 of file fe_l2_hierarchic_shape_2D.C.

{
  libmesh_error_msg("Hierarchic polynomials require the element type \nbecause edge orientation is needed.");
  return 0.;
}

shape() [10/109]

Real libMesh::FE< 0, BERNSTEIN >::shape ( const ElemType  , const Order  , const unsigned int  libmesh_dbg_vari, const Point &    )

inherited

Definition at line 32 of file fe_bernstein_shape_0D.C.

{
  libmesh_assert_less (i, 1);
  return 1.;
}

shape() [11/109]

Real libMesh::FE< 0, L2_LAGRANGE >::shape ( const ElemType  , const Order  , const unsigned int  libmesh_dbg_vari, const Point &    )

inherited

Definition at line 32 of file fe_l2_lagrange_shape_0D.C.

{
  libmesh_assert_less (i, 1);
  return 1.;
}

shape() [12/109]

Real libMesh::FE< 0, HIERARCHIC >::shape ( const ElemType  , const Order  , const unsigned int  libmesh_dbg_vari, const Point &    )

inherited

Definition at line 32 of file fe_hierarchic_shape_0D.C.

{
  libmesh_assert_less (i, 1);
  return 1.;
}

shape() [13/109]

Real libMesh::FE< 2, HIERARCHIC >::shape ( const ElemType  , const Order  , const unsigned  int, const Point &    )

inherited

Definition at line 32 of file fe_hierarchic_shape_2D.C.

{
  libmesh_error_msg("Hierarchic polynomials require the element type \nbecause edge orientation is needed.");
  return 0.;
}

shape() [14/109]

Real libMesh::FE< 0, MONOMIAL >::shape ( const ElemType  , const Order  , const unsigned int  libmesh_dbg_vari, const Point &    )

inherited

Definition at line 32 of file fe_monomial_shape_0D.C.

{
  libmesh_assert_less (i, 1);
  return 1.;
}

shape() [15/109]

Real libMesh::FE< 1, MONOMIAL >::shape ( const ElemType  , const Order  libmesh_dbg_varorder, const unsigned int  i, const Point &  p  )

inherited

Definition at line 32 of file fe_monomial_shape_1D.C.

{
  const Real xi = p(0);

  libmesh_assert_less_equal (i, static_cast<unsigned int>(order));

  // monomials. since they are hierarchic we only need one case block.
  switch (i)
    {
    case 0:
      return 1.;

    case 1:
      return xi;

    case 2:
      return xi*xi;

    case 3:
      return xi*xi*xi;

    case 4:
      return xi*xi*xi*xi;

    default:
      Real val = 1.;
      for (unsigned int index = 0; index != i; ++index)
        val *= xi;
      return val;
    }
}

shape() [16/109]

Real libMesh::FE< 2, MONOMIAL >::shape ( const ElemType  , const Order  libmesh_dbg_varorder, const unsigned int  i, const Point &  p  )

inherited

Definition at line 32 of file fe_monomial_shape_2D.C.

{
#if LIBMESH_DIM > 1

  libmesh_assert_less (i, (static_cast<unsigned int>(order)+1)*
                       (static_cast<unsigned int>(order)+2)/2);

  const Real xi  = p(0);
  const Real eta = p(1);

  switch (i)
    {
      // constant
    case 0:
      return 1.;

      // linear
    case 1:
      return xi;

    case 2:
      return eta;

      // quadratics
    case 3:
      return xi*xi;

    case 4:
      return xi*eta;

    case 5:
      return eta*eta;

      // cubics
    case 6:
      return xi*xi*xi;

    case 7:
      return xi*xi*eta;

    case 8:
      return xi*eta*eta;

    case 9:
      return eta*eta*eta;

      // quartics
    case 10:
      return xi*xi*xi*xi;

    case 11:
      return xi*xi*xi*eta;

    case 12:
      return xi*xi*eta*eta;

    case 13:
      return xi*eta*eta*eta;

    case 14:
      return eta*eta*eta*eta;

    default:
      unsigned int o = 0;
      for (; i >= (o+1)*(o+2)/2; o++) { }
      unsigned int ny = i - (o*(o+1)/2);
      unsigned int nx = o - ny;
      Real val = 1.;
      for (unsigned int index=0; index != nx; index++)
        val *= xi;
      for (unsigned int index=0; index != ny; index++)
        val *= eta;
      return val;
    }

#else
  return 0.;
#endif
}

shape() [17/109]

Real libMesh::FE< 0, CLOUGH >::shape ( const ElemType  , const Order  , const unsigned int  libmesh_dbg_vari, const Point &    )

inherited

Definition at line 32 of file fe_clough_shape_0D.C.

{
  libmesh_assert_less (i, 1);
  return 1.;
}

shape() [18/109]

Real libMesh::FE< 3, MONOMIAL >::shape ( const ElemType  , const Order  libmesh_dbg_varorder, const unsigned int  i, const Point &  p  )

inherited

Definition at line 32 of file fe_monomial_shape_3D.C.

{
#if LIBMESH_DIM == 3

  const Real xi   = p(0);
  const Real eta  = p(1);
  const Real zeta = p(2);

  libmesh_assert_less (i, (static_cast<unsigned int>(order)+1)*
                       (static_cast<unsigned int>(order)+2)*
                       (static_cast<unsigned int>(order)+3)/6);

  // monomials. since they are hierarchic we only need one case block.
  switch (i)
    {
      // constant
    case 0:
      return 1.;

      // linears
    case 1:
      return xi;

    case 2:
      return eta;

    case 3:
      return zeta;

      // quadratics
    case 4:
      return xi*xi;

    case 5:
      return xi*eta;

    case 6:
      return eta*eta;

    case 7:
      return xi*zeta;

    case 8:
      return zeta*eta;

    case 9:
      return zeta*zeta;

      // cubics
    case 10:
      return xi*xi*xi;

    case 11:
      return xi*xi*eta;

    case 12:
      return xi*eta*eta;

    case 13:
      return eta*eta*eta;

    case 14:
      return xi*xi*zeta;

    case 15:
      return xi*eta*zeta;

    case 16:
      return eta*eta*zeta;

    case 17:
      return xi*zeta*zeta;

    case 18:
      return eta*zeta*zeta;

    case 19:
      return zeta*zeta*zeta;

      // quartics
    case 20:
      return xi*xi*xi*xi;

    case 21:
      return xi*xi*xi*eta;

    case 22:
      return xi*xi*eta*eta;

    case 23:
      return xi*eta*eta*eta;

    case 24:
      return eta*eta*eta*eta;

    case 25:
      return xi*xi*xi*zeta;

    case 26:
      return xi*xi*eta*zeta;

    case 27:
      return xi*eta*eta*zeta;

    case 28:
      return eta*eta*eta*zeta;

    case 29:
      return xi*xi*zeta*zeta;

    case 30:
      return xi*eta*zeta*zeta;

    case 31:
      return eta*eta*zeta*zeta;

    case 32:
      return xi*zeta*zeta*zeta;

    case 33:
      return eta*zeta*zeta*zeta;

    case 34:
      return zeta*zeta*zeta*zeta;

    default:
      unsigned int o = 0;
      for (; i >= (o+1)*(o+2)*(o+3)/6; o++) { }
      unsigned int i2 = i - (o*(o+1)*(o+2)/6);
      unsigned int block=o, nz = 0;
      for (; block < i2; block += (o-nz+1)) { nz++; }
      const unsigned int nx = block - i2;
      const unsigned int ny = o - nx - nz;
      Real val = 1.;
      for (unsigned int index=0; index != nx; index++)
        val *= xi;
      for (unsigned int index=0; index != ny; index++)
        val *= eta;
      for (unsigned int index=0; index != nz; index++)
        val *= zeta;
      return val;
    }

#else
  return 0.;
#endif
}

shape() [19/109]

Real libMesh::FE< 0, L2_HIERARCHIC >::shape ( const ElemType  , const Order  , const unsigned int  libmesh_dbg_vari, const Point &    )

inherited

Definition at line 32 of file fe_l2_hierarchic_shape_0D.C.

{
  libmesh_assert_less (i, 1);
  return 1.;
}

shape() [20/109]

Real libMesh::FE< 0, SZABAB >::shape ( const ElemType  , const Order  , const unsigned int  libmesh_dbg_vari, const Point &    )

inherited

Definition at line 32 of file fe_szabab_shape_0D.C.

{
  libmesh_assert_less (i, 1);
  return 1.;
}

shape() [21/109]

Real libMesh::FE< 3, LAGRANGE >::shape ( const ElemType  type, const Order  order, const unsigned int  i, const Point &  p  )

inherited

Definition at line 32 of file fe_lagrange_shape_3D.C.

{
#if LIBMESH_DIM == 3


  switch (order)
    {
      // linear Lagrange shape functions
    case FIRST:
      {
        switch (type)
          {
            // trilinear hexahedral shape functions
          case HEX8:
          case HEX20:
          case HEX27:
            {
              libmesh_assert_less (i, 8);

              // Compute hex shape functions as a tensor-product
              const Real xi   = p(0);
              const Real eta  = p(1);
              const Real zeta = p(2);

              //                                0  1  2  3  4  5  6  7
              static const unsigned int i0[] = {0, 1, 1, 0, 0, 1, 1, 0};
              static const unsigned int i1[] = {0, 0, 1, 1, 0, 0, 1, 1};
              static const unsigned int i2[] = {0, 0, 0, 0, 1, 1, 1, 1};

              return (FE<1,LAGRANGE>::shape(EDGE2, FIRST, i0[i], xi)*
                      FE<1,LAGRANGE>::shape(EDGE2, FIRST, i1[i], eta)*
                      FE<1,LAGRANGE>::shape(EDGE2, FIRST, i2[i], zeta));
            }

            // linear tetrahedral shape functions
          case TET4:
          case TET10:
            {
              libmesh_assert_less (i, 4);

              // Area coordinates, pg. 205, Vol. I, Carey, Oden, Becker FEM
              const Real zeta1 = p(0);
              const Real zeta2 = p(1);
              const Real zeta3 = p(2);
              const Real zeta0 = 1. - zeta1 - zeta2 - zeta3;

              switch(i)
                {
                case 0:
                  return zeta0;

                case 1:
                  return zeta1;

                case 2:
                  return zeta2;

                case 3:
                  return zeta3;

                default:
                  libmesh_error_msg("Invalid i = " << i);
                }
            }

            // linear prism shape functions
          case PRISM6:
          case PRISM15:
          case PRISM18:
            {
              libmesh_assert_less (i, 6);

              // Compute prism shape functions as a tensor-product
              // of a triangle and an edge

              Point p2d(p(0),p(1));
              Point p1d(p(2));

              //                                0  1  2  3  4  5
              static const unsigned int i0[] = {0, 0, 0, 1, 1, 1};
              static const unsigned int i1[] = {0, 1, 2, 0, 1, 2};

              return (FE<2,LAGRANGE>::shape(TRI3,  FIRST, i1[i], p2d)*
                      FE<1,LAGRANGE>::shape(EDGE2, FIRST, i0[i], p1d));
            }

            // linear pyramid shape functions
          case PYRAMID5:
          case PYRAMID13:
          case PYRAMID14:
            {
              libmesh_assert_less (i, 5);

              const Real xi   = p(0);
              const Real eta  = p(1);
              const Real zeta = p(2);
              const Real eps  = 1.e-35;

              switch(i)
                {
                case 0:
                  return .25*(zeta + xi - 1.)*(zeta + eta - 1.)/((1. - zeta) + eps);

                case 1:
                  return .25*(zeta - xi - 1.)*(zeta + eta - 1.)/((1. - zeta) + eps);

                case 2:
                  return .25*(zeta - xi - 1.)*(zeta - eta - 1.)/((1. - zeta) + eps);

                case 3:
                  return .25*(zeta + xi - 1.)*(zeta - eta - 1.)/((1. - zeta) + eps);

                case 4:
                  return zeta;

                default:
                  libmesh_error_msg("Invalid i = " << i);
                }
            }


          default:
            libmesh_error_msg("ERROR: Unsupported 3D element type!: " << type);
          }
      }


      // quadratic Lagrange shape functions
    case SECOND:
      {
        switch (type)
          {

            // serendipity hexahedral quadratic shape functions
          case HEX20:
            {
              libmesh_assert_less (i, 20);

              const Real xi   = p(0);
              const Real eta  = p(1);
              const Real zeta = p(2);

              // these functions are defined for (x,y,z) in [0,1]^3
              // so transform the locations
              const Real x = .5*(xi   + 1.);
              const Real y = .5*(eta  + 1.);
              const Real z = .5*(zeta + 1.);

              switch (i)
                {
                case 0:
                  return (1. - x)*(1. - y)*(1. - z)*(1. - 2.*x - 2.*y - 2.*z);

                case 1:
                  return x*(1. - y)*(1. - z)*(2.*x - 2.*y - 2.*z - 1.);

                case 2:
                  return x*y*(1. - z)*(2.*x + 2.*y - 2.*z - 3.);

                case 3:
                  return (1. - x)*y*(1. - z)*(2.*y - 2.*x - 2.*z - 1.);

                case 4:
                  return (1. - x)*(1. - y)*z*(2.*z - 2.*x - 2.*y - 1.);

                case 5:
                  return x*(1. - y)*z*(2.*x - 2.*y + 2.*z - 3.);

                case 6:
                  return x*y*z*(2.*x + 2.*y + 2.*z - 5.);

                case 7:
                  return (1. - x)*y*z*(2.*y - 2.*x + 2.*z - 3.);

                case 8:
                  return 4.*x*(1. - x)*(1. - y)*(1. - z);

                case 9:
                  return 4.*x*y*(1. - y)*(1. - z);

                case 10:
                  return 4.*x*(1. - x)*y*(1. - z);

                case 11:
                  return 4.*(1. - x)*y*(1. - y)*(1. - z);

                case 12:
                  return 4.*(1. - x)*(1. - y)*z*(1. - z);

                case 13:
                  return 4.*x*(1. - y)*z*(1. - z);

                case 14:
                  return 4.*x*y*z*(1. - z);

                case 15:
                  return 4.*(1. - x)*y*z*(1. - z);

                case 16:
                  return 4.*x*(1. - x)*(1. - y)*z;

                case 17:
                  return 4.*x*y*(1. - y)*z;

                case 18:
                  return 4.*x*(1. - x)*y*z;

                case 19:
                  return 4.*(1. - x)*y*(1. - y)*z;

                default:
                  libmesh_error_msg("Invalid i = " << i);
                }
            }

            // triquadratic hexahedral shape functions
          case HEX27:
            {
              libmesh_assert_less (i, 27);

              // Compute hex shape functions as a tensor-product
              const Real xi   = p(0);
              const Real eta  = p(1);
              const Real zeta = p(2);

              // The only way to make any sense of this
              // is to look at the mgflo/mg2/mgf documentation
              // and make the cut-out cube!
              //                                0  1  2  3  4  5  6  7  8  9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26
              static const unsigned int i0[] = {0, 1, 1, 0, 0, 1, 1, 0, 2, 1, 2, 0, 0, 1, 1, 0, 2, 1, 2, 0, 2, 2, 1, 2, 0, 2, 2};
              static const unsigned int i1[] = {0, 0, 1, 1, 0, 0, 1, 1, 0, 2, 1, 2, 0, 0, 1, 1, 0, 2, 1, 2, 2, 0, 2, 1, 2, 2, 2};
              static const unsigned int i2[] = {0, 0, 0, 0, 1, 1, 1, 1, 0, 0, 0, 0, 2, 2, 2, 2, 1, 1, 1, 1, 0, 2, 2, 2, 2, 1, 2};

              return (FE<1,LAGRANGE>::shape(EDGE3, SECOND, i0[i], xi)*
                      FE<1,LAGRANGE>::shape(EDGE3, SECOND, i1[i], eta)*
                      FE<1,LAGRANGE>::shape(EDGE3, SECOND, i2[i], zeta));
            }

            // quadratic tetrahedral shape functions
          case TET10:
            {
              libmesh_assert_less (i, 10);

              // Area coordinates, pg. 205, Vol. I, Carey, Oden, Becker FEM
              const Real zeta1 = p(0);
              const Real zeta2 = p(1);
              const Real zeta3 = p(2);
              const Real zeta0 = 1. - zeta1 - zeta2 - zeta3;

              switch(i)
                {
                case 0:
                  return zeta0*(2.*zeta0 - 1.);

                case 1:
                  return zeta1*(2.*zeta1 - 1.);

                case 2:
                  return zeta2*(2.*zeta2 - 1.);

                case 3:
                  return zeta3*(2.*zeta3 - 1.);

                case 4:
                  return 4.*zeta0*zeta1;

                case 5:
                  return 4.*zeta1*zeta2;

                case 6:
                  return 4.*zeta2*zeta0;

                case 7:
                  return 4.*zeta0*zeta3;

                case 8:
                  return 4.*zeta1*zeta3;

                case 9:
                  return 4.*zeta2*zeta3;

                default:
                  libmesh_error_msg("Invalid i = " << i);
                }
            }

            // "serendipity" prism
          case PRISM15:
            {
              libmesh_assert_less (i, 15);

              const Real xi   = p(0);
              const Real eta  = p(1);
              const Real zeta = p(2);

              switch(i)
                {
                case 0:
                  return (1. - zeta)*(xi + eta - 1.)*(xi + eta + 0.5*zeta);

                case 1:
                  return (1. - zeta)*xi*(xi - 1. - 0.5*zeta);

                case 2: // phi1 with xi <- eta
                  return (1. - zeta)*eta*(eta - 1. - 0.5*zeta);

                case 3: // phi0 with zeta <- (-zeta)
                  return (1. + zeta)*(xi + eta - 1.)*(xi + eta - 0.5*zeta);

                case 4: // phi1 with zeta <- (-zeta)
                  return (1. + zeta)*xi*(xi - 1. + 0.5*zeta);

                case 5: // phi4 with xi <- eta
                  return (1. + zeta)*eta*(eta - 1. + 0.5*zeta);

                case 6:
                  return 2.*(1. - zeta)*xi*(1. - xi - eta);

                case 7:
                  return 2.*(1. - zeta)*xi*eta;

                case 8:
                  return 2.*(1. - zeta)*eta*(1. - xi - eta);

                case 9:
                  return (1. - zeta)*(1. + zeta)*(1. - xi - eta);

                case 10:
                  return (1. - zeta)*(1. + zeta)*xi;

                case 11: // phi10 with xi <-> eta
                  return (1. - zeta)*(1. + zeta)*eta;

                case 12: // phi6 with zeta <- (-zeta)
                  return 2.*(1. + zeta)*xi*(1. - xi - eta);

                case 13: // phi7 with zeta <- (-zeta)
                  return 2.*(1. + zeta)*xi*eta;

                case 14: // phi8 with zeta <- (-zeta)
                  return 2.*(1. + zeta)*eta*(1. - xi - eta);

                default:
                  libmesh_error_msg("Invalid i = " << i);
                }
            }

            // quadratic prism shape functions
          case PRISM18:
            {
              libmesh_assert_less (i, 18);

              // Compute prism shape functions as a tensor-product
              // of a triangle and an edge

              Point p2d(p(0),p(1));
              Point p1d(p(2));

              //                                0  1  2  3  4  5  6  7  8  9 10 11 12 13 14 15 16 17
              static const unsigned int i0[] = {0, 0, 0, 1, 1, 1, 0, 0, 0, 2, 2, 2, 1, 1, 1, 2, 2, 2};
              static const unsigned int i1[] = {0, 1, 2, 0, 1, 2, 3, 4, 5, 0, 1, 2, 3, 4, 5, 3, 4, 5};

              return (FE<2,LAGRANGE>::shape(TRI6,  SECOND, i1[i], p2d)*
                      FE<1,LAGRANGE>::shape(EDGE3, SECOND, i0[i], p1d));
            }

            // G. Bedrosian, "Shape functions and integration formulas for
            // three-dimensional finite element analysis", Int. J. Numerical
            // Methods Engineering, vol 35, p. 95-108, 1992.
          case PYRAMID13:
            {
              libmesh_assert_less (i, 13);

              const Real xi   = p(0);
              const Real eta  = p(1);
              const Real zeta = p(2);
              const Real eps  = 1.e-35;

              // Denominators are perturbed by epsilon to avoid
              // divide-by-zero issues.
              Real den = (1. - zeta + eps);

              switch(i)
                {
                case 0:
                  return 0.25*(-xi - eta - 1.)*((1. - xi)*(1. - eta) - zeta + xi*eta*zeta/den);

                case 1:
                  return 0.25*(-eta + xi - 1.)*((1. + xi)*(1. - eta) - zeta - xi*eta*zeta/den);

                case 2:
                  return 0.25*(xi + eta - 1.)*((1. + xi)*(1. + eta) - zeta + xi*eta*zeta/den);

                case 3:
                  return 0.25*(eta - xi - 1.)*((1. - xi)*(1. + eta) - zeta - xi*eta*zeta/den);

                case 4:
                  return zeta*(2.*zeta - 1.);

                case 5:
                  return 0.5*(1. + xi - zeta)*(1. - xi - zeta)*(1. - eta - zeta)/den;

                case 6:
                  return 0.5*(1. + eta - zeta)*(1. - eta - zeta)*(1. + xi - zeta)/den;

                case 7:
                  return 0.5*(1. + xi - zeta)*(1. - xi - zeta)*(1. + eta - zeta)/den;

                case 8:
                  return 0.5*(1. + eta - zeta)*(1. - eta - zeta)*(1. - xi - zeta)/den;

                case 9:
                  return zeta*(1. - xi - zeta)*(1. - eta - zeta)/den;

                case 10:
                  return zeta*(1. + xi - zeta)*(1. - eta - zeta)/den;

                case 11:
                  return zeta*(1. + eta - zeta)*(1. + xi - zeta)/den;

                case 12:
                  return zeta*(1. - xi - zeta)*(1. + eta - zeta)/den;

                default:
                  libmesh_error_msg("Invalid i = " << i);
                }
            }

            // Quadratic shape functions, as defined in R. Graglia, "Higher order
            // bases on pyramidal elements", IEEE Trans Antennas and Propagation,
            // vol 47, no 5, May 1999.
          case PYRAMID14:
            {
              libmesh_assert_less (i, 14);

              const Real xi   = p(0);
              const Real eta  = p(1);
              const Real zeta = p(2);
              const Real eps  = 1.e-35;

              // The "normalized coordinates" defined by Graglia.  These are
              // the planes which define the faces of the pyramid.
              Real
                p1 = 0.5*(1. - eta - zeta), // back
                p2 = 0.5*(1. + xi  - zeta), // left
                p3 = 0.5*(1. + eta - zeta), // front
                p4 = 0.5*(1. - xi  - zeta); // right

              // Denominators are perturbed by epsilon to avoid
              // divide-by-zero issues.
              Real
                den = (-1. + zeta + eps),
                den2 = den*den;

              switch(i)
                {
                case 0:
                  return p4*p1*(xi*eta - zeta + zeta*zeta)/den2;

                case 1:
                  return -p1*p2*(xi*eta + zeta - zeta*zeta)/den2;

                case 2:
                  return p2*p3*(xi*eta - zeta + zeta*zeta)/den2;

                case 3:
                  return -p3*p4*(xi*eta + zeta - zeta*zeta)/den2;

                case 4:
                  return zeta*(2.*zeta - 1.);

                case 5:
                  return -4.*p2*p1*p4*eta/den2;

                case 6:
                  return 4.*p1*p2*p3*xi/den2;

                case 7:
                  return 4.*p2*p3*p4*eta/den2;

                case 8:
                  return -4.*p3*p4*p1*xi/den2;

                case 9:
                  return -4.*p1*p4*zeta/den;

                case 10:
                  return -4.*p2*p1*zeta/den;

                case 11:
                  return -4.*p3*p2*zeta/den;

                case 12:
                  return -4.*p4*p3*zeta/den;

                case 13:
                  return 16.*p1*p2*p3*p4/den2;

                default:
                  libmesh_error_msg("Invalid i = " << i);
                }
            }


          default:
            libmesh_error_msg("ERROR: Unsupported 3D element type!: " << type);
          }
      }


      // unsupported order
    default:
      libmesh_error_msg("ERROR: Unsupported 3D FE order!: " << order);
    }

#else
  return 0.;
#endif
}

shape() [22/109]

Real libMesh::FE< 3, SZABAB >::shape ( const ElemType  , const Order  , const unsigned  int, const Point &    )

inherited

Definition at line 32 of file fe_szabab_shape_3D.C.

{
  libmesh_error_msg("Szabo-Babuska polynomials are not defined in 3D");
  return 0.;
}

shape() [23/109]

Real libMesh::FE< 1, L2_LAGRANGE >::shape ( const ElemType  , const Order  order, const unsigned int  i, const Point &  p  )

inherited

Definition at line 32 of file fe_l2_lagrange_shape_1D.C.

{
  const Real xi = p(0);


  switch (order)
    {
      // Lagrange linears
    case FIRST:
      {
        libmesh_assert_less (i, 2);

        switch (i)
          {
          case 0:
            return .5*(1. - xi);

          case 1:
            return .5*(1. + xi);

          default:
            libmesh_error_msg("Invalid shape function index i = " << i);
          }
      }



      // Lagrange quadratics
    case SECOND:
      {
        libmesh_assert_less (i, 3);

        switch (i)
          {
          case 0:
            return .5*xi*(xi - 1.);

          case 1:
            return .5*xi*(xi + 1);

          case 2:
            return (1. - xi*xi);

          default:
            libmesh_error_msg("Invalid shape function index i = " << i);
          }
      }



      // Lagrange cubics
    case THIRD:
      {
        libmesh_assert_less (i, 4);

        switch (i)
          {
          case 0:
            return 9./16.*(1./9.-xi*xi)*(xi-1.);

          case 1:
            return -9./16.*(1./9.-xi*xi)*(xi+1.);

          case 2:
            return 27./16.*(1.-xi*xi)*(1./3.-xi);

          case 3:
            return 27./16.*(1.-xi*xi)*(1./3.+xi);

          default:
            libmesh_error_msg("Invalid shape function index i = " << i);
          }
      }

    default:
      libmesh_error_msg("ERROR: Unsupported polynomial order = " << order);
    }
}

shape() [24/109]

Real libMesh::FE< 0, XYZ >::shape ( const ElemType  , const Order  , const unsigned int  libmesh_dbg_vari, const Point &    )

inherited

Definition at line 32 of file fe_xyz_shape_0D.C.

{
  libmesh_assert_less (i, 1);
  return 1.;
}

shape() [25/109]

Real libMesh::FE< 1, XYZ >::shape ( const ElemType  , const Order  , const unsigned  int, const Point &    )

inherited

Definition at line 32 of file fe_xyz_shape_1D.C.

{
  libmesh_error_msg("XYZ polynomials require the element \n because the centroid is needed.");
  return 0.;
}

shape() [26/109]

Real libMesh::FE< 0, HERMITE >::shape ( const ElemType  , const Order  , const unsigned int  libmesh_dbg_vari, const Point &    )

inherited

Definition at line 32 of file fe_hermite_shape_0D.C.

{
  libmesh_assert_less (i, 1);
  return 1.;
}

shape() [27/109]

Real libMesh::FE< 2, L2_LAGRANGE >::shape ( const ElemType  type, const Order  order, const unsigned int  i, const Point &  p  )

inherited

Definition at line 32 of file fe_l2_lagrange_shape_2D.C.

{
#if LIBMESH_DIM > 1

  switch (order)
    {
      // linear Lagrange shape functions
    case FIRST:
      {
        switch (type)
          {
          case QUAD4:
          case QUADSHELL4:
          case QUAD8:
          case QUADSHELL8:
          case QUAD9:
            {
              // Compute quad shape functions as a tensor-product
              const Real xi  = p(0);
              const Real eta = p(1);

              libmesh_assert_less (i, 4);

              //                                0  1  2  3
              static const unsigned int i0[] = {0, 1, 1, 0};
              static const unsigned int i1[] = {0, 0, 1, 1};

              return (FE<1,L2_LAGRANGE>::shape(EDGE2, FIRST, i0[i], xi)*
                      FE<1,L2_LAGRANGE>::shape(EDGE2, FIRST, i1[i], eta));
            }

          case TRI3:
          case TRISHELL3:
          case TRI6:
            {
              const Real zeta1 = p(0);
              const Real zeta2 = p(1);
              const Real zeta0 = 1. - zeta1 - zeta2;

              libmesh_assert_less (i, 3);

              switch(i)
                {
                case 0:
                  return zeta0;

                case 1:
                  return zeta1;

                case 2:
                  return zeta2;

                default:
                  libmesh_error_msg("Invalid shape function index i = " << i);
                }
            }

          default:
            libmesh_error_msg("ERROR: Unsupported 2D element type: " << type);
          }
      }


      // quadratic Lagrange shape functions
    case SECOND:
      {
        switch (type)
          {
          case QUAD8:
          case QUADSHELL8:
            {
              const Real xi  = p(0);
              const Real eta = p(1);

              libmesh_assert_less (i, 8);

              switch (i)
                {
                case 0:
                  return .25*(1. - xi)*(1. - eta)*(-1. - xi - eta);

                case 1:
                  return .25*(1. + xi)*(1. - eta)*(-1. + xi - eta);

                case 2:
                  return .25*(1. + xi)*(1. + eta)*(-1. + xi + eta);

                case 3:
                  return .25*(1. - xi)*(1. + eta)*(-1. - xi + eta);

                case 4:
                  return .5*(1. - xi*xi)*(1. - eta);

                case 5:
                  return .5*(1. + xi)*(1. - eta*eta);

                case 6:
                  return .5*(1. - xi*xi)*(1. + eta);

                case 7:
                  return .5*(1. - xi)*(1. - eta*eta);

                default:
                  libmesh_error_msg("Invalid shape function index i = " << i);
                }
            }

          case QUAD9:
            {
              // Compute quad shape functions as a tensor-product
              const Real xi  = p(0);
              const Real eta = p(1);

              libmesh_assert_less (i, 9);

              //                                0  1  2  3  4  5  6  7  8
              static const unsigned int i0[] = {0, 1, 1, 0, 2, 1, 2, 0, 2};
              static const unsigned int i1[] = {0, 0, 1, 1, 0, 2, 1, 2, 2};

              return (FE<1,L2_LAGRANGE>::shape(EDGE3, SECOND, i0[i], xi)*
                      FE<1,L2_LAGRANGE>::shape(EDGE3, SECOND, i1[i], eta));
            }

          case TRI6:
            {
              const Real zeta1 = p(0);
              const Real zeta2 = p(1);
              const Real zeta0 = 1. - zeta1 - zeta2;

              libmesh_assert_less (i, 6);

              switch(i)
                {
                case 0:
                  return 2.*zeta0*(zeta0-0.5);

                case 1:
                  return 2.*zeta1*(zeta1-0.5);

                case 2:
                  return 2.*zeta2*(zeta2-0.5);

                case 3:
                  return 4.*zeta0*zeta1;

                case 4:
                  return 4.*zeta1*zeta2;

                case 5:
                  return 4.*zeta2*zeta0;

                default:
                  libmesh_error_msg("Invalid shape function index i = " << i);
                }
            }

          default:
            libmesh_error_msg("ERROR: Unsupported 2D element type: " << type);
          }
      }



      // unsupported order
    default:
      libmesh_error_msg("ERROR: Unsupported 2D FE order: " << order);
    }
#endif
}

shape() [28/109]

Real libMesh::FE< 2, XYZ >::shape ( const ElemType  , const Order  , const unsigned  int, const Point &    )

inherited

Definition at line 32 of file fe_xyz_shape_2D.C.

{
  libmesh_error_msg("XYZ polynomials require the element \nbecause the centroid is needed.");
  return 0.;
}

shape() [29/109]

Real libMesh::FE< 3, XYZ >::shape ( const ElemType  , const Order  , const unsigned  int, const Point &    )

inherited

Definition at line 32 of file fe_xyz_shape_3D.C.

{
  libmesh_error_msg("XYZ polynomials require the element because the centroid is needed.");
  return 0.;
}

shape() [30/109]

Real libMesh::FE< 3, L2_LAGRANGE >::shape ( const ElemType  type, const Order  order, const unsigned int  i, const Point &  p  )

inherited

Definition at line 32 of file fe_l2_lagrange_shape_3D.C.

{
#if LIBMESH_DIM == 3


  switch (order)
    {
      // linear Lagrange shape functions
    case FIRST:
      {
        switch (type)
          {
            // trilinear hexahedral shape functions
          case HEX8:
          case HEX20:
          case HEX27:
            {
              libmesh_assert_less (i, 8);

              // Compute hex shape functions as a tensor-product
              const Real xi   = p(0);
              const Real eta  = p(1);
              const Real zeta = p(2);

              //                                0  1  2  3  4  5  6  7
              static const unsigned int i0[] = {0, 1, 1, 0, 0, 1, 1, 0};
              static const unsigned int i1[] = {0, 0, 1, 1, 0, 0, 1, 1};
              static const unsigned int i2[] = {0, 0, 0, 0, 1, 1, 1, 1};

              return (FE<1,L2_LAGRANGE>::shape(EDGE2, FIRST, i0[i], xi)*
                      FE<1,L2_LAGRANGE>::shape(EDGE2, FIRST, i1[i], eta)*
                      FE<1,L2_LAGRANGE>::shape(EDGE2, FIRST, i2[i], zeta));
            }

            // linear tetrahedral shape functions
          case TET4:
          case TET10:
            {
              libmesh_assert_less (i, 4);

              // Area coordinates, pg. 205, Vol. I, Carey, Oden, Becker FEM
              const Real zeta1 = p(0);
              const Real zeta2 = p(1);
              const Real zeta3 = p(2);
              const Real zeta0 = 1. - zeta1 - zeta2 - zeta3;

              switch(i)
                {
                case 0:
                  return zeta0;

                case 1:
                  return zeta1;

                case 2:
                  return zeta2;

                case 3:
                  return zeta3;

                default:
                  libmesh_error_msg("Invalid i = " << i);
                }
            }

            // linear prism shape functions
          case PRISM6:
          case PRISM15:
          case PRISM18:
            {
              libmesh_assert_less (i, 6);

              // Compute prism shape functions as a tensor-product
              // of a triangle and an edge

              Point p2d(p(0),p(1));
              Point p1d(p(2));

              //                                0  1  2  3  4  5
              static const unsigned int i0[] = {0, 0, 0, 1, 1, 1};
              static const unsigned int i1[] = {0, 1, 2, 0, 1, 2};

              return (FE<2,L2_LAGRANGE>::shape(TRI3,  FIRST, i1[i], p2d)*
                      FE<1,L2_LAGRANGE>::shape(EDGE2, FIRST, i0[i], p1d));
            }

            // linear pyramid shape functions
          case PYRAMID5:
            {
              libmesh_assert_less (i, 5);

              const Real xi   = p(0);
              const Real eta  = p(1);
              const Real zeta = p(2);
              const Real eps  = 1.e-35;

              switch(i)
                {
                case 0:
                  return .25*(zeta + xi - 1.)*(zeta + eta - 1.)/((1. - zeta) + eps);

                case 1:
                  return .25*(zeta - xi - 1.)*(zeta + eta - 1.)/((1. - zeta) + eps);

                case 2:
                  return .25*(zeta - xi - 1.)*(zeta - eta - 1.)/((1. - zeta) + eps);

                case 3:
                  return .25*(zeta + xi - 1.)*(zeta - eta - 1.)/((1. - zeta) + eps);

                case 4:
                  return zeta;

                default:
                  libmesh_error_msg("Invalid i = " << i);
                }
            }


          default:
            libmesh_error_msg("ERROR: Unsupported 3D element type!: " << type);
          }
      }


      // quadratic Lagrange shape functions
    case SECOND:
      {
        switch (type)
          {

            // serendipity hexahedral quadratic shape functions
          case HEX20:
            {
              libmesh_assert_less (i, 20);

              const Real xi   = p(0);
              const Real eta  = p(1);
              const Real zeta = p(2);

              // these functions are defined for (x,y,z) in [0,1]^3
              // so transform the locations
              const Real x = .5*(xi   + 1.);
              const Real y = .5*(eta  + 1.);
              const Real z = .5*(zeta + 1.);

              switch (i)
                {
                case 0:
                  return (1. - x)*(1. - y)*(1. - z)*(1. - 2.*x - 2.*y - 2.*z);

                case 1:
                  return x*(1. - y)*(1. - z)*(2.*x - 2.*y - 2.*z - 1.);

                case 2:
                  return x*y*(1. - z)*(2.*x + 2.*y - 2.*z - 3.);

                case 3:
                  return (1. - x)*y*(1. - z)*(2.*y - 2.*x - 2.*z - 1.);

                case 4:
                  return (1. - x)*(1. - y)*z*(2.*z - 2.*x - 2.*y - 1.);

                case 5:
                  return x*(1. - y)*z*(2.*x - 2.*y + 2.*z - 3.);

                case 6:
                  return x*y*z*(2.*x + 2.*y + 2.*z - 5.);

                case 7:
                  return (1. - x)*y*z*(2.*y - 2.*x + 2.*z - 3.);

                case 8:
                  return 4.*x*(1. - x)*(1. - y)*(1. - z);

                case 9:
                  return 4.*x*y*(1. - y)*(1. - z);

                case 10:
                  return 4.*x*(1. - x)*y*(1. - z);

                case 11:
                  return 4.*(1. - x)*y*(1. - y)*(1. - z);

                case 12:
                  return 4.*(1. - x)*(1. - y)*z*(1. - z);

                case 13:
                  return 4.*x*(1. - y)*z*(1. - z);

                case 14:
                  return 4.*x*y*z*(1. - z);

                case 15:
                  return 4.*(1. - x)*y*z*(1. - z);

                case 16:
                  return 4.*x*(1. - x)*(1. - y)*z;

                case 17:
                  return 4.*x*y*(1. - y)*z;

                case 18:
                  return 4.*x*(1. - x)*y*z;

                case 19:
                  return 4.*(1. - x)*y*(1. - y)*z;

                default:
                  libmesh_error_msg("Invalid i = " << i);
                }
            }

            // triquadratic hexahedral shape functions
          case HEX27:
            {
              libmesh_assert_less (i, 27);

              // Compute hex shape functions as a tensor-product
              const Real xi   = p(0);
              const Real eta  = p(1);
              const Real zeta = p(2);

              // The only way to make any sense of this
              // is to look at the mgflo/mg2/mgf documentation
              // and make the cut-out cube!
              //                                0  1  2  3  4  5  6  7  8  9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26
              static const unsigned int i0[] = {0, 1, 1, 0, 0, 1, 1, 0, 2, 1, 2, 0, 0, 1, 1, 0, 2, 1, 2, 0, 2, 2, 1, 2, 0, 2, 2};
              static const unsigned int i1[] = {0, 0, 1, 1, 0, 0, 1, 1, 0, 2, 1, 2, 0, 0, 1, 1, 0, 2, 1, 2, 2, 0, 2, 1, 2, 2, 2};
              static const unsigned int i2[] = {0, 0, 0, 0, 1, 1, 1, 1, 0, 0, 0, 0, 2, 2, 2, 2, 1, 1, 1, 1, 0, 2, 2, 2, 2, 1, 2};

              return (FE<1,L2_LAGRANGE>::shape(EDGE3, SECOND, i0[i], xi)*
                      FE<1,L2_LAGRANGE>::shape(EDGE3, SECOND, i1[i], eta)*
                      FE<1,L2_LAGRANGE>::shape(EDGE3, SECOND, i2[i], zeta));
            }

            // quadratic tetrahedral shape functions
          case TET10:
            {
              libmesh_assert_less (i, 10);

              // Area coordinates, pg. 205, Vol. I, Carey, Oden, Becker FEM
              const Real zeta1 = p(0);
              const Real zeta2 = p(1);
              const Real zeta3 = p(2);
              const Real zeta0 = 1. - zeta1 - zeta2 - zeta3;

              switch(i)
                {
                case 0:
                  return zeta0*(2.*zeta0 - 1.);

                case 1:
                  return zeta1*(2.*zeta1 - 1.);

                case 2:
                  return zeta2*(2.*zeta2 - 1.);

                case 3:
                  return zeta3*(2.*zeta3 - 1.);

                case 4:
                  return 4.*zeta0*zeta1;

                case 5:
                  return 4.*zeta1*zeta2;

                case 6:
                  return 4.*zeta2*zeta0;

                case 7:
                  return 4.*zeta0*zeta3;

                case 8:
                  return 4.*zeta1*zeta3;

                case 9:
                  return 4.*zeta2*zeta3;

                default:
                  libmesh_error_msg("Invalid i = " << i);
                }
            }

            // quadratic prism shape functions
          case PRISM18:
            {
              libmesh_assert_less (i, 18);

              // Compute prism shape functions as a tensor-product
              // of a triangle and an edge

              Point p2d(p(0),p(1));
              Point p1d(p(2));

              //                                0  1  2  3  4  5  6  7  8  9 10 11 12 13 14 15 16 17
              static const unsigned int i0[] = {0, 0, 0, 1, 1, 1, 0, 0, 0, 2, 2, 2, 1, 1, 1, 2, 2, 2};
              static const unsigned int i1[] = {0, 1, 2, 0, 1, 2, 3, 4, 5, 0, 1, 2, 3, 4, 5, 3, 4, 5};

              return (FE<2,L2_LAGRANGE>::shape(TRI6,  SECOND, i1[i], p2d)*
                      FE<1,L2_LAGRANGE>::shape(EDGE3, SECOND, i0[i], p1d));
            }


          default:
            libmesh_error_msg("ERROR: Unsupported 3D element type!: " << type);
          }
      }


      // unsupported order
    default:
      libmesh_error_msg("ERROR: Unsupported 3D FE order!: " << order);
    }

#endif
}

shape() [31/109]

Real libMesh::FE< 0, LAGRANGE >::shape ( const ElemType  , const Order  , const unsigned int  libmesh_dbg_vari, const Point &    )

inherited

Definition at line 32 of file fe_lagrange_shape_0D.C.

{
  libmesh_assert_less (i, 1);
  return 1.;
}

shape() [32/109]

Real libMesh::FE< 3, CLOUGH >::shape ( const ElemType  , const Order  , const unsigned  int, const Point &    )

inherited

Definition at line 33 of file fe_clough_shape_3D.C.

{
  libmesh_error_msg("Clough-Tocher elements require the real element \nto construct gradient-based degrees of freedom.");
  return 0.;
}

shape() [33/109]

Real libMesh::FE< 1, HIERARCHIC >::shape ( const ElemType  , const Order  libmesh_dbg_varorder, const unsigned int  i, const Point &  p  )

inherited

Definition at line 33 of file fe_hierarchic_shape_1D.C.

{
  libmesh_assert_less (i, order+1u);

  // Declare that we are using our own special power function
  // from the Utility namespace.  This saves typing later.
  using Utility::pow;

  const Real xi = p(0);

  Real returnval = 1.;

  switch (i)
    {
    case 0:
      returnval = .5*(1. - xi);
      break;
    case 1:
      returnval = .5*(1.  + xi);
      break;
      // All even-terms have the same form.
      // (xi^p - 1.)/p!
    case 2:
      returnval = (xi*xi - 1.)/2.;
      break;
    case 4:
      returnval = (pow<4>(xi) - 1.)/24.;
      break;
    case 6:
      returnval = (pow<6>(xi) - 1.)/720.;
      break;

      // All odd-terms have the same form.
      // (xi^p - xi)/p!
    case 3:
      returnval = (xi*xi*xi - xi)/6.;
      break;
    case 5:
      returnval = (pow<5>(xi) - xi)/120.;
      break;
    case 7:
      returnval = (pow<7>(xi) - xi)/5040.;
      break;
    default:
      Real denominator = 1.;
      for (unsigned int n=1; n <= i; ++n)
        {
          returnval *= xi;
          denominator *= n;
        }
      // Odd:
      if (i % 2)
        returnval = (returnval - xi)/denominator;
      // Even:
      else
        returnval = (returnval - 1.)/denominator;
      break;
    }

  return returnval;
}

shape() [34/109]

Real libMesh::FE< 1, L2_HIERARCHIC >::shape ( const ElemType  , const Order  libmesh_dbg_varorder, const unsigned int  i, const Point &  p  )

inherited

Definition at line 33 of file fe_l2_hierarchic_shape_1D.C.

{
  libmesh_assert_less (i, order+1u);

  // Declare that we are using our own special power function
  // from the Utility namespace.  This saves typing later.
  using Utility::pow;

  const Real xi = p(0);

  Real returnval = 1.;

  switch (i)
    {
    case 0:
      returnval = .5*(1. - xi);
      break;
    case 1:
      returnval = .5*(1.  + xi);
      break;
      // All even-terms have the same form.
      // (xi^p - 1.)/p!
    case 2:
      returnval = (xi*xi - 1.)/2.;
      break;
    case 4:
      returnval = (pow<4>(xi) - 1.)/24.;
      break;
    case 6:
      returnval = (pow<6>(xi) - 1.)/720.;
      break;

      // All odd-terms have the same form.
      // (xi^p - xi)/p!
    case 3:
      returnval = (xi*xi*xi - xi)/6.;
      break;
    case 5:
      returnval = (pow<5>(xi) - xi)/120.;
      break;
    case 7:
      returnval = (pow<7>(xi) - xi)/5040.;
      break;
    default:
      Real denominator = 1.;
      for (unsigned int n=1; n <= i; ++n)
        {
          returnval *= xi;
          denominator *= n;
        }
      // Odd:
      if (i % 2)
        returnval = (returnval - xi)/denominator;
      // Even:
      else
        returnval = (returnval - 1.)/denominator;
      break;
    }

  return returnval;
}

shape() [35/109]

Real libMesh::FE< 2, BERNSTEIN >::shape ( const ElemType  , const Order  , const unsigned  int, const Point &    )

inherited

Definition at line 34 of file fe_bernstein_shape_2D.C.

{
  libmesh_error_msg("Bernstein polynomials require the element type \nbecause edge orientation is needed.");
  return 0.;
}

shape() [36/109]

Real libMesh::FE< 1, SZABAB >::shape ( const ElemType  , const Order  libmesh_dbg_varorder, const unsigned int  i, const Point &  p  )

inherited

Definition at line 35 of file fe_szabab_shape_1D.C.

{
  const Real xi  = p(0);
  const Real xi2 = xi*xi;


  // Use this libmesh_assert rather than a switch with a single entry...
  // It will go away in optimized mode, essentially has the same effect.
  libmesh_assert_less_equal (order, SEVENTH);

  //   switch (order)
  //     {
  //     case FIRST:
  //     case SECOND:
  //     case THIRD:
  //     case FOURTH:
  //     case FIFTH:
  //     case SIXTH:
  //     case SEVENTH:

  switch(i)
    {
      //nodal shape functions
    case 0: return 1./2.-1./2.*xi;
    case 1: return 1./2.+1./2.*xi;
    case 2: return 1./4.  *2.4494897427831780982*(xi2-1.);
    case 3: return 1./4.  *3.1622776601683793320*(xi2-1.)*xi;
    case 4: return 1./16. *3.7416573867739413856*((5.*xi2-6.)*xi2+1.);
    case 5: return 3./16. *1.4142135623730950488*(3.+(-10.+7.*xi2)*xi2)*xi;
    case 6: return 1./32. *4.6904157598234295546*(-1.+(15.+(-35.+21.*xi2)*xi2)*xi2);
    case 7: return 1./32. *5.0990195135927848300*(-5.+(35.+(-63.+33.*xi2)*xi2)*xi2)*xi;
    case 8: return 1./256.*5.4772255750516611346*(5.+(-140.+(630.+(-924.+429.*xi2)*xi2)*xi2)*xi2);

    default:
      libmesh_error_msg("Invalid shape function index!");
    }
}

shape() [37/109]

Real libMesh::FE< 3, BERNSTEIN >::shape ( const ElemType  , const Order  , const unsigned  int, const Point &    )

inherited

Definition at line 36 of file fe_bernstein_shape_3D.C.

{
  libmesh_error_msg("Bernstein polynomials require the element type \nbecause edge and face orientation is needed.");
  return 0.;
}

shape() [38/109]

Real libMesh::FE< 2, SCALAR >::shape ( const Elem *  , const Order  , const unsigned  int, const Point &    )

inherited

Definition at line 38 of file fe_scalar_shape_2D.C.

{
  return 1.;
}

shape() [39/109]

Real libMesh::FE< 3, SCALAR >::shape ( const Elem *  , const Order  , const unsigned  int, const Point &    )

inherited

Definition at line 38 of file fe_scalar_shape_3D.C.

{
  return 1.;
}

shape() [40/109]

Real libMesh::FE< 0, SCALAR >::shape ( const Elem *  , const Order  , const unsigned  int, const Point &    )

inherited

Definition at line 38 of file fe_scalar_shape_0D.C.

{
  return 1.;
}

shape() [41/109]

Real libMesh::FE< 1, SCALAR >::shape ( const Elem *  , const Order  , const unsigned  int, const Point &    )

inherited

Definition at line 38 of file fe_scalar_shape_1D.C.

{
  return 1.;
}

shape() [42/109]

Real libMesh::FE< 1, BERNSTEIN >::shape ( const ElemType  , const Order  order, const unsigned int  i, const Point &  p  )

inherited

Definition at line 38 of file fe_bernstein_shape_1D.C.

{
  const Real xi = p(0);
  using Utility::pow;

  switch (order)
    {
    case FIRST:

      switch(i)
        {
        case 0:
          return (1.-xi)/2.;
        case 1:
          return (1.+xi)/2.;
        default:
          libmesh_error_msg("Invalid shape function index i = " << i);
        }

    case SECOND:

      switch(i)
        {
        case 0:
          return (1./4.)*pow<2>(1.-xi);
        case 1:
          return (1./4.)*pow<2>(1.+xi);
        case 2:
          return (1./2.)*(1.-xi)*(1.+xi);
        default:
          libmesh_error_msg("Invalid shape function index i = " << i);
        }

    case THIRD:

      switch(i)
        {
        case 0:
          return (1./8.)*pow<3>(1.-xi);
        case 1:
          return (1./8.)*pow<3>(1.+xi);
        case 2:
          return (3./8.)*(1.+xi)*pow<2>(1.-xi);
        case 3:
          return (3./8.)*pow<2>(1.+xi)*(1.-xi);
        default:
          libmesh_error_msg("Invalid shape function index i = " << i);
        }

    case FOURTH:

      switch(i)
        {
        case 0:
          return (1./16.)*pow<4>(1.-xi);
        case 1:
          return (1./16.)*pow<4>(1.+xi);
        case 2:
          return (1./ 4.)*(1.+xi)*pow<3>(1.-xi);
        case 3:
          return (3./ 8.)*pow<2>(1.+xi)*pow<2>(1.-xi);
        case 4:
          return (1./ 4.)*pow<3>(1.+xi)*(1.-xi);
        default:
          libmesh_error_msg("Invalid shape function index i = " << i);
        }


    case FIFTH:

      switch(i)
        {
        case 0:
          return (1./32.)*pow<5>(1.-xi);
        case 1:
          return (1./32.)*pow<5>(1.+xi);
        case 2:
          return (5./32.)*(1.+xi)*pow<4>(1.-xi);
        case 3:
          return (5./16.)*pow<2>(1.+xi)*pow<3>(1.-xi);
        case 4:
          return (5./16.)*pow<3>(1.+xi)*pow<2>(1.-xi);
        case 5:
          return (5./32.)*pow<4>(1.+xi)*(1.-xi);
        default:
          libmesh_error_msg("Invalid shape function index i = " << i);
        }


    case SIXTH:

      switch (i)
        {
        case 0:
          return ( 1./64.)*pow<6>(1.-xi);
        case 1:
          return ( 1./64.)*pow<6>(1.+xi);
        case 2:
          return ( 3./32.)*(1.+xi)*pow<5>(1.-xi);
        case 3:
          return (15./64.)*pow<2>(1.+xi)*pow<4>(1.-xi);
        case 4:
          return ( 5./16.)*pow<3>(1.+xi)*pow<3>(1.-xi);
        case 5:
          return (15./64.)*pow<4>(1.+xi)*pow<2>(1.-xi);
        case 6:
          return ( 3./32.)*pow<5>(1.+xi)*(1.-xi);
        default:
          libmesh_error_msg("Invalid shape function index i = " << i);
        }

    default:
      {
        libmesh_assert (order>6);

        // Use this for arbitrary orders.
        // Note that this implementation is less efficient.
        const int p_order = static_cast<int>(order);
        const int m       = p_order-i+1;
        const int n       = (i-1);

        Real binomial_p_i = 1;

        // the binomial coefficient (p choose n)
        // Using an unsigned long here will work for any of the orders we support.
        // Explicitly construct a Real to prevent conversion warnings
        if (i>1)
          binomial_p_i = Real(Utility::binomial(static_cast<unsigned long>(p_order),
                                                static_cast<unsigned long>(n)));

        switch(i)
          {
          case 0:
            return binomial_p_i * std::pow((1-xi)/2, p_order);
          case 1:
            return binomial_p_i * std::pow((1+xi)/2, p_order);
          default:
            {
              return binomial_p_i * std::pow((1+xi)/2,n)
                * std::pow((1-xi)/2,m);
            }
          }
      }
    }
}

shape() [43/109]

RealGradient libMesh::FE< 3, NEDELEC_ONE >::shape ( const Elem *  elem, const Order  order, const unsigned int  i, const Point &  p  )

inherited

Definition at line 41 of file fe_nedelec_one_shape_3D.C.

{
#if LIBMESH_DIM == 3
  libmesh_assert(elem);

  const Order totalorder = static_cast<Order>(order + elem->p_level());

  switch (totalorder)
    {
      // linear Lagrange shape functions
    case FIRST:
      {
        switch (elem->type())
          {
          case HEX20:
          case HEX27:
            {
              libmesh_assert_less (i, 12);

              const Real xi   = p(0);
              const Real eta  = p(1);
              const Real zeta = p(2);

              // Even with a loose inverse_map tolerance we ought to
              // be nearly on the element interior in master
              // coordinates
              libmesh_assert_less_equal ( std::fabs(xi),   1.0+10*TOLERANCE );
              libmesh_assert_less_equal ( std::fabs(eta),  1.0+10*TOLERANCE );
              libmesh_assert_less_equal ( std::fabs(zeta), 1.0+10*TOLERANCE );

              switch(i)
                {
                case 0:
                  {
                    if (elem->point(0) > elem->point(1))
                      return RealGradient( -0.125*(1.0-eta-zeta+eta*zeta), 0.0, 0.0 );
                    else
                      return RealGradient(  0.125*(1.0-eta-zeta+eta*zeta), 0.0, 0.0 );
                  }
                case 1:
                  {
                    if (elem->point(1) > elem->point(2))
                      return RealGradient( 0.0, -0.125*(1.0+xi-zeta-xi*zeta), 0.0 );
                    else
                      return RealGradient( 0.0,  0.125*(1.0+xi-zeta-xi*zeta), 0.0 );
                  }
                case 2:
                  {
                    if (elem->point(2) > elem->point(3))
                      return RealGradient(  0.125*(1.0+eta-zeta-eta*zeta), 0.0, 0.0 );
                    else
                      return RealGradient( -0.125*(1.0+eta-zeta-eta*zeta), 0.0, 0.0 );
                  }
                case 3:
                  {
                    if (elem->point(3) > elem->point(0))
                      return RealGradient( 0.0,  0.125*(1.0-xi-zeta+xi*zeta), 0.0 );
                    else
                      return RealGradient( 0.0, -0.125*(1.0-xi-zeta+xi*zeta), 0.0 );
                  }
                case 4:
                  {
                    if (elem->point(0) > elem->point(4))
                      return RealGradient( 0.0, 0.0, -0.125*(1.0-xi-eta+xi*eta) );
                    else
                      return RealGradient( 0.0, 0.0,  0.125*(1.0-xi-eta+xi*eta) );
                  }
                case 5:
                  {
                    if (elem->point(1) > elem->point(5))
                      return RealGradient( 0.0, 0.0, -0.125*(1.0+xi-eta-xi*eta) );
                    else
                      return RealGradient( 0.0, 0.0,  0.125*(1.0+xi-eta-xi*eta) );
                  }
                case 6:
                  {
                    if (elem->point(2) > elem->point(6))
                      return RealGradient( 0.0, 0.0, -0.125*(1.0+xi+eta+xi*eta) );
                    else
                      return RealGradient( 0.0, 0.0,  0.125*(1.0+xi+eta+xi*eta) );
                  }
                case 7:
                  {
                    if (elem->point(3) > elem->point(7))
                      return RealGradient( 0.0, 0.0, -0.125*(1.0-xi+eta-xi*eta) );
                    else
                      return RealGradient( 0.0, 0.0,  0.125*(1.0-xi+eta-xi*eta) );
                  }
                case 8:
                  {
                    if (elem->point(4) > elem->point(5))
                      return RealGradient( -0.125*(1.0-eta+zeta-eta*zeta), 0.0, 0.0 );
                    else
                      return RealGradient(  0.125*(1.0-eta+zeta-eta*zeta), 0.0, 0.0 );
                  }
                case 9:
                  {
                    if (elem->point(5) > elem->point(6))
                      return RealGradient( 0.0, -0.125*(1.0+xi+zeta+xi*zeta), 0.0 );
                    else
                      return RealGradient( 0.0,  0.125*(1.0+xi+zeta+xi*zeta), 0.0 );
                  }
                case 10:
                  {
                    if (elem->point(7) > elem->point(6))
                      return RealGradient( -0.125*(1.0+eta+zeta+eta*zeta), 0.0, 0.0 );
                    else
                      return RealGradient(  0.125*(1.0+eta+zeta+eta*zeta), 0.0, 0.0 );
                  }
                case 11:
                  {
                    if (elem->point(4) > elem->point(7))
                      return RealGradient( 0.0, -0.125*(1.0-xi+zeta-xi*zeta), 0.0 );
                    else
                      return RealGradient( 0.0,  0.125*(1.0-xi+zeta-xi*zeta), 0.0 );
                  }

                default:
                  libmesh_error_msg("Invalid i = " << i);
                }

              return RealGradient();
            }

          case TET10:
            {
              libmesh_assert_less (i, 6);

              libmesh_not_implemented();
              return RealGradient();
            }

          default:
            libmesh_error_msg("ERROR: Unsupported 3D element type!: " << elem->type());
          }
      }

      // unsupported order
    default:
      libmesh_error_msg("ERROR: Unsupported 3D FE order!: " << totalorder);
    }
#else
  return RealGradient();
#endif
}

shape() [44/109]

RealGradient libMesh::FE< 2, NEDELEC_ONE >::shape ( const Elem *  elem, const Order  order, const unsigned int  i, const Point &  p  )

inherited

Definition at line 42 of file fe_nedelec_one_shape_2D.C.

{
#if LIBMESH_DIM > 1
  libmesh_assert(elem);

  const Order total_order = static_cast<Order>(order + elem->p_level());

  switch (total_order)
    {
    case FIRST:
      {
        switch (elem->type())
          {
          case QUAD8:
          case QUAD9:
            {
              libmesh_assert_less (i, 4);

              const Real xi  = p(0);
              const Real eta = p(1);

              // Even with a loose inverse_map tolerance we ought to
              // be nearly on the element interior in master
              // coordinates
              libmesh_assert_less_equal ( std::fabs(xi), 1.0+10*TOLERANCE );
              libmesh_assert_less_equal ( std::fabs(eta), 1.0+10*TOLERANCE );

              switch(i)
                {
                case 0:
                  {
                    if (elem->point(0) > elem->point(1))
                      return RealGradient( -0.25*(1.0-eta), 0.0 );
                    else
                      return RealGradient( 0.25*(1.0-eta), 0.0 );
                  }
                case 1:
                  {
                    if (elem->point(1) > elem->point(2))
                      return RealGradient( 0.0, -0.25*(1.0+xi) );
                    else
                      return RealGradient( 0.0, 0.25*(1.0+xi) );
                  }

                case 2:
                  {
                    if (elem->point(2) > elem->point(3))
                      return RealGradient( 0.25*(1.0+eta), 0.0 );
                    else
                      return RealGradient( -0.25*(1.0+eta), 0.0 );
                  }
                case 3:
                  {
                    if (elem->point(3) > elem->point(0))
                      return RealGradient( 0.0, -0.25*(xi-1.0) );
                    else
                      return RealGradient( 0.0, 0.25*(xi-1.0) );
                  }

                default:
                  libmesh_error_msg("Invalid i = " << i);
                }

              return RealGradient();
            }

          case TRI6:
            {
              const Real xi  = p(0);
              const Real eta = p(1);

              libmesh_assert_less (i, 3);

              switch(i)
                {
                case 0:
                  {
                    if (elem->point(0) > elem->point(1))
                      return RealGradient( -1.0+eta, -xi );
                    else
                      return RealGradient( 1.0-eta, xi );
                  }
                case 1:
                  {
                    if (elem->point(1) > elem->point(2))
                      return RealGradient( eta, -xi );
                    else
                      return RealGradient( -eta, xi );
                  }

                case 2:
                  {
                    if (elem->point(2) > elem->point(0))
                      return RealGradient( eta, -xi+1.0 );
                    else
                      return RealGradient( -eta, xi-1.0 );
                  }

                default:
                  libmesh_error_msg("Invalid i = " << i);
                }
            }

          default:
            libmesh_error_msg("ERROR: Unsupported 2D element type!: " << elem->type());
          }
      }

      // unsupported order
    default:
      libmesh_error_msg("ERROR: Unsupported 2D FE order!: " << total_order);
    }
#else // LIBMESH_DIM > 1
  return RealGradient();
#endif
}

shape() [45/109]

Real libMesh::FE< 2, L2_HIERARCHIC >::shape ( const Elem *  elem, const Order  order, const unsigned int  i, const Point &  p  )

inherited

Definition at line 44 of file fe_l2_hierarchic_shape_2D.C.

{
  libmesh_assert(elem);

  const Order totalorder = static_cast<Order>(order+elem->p_level());
  libmesh_assert_greater (totalorder, 0);

  switch (elem->type())
    {
    case TRI3:
    case TRISHELL3:
    case TRI6:
      {
        const Real zeta1 = p(0);
        const Real zeta2 = p(1);
        const Real zeta0 = 1. - zeta1 - zeta2;

        libmesh_assert_less (i, (totalorder+1u)*(totalorder+2u)/2);
        libmesh_assert (elem->type() == TRI6 || totalorder < 2);

        // Vertex DoFs
        if (i == 0)
          return zeta0;
        else if (i == 1)
          return zeta1;
        else if (i == 2)
          return zeta2;
        // Edge DoFs
        else if (i < totalorder + 2u)
          {
            // Avoid returning NaN on vertices!
            if (zeta0 + zeta1 == 0.)
              return 0.;

            const unsigned int basisorder = i - 1;
            // Get factors to account for edge-flipping
            Real f0 = 1;
            if (basisorder%2 && (elem->point(0) > elem->point(1)))
              f0 = -1.;

            Real edgeval = (zeta1 - zeta0) / (zeta1 + zeta0);
            Real crossfunc = zeta0 + zeta1;
            for (unsigned int n=1; n != basisorder; ++n)
              crossfunc *= (zeta0 + zeta1);

            return f0 * crossfunc *
              FE<1,L2_HIERARCHIC>::shape(EDGE3, totalorder,
                                         basisorder, edgeval);
          }
        else if (i < 2u*totalorder + 1)
          {
            // Avoid returning NaN on vertices!
            if (zeta1 + zeta2 == 0.)
              return 0.;

            const unsigned int basisorder = i - totalorder;
            // Get factors to account for edge-flipping
            Real f1 = 1;
            if (basisorder%2 && (elem->point(1) > elem->point(2)))
              f1 = -1.;

            Real edgeval = (zeta2 - zeta1) / (zeta2 + zeta1);
            Real crossfunc = zeta2 + zeta1;
            for (unsigned int n=1; n != basisorder; ++n)
              crossfunc *= (zeta2 + zeta1);

            return f1 * crossfunc *
              FE<1,L2_HIERARCHIC>::shape(EDGE3, totalorder,
                                         basisorder, edgeval);
          }
        else if (i < 3u*totalorder)
          {
            // Avoid returning NaN on vertices!
            if (zeta0 + zeta2 == 0.)
              return 0.;

            const unsigned int basisorder = i - (2u*totalorder) + 1;
            // Get factors to account for edge-flipping
            Real f2 = 1;
            if (basisorder%2 && (elem->point(2) > elem->point(0)))
              f2 = -1.;

            Real edgeval = (zeta0 - zeta2) / (zeta0 + zeta2);
            Real crossfunc = zeta0 + zeta2;
            for (unsigned int n=1; n != basisorder; ++n)
              crossfunc *= (zeta0 + zeta2);

            return f2 * crossfunc *
              FE<1,L2_HIERARCHIC>::shape(EDGE3, totalorder,
                                         basisorder, edgeval);
          }
        // Interior DoFs
        else
          {
            const unsigned int basisnum = i - (3u*totalorder);
            unsigned int exp0 = triangular_number_column[basisnum] + 1;
            unsigned int exp1 = triangular_number_row[basisnum] + 1 -
              triangular_number_column[basisnum];

            Real returnval = 1;
            for (unsigned int n = 0; n != exp0; ++n)
              returnval *= zeta0;
            for (unsigned int n = 0; n != exp1; ++n)
              returnval *= zeta1;
            returnval *= zeta2;
            return returnval;
          }
      }

      // Hierarchic shape functions on the quadrilateral.
    case QUAD4:
    case QUADSHELL4:
      libmesh_assert_less (totalorder, 2);
      libmesh_fallthrough();
    case QUAD8:
    case QUADSHELL8:
    case QUAD9:
      {
        // Compute quad shape functions as a tensor-product
        const Real xi  = p(0);
        const Real eta = p(1);

        libmesh_assert_less (i, (totalorder+1u)*(totalorder+1u));

        // Example i, i0, i1 values for totalorder = 5:
        //                                    0  1  2  3  4  5  6  7  8  9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35
        //  static const unsigned int i0[] = {0, 1, 1, 0, 2, 3, 4, 5, 1, 1, 1, 1, 2, 3, 4, 5, 0, 0, 0, 0, 2, 3, 3, 2, 4, 4, 4, 3, 2, 5, 5, 5, 5, 4, 3, 2};
        //  static const unsigned int i1[] = {0, 0, 1, 1, 0, 0, 0, 0, 2, 3, 4, 5, 1, 1, 1, 1, 2, 3, 4, 5, 2, 2, 3, 3, 2, 3, 4, 4, 4, 2, 3, 4, 5, 5, 5, 5};

        unsigned int i0, i1;

        // Vertex DoFs
        if (i == 0)
          { i0 = 0; i1 = 0; }
        else if (i == 1)
          { i0 = 1; i1 = 0; }
        else if (i == 2)
          { i0 = 1; i1 = 1; }
        else if (i == 3)
          { i0 = 0; i1 = 1; }
        // Edge DoFs
        else if (i < totalorder + 3u)
          { i0 = i - 2; i1 = 0; }
        else if (i < 2u*totalorder + 2)
          { i0 = 1; i1 = i - totalorder - 1; }
        else if (i < 3u*totalorder + 1)
          { i0 = i - 2u*totalorder; i1 = 1; }
        else if (i < 4u*totalorder)
          { i0 = 0; i1 = i - 3u*totalorder + 1; }
        // Interior DoFs
        else
          {
            unsigned int basisnum = i - 4*totalorder;
            i0 = square_number_column[basisnum] + 2;
            i1 = square_number_row[basisnum] + 2;
          }

        // Flip odd degree of freedom values if necessary
        // to keep continuity on sides
        Real f = 1.;

        if ((i0%2) && (i0 > 2) && (i1 == 0))
          f = (elem->point(0) > elem->point(1))?-1.:1.;
        else if ((i0%2) && (i0>2) && (i1 == 1))
          f = (elem->point(3) > elem->point(2))?-1.:1.;
        else if ((i0 == 0) && (i1%2) && (i1>2))
          f = (elem->point(0) > elem->point(3))?-1.:1.;
        else if ((i0 == 1) && (i1%2) && (i1>2))
          f = (elem->point(1) > elem->point(2))?-1.:1.;

        return f*(FE<1,L2_HIERARCHIC>::shape(EDGE3, totalorder, i0, xi)*
                  FE<1,L2_HIERARCHIC>::shape(EDGE3, totalorder, i1, eta));
      }

    default:
      libmesh_error_msg("ERROR: Unsupported element type = " << elem->type());
    }

  return 0.;
}

shape() [46/109]

Real libMesh::FE< 0, CLOUGH >::shape ( const Elem *  , const Order  , const unsigned int  libmesh_dbg_vari, const Point &    )

inherited

Definition at line 44 of file fe_clough_shape_0D.C.

{
  libmesh_assert_less (i, 1);
  return 1.;
}

shape() [47/109]

Real libMesh::FE< 0, HIERARCHIC >::shape ( const Elem *  , const Order  , const unsigned int  libmesh_dbg_vari, const Point &    )

inherited

Definition at line 44 of file fe_hierarchic_shape_0D.C.

{
  libmesh_assert_less (i, 1);
  return 1.;
}

shape() [48/109]

Real libMesh::FE< 0, MONOMIAL >::shape ( const Elem *  , const Order  , const unsigned int  libmesh_dbg_vari, const Point &    )

inherited

Definition at line 44 of file fe_monomial_shape_0D.C.

{
  libmesh_assert_less (i, 1);
  return 1.;
}

shape() [49/109]

Real libMesh::FE< 2, HIERARCHIC >::shape ( const Elem *  elem, const Order  order, const unsigned int  i, const Point &  p  )

inherited

Definition at line 44 of file fe_hierarchic_shape_2D.C.

{
  libmesh_assert(elem);

  const Order totalorder = static_cast<Order>(order+elem->p_level());
  libmesh_assert_greater (totalorder, 0);

  switch (elem->type())
    {
    case TRI3:
    case TRISHELL3:
    case TRI6:
      {
        const Real zeta1 = p(0);
        const Real zeta2 = p(1);
        const Real zeta0 = 1. - zeta1 - zeta2;

        libmesh_assert_less (i, (totalorder+1u)*(totalorder+2u)/2);
        libmesh_assert (elem->type() == TRI6 || totalorder < 2);

        // Vertex DoFs
        if (i == 0)
          return zeta0;
        else if (i == 1)
          return zeta1;
        else if (i == 2)
          return zeta2;
        // Edge DoFs
        else if (i < totalorder + 2u)
          {
            // Avoid returning NaN on vertices!
            if (zeta0 + zeta1 == 0.)
              return 0.;

            const unsigned int basisorder = i - 1;
            // Get factors to account for edge-flipping
            Real f0 = 1;
            if (basisorder%2 && (elem->point(0) > elem->point(1)))
              f0 = -1.;

            Real edgeval = (zeta1 - zeta0) / (zeta1 + zeta0);
            Real crossfunc = zeta0 + zeta1;
            for (unsigned int n=1; n != basisorder; ++n)
              crossfunc *= (zeta0 + zeta1);

            return f0 * crossfunc *
              FE<1,HIERARCHIC>::shape(EDGE3, totalorder,
                                      basisorder, edgeval);
          }
        else if (i < 2u*totalorder + 1)
          {
            // Avoid returning NaN on vertices!
            if (zeta1 + zeta2 == 0.)
              return 0.;

            const unsigned int basisorder = i - totalorder;
            // Get factors to account for edge-flipping
            Real f1 = 1;
            if (basisorder%2 && (elem->point(1) > elem->point(2)))
              f1 = -1.;

            Real edgeval = (zeta2 - zeta1) / (zeta2 + zeta1);
            Real crossfunc = zeta2 + zeta1;
            for (unsigned int n=1; n != basisorder; ++n)
              crossfunc *= (zeta2 + zeta1);

            return f1 * crossfunc *
              FE<1,HIERARCHIC>::shape(EDGE3, totalorder,
                                      basisorder, edgeval);
          }
        else if (i < 3u*totalorder)
          {
            // Avoid returning NaN on vertices!
            if (zeta0 + zeta2 == 0.)
              return 0.;

            const unsigned int basisorder = i - (2u*totalorder) + 1;
            // Get factors to account for edge-flipping
            Real f2 = 1;
            if (basisorder%2 && (elem->point(2) > elem->point(0)))
              f2 = -1.;

            Real edgeval = (zeta0 - zeta2) / (zeta0 + zeta2);
            Real crossfunc = zeta0 + zeta2;
            for (unsigned int n=1; n != basisorder; ++n)
              crossfunc *= (zeta0 + zeta2);

            return f2 * crossfunc *
              FE<1,HIERARCHIC>::shape(EDGE3, totalorder,
                                      basisorder, edgeval);
          }
        // Interior DoFs
        else
          {
            const unsigned int basisnum = i - (3u*totalorder);
            unsigned int exp0 = triangular_number_column[basisnum] + 1;
            unsigned int exp1 = triangular_number_row[basisnum] + 1 -
              triangular_number_column[basisnum];

            Real returnval = 1;
            for (unsigned int n = 0; n != exp0; ++n)
              returnval *= zeta0;
            for (unsigned int n = 0; n != exp1; ++n)
              returnval *= zeta1;
            returnval *= zeta2;
            return returnval;
          }
      }

      // Hierarchic shape functions on the quadrilateral.
    case QUAD4:
    case QUADSHELL4:
      libmesh_assert_less (totalorder, 2);
      libmesh_fallthrough();
    case QUAD8:
    case QUADSHELL8:
    case QUAD9:
      {
        // Compute quad shape functions as a tensor-product
        const Real xi  = p(0);
        const Real eta = p(1);

        libmesh_assert_less (i, (totalorder+1u)*(totalorder+1u));

        // Example i, i0, i1 values for totalorder = 5:
        //                                    0  1  2  3  4  5  6  7  8  9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35
        //  static const unsigned int i0[] = {0, 1, 1, 0, 2, 3, 4, 5, 1, 1, 1, 1, 2, 3, 4, 5, 0, 0, 0, 0, 2, 3, 3, 2, 4, 4, 4, 3, 2, 5, 5, 5, 5, 4, 3, 2};
        //  static const unsigned int i1[] = {0, 0, 1, 1, 0, 0, 0, 0, 2, 3, 4, 5, 1, 1, 1, 1, 2, 3, 4, 5, 2, 2, 3, 3, 2, 3, 4, 4, 4, 2, 3, 4, 5, 5, 5, 5};

        unsigned int i0, i1;

        // Vertex DoFs
        if (i == 0)
          { i0 = 0; i1 = 0; }
        else if (i == 1)
          { i0 = 1; i1 = 0; }
        else if (i == 2)
          { i0 = 1; i1 = 1; }
        else if (i == 3)
          { i0 = 0; i1 = 1; }
        // Edge DoFs
        else if (i < totalorder + 3u)
          { i0 = i - 2; i1 = 0; }
        else if (i < 2u*totalorder + 2)
          { i0 = 1; i1 = i - totalorder - 1; }
        else if (i < 3u*totalorder + 1)
          { i0 = i - 2u*totalorder; i1 = 1; }
        else if (i < 4u*totalorder)
          { i0 = 0; i1 = i - 3u*totalorder + 1; }
        // Interior DoFs
        else
          {
            unsigned int basisnum = i - 4*totalorder;
            i0 = square_number_column[basisnum] + 2;
            i1 = square_number_row[basisnum] + 2;
          }

        // Flip odd degree of freedom values if necessary
        // to keep continuity on sides
        Real f = 1.;

        if ((i0%2) && (i0 > 2) && (i1 == 0))
          f = (elem->point(0) > elem->point(1))?-1.:1.;
        else if ((i0%2) && (i0>2) && (i1 == 1))
          f = (elem->point(3) > elem->point(2))?-1.:1.;
        else if ((i0 == 0) && (i1%2) && (i1>2))
          f = (elem->point(0) > elem->point(3))?-1.:1.;
        else if ((i0 == 1) && (i1%2) && (i1>2))
          f = (elem->point(1) > elem->point(2))?-1.:1.;

        return f*(FE<1,HIERARCHIC>::shape(EDGE3, totalorder, i0, xi)*
                  FE<1,HIERARCHIC>::shape(EDGE3, totalorder, i1, eta));
      }

    default:
      libmesh_error_msg("ERROR: Unsupported element type = " << elem->type());
    }

  return 0.;
}

shape() [50/109]

Real libMesh::FE< 0, SZABAB >::shape ( const Elem *  , const Order  , const unsigned int  libmesh_dbg_vari, const Point &    )

inherited

Definition at line 44 of file fe_szabab_shape_0D.C.

{
  libmesh_assert_less (i, 1);
  return 1.;
}

shape() [51/109]

Real libMesh::FE< 0, BERNSTEIN >::shape ( const Elem *  , const Order  , const unsigned int  libmesh_dbg_vari, const Point &    )

inherited

Definition at line 44 of file fe_bernstein_shape_0D.C.

{
  libmesh_assert_less (i, 1);
  return 1.;
}

shape() [52/109]

Real libMesh::FE< 0, L2_HIERARCHIC >::shape ( const Elem *  , const Order  , const unsigned int  libmesh_dbg_vari, const Point &    )

inherited

Definition at line 44 of file fe_l2_hierarchic_shape_0D.C.

{
  libmesh_assert_less (i, 1);
  return 1.;
}

shape() [53/109]

Real libMesh::FE< 0, L2_LAGRANGE >::shape ( const Elem *  , const Order  , const unsigned int  libmesh_dbg_vari, const Point &    )

inherited

Definition at line 44 of file fe_l2_lagrange_shape_0D.C.

{
  libmesh_assert_less (i, 1);
  return 1.;
}

shape() [54/109]

Real libMesh::FE< 3, SZABAB >::shape ( const Elem *  , const Order  , const unsigned  int, const Point &    )

inherited

Definition at line 44 of file fe_szabab_shape_3D.C.

{
  libmesh_error_msg("Szabo-Babuska polynomials are not defined in 3D");
  return 0.;
}

shape() [55/109]

Real libMesh::FE< 0, XYZ >::shape ( const Elem *  , const Order  , const unsigned int  libmesh_dbg_vari, const Point &    )

inherited

Definition at line 44 of file fe_xyz_shape_0D.C.

{
  libmesh_assert_less (i, 1);
  return 1.;
}

shape() [56/109]

Real libMesh::FE< 1, XYZ >::shape ( const Elem *  elem, const Order  libmesh_dbg_varorder, const unsigned int  i, const Point &  point_in  )

inherited

Definition at line 44 of file fe_xyz_shape_1D.C.

{
  libmesh_assert(elem);
  libmesh_assert_less_equal (i, order + elem->p_level());

  Point centroid = elem->centroid();
  Real max_distance = 0.;
  for (unsigned int p = 0; p < elem->n_nodes(); p++)
    {
      const Real distance = std::abs(centroid(0) - elem->point(p)(0));
      max_distance = std::max(distance, max_distance);
    }

  const Real x  = point_in(0);
  const Real xc = centroid(0);
  const Real dx = (x - xc)/max_distance;

  // monomials. since they are hierarchic we only need one case block.
  switch (i)
    {
    case 0:
      return 1.;

    case 1:
      return dx;

    case 2:
      return dx*dx;

    case 3:
      return dx*dx*dx;

    case 4:
      return dx*dx*dx*dx;

    default:
      Real val = 1.;
      for (unsigned int index = 0; index != i; ++index)
        val *= dx;
      return val;
    }
}

shape() [57/109]

Real libMesh::FE< 2, XYZ >::shape ( const Elem *  elem, const Order  libmesh_dbg_varorder, const unsigned int  i, const Point &  point_in  )

inherited

Definition at line 44 of file fe_xyz_shape_2D.C.

{
#if LIBMESH_DIM > 1

  libmesh_assert(elem);

  Point centroid = elem->centroid();
  Point max_distance = Point(0.,0.,0.);
  for (unsigned int p = 0; p < elem->n_nodes(); p++)
    for (unsigned int d = 0; d < 2; d++)
      {
        const Real distance = std::abs(centroid(d) - elem->point(p)(d));
        max_distance(d) = std::max(distance, max_distance(d));
      }

  const Real x  = point_in(0);
  const Real y  = point_in(1);
  const Real xc = centroid(0);
  const Real yc = centroid(1);
  const Real distx = max_distance(0);
  const Real disty = max_distance(1);
  const Real dx = (x - xc)/distx;
  const Real dy = (y - yc)/disty;

#ifndef NDEBUG
  // totalorder is only used in the assertion below, so
  // we avoid declaring it when asserts are not active.
  const unsigned int totalorder = order + elem->p_level();
#endif
  libmesh_assert_less (i, (totalorder+1)*(totalorder+2)/2);


  // monomials. since they are hierarchic we only need one case block.
  switch (i)
    {
      // constant
    case 0:
      return 1.;

      // linear
    case 1:
      return dx;

    case 2:
      return dy;

      // quadratics
    case 3:
      return dx*dx;

    case 4:
      return dx*dy;

    case 5:
      return dy*dy;

      // cubics
    case 6:
      return dx*dx*dx;

    case 7:
      return dx*dx*dy;

    case 8:
      return dx*dy*dy;

    case 9:
      return dy*dy*dy;

      // quartics
    case 10:
      return dx*dx*dx*dx;

    case 11:
      return dx*dx*dx*dy;

    case 12:
      return dx*dx*dy*dy;

    case 13:
      return dx*dy*dy*dy;

    case 14:
      return dy*dy*dy*dy;

    default:
      unsigned int o = 0;
      for (; i >= (o+1)*(o+2)/2; o++) { }
      unsigned int i2 = i - (o*(o+1)/2);
      Real val = 1.;
      for (unsigned int index=i2; index != o; index++)
        val *= dx;
      for (unsigned int index=0; index != i2; index++)
        val *= dy;
      return val;
    }

#else
  return 0.;
#endif
}

shape() [58/109]

Real libMesh::FE< 0, HERMITE >::shape ( const Elem *  , const Order  , const unsigned int  libmesh_dbg_vari, const Point &    )

inherited

Definition at line 44 of file fe_hermite_shape_0D.C.

{
  libmesh_assert_less (i, 1);
  return 1.;
}

shape() [59/109]

Real libMesh::FE< 3, XYZ >::shape ( const Elem *  elem, const Order  libmesh_dbg_varorder, const unsigned int  i, const Point &  point_in  )

inherited

Definition at line 44 of file fe_xyz_shape_3D.C.

{
#if LIBMESH_DIM == 3
  libmesh_assert(elem);

  Point centroid = elem->centroid();
  Point max_distance = Point(0.,0.,0.);
  for (unsigned int p = 0; p < elem->n_nodes(); p++)
    for (unsigned int d = 0; d < 3; d++)
      {
        const Real distance = std::abs(centroid(d) - elem->point(p)(d));
        max_distance(d) = std::max(distance, max_distance(d));
      }

  const Real x  = point_in(0);
  const Real y  = point_in(1);
  const Real z  = point_in(2);
  const Real xc = centroid(0);
  const Real yc = centroid(1);
  const Real zc = centroid(2);
  const Real distx = max_distance(0);
  const Real disty = max_distance(1);
  const Real distz = max_distance(2);
  const Real dx = (x - xc)/distx;
  const Real dy = (y - yc)/disty;
  const Real dz = (z - zc)/distz;

#ifndef NDEBUG
  // totalorder is only used in the assertion below, so
  // we avoid declaring it when asserts are not active.
  const unsigned int totalorder = order + elem->p_level();
#endif
  libmesh_assert_less (i, (static_cast<unsigned int>(totalorder)+1)*
                       (static_cast<unsigned int>(totalorder)+2)*
                       (static_cast<unsigned int>(totalorder)+3)/6);

  // monomials. since they are hierarchic we only need one case block.
  switch (i)
    {
      // constant
    case 0:
      return 1.;

      // linears
    case 1:
      return dx;

    case 2:
      return dy;

    case 3:
      return dz;

      // quadratics
    case 4:
      return dx*dx;

    case 5:
      return dx*dy;

    case 6:
      return dy*dy;

    case 7:
      return dx*dz;

    case 8:
      return dz*dy;

    case 9:
      return dz*dz;

      // cubics
    case 10:
      return dx*dx*dx;

    case 11:
      return dx*dx*dy;

    case 12:
      return dx*dy*dy;

    case 13:
      return dy*dy*dy;

    case 14:
      return dx*dx*dz;

    case 15:
      return dx*dy*dz;

    case 16:
      return dy*dy*dz;

    case 17:
      return dx*dz*dz;

    case 18:
      return dy*dz*dz;

    case 19:
      return dz*dz*dz;

      // quartics
    case 20:
      return dx*dx*dx*dx;

    case 21:
      return dx*dx*dx*dy;

    case 22:
      return dx*dx*dy*dy;

    case 23:
      return dx*dy*dy*dy;

    case 24:
      return dy*dy*dy*dy;

    case 25:
      return dx*dx*dx*dz;

    case 26:
      return dx*dx*dy*dz;

    case 27:
      return dx*dy*dy*dz;

    case 28:
      return dy*dy*dy*dz;

    case 29:
      return dx*dx*dz*dz;

    case 30:
      return dx*dy*dz*dz;

    case 31:
      return dy*dy*dz*dz;

    case 32:
      return dx*dz*dz*dz;

    case 33:
      return dy*dz*dz*dz;

    case 34:
      return dz*dz*dz*dz;

    default:
      unsigned int o = 0;
      for (; i >= (o+1)*(o+2)*(o+3)/6; o++) { }
      unsigned int i2 = i - (o*(o+1)*(o+2)/6);
      unsigned int block=o, nz = 0;
      for (; block < i2; block += (o-nz+1)) { nz++; }
      const unsigned int nx = block - i2;
      const unsigned int ny = o - nx - nz;
      Real val = 1.;
      for (unsigned int index=0; index != nx; index++)
        val *= dx;
      for (unsigned int index=0; index != ny; index++)
        val *= dy;
      for (unsigned int index=0; index != nz; index++)
        val *= dz;
      return val;
    }

#else
  return 0.;
#endif
}

shape() [60/109]

Real libMesh::FE< 0, LAGRANGE >::shape ( const Elem *  , const Order  , const unsigned int  libmesh_dbg_vari, const Point &    )

inherited

Definition at line 44 of file fe_lagrange_shape_0D.C.

{
  libmesh_assert_less (i, 1);
  return 1.;
}

shape() [61/109]

Real libMesh::FE< 3, CLOUGH >::shape ( const Elem *  libmesh_dbg_varelem, const Order  , const unsigned  int, const Point &    )

inherited

Definition at line 45 of file fe_clough_shape_3D.C.

{
  libmesh_assert(elem);

  libmesh_not_implemented();
  return 0.;
}

shape() [62/109]

Real libMesh::FE< 2, BERNSTEIN >::shape ( const Elem *  elem, const Order  order, const unsigned int  i, const Point &  p  )

inherited

Definition at line 46 of file fe_bernstein_shape_2D.C.

{
  libmesh_assert(elem);

  const ElemType type = elem->type();

  const Order totalorder = static_cast<Order>(order + elem->p_level());

  // Declare that we are using our own special power function
  // from the Utility namespace.  This saves typing later.
  using Utility::pow;

  switch (type)
    {
      // Hierarchic shape functions on the quadrilateral.
    case QUAD4:
    case QUADSHELL4:
    case QUAD9:
      {
        // Compute quad shape functions as a tensor-product
        const Real xi  = p(0);
        const Real eta = p(1);

        libmesh_assert_less (i, (totalorder+1u)*(totalorder+1u));

        // Example i, i0, i1 values for totalorder = 5:
        //                                    0  1  2  3  4  5  6  7  8  9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35
        //  static const unsigned int i0[] = {0, 1, 1, 0, 2, 3, 4, 5, 1, 1, 1, 1, 2, 3, 4, 5, 0, 0, 0, 0, 2, 3, 3, 2, 4, 4, 4, 3, 2, 5, 5, 5, 5, 4, 3, 2};
        //  static const unsigned int i1[] = {0, 0, 1, 1, 0, 0, 0, 0, 2, 3, 4, 5, 1, 1, 1, 1, 2, 3, 4, 5, 2, 2, 3, 3, 2, 3, 4, 4, 4, 2, 3, 4, 5, 5, 5, 5};

        unsigned int i0, i1;

        // Vertex DoFs
        if (i == 0)
          { i0 = 0; i1 = 0; }
        else if (i == 1)
          { i0 = 1; i1 = 0; }
        else if (i == 2)
          { i0 = 1; i1 = 1; }
        else if (i == 3)
          { i0 = 0; i1 = 1; }


        // Edge DoFs
        else if (i < totalorder + 3u)
          { i0 = i - 2; i1 = 0; }
        else if (i < 2u*totalorder + 2)
          { i0 = 1; i1 = i - totalorder - 1; }
        else if (i < 3u*totalorder + 1)
          { i0 = i - 2u*totalorder; i1 = 1; }
        else if (i < 4u*totalorder)
          { i0 = 0; i1 = i - 3u*totalorder + 1; }
        // Interior DoFs. Use Roy's number look up
        else
          {
            unsigned int basisnum = i - 4*totalorder;
            i0 = square_number_column[basisnum] + 2;
            i1 = square_number_row[basisnum] + 2;
          }


        // Flip odd degree of freedom values if necessary
        // to keep continuity on sides.
        if     ((i>= 4                 && i<= 4+  totalorder-2u) && elem->point(0) > elem->point(1)) i0=totalorder+2-i0;//
        else if ((i>= 4+  totalorder-1u && i<= 4+2*totalorder-3u) && elem->point(1) > elem->point(2)) i1=totalorder+2-i1;
        else if ((i>= 4+2*totalorder-2u && i<= 4+3*totalorder-4u) && elem->point(3) > elem->point(2)) i0=totalorder+2-i0;
        else if ((i>= 4+3*totalorder-3u && i<= 4+4*totalorder-5u) && elem->point(0) > elem->point(3)) i1=totalorder+2-i1;


        return (FE<1,BERNSTEIN>::shape(EDGE3, totalorder, i0, xi)*
                FE<1,BERNSTEIN>::shape(EDGE3, totalorder, i1, eta));
      }
      // handle serendipity QUAD8 element separately
    case QUAD8:
    case QUADSHELL8:
      {
        libmesh_assert_less (totalorder, 3);

        const Real xi  = p(0);
        const Real eta = p(1);

        libmesh_assert_less (i, 8);

        //                                0  1  2  3  4  5  6  7  8
        static const unsigned int i0[] = {0, 1, 1, 0, 2, 1, 2, 0, 2};
        static const unsigned int i1[] = {0, 0, 1, 1, 0, 2, 1, 2, 2};
        static const Real scal[] = {-0.25, -0.25, -0.25, -0.25, 0.5, 0.5, 0.5, 0.5};

        //B_t,i0(i)|xi * B_s,i1(i)|eta + scal(i) * B_t,2|xi * B_t,2|eta
        return (FE<1,BERNSTEIN>::shape(EDGE3, totalorder, i0[i], xi)*
                FE<1,BERNSTEIN>::shape(EDGE3, totalorder, i1[i], eta)
                +scal[i]*
                FE<1,BERNSTEIN>::shape(EDGE3, totalorder, i0[8], xi)*
                FE<1,BERNSTEIN>::shape(EDGE3, totalorder, i1[8], eta));

      }

    case TRI3:
    case TRISHELL3:
      libmesh_assert_less (totalorder, 2);
      libmesh_fallthrough();
    case TRI6:
      switch (totalorder)
        {
        case FIRST:
          {
            const Real x=p(0);
            const Real y=p(1);
            const Real r=1.-x-y;

            libmesh_assert_less (i, 3);

            switch(i)
              {
              case 0: return r;  //f0,0,1
              case 1: return x;  //f0,1,1
              case 2: return y;  //f1,0,1

              default:
                libmesh_error_msg("Invalid shape function index i = " << i);
              }
          }
        case SECOND:
          {
            const Real x=p(0);
            const Real y=p(1);
            const Real r=1.-x-y;

            libmesh_assert_less (i, 6);

            switch(i)
              {
              case 0: return r*r;
              case 1: return x*x;
              case 2: return y*y;

              case 3: return 2.*x*r;
              case 4: return 2.*x*y;
              case 5: return 2.*r*y;

              default:
                libmesh_error_msg("Invalid shape function index i = " << i);
              }
          }
        case THIRD:
          {
            const Real x=p(0);
            const Real y=p(1);
            const Real r=1.-x-y;
            libmesh_assert_less (i, 10);

            unsigned int shape=i;


            if ((i==3||i==4) && elem->point(0) > elem->point(1)) shape=7-i;
            if ((i==5||i==6) && elem->point(1) > elem->point(2)) shape=11-i;
            if ((i==7||i==8) && elem->point(0) > elem->point(2)) shape=15-i;

            switch(shape)
              {
              case 0: return r*r*r;
              case 1: return x*x*x;
              case 2: return y*y*y;

              case 3: return 3.*x*r*r;
              case 4: return 3.*x*x*r;

              case 5: return 3.*y*x*x;
              case 6: return 3.*y*y*x;

              case 7: return 3.*y*r*r;
              case 8: return 3.*y*y*r;

              case 9: return 6.*x*y*r;

              default:
                libmesh_error_msg("Invalid shape function index shape = " << shape);
              }
          }
        case FOURTH:
          {
            const Real x=p(0);
            const Real y=p(1);
            const Real r=1-x-y;
            unsigned int shape=i;

            libmesh_assert_less (i, 15);

            if ((i==3||i== 5) && elem->point(0) > elem->point(1)) shape=8-i;
            if ((i==6||i== 8) && elem->point(1) > elem->point(2)) shape=14-i;
            if ((i==9||i==11) && elem->point(0) > elem->point(2)) shape=20-i;


            switch(shape)
              {
                // point functions
              case  0: return r*r*r*r;
              case  1: return x*x*x*x;
              case  2: return y*y*y*y;

                // edge functions
              case  3: return 4.*x*r*r*r;
              case  4: return 6.*x*x*r*r;
              case  5: return 4.*x*x*x*r;

              case  6: return 4.*y*x*x*x;
              case  7: return 6.*y*y*x*x;
              case  8: return 4.*y*y*y*x;

              case  9: return 4.*y*r*r*r;
              case 10: return 6.*y*y*r*r;
              case 11: return 4.*y*y*y*r;

                // inner functions
              case 12: return 12.*x*y*r*r;
              case 13: return 12.*x*x*y*r;
              case 14: return 12.*x*y*y*r;

              default:
                libmesh_error_msg("Invalid shape function index shape = " << shape);
              }
          }
        case FIFTH:
          {
            const Real x=p(0);
            const Real y=p(1);
            const Real r=1-x-y;
            unsigned int shape=i;

            libmesh_assert_less (i, 21);

            if ((i>= 3&&i<= 6) && elem->point(0) > elem->point(1)) shape=9-i;
            if ((i>= 7&&i<=10) && elem->point(1) > elem->point(2)) shape=17-i;
            if ((i>=11&&i<=14) && elem->point(0) > elem->point(2)) shape=25-i;

            switch(shape)
              {
                //point functions
              case  0: return pow<5>(r);
              case  1: return pow<5>(x);
              case  2: return pow<5>(y);

                //edge functions
              case  3: return  5.*x        *pow<4>(r);
              case  4: return 10.*pow<2>(x)*pow<3>(r);
              case  5: return 10.*pow<3>(x)*pow<2>(r);
              case  6: return  5.*pow<4>(x)*r;

              case  7: return  5.*y   *pow<4>(x);
              case  8: return 10.*pow<2>(y)*pow<3>(x);
              case  9: return 10.*pow<3>(y)*pow<2>(x);
              case 10: return  5.*pow<4>(y)*x;

              case 11: return  5.*y   *pow<4>(r);
              case 12: return 10.*pow<2>(y)*pow<3>(r);
              case 13: return 10.*pow<3>(y)*pow<2>(r);
              case 14: return  5.*pow<4>(y)*r;

                //inner functions
              case 15: return 20.*x*y*pow<3>(r);
              case 16: return 30.*x*pow<2>(y)*pow<2>(r);
              case 17: return 30.*pow<2>(x)*y*pow<2>(r);
              case 18: return 20.*x*pow<3>(y)*r;
              case 19: return 20.*pow<3>(x)*y*r;
              case 20: return 30.*pow<2>(x)*pow<2>(y)*r;

              default:
                libmesh_error_msg("Invalid shape function index shape = " << shape);
              }
          }
        case SIXTH:
          {
            const Real x=p(0);
            const Real y=p(1);
            const Real r=1-x-y;
            unsigned int shape=i;

            libmesh_assert_less (i, 28);

            if ((i>= 3&&i<= 7) && elem->point(0) > elem->point(1)) shape=10-i;
            if ((i>= 8&&i<=12) && elem->point(1) > elem->point(2)) shape=20-i;
            if ((i>=13&&i<=17) && elem->point(0) > elem->point(2)) shape=30-i;

            switch(shape)
              {
                //point functions
              case  0: return pow<6>(r);
              case  1: return pow<6>(x);
              case  2: return pow<6>(y);

                //edge functions
              case  3: return  6.*x        *pow<5>(r);
              case  4: return 15.*pow<2>(x)*pow<4>(r);
              case  5: return 20.*pow<3>(x)*pow<3>(r);
              case  6: return 15.*pow<4>(x)*pow<2>(r);
              case  7: return  6.*pow<5>(x)*r;

              case  8: return  6.*y        *pow<5>(x);
              case  9: return 15.*pow<2>(y)*pow<4>(x);
              case 10: return 20.*pow<3>(y)*pow<3>(x);
              case 11: return 15.*pow<4>(y)*pow<2>(x);
              case 12: return  6.*pow<5>(y)*x;

              case 13: return  6.*y        *pow<5>(r);
              case 14: return 15.*pow<2>(y)*pow<4>(r);
              case 15: return 20.*pow<3>(y)*pow<3>(r);
              case 16: return 15.*pow<4>(y)*pow<2>(r);
              case 17: return  6.*pow<5>(y)*r;

                //inner functions
              case 18: return 30.*x*y*pow<4>(r);
              case 19: return 60.*x*pow<2>(y)*pow<3>(r);
              case 20: return 60.*  pow<2>(x)*y*pow<3>(r);
              case 21: return 60.*x*pow<3>(y)*pow<2>(r);
              case 22: return 60.*pow<3>(x)*y*pow<2>(r);
              case 23: return 90.*pow<2>(x)*pow<2>(y)*pow<2>(r);
              case 24: return 30.*x*pow<4>(y)*r;
              case 25: return 60.*pow<2>(x)*pow<3>(y)*r;
              case 26: return 60.*pow<3>(x)*pow<2>(y)*r;
              case 27: return 30.*pow<4>(x)*y*r;

              default:
                libmesh_error_msg("Invalid shape function index shape = " << shape);
              } // switch shape
          } // case TRI6
        default:
          libmesh_error_msg("Invalid totalorder = " << totalorder);
        } // switch order

    default:
      libmesh_error_msg("ERROR: Unsupported element type = " << type);
    } // switch type
}

shape() [63/109]

Real libMesh::FE< 3, BERNSTEIN >::shape ( const Elem *  elem, const Order  order, const unsigned int  i, const Point &  p  )

inherited

Definition at line 48 of file fe_bernstein_shape_3D.C.

{

#if LIBMESH_DIM == 3

  libmesh_assert(elem);
  const ElemType type = elem->type();

  const Order totalorder = static_cast<Order>(order + elem->p_level());

  switch (totalorder)
    {
      // 1st order Bernstein.
    case FIRST:
      {
        switch (type)
          {

            // Bernstein shape functions on the tetrahedron.
          case TET4:
          case TET10:
            {
              libmesh_assert_less (i, 4);

              // Area coordinates
              const Real zeta1 = p(0);
              const Real zeta2 = p(1);
              const Real zeta3 = p(2);
              const Real zeta0 = 1. - zeta1 - zeta2 - zeta3;

              switch(i)
                {
                case  0:  return zeta0;
                case  1:  return zeta1;
                case  2:  return zeta2;
                case  3:  return zeta3;

                default:
                  libmesh_error_msg("Invalid shape function index i = " << i);
                }
            }

            // Bernstein shape functions on the hexahedral.
          case HEX8:
          case HEX20:
          case HEX27:
            {
              libmesh_assert_less (i, 8);

              // Compute hex shape functions as a tensor-product
              const Real xi   = p(0);
              const Real eta  = p(1);
              const Real zeta = p(2);

              // The only way to make any sense of this
              // is to look at the mgflo/mg2/mgf documentation
              // and make the cut-out cube!
              //                                0  1  2  3  4  5  6  7
              static const unsigned int i0[] = {0, 1, 1, 0, 0, 1, 1, 0};
              static const unsigned int i1[] = {0, 0, 1, 1, 0, 0, 1, 1};
              static const unsigned int i2[] = {0, 0, 0, 0, 1, 1, 1, 1};

              return (FE<1,BERNSTEIN>::shape(EDGE3, totalorder, i0[i], xi)*
                      FE<1,BERNSTEIN>::shape(EDGE3, totalorder, i1[i], eta)*
                      FE<1,BERNSTEIN>::shape(EDGE3, totalorder, i2[i], zeta));
            }


          default:
            libmesh_error_msg("Invalid element type = " << type);
          }
      }




    case SECOND:
      {
        switch (type)
          {

            // Bernstein shape functions on the tetrahedron.
          case TET10:
            {
              libmesh_assert_less (i, 10);

              // Area coordinates
              const Real zeta1 = p(0);
              const Real zeta2 = p(1);
              const Real zeta3 = p(2);
              const Real zeta0 = 1. - zeta1 - zeta2 - zeta3;

              switch(i)
                {
                case  0:  return zeta0*zeta0;
                case  1:  return zeta1*zeta1;
                case  2:  return zeta2*zeta2;
                case  3:  return zeta3*zeta3;
                case  4:  return 2.*zeta0*zeta1;
                case  5:  return 2.*zeta1*zeta2;
                case  6:  return 2.*zeta0*zeta2;
                case  7:  return 2.*zeta3*zeta0;
                case  8:  return 2.*zeta1*zeta3;
                case  9:  return 2.*zeta2*zeta3;

                default:
                  libmesh_error_msg("Invalid shape function index i = " << i);
                }
            }

            // Bernstein shape functions on the 20-noded hexahedral.
          case HEX20:
            {
              libmesh_assert_less (i, 20);

              // Compute hex shape functions as a tensor-product
              const Real xi   = p(0);
              const Real eta  = p(1);
              const Real zeta = p(2);

              // The only way to make any sense of this
              // is to look at the mgflo/mg2/mgf documentation
              // and make the cut-out cube!
              //                                0      1      2      3      4      5      6      7      8      9      10     11     12     13     14     15     16     17     18     19  20 21 22 23 24 25 26
              static const unsigned int i0[] = {0,     1,     1,     0,     0,     1,     1,     0,     2,     1,     2,     0,     0,     1,     1,     0,     2,     1,     2,     0,  2, 2, 1, 2, 0, 2, 2};
              static const unsigned int i1[] = {0,     0,     1,     1,     0,     0,     1,     1,     0,     2,     1,     2,     0,     0,     1,     1,     0,     2,     1,     2,  2, 0, 2, 1, 2, 2, 2};
              static const unsigned int i2[] = {0,     0,     0,     0,     1,     1,     1,     1,     0,     0,     0,     0,     2,     2,     2,     2,     1,     1,     1,     1,  0, 2, 2, 2, 2, 1, 2};
              //To compute the hex20 shape functions the original shape functions for hex27 are used.
              //scalx[i] tells how often the original x-th shape function has to be added to the original i-th shape function
              //to compute the new i-th shape function for hex20
              //example: B_0^HEX20 = B_0^HEX27 - 0.25*B_20^HEX27 - 0.25*B_21^HEX27 + 0*B_22^HEX27 + 0*B_23^HEX27 - 0.25*B_24^HEX27 + 0*B_25^HEX27 - 0.25*B_26^HEX27
              //         B_0^HEX20 = B_0^HEX27 + scal20[0]*B_20^HEX27 + scal21[0]*B_21^HEX27 + ...
              static const Real scal20[] =     {-0.25, -0.25, -0.25, -0.25, 0,     0,     0,     0,     0.5,   0.5,   0.5,   0.5,   0,     0,     0,     0,     0,     0,     0,     0};
              static const Real scal21[] =     {-0.25, -0.25, 0,     0,     -0.25, -0.25, 0,     0,     0.5,   0,     0,     0,     0.5,   0.5,   0,     0,     0.5,   0,     0,     0};
              static const Real scal22[] =     {0,     -0.25, -0.25, 0,     0,     -0.25, -0.25, 0,     0,     0.5,   0,     0,     0,     0.5,   0.5,   0,     0,     0.5,   0,     0};
              static const Real scal23[] =     {0,     0,     -0.25, -0.25, 0,     0,     -0.25, -0.25, 0,     0,     0.5,   0,     0,     0,     0.5,   0.5,   0,     0,     0.5,   0};
              static const Real scal24[] =     {-0.25, 0,     0,     -0.25, -0.25, 0,     0,     -0.25, 0,     0,     0,     0.5,   0.5,   0,     0,     0.5,   0,     0,     0,     0.5};
              static const Real scal25[] =     {0,     0,     0,     0,     -0.25, -0.25, -0.25, -0.25, 0,     0,     0,     0,     0,     0,     0,     0,     0.5,   0.5,   0.5,   0.5};
              static const Real scal26[] =     {-0.25, -0.25, -0.25, -0.25, -0.25, -0.25, -0.25, -0.25, 0.25,  0.25,  0.25,  0.25,  0.25,  0.25,  0.25,  0.25,  0.25,  0.25,  0.25,  0.25};

              return (FE<1,BERNSTEIN>::shape(EDGE3, totalorder, i0[i], xi)*
                      FE<1,BERNSTEIN>::shape(EDGE3, totalorder, i1[i], eta)*
                      FE<1,BERNSTEIN>::shape(EDGE3, totalorder, i2[i], zeta)
                      +scal20[i]*
                      FE<1,BERNSTEIN>::shape(EDGE3, totalorder, i0[20], xi)*
                      FE<1,BERNSTEIN>::shape(EDGE3, totalorder, i1[20], eta)*
                      FE<1,BERNSTEIN>::shape(EDGE3, totalorder, i2[20], zeta)
                      +scal21[i]*
                      FE<1,BERNSTEIN>::shape(EDGE3, totalorder, i0[21], xi)*
                      FE<1,BERNSTEIN>::shape(EDGE3, totalorder, i1[21], eta)*
                      FE<1,BERNSTEIN>::shape(EDGE3, totalorder, i2[21], zeta)
                      +scal22[i]*
                      FE<1,BERNSTEIN>::shape(EDGE3, totalorder, i0[22], xi)*
                      FE<1,BERNSTEIN>::shape(EDGE3, totalorder, i1[22], eta)*
                      FE<1,BERNSTEIN>::shape(EDGE3, totalorder, i2[22], zeta)
                      +scal23[i]*
                      FE<1,BERNSTEIN>::shape(EDGE3, totalorder, i0[23], xi)*
                      FE<1,BERNSTEIN>::shape(EDGE3, totalorder, i1[23], eta)*
                      FE<1,BERNSTEIN>::shape(EDGE3, totalorder, i2[23], zeta)
                      +scal24[i]*
                      FE<1,BERNSTEIN>::shape(EDGE3, totalorder, i0[24], xi)*
                      FE<1,BERNSTEIN>::shape(EDGE3, totalorder, i1[24], eta)*
                      FE<1,BERNSTEIN>::shape(EDGE3, totalorder, i2[24], zeta)
                      +scal25[i]*
                      FE<1,BERNSTEIN>::shape(EDGE3, totalorder, i0[25], xi)*
                      FE<1,BERNSTEIN>::shape(EDGE3, totalorder, i1[25], eta)*
                      FE<1,BERNSTEIN>::shape(EDGE3, totalorder, i2[25], zeta)
                      +scal26[i]*
                      FE<1,BERNSTEIN>::shape(EDGE3, totalorder, i0[26], xi)*
                      FE<1,BERNSTEIN>::shape(EDGE3, totalorder, i1[26], eta)*
                      FE<1,BERNSTEIN>::shape(EDGE3, totalorder, i2[26], zeta));
            }

            // Bernstein shape functions on the hexahedral.
          case HEX27:
            {
              libmesh_assert_less (i, 27);

              // Compute hex shape functions as a tensor-product
              const Real xi   = p(0);
              const Real eta  = p(1);
              const Real zeta = p(2);

              // The only way to make any sense of this
              // is to look at the mgflo/mg2/mgf documentation
              // and make the cut-out cube!
              //                                0  1  2  3  4  5  6  7  8  9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26
              static const unsigned int i0[] = {0, 1, 1, 0, 0, 1, 1, 0, 2, 1, 2, 0, 0, 1, 1, 0, 2, 1, 2, 0, 2, 2, 1, 2, 0, 2, 2};
              static const unsigned int i1[] = {0, 0, 1, 1, 0, 0, 1, 1, 0, 2, 1, 2, 0, 0, 1, 1, 0, 2, 1, 2, 2, 0, 2, 1, 2, 2, 2};
              static const unsigned int i2[] = {0, 0, 0, 0, 1, 1, 1, 1, 0, 0, 0, 0, 2, 2, 2, 2, 1, 1, 1, 1, 0, 2, 2, 2, 2, 1, 2};

              return (FE<1,BERNSTEIN>::shape(EDGE3, totalorder, i0[i], xi)*
                      FE<1,BERNSTEIN>::shape(EDGE3, totalorder, i1[i], eta)*
                      FE<1,BERNSTEIN>::shape(EDGE3, totalorder, i2[i], zeta));
            }


          default:
            libmesh_error_msg("Invalid element type = " << type);
          }

      }



      // 3rd-order Bernstein.
    case THIRD:
      {
        switch (type)
          {

            //     // Bernstein shape functions on the tetrahedron.
            //   case TET10:
            //     {
            //       libmesh_assert_less (i, 20);

            //       // Area coordinates
            //       const Real zeta1 = p(0);
            //       const Real zeta2 = p(1);
            //       const Real zeta3 = p(2);
            //       const Real zeta0 = 1. - zeta1 - zeta2 - zeta3;


            //       unsigned int shape=i;

            //       // handle the edge orientation

            //       if ((i== 4||i== 5) && elem->node_id(0) > elem->node_id(1))shape= 9-i;   //Edge 0
            //       if ((i== 6||i== 7) && elem->node_id(1) > elem->node_id(2))shape=13-i;   //Edge 1
            //       if ((i== 8||i== 9) && elem->node_id(0) > elem->node_id(2))shape=17-i;   //Edge 2
            //       if ((i==10||i==11) && elem->node_id(0) > elem->node_id(3))shape=21-i;   //Edge 3
            //       if ((i==12||i==13) && elem->node_id(1) > elem->node_id(3))shape=25-i;   //Edge 4
            //       if ((i==14||i==15) && elem->node_id(2) > elem->node_id(3))shape=29-i;   //Edge 5

            //       // No need to handle face orientation in 3rd order.


            //       switch(shape)
            // {
            //   //point function
            // case  0:  return zeta0*zeta0*zeta0;
            // case  1:  return zeta1*zeta1*zeta1;
            // case  2:  return zeta2*zeta2*zeta2;
            // case  3:  return zeta3*zeta3*zeta3;

            //   //edge functions
            // case  4:  return 3.*zeta0*zeta0*zeta1;
            // case  5:  return 3.*zeta1*zeta1*zeta0;

            // case  6:  return 3.*zeta1*zeta1*zeta2;
            // case  7:  return 3.*zeta2*zeta2*zeta1;

            // case  8:  return 3.*zeta0*zeta0*zeta2;
            // case  9:  return 3.*zeta2*zeta2*zeta0;

            // case 10:  return 3.*zeta0*zeta0*zeta3;
            // case 11:  return 3.*zeta3*zeta3*zeta0;

            // case 12:  return 3.*zeta1*zeta1*zeta3;
            // case 13:  return 3.*zeta3*zeta3*zeta1;

            // case 14:  return 3.*zeta2*zeta2*zeta3;
            // case 15:  return 3.*zeta3*zeta3*zeta2;

            //   //face functions
            // case 16:  return 6.*zeta0*zeta1*zeta2;
            // case 17:  return 6.*zeta0*zeta1*zeta3;
            // case 18:  return 6.*zeta1*zeta2*zeta3;
            // case 19:  return 6.*zeta2*zeta0*zeta3;

            // default:
            // libmesh_error_msg("Invalid shape function index i = " << i);
            // }
            //     }


            // Bernstein shape functions on the hexahedral.
          case HEX27:
            {
              libmesh_assert_less (i, 64);

              // Compute hex shape functions as a tensor-product
              const Real xi    = p(0);
              const Real eta   = p(1);
              const Real zeta  = p(2);
              Real xi_mapped   = p(0);
              Real eta_mapped  = p(1);
              Real zeta_mapped = p(2);

              // The only way to make any sense of this
              // is to look at the mgflo/mg2/mgf documentation
              // and make the cut-out cube!
              //  Nodes                         0  1  2  3  4  5  6  7  8  8  9  9 10 10 11 11 12 12 13 13 14 14 15 15 16 16 17 17 18 18 19 19 20 20 20 20 21 21 21 21 22 22 22 22 23 23 23 23 24 24 24 24 25 25 25 25 26 26 26 26 26 26 26 26
              //  DOFS                          0  1  2  3  4  5  6  7  8  9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 18 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 60 62 63
              static const unsigned int i0[] = {0, 1, 1, 0, 0, 1, 1, 0, 2, 3, 1, 1, 2, 3, 0, 0, 0, 0, 1, 1, 1, 1, 0, 0, 2, 3, 1, 1, 2, 3, 0, 0, 2, 3, 2, 3, 2, 3, 2, 3, 1, 1, 1, 1, 2, 3, 2, 3, 0, 0, 0, 0, 2, 3, 2, 3, 2, 3, 2, 3, 2, 3, 2, 3};
              static const unsigned int i1[] = {0, 0, 1, 1, 0, 0, 1, 1, 0, 0, 2, 3, 1, 1, 2, 3, 0, 0, 0, 0, 1, 1, 1, 1, 0, 0, 2, 3, 1, 1, 2, 3, 2, 2, 3, 3, 0, 0, 0, 0, 2, 3, 2, 3, 1, 1, 1, 1, 2, 3, 2, 3, 2, 2, 3, 3, 2, 2, 3, 3, 2, 2, 3, 3};
              static const unsigned int i2[] = {0, 0, 0, 0, 1, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 2, 3, 2, 3, 2, 3, 2, 3, 1, 1, 1, 1, 1, 1, 1, 1, 0, 0, 0, 0, 2, 2, 3, 3, 2, 2, 3, 3, 2, 2, 3, 3, 2, 2, 3, 3, 1, 1, 1, 1, 2, 2, 2, 2, 3, 3, 3, 3};



              // handle the edge orientation
              {
                // Edge 0
                if ((i0[i] >= 2) && (i1[i] == 0) && (i2[i] == 0))
                  {
                    if (elem->point(0) != std::min(elem->point(0), elem->point(1)))
                      xi_mapped = -xi;
                  }
                // Edge 1
                else if ((i0[i] == 1) && (i1[i] >= 2) && (i2[i] == 0))
                  {
                    if (elem->point(1) != std::min(elem->point(1), elem->point(2)))
                      eta_mapped = -eta;
                  }
                // Edge 2
                else if ((i0[i] >= 2) && (i1[i] == 1) && (i2[i] == 0))
                  {
                    if (elem->point(3) != std::min(elem->point(3), elem->point(2)))
                      xi_mapped = -xi;
                  }
                // Edge 3
                else if ((i0[i] == 0) && (i1[i] >= 2) && (i2[i] == 0))
                  {
                    if (elem->point(0) != std::min(elem->point(0), elem->point(3)))
                      eta_mapped = -eta;
                  }
                // Edge 4
                else if ((i0[i] == 0) && (i1[i] == 0) && (i2[i] >=2 ))
                  {
                    if (elem->point(0) != std::min(elem->point(0), elem->point(4)))
                      zeta_mapped = -zeta;
                  }
                // Edge 5
                else if ((i0[i] == 1) && (i1[i] == 0) && (i2[i] >=2 ))
                  {
                    if (elem->point(1) != std::min(elem->point(1), elem->point(5)))
                      zeta_mapped = -zeta;
                  }
                // Edge 6
                else if ((i0[i] == 1) && (i1[i] == 1) && (i2[i] >=2 ))
                  {
                    if (elem->point(2) != std::min(elem->point(2), elem->point(6)))
                      zeta_mapped = -zeta;
                  }
                // Edge 7
                else if ((i0[i] == 0) && (i1[i] == 1) && (i2[i] >=2 ))
                  {
                    if (elem->point(3) != std::min(elem->point(3), elem->point(7)))
                      zeta_mapped = -zeta;
                  }
                // Edge 8
                else if ((i0[i] >=2 ) && (i1[i] == 0) && (i2[i] == 1))
                  {
                    if (elem->point(4) != std::min(elem->point(4), elem->point(5)))
                      xi_mapped = -xi;
                  }
                // Edge 9
                else if ((i0[i] == 1) && (i1[i] >=2 ) && (i2[i] == 1))
                  {
                    if (elem->point(5) != std::min(elem->point(5), elem->point(6)))
                      eta_mapped = -eta;
                  }
                // Edge 10
                else if ((i0[i] >=2 ) && (i1[i] == 1) && (i2[i] == 1))
                  {
                    if (elem->point(7) != std::min(elem->point(7), elem->point(6)))
                      xi_mapped = -xi;
                  }
                // Edge 11
                else if ((i0[i] == 0) && (i1[i] >=2 ) && (i2[i] == 1))
                  {
                    if (elem->point(4) != std::min(elem->point(4), elem->point(7)))
                      eta_mapped = -eta;
                  }
              }


              // handle the face orientation
              {
                // Face 0
                if ((i2[i] == 0) && (i0[i] >= 2) && (i1[i] >= 2))
                  {
                    const Point min_point = std::min(elem->point(1),
                                                     std::min(elem->point(2),
                                                              std::min(elem->point(0),
                                                                       elem->point(3))));
                    if (elem->point(0) == min_point)
                      if (elem->point(1) == std::min(elem->point(1), elem->point(3)))
                        {
                          // Case 1
                          xi_mapped  = xi;
                          eta_mapped = eta;
                        }
                      else
                        {
                          // Case 2
                          xi_mapped  = eta;
                          eta_mapped = xi;
                        }

                    else if (elem->point(3) == min_point)
                      if (elem->point(0) == std::min(elem->point(0), elem->point(2)))
                        {
                          // Case 3
                          xi_mapped  = -eta;
                          eta_mapped = xi;
                        }
                      else
                        {
                          // Case 4
                          xi_mapped  = xi;
                          eta_mapped = -eta;
                        }

                    else if (elem->point(2) == min_point)
                      if (elem->point(3) == std::min(elem->point(3), elem->point(1)))
                        {
                          // Case 5
                          xi_mapped  = -xi;
                          eta_mapped = -eta;
                        }
                      else
                        {
                          // Case 6
                          xi_mapped  = -eta;
                          eta_mapped = -xi;
                        }

                    else if (elem->point(1) == min_point)
                      {
                        if (elem->point(2) == std::min(elem->point(2), elem->point(0)))
                          {
                            // Case 7
                            xi_mapped  = eta;
                            eta_mapped = -xi;
                          }
                        else
                          {
                            // Case 8
                            xi_mapped  = -xi;
                            eta_mapped = eta;
                          }
                      }
                  }


                // Face 1
                else if ((i1[i] == 0) && (i0[i] >= 2) && (i2[i] >= 2))
                  {
                    const Point min_point = std::min(elem->point(0),
                                                     std::min(elem->point(1),
                                                              std::min(elem->point(5),
                                                                       elem->point(4))));
                    if (elem->point(0) == min_point)
                      if (elem->point(1) == std::min(elem->point(1), elem->point(4)))
                        {
                          // Case 1
                          xi_mapped   = xi;
                          zeta_mapped = zeta;
                        }
                      else
                        {
                          // Case 2
                          xi_mapped   = zeta;
                          zeta_mapped = xi;
                        }

                    else if (elem->point(1) == min_point)
                      if (elem->point(5) == std::min(elem->point(5), elem->point(0)))
                        {
                          // Case 3
                          xi_mapped   = zeta;
                          zeta_mapped = -xi;
                        }
                      else
                        {
                          // Case 4
                          xi_mapped   = -xi;
                          zeta_mapped = zeta;
                        }

                    else if (elem->point(5) == min_point)
                      if (elem->point(4) == std::min(elem->point(4), elem->point(1)))
                        {
                          // Case 5
                          xi_mapped   = -xi;
                          zeta_mapped = -zeta;
                        }
                      else
                        {
                          // Case 6
                          xi_mapped   = -zeta;
                          zeta_mapped = -xi;
                        }

                    else if (elem->point(4) == min_point)
                      {
                        if (elem->point(0) == std::min(elem->point(0), elem->point(5)))
                          {
                            // Case 7
                            xi_mapped   = -xi;
                            zeta_mapped = zeta;
                          }
                        else
                          {
                            // Case 8
                            xi_mapped   = xi;
                            zeta_mapped = -zeta;
                          }
                      }
                  }


                // Face 2
                else if ((i0[i] == 1) && (i1[i] >= 2) && (i2[i] >= 2))
                  {
                    const Point min_point = std::min(elem->point(1),
                                                     std::min(elem->point(2),
                                                              std::min(elem->point(6),
                                                                       elem->point(5))));
                    if (elem->point(1) == min_point)
                      if (elem->point(2) == std::min(elem->point(2), elem->point(5)))
                        {
                          // Case 1
                          eta_mapped  = eta;
                          zeta_mapped = zeta;
                        }
                      else
                        {
                          // Case 2
                          eta_mapped  = zeta;
                          zeta_mapped = eta;
                        }

                    else if (elem->point(2) == min_point)
                      if (elem->point(6) == std::min(elem->point(6), elem->point(1)))
                        {
                          // Case 3
                          eta_mapped  = zeta;
                          zeta_mapped = -eta;
                        }
                      else
                        {
                          // Case 4
                          eta_mapped  = -eta;
                          zeta_mapped = zeta;
                        }

                    else if (elem->point(6) == min_point)
                      if (elem->point(5) == std::min(elem->point(5), elem->point(2)))
                        {
                          // Case 5
                          eta_mapped  = -eta;
                          zeta_mapped = -zeta;
                        }
                      else
                        {
                          // Case 6
                          eta_mapped  = -zeta;
                          zeta_mapped = -eta;
                        }

                    else if (elem->point(5) == min_point)
                      {
                        if (elem->point(1) == std::min(elem->point(1), elem->point(6)))
                          {
                            // Case 7
                            eta_mapped  = -zeta;
                            zeta_mapped = eta;
                          }
                        else
                          {
                            // Case 8
                            eta_mapped   = eta;
                            zeta_mapped = -zeta;
                          }
                      }
                  }


                // Face 3
                else if ((i1[i] == 1) && (i0[i] >= 2) && (i2[i] >= 2))
                  {
                    const Point min_point = std::min(elem->point(2),
                                                     std::min(elem->point(3),
                                                              std::min(elem->point(7),
                                                                       elem->point(6))));
                    if (elem->point(3) == min_point)
                      if (elem->point(2) == std::min(elem->point(2), elem->point(7)))
                        {
                          // Case 1
                          xi_mapped   = xi;
                          zeta_mapped = zeta;
                        }
                      else
                        {
                          // Case 2
                          xi_mapped   = zeta;
                          zeta_mapped = xi;
                        }

                    else if (elem->point(7) == min_point)
                      if (elem->point(3) == std::min(elem->point(3), elem->point(6)))
                        {
                          // Case 3
                          xi_mapped   = -zeta;
                          zeta_mapped = xi;
                        }
                      else
                        {
                          // Case 4
                          xi_mapped   = xi;
                          zeta_mapped = -zeta;
                        }

                    else if (elem->point(6) == min_point)
                      if (elem->point(7) == std::min(elem->point(7), elem->point(2)))
                        {
                          // Case 5
                          xi_mapped   = -xi;
                          zeta_mapped = -zeta;
                        }
                      else
                        {
                          // Case 6
                          xi_mapped   = -zeta;
                          zeta_mapped = -xi;
                        }

                    else if (elem->point(2) == min_point)
                      {
                        if (elem->point(6) == std::min(elem->point(3), elem->point(6)))
                          {
                            // Case 7
                            xi_mapped   = zeta;
                            zeta_mapped = -xi;
                          }
                        else
                          {
                            // Case 8
                            xi_mapped   = -xi;
                            zeta_mapped = zeta;
                          }
                      }
                  }


                // Face 4
                else if ((i0[i] == 0) && (i1[i] >= 2) && (i2[i] >= 2))
                  {
                    const Point min_point = std::min(elem->point(3),
                                                     std::min(elem->point(0),
                                                              std::min(elem->point(4),
                                                                       elem->point(7))));
                    if (elem->point(0) == min_point)
                      if (elem->point(3) == std::min(elem->point(3), elem->point(4)))
                        {
                          // Case 1
                          eta_mapped  = eta;
                          zeta_mapped = zeta;
                        }
                      else
                        {
                          // Case 2
                          eta_mapped  = zeta;
                          zeta_mapped = eta;
                        }

                    else if (elem->point(4) == min_point)
                      if (elem->point(0) == std::min(elem->point(0), elem->point(7)))
                        {
                          // Case 3
                          eta_mapped  = -zeta;
                          zeta_mapped = eta;
                        }
                      else
                        {
                          // Case 4
                          eta_mapped  = eta;
                          zeta_mapped = -zeta;
                        }

                    else if (elem->point(7) == min_point)
                      if (elem->point(4) == std::min(elem->point(4), elem->point(3)))
                        {
                          // Case 5
                          eta_mapped  = -eta;
                          zeta_mapped = -zeta;
                        }
                      else
                        {
                          // Case 6
                          eta_mapped  = -zeta;
                          zeta_mapped = -eta;
                        }

                    else if (elem->point(3) == min_point)
                      {
                        if (elem->point(7) == std::min(elem->point(7), elem->point(0)))
                          {
                            // Case 7
                            eta_mapped   = zeta;
                            zeta_mapped = -eta;
                          }
                        else
                          {
                            // Case 8
                            eta_mapped  = -eta;
                            zeta_mapped = zeta;
                          }
                      }
                  }


                // Face 5
                else if ((i2[i] == 1) && (i0[i] >= 2) && (i1[i] >= 2))
                  {
                    const Point min_point = std::min(elem->point(4),
                                                     std::min(elem->point(5),
                                                              std::min(elem->point(6),
                                                                       elem->point(7))));
                    if (elem->point(4) == min_point)
                      if (elem->point(5) == std::min(elem->point(5), elem->point(7)))
                        {
                          // Case 1
                          xi_mapped  = xi;
                          eta_mapped = eta;
                        }
                      else
                        {
                          // Case 2
                          xi_mapped  = eta;
                          eta_mapped = xi;
                        }

                    else if (elem->point(5) == min_point)
                      if (elem->point(6) == std::min(elem->point(6), elem->point(4)))
                        {
                          // Case 3
                          xi_mapped  = eta;
                          eta_mapped = -xi;
                        }
                      else
                        {
                          // Case 4
                          xi_mapped  = -xi;
                          eta_mapped = eta;
                        }

                    else if (elem->point(6) == min_point)
                      if (elem->point(7) == std::min(elem->point(7), elem->point(5)))
                        {
                          // Case 5
                          xi_mapped  = -xi;
                          eta_mapped = -eta;
                        }
                      else
                        {
                          // Case 6
                          xi_mapped  = -eta;
                          eta_mapped = -xi;
                        }

                    else if (elem->point(7) == min_point)
                      {
                        if (elem->point(4) == std::min(elem->point(4), elem->point(6)))
                          {
                            // Case 7
                            xi_mapped  = -eta;
                            eta_mapped = xi;
                          }
                        else
                          {
                            // Case 8
                            xi_mapped  = xi;
                            eta_mapped = eta;
                          }
                      }
                  }
              }

              return (FE<1,BERNSTEIN>::shape(EDGE3, totalorder, i0[i], xi_mapped)*
                      FE<1,BERNSTEIN>::shape(EDGE3, totalorder, i1[i], eta_mapped)*
                      FE<1,BERNSTEIN>::shape(EDGE3, totalorder, i2[i], zeta_mapped));
            }


          default:
            libmesh_error_msg("Invalid element type = " << type);
          } //case HEX27

      }//case THIRD


      // 4th-order Bernstein.
    case FOURTH:
      {
        switch (type)
          {

            // Bernstein shape functions on the hexahedral.
          case HEX27:
            {
              libmesh_assert_less (i, 125);

              // Compute hex shape functions as a tensor-product
              const Real xi    = p(0);
              const Real eta   = p(1);
              const Real zeta  = p(2);
              Real xi_mapped   = p(0);
              Real eta_mapped  = p(1);
              Real zeta_mapped = p(2);

              // The only way to make any sense of this
              // is to look at the mgflo/mg2/mgf documentation
              // and make the cut-out cube!
              //  Nodes                         0  1  2  3  4  5  6  7  8  8  9  9 10 10 11 11 12 12 13 13 14 14 15 15 16 16 17 17 18 18 19 19 20 20 20 20 21 21 21 21 22 22 22 22 23 23 23 23 24 24 24 24 25 25 25 25 26 26 26 26 26 26 26 26
              //  DOFS                          0  1  2  3  4  5  6  7  8  9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 18 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 60 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99  0  1  2  3  4  5  6  7  8  9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24
              static const unsigned int i0[] = {0, 1, 1, 0, 0, 1, 1, 0, 2, 3, 4, 1, 1, 1, 2, 3, 4, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 0, 0, 0, 2, 3, 4, 1, 1, 1, 2, 3, 4, 0, 0, 0, 2, 3, 4, 2, 3, 4, 2, 3, 4, 2, 3, 4, 2, 3, 4, 2, 3, 4, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 3, 4, 2, 3, 4, 2, 3, 4, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 3, 4, 2, 3, 4, 2, 3, 4, 2, 3, 4, 2, 3, 4, 2, 3, 4, 2, 3, 4, 2, 3, 4, 2, 3, 4, 2, 3, 4, 2, 3, 4, 2, 3, 4};
              static const unsigned int i1[] = {0, 0, 1, 1, 0, 0, 1, 1, 0, 0, 0, 2, 3, 4, 1, 1, 1, 2, 3, 4, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 0, 0, 0, 2, 3, 4, 1, 1, 1, 2, 3, 4, 2, 2, 2, 3, 3, 3, 4, 4, 4, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 3, 4, 2, 3, 4, 2, 3, 4, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 3, 4, 2, 3, 4, 2, 3, 4, 2, 2, 2, 3, 3, 3, 4, 4, 4, 2, 2, 2, 3, 3, 3, 4, 4, 4, 2, 2, 2, 3, 3, 3, 4, 4, 4, 2, 2, 2, 3, 3, 3, 4, 4, 4};
              static const unsigned int i2[] = {0, 0, 0, 0, 1, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 3, 4, 2, 3, 4, 2, 3, 4, 2, 3, 4, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 2, 2, 3, 3, 3, 4, 4, 4, 2, 2, 2, 3, 3, 3, 4, 4, 4, 2, 2, 2, 3, 3, 3, 4, 4, 4, 2, 2, 2, 3, 3, 3, 4, 4, 4, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 3, 3, 3, 3, 3, 3, 3, 3, 3, 4, 4, 4, 4, 4, 4, 4, 4, 4};



              // handle the edge orientation
              {
                // Edge 0
                if ((i0[i] >= 2) && (i1[i] == 0) && (i2[i] == 0))
                  {
                    if (elem->point(0) != std::min(elem->point(0), elem->point(1)))
                      xi_mapped = -xi;
                  }
                // Edge 1
                else if ((i0[i] == 1) && (i1[i] >= 2) && (i2[i] == 0))
                  {
                    if (elem->point(1) != std::min(elem->point(1), elem->point(2)))
                      eta_mapped = -eta;
                  }
                // Edge 2
                else if ((i0[i] >= 2) && (i1[i] == 1) && (i2[i] == 0))
                  {
                    if (elem->point(3) != std::min(elem->point(3), elem->point(2)))
                      xi_mapped = -xi;
                  }
                // Edge 3
                else if ((i0[i] == 0) && (i1[i] >= 2) && (i2[i] == 0))
                  {
                    if (elem->point(0) != std::min(elem->point(0), elem->point(3)))
                      eta_mapped = -eta;
                  }
                // Edge 4
                else if ((i0[i] == 0) && (i1[i] == 0) && (i2[i] >=2 ))
                  {
                    if (elem->point(0) != std::min(elem->point(0), elem->point(4)))
                      zeta_mapped = -zeta;
                  }
                // Edge 5
                else if ((i0[i] == 1) && (i1[i] == 0) && (i2[i] >=2 ))
                  {
                    if (elem->point(1) != std::min(elem->point(1), elem->point(5)))
                      zeta_mapped = -zeta;
                  }
                // Edge 6
                else if ((i0[i] == 1) && (i1[i] == 1) && (i2[i] >=2 ))
                  {
                    if (elem->point(2) != std::min(elem->point(2), elem->point(6)))
                      zeta_mapped = -zeta;
                  }
                // Edge 7
                else if ((i0[i] == 0) && (i1[i] == 1) && (i2[i] >=2 ))
                  {
                    if (elem->point(3) != std::min(elem->point(3), elem->point(7)))
                      zeta_mapped = -zeta;
                  }
                // Edge 8
                else if ((i0[i] >=2 ) && (i1[i] == 0) && (i2[i] == 1))
                  {
                    if (elem->point(4) != std::min(elem->point(4), elem->point(5)))
                      xi_mapped = -xi;
                  }
                // Edge 9
                else if ((i0[i] == 1) && (i1[i] >=2 ) && (i2[i] == 1))
                  {
                    if (elem->point(5) != std::min(elem->point(5), elem->point(6)))
                      eta_mapped = -eta;
                  }
                // Edge 10
                else if ((i0[i] >=2 ) && (i1[i] == 1) && (i2[i] == 1))
                  {
                    if (elem->point(7) != std::min(elem->point(7), elem->point(6)))
                      xi_mapped = -xi;
                  }
                // Edge 11
                else if ((i0[i] == 0) && (i1[i] >=2 ) && (i2[i] == 1))
                  {
                    if (elem->point(4) != std::min(elem->point(4), elem->point(7)))
                      eta_mapped = -eta;
                  }
              }


              // handle the face orientation
              {
                // Face 0
                if ((i2[i] == 0) && (i0[i] >= 2) && (i1[i] >= 2))
                  {
                    const Point min_point = std::min(elem->point(1),
                                                     std::min(elem->point(2),
                                                              std::min(elem->point(0),
                                                                       elem->point(3))));
                    if (elem->point(0) == min_point)
                      if (elem->point(1) == std::min(elem->point(1), elem->point(3)))
                        {
                          // Case 1
                          xi_mapped  = xi;
                          eta_mapped = eta;
                        }
                      else
                        {
                          // Case 2
                          xi_mapped  = eta;
                          eta_mapped = xi;
                        }

                    else if (elem->point(3) == min_point)
                      if (elem->point(0) == std::min(elem->point(0), elem->point(2)))
                        {
                          // Case 3
                          xi_mapped  = -eta;
                          eta_mapped = xi;
                        }
                      else
                        {
                          // Case 4
                          xi_mapped  = xi;
                          eta_mapped = -eta;
                        }

                    else if (elem->point(2) == min_point)
                      if (elem->point(3) == std::min(elem->point(3), elem->point(1)))
                        {
                          // Case 5
                          xi_mapped  = -xi;
                          eta_mapped = -eta;
                        }
                      else
                        {
                          // Case 6
                          xi_mapped  = -eta;
                          eta_mapped = -xi;
                        }

                    else if (elem->point(1) == min_point)
                      {
                        if (elem->point(2) == std::min(elem->point(2), elem->point(0)))
                          {
                            // Case 7
                            xi_mapped  = eta;
                            eta_mapped = -xi;
                          }
                        else
                          {
                            // Case 8
                            xi_mapped  = -xi;
                            eta_mapped = eta;
                          }
                      }
                  }


                // Face 1
                else if ((i1[i] == 0) && (i0[i] >= 2) && (i2[i] >= 2))
                  {
                    const Point min_point = std::min(elem->point(0),
                                                     std::min(elem->point(1),
                                                              std::min(elem->point(5),
                                                                       elem->point(4))));
                    if (elem->point(0) == min_point)
                      if (elem->point(1) == std::min(elem->point(1), elem->point(4)))
                        {
                          // Case 1
                          xi_mapped   = xi;
                          zeta_mapped = zeta;
                        }
                      else
                        {
                          // Case 2
                          xi_mapped   = zeta;
                          zeta_mapped = xi;
                        }

                    else if (elem->point(1) == min_point)
                      if (elem->point(5) == std::min(elem->point(5), elem->point(0)))
                        {
                          // Case 3
                          xi_mapped   = zeta;
                          zeta_mapped = -xi;
                        }
                      else
                        {
                          // Case 4
                          xi_mapped   = -xi;
                          zeta_mapped = zeta;
                        }

                    else if (elem->point(5) == min_point)
                      if (elem->point(4) == std::min(elem->point(4), elem->point(1)))
                        {
                          // Case 5
                          xi_mapped   = -xi;
                          zeta_mapped = -zeta;
                        }
                      else
                        {
                          // Case 6
                          xi_mapped   = -zeta;
                          zeta_mapped = -xi;
                        }

                    else if (elem->point(4) == min_point)
                      {
                        if (elem->point(0) == std::min(elem->point(0), elem->point(5)))
                          {
                            // Case 7
                            xi_mapped   = -xi;
                            zeta_mapped = zeta;
                          }
                        else
                          {
                            // Case 8
                            xi_mapped   = xi;
                            zeta_mapped = -zeta;
                          }
                      }
                  }


                // Face 2
                else if ((i0[i] == 1) && (i1[i] >= 2) && (i2[i] >= 2))
                  {
                    const Point min_point = std::min(elem->point(1),
                                                     std::min(elem->point(2),
                                                              std::min(elem->point(6),
                                                                       elem->point(5))));
                    if (elem->point(1) == min_point)
                      if (elem->point(2) == std::min(elem->point(2), elem->point(5)))
                        {
                          // Case 1
                          eta_mapped  = eta;
                          zeta_mapped = zeta;
                        }
                      else
                        {
                          // Case 2
                          eta_mapped  = zeta;
                          zeta_mapped = eta;
                        }

                    else if (elem->point(2) == min_point)
                      if (elem->point(6) == std::min(elem->point(6), elem->point(1)))
                        {
                          // Case 3
                          eta_mapped  = zeta;
                          zeta_mapped = -eta;
                        }
                      else
                        {
                          // Case 4
                          eta_mapped  = -eta;
                          zeta_mapped = zeta;
                        }

                    else if (elem->point(6) == min_point)
                      if (elem->point(5) == std::min(elem->point(5), elem->point(2)))
                        {
                          // Case 5
                          eta_mapped  = -eta;
                          zeta_mapped = -zeta;
                        }
                      else
                        {
                          // Case 6
                          eta_mapped  = -zeta;
                          zeta_mapped = -eta;
                        }

                    else if (elem->point(5) == min_point)
                      {
                        if (elem->point(1) == std::min(elem->point(1), elem->point(6)))
                          {
                            // Case 7
                            eta_mapped  = -zeta;
                            zeta_mapped = eta;
                          }
                        else
                          {
                            // Case 8
                            eta_mapped   = eta;
                            zeta_mapped = -zeta;
                          }
                      }
                  }


                // Face 3
                else if ((i1[i] == 1) && (i0[i] >= 2) && (i2[i] >= 2))
                  {
                    const Point min_point = std::min(elem->point(2),
                                                     std::min(elem->point(3),
                                                              std::min(elem->point(7),
                                                                       elem->point(6))));
                    if (elem->point(3) == min_point)
                      if (elem->point(2) == std::min(elem->point(2), elem->point(7)))
                        {
                          // Case 1
                          xi_mapped   = xi;
                          zeta_mapped = zeta;
                        }
                      else
                        {
                          // Case 2
                          xi_mapped   = zeta;
                          zeta_mapped = xi;
                        }

                    else if (elem->point(7) == min_point)
                      if (elem->point(3) == std::min(elem->point(3), elem->point(6)))
                        {
                          // Case 3
                          xi_mapped   = -zeta;
                          zeta_mapped = xi;
                        }
                      else
                        {
                          // Case 4
                          xi_mapped   = xi;
                          zeta_mapped = -zeta;
                        }

                    else if (elem->point(6) == min_point)
                      if (elem->point(7) == std::min(elem->point(7), elem->point(2)))
                        {
                          // Case 5
                          xi_mapped   = -xi;
                          zeta_mapped = -zeta;
                        }
                      else
                        {
                          // Case 6
                          xi_mapped   = -zeta;
                          zeta_mapped = -xi;
                        }

                    else if (elem->point(2) == min_point)
                      {
                        if (elem->point(6) == std::min(elem->point(3), elem->point(6)))
                          {
                            // Case 7
                            xi_mapped   = zeta;
                            zeta_mapped = -xi;
                          }
                        else
                          {
                            // Case 8
                            xi_mapped   = -xi;
                            zeta_mapped = zeta;
                          }
                      }
                  }


                // Face 4
                else if ((i0[i] == 0) && (i1[i] >= 2) && (i2[i] >= 2))
                  {
                    const Point min_point = std::min(elem->point(3),
                                                     std::min(elem->point(0),
                                                              std::min(elem->point(4),
                                                                       elem->point(7))));
                    if (elem->point(0) == min_point)
                      if (elem->point(3) == std::min(elem->point(3), elem->point(4)))
                        {
                          // Case 1
                          eta_mapped  = eta;
                          zeta_mapped = zeta;
                        }
                      else
                        {
                          // Case 2
                          eta_mapped  = zeta;
                          zeta_mapped = eta;
                        }

                    else if (elem->point(4) == min_point)
                      if (elem->point(0) == std::min(elem->point(0), elem->point(7)))
                        {
                          // Case 3
                          eta_mapped  = -zeta;
                          zeta_mapped = eta;
                        }
                      else
                        {
                          // Case 4
                          eta_mapped  = eta;
                          zeta_mapped = -zeta;
                        }

                    else if (elem->point(7) == min_point)
                      if (elem->point(4) == std::min(elem->point(4), elem->point(3)))
                        {
                          // Case 5
                          eta_mapped  = -eta;
                          zeta_mapped = -zeta;
                        }
                      else
                        {
                          // Case 6
                          eta_mapped  = -zeta;
                          zeta_mapped = -eta;
                        }

                    else if (elem->point(3) == min_point)
                      {
                        if (elem->point(7) == std::min(elem->point(7), elem->point(0)))
                          {
                            // Case 7
                            eta_mapped   = zeta;
                            zeta_mapped = -eta;
                          }
                        else
                          {
                            // Case 8
                            eta_mapped  = -eta;
                            zeta_mapped = zeta;
                          }
                      }
                  }


                // Face 5
                else if ((i2[i] == 1) && (i0[i] >= 2) && (i1[i] >= 2))
                  {
                    const Point min_point = std::min(elem->point(4),
                                                     std::min(elem->point(5),
                                                              std::min(elem->point(6),
                                                                       elem->point(7))));
                    if (elem->point(4) == min_point)
                      if (elem->point(5) == std::min(elem->point(5), elem->point(7)))
                        {
                          // Case 1
                          xi_mapped  = xi;
                          eta_mapped = eta;
                        }
                      else
                        {
                          // Case 2
                          xi_mapped  = eta;
                          eta_mapped = xi;
                        }

                    else if (elem->point(5) == min_point)
                      if (elem->point(6) == std::min(elem->point(6), elem->point(4)))
                        {
                          // Case 3
                          xi_mapped  = eta;
                          eta_mapped = -xi;
                        }
                      else
                        {
                          // Case 4
                          xi_mapped  = -xi;
                          eta_mapped = eta;
                        }

                    else if (elem->point(6) == min_point)
                      if (elem->point(7) == std::min(elem->point(7), elem->point(5)))
                        {
                          // Case 5
                          xi_mapped  = -xi;
                          eta_mapped = -eta;
                        }
                      else
                        {
                          // Case 6
                          xi_mapped  = -eta;
                          eta_mapped = -xi;
                        }

                    else if (elem->point(7) == min_point)
                      {
                        if (elem->point(4) == std::min(elem->point(4), elem->point(6)))
                          {
                            // Case 7
                            xi_mapped  = -eta;
                            eta_mapped = xi;
                          }
                        else
                          {
                            // Case 8
                            xi_mapped  = xi;
                            eta_mapped = eta;
                          }
                      }
                  }


              }


              return (FE<1,BERNSTEIN>::shape(EDGE3, totalorder, i0[i], xi_mapped)*
                      FE<1,BERNSTEIN>::shape(EDGE3, totalorder, i1[i], eta_mapped)*
                      FE<1,BERNSTEIN>::shape(EDGE3, totalorder, i2[i], zeta_mapped));
            }


          default:
            libmesh_error_msg("Invalid element type = " << type);
          }
      }


    default:
      libmesh_error_msg("Invalid totalorder = " << totalorder);
    }

#endif
}

shape() [64/109]

Real libMesh::FE< 2, SZABAB >::shape ( const ElemType  , const Order  , const unsigned  int, const Point &    )

inherited

Definition at line 53 of file fe_szabab_shape_2D.C.

{
  libmesh_error_msg("Szabo-Babuska polynomials require the element type \nbecause edge orientation is needed.");
  return 0.;
}

shape() [65/109]

Real libMesh::FE< 2, SZABAB >::shape ( const Elem *  elem, const Order  order, const unsigned int  i, const Point &  p  )

inherited

Definition at line 65 of file fe_szabab_shape_2D.C.

{
  libmesh_assert(elem);

  const ElemType type = elem->type();

  const Order totalorder = static_cast<Order>(order + elem->p_level());

  // Declare that we are using our own special power function
  // from the Utility namespace.  This saves typing later.
  using Utility::pow;

  switch (totalorder)
    {
      // 1st & 2nd-order Szabo-Babuska.
    case FIRST:
    case SECOND:
      {
        switch (type)
          {

            // Szabo-Babuska shape functions on the triangle.
          case TRI3:
          case TRI6:
            {
              const Real l1 = 1-p(0)-p(1);
              const Real l2 = p(0);
              const Real l3 = p(1);

              libmesh_assert_less (i, 6);

              switch (i)
                {
                case 0: return l1;
                case 1: return l2;
                case 2: return l3;

                case 3: return l1*l2*(-4.*sqrt6);
                case 4: return l2*l3*(-4.*sqrt6);
                case 5: return l3*l1*(-4.*sqrt6);

                default:
                  libmesh_error_msg("Invalid i = " << i);
                }
            }


            // Szabo-Babuska shape functions on the quadrilateral.
          case QUAD4:
          case QUAD8:
          case QUAD9:
            {
              // Compute quad shape functions as a tensor-product
              const Real xi  = p(0);
              const Real eta = p(1);

              libmesh_assert_less (i, 9);

              //                                0  1  2  3  4  5  6  7  8
              static const unsigned int i0[] = {0, 1, 1, 0, 2, 1, 2, 0, 2};
              static const unsigned int i1[] = {0, 0, 1, 1, 0, 2, 1, 2, 2};

              return (FE<1,SZABAB>::shape(EDGE3, totalorder, i0[i], xi)*
                      FE<1,SZABAB>::shape(EDGE3, totalorder, i1[i], eta));

            }

          default:
            libmesh_error_msg("Invalid element type = " << type);
          }
      }


      // 3rd-order Szabo-Babuska.
    case THIRD:
      {
        switch (type)
          {

            // Szabo-Babuska shape functions on the triangle.
          case TRI6:
            {
              Real l1 = 1-p(0)-p(1);
              Real l2 = p(0);
              Real l3 = p(1);

              Real f=1;

              libmesh_assert_less (i, 10);


              if (i==4 && (elem->point(0) > elem->point(1)))f=-1;
              if (i==6 && (elem->point(1) > elem->point(2)))f=-1;
              if (i==8 && (elem->point(2) > elem->point(0)))f=-1;


              switch (i)
                {
                  //nodal modes
                case 0: return l1;
                case 1: return l2;
                case 2: return l3;

                  //side modes
                case 3: return   l1*l2*(-4.*sqrt6);
                case 4: return f*l1*l2*(-4.*sqrt10)*(l2-l1);

                case 5: return   l2*l3*(-4.*sqrt6);
                case 6: return f*l2*l3*(-4.*sqrt10)*(l3-l2);

                case 7: return   l3*l1*(-4.*sqrt6);
                case 8: return f*l3*l1*(-4.*sqrt10)*(l1-l3);

                  //internal modes
                case 9: return l1*l2*l3;

                default:
                  libmesh_error_msg("Invalid i = " << i);
                }
            }


            // Szabo-Babuska shape functions on the quadrilateral.
          case QUAD8:
          case QUAD9:
            {
              // Compute quad shape functions as a tensor-product
              const Real xi  = p(0);
              const Real eta = p(1);

              libmesh_assert_less (i, 16);

              //                                0   1   2   3   4   5   6   7   8   9  10  11  12  13  14  15
              static const unsigned int i0[] = {0,  1,  1,  0,  2,  3,  1,  1,  2,  3,  0,  0,  2,  3,  2,  3};
              static const unsigned int i1[] = {0,  0,  1,  1,  0,  0,  2,  3,  1,  1,  2,  3,  2,  2,  3,  3};

              Real f=1.;

              // take care of edge orientation, this is needed at
              // edge shapes with (y=0)-asymmetric 1D shapes, these have
              // one 1D shape index being 0 or 1, the other one being odd and >=3

              switch(i)
                {
                case  5: // edge 0 points
                  if (elem->point(0) > elem->point(1))f = -1.;
                  break;
                case  7: // edge 1 points
                  if (elem->point(1) > elem->point(2))f = -1.;
                  break;
                case  9: // edge 2 points
                  if (elem->point(3) > elem->point(2))f = -1.;
                  break;
                case 11: // edge 3 points
                  if (elem->point(0) > elem->point(3))f = -1.;
                  break;

                default:
                  // Everything else keeps f=1
                  break;
                }

              return f*(FE<1,SZABAB>::shape(EDGE3, totalorder, i0[i], xi)*
                        FE<1,SZABAB>::shape(EDGE3, totalorder, i1[i], eta));
            }

          default:
            libmesh_error_msg("Invalid element type = " << type);
          }
      }




      // 4th-order Szabo-Babuska.
    case FOURTH:
      {
        switch (type)
          {
            // Szabo-Babuska shape functions on the triangle.
          case TRI6:
            {
              Real l1 = 1-p(0)-p(1);
              Real l2 = p(0);
              Real l3 = p(1);

              Real f=1;

              libmesh_assert_less (i, 15);


              if (i== 4 && (elem->point(0) > elem->point(1)))f=-1;
              if (i== 7 && (elem->point(1) > elem->point(2)))f=-1;
              if (i==10 && (elem->point(2) > elem->point(0)))f=-1;


              switch (i)
                {
                  //nodal modes
                case  0: return l1;
                case  1: return l2;
                case  2: return l3;

                  //side modes
                case  3: return   l1*l2*(-4.*sqrt6);
                case  4: return f*l1*l2*(-4.*sqrt10)*(l2-l1);
                case  5: return   l1*l2*(-sqrt14)*(5.*pow<2>(l2-l1)-1);

                case  6: return   l2*l3*(-4.*sqrt6);
                case  7: return f*l2*l3*(-4.*sqrt10)*(l3-l2);
                case  8: return   l2*l3*(-sqrt14)*(5.*pow<2>(l3-l2)-1);

                case  9: return   l3*l1*(-4.*sqrt6);
                case 10: return f*l3*l1*(-4.*sqrt10)*(l1-l3);
                case 11: return   l3*l1*(-sqrt14)*(5.*pow<2>(l1-l3)-1);

                  //internal modes
                case 12: return l1*l2*l3;

                case 13: return l1*l2*l3*(l2-l1);
                case 14: return l1*l2*l3*(2*l3-1);

                default:
                  libmesh_error_msg("Invalid i = " << i);
                }
            }


            // Szabo-Babuska shape functions on the quadrilateral.
          case QUAD8:
          case QUAD9:
            {
              // Compute quad shape functions as a tensor-product
              const Real xi  = p(0);
              const Real eta = p(1);

              libmesh_assert_less (i, 25);

              //                                0  1  2  3  4  5  6  7  8  9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24
              static const unsigned int i0[] = {0, 1, 1, 0, 2, 3, 4, 1, 1, 1, 2, 3, 4, 0, 0, 0, 2, 3, 4, 2, 3, 4, 2, 3, 4};
              static const unsigned int i1[] = {0, 0, 1, 1, 0, 0, 0, 2, 3, 4, 1, 1, 1, 2, 3, 4, 2, 2, 2, 3, 3, 3, 4, 4, 4};

              Real f=1.;

              switch(i)
                {
                case  5: // edge 0 points
                  if (elem->point(0) > elem->point(1))f = -1.;
                  break;
                case  8: // edge 1 points
                  if (elem->point(1) > elem->point(2))f = -1.;
                  break;
                case 11: // edge 2 points
                  if (elem->point(3) > elem->point(2))f = -1.;
                  break;
                case 14: // edge 3 points
                  if (elem->point(0) > elem->point(3))f = -1.;
                  break;

                default:
                  // Everything else keeps f=1
                  break;
                }

              return f*(FE<1,SZABAB>::shape(EDGE3, totalorder, i0[i], xi)*
                        FE<1,SZABAB>::shape(EDGE3, totalorder, i1[i], eta));
            }

          default:
            libmesh_error_msg("Invalid element type = " << type);
          }
      }




      // 5th-order Szabo-Babuska.
    case FIFTH:
      {
        switch (type)
          {
            // Szabo-Babuska shape functions on the triangle.
          case TRI6:
            {
              Real l1 = 1-p(0)-p(1);
              Real l2 = p(0);
              Real l3 = p(1);

              const Real x=l2-l1;
              const Real y=2.*l3-1;

              Real f=1;

              libmesh_assert_less (i, 21);


              if ((i== 4||i== 6) && (elem->point(0) > elem->point(1)))f=-1;
              if ((i== 8||i==10) && (elem->point(1) > elem->point(2)))f=-1;
              if ((i==12||i==14) && (elem->point(2) > elem->point(0)))f=-1;


              switch (i)
                {
                  //nodal modes
                case  0: return l1;
                case  1: return l2;
                case  2: return l3;

                  //side modes
                case  3: return   l1*l2*(-4.*sqrt6);
                case  4: return f*l1*l2*(-4.*sqrt10)*(l2-l1);
                case  5: return   -sqrt14*l1*l2*(5.0*l1*l1-1.0+(-10.0*l1+5.0*l2)*l2);
                case  6: return f*(-sqrt2)*l1*l2*((9.-21.*l1*l1)*l1+(-9.+63.*l1*l1+(-63.*l1+21.*l2)*l2)*l2);

                case  7: return   l2*l3*(-4.*sqrt6);
                case  8: return f*l2*l3*(-4.*sqrt10)*(l3-l2);
                case  9: return   -sqrt14*l2*l3*(5.0*l3*l3-1.0+(-10.0*l3+5.0*l2)*l2);
                case 10: return -f*sqrt2*l2*l3*((-9.0+21.0*l3*l3)*l3+(-63.0*l3*l3+9.0+(63.0*l3-21.0*l2)*l2)*l2);

                case 11: return   l3*l1*(-4.*sqrt6);
                case 12: return f*l3*l1*(-4.*sqrt10)*(l1-l3);
                case 13: return -sqrt14*l3*l1*(5.0*l3*l3-1.0+(-10.0*l3+5.0*l1)*l1);
                case 14: return f*(-sqrt2)*l3*l1*((9.0-21.0*l3*l3)*l3+(-9.0+63.0*l3*l3+(-63.0*l3+21.0*l1)*l1)*l1);

                  //internal modes
                case 15: return l1*l2*l3;

                case 16: return l1*l2*l3*x;
                case 17: return l1*l2*l3*y;

                case 18: return l1*l2*l3*(1.5*l1*l1-.5+(-3.0*l1+1.5*l2)*l2);
                case 19: return l1*l2*l3*x*y;
                case 20: return l1*l2*l3*(1.0+(-6.0+6.0*l3)*l3);

                default:
                  libmesh_error_msg("Invalid i = " << i);
                }
            } // case TRI6

            // Szabo-Babuska shape functions on the quadrilateral.
          case QUAD8:
          case QUAD9:
            {
              // Compute quad shape functions as a tensor-product
              const Real xi  = p(0);
              const Real eta = p(1);

              libmesh_assert_less (i, 36);

              //                                0  1  2  3  4  5  6  7  8  9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35
              static const unsigned int i0[] = {0, 1, 1, 0, 2, 3, 4, 5, 1, 1, 1, 1, 2, 3, 4, 5, 0, 0, 0, 0, 2, 3, 4, 5, 2, 3, 4, 5, 2, 3, 4, 5, 2, 3, 4, 5};
              static const unsigned int i1[] = {0, 0, 1, 1, 0, 0, 0, 0, 2, 3, 4, 5, 1, 1, 1, 1, 2, 3, 4, 5, 2, 2, 2, 2, 3, 3, 3, 3, 4, 4, 4, 4, 5, 5, 5, 5};

              Real f=1.;

              switch(i)
                {
                case  5: // edge 0 points
                case  7:
                  if (elem->point(0) > elem->point(1))f = -1.;
                  break;
                case  9: // edge 1 points
                case 11:
                  if (elem->point(1) > elem->point(2))f = -1.;
                  break;
                case 13: // edge 2 points
                case 15:
                  if (elem->point(3) > elem->point(2))f = -1.;
                  break;
                case 14: // edge 3 points
                case 19:
                  if (elem->point(0) > elem->point(3))f = -1.;
                  break;

                default:
                  // Everything else keeps f=1
                  break;
                }

              return f*(FE<1,SZABAB>::shape(EDGE3, totalorder, i0[i], xi)*
                        FE<1,SZABAB>::shape(EDGE3, totalorder, i1[i], eta));

            } // case QUAD8/QUAD9

          default:
            libmesh_error_msg("Invalid element type = " << type);

          } // switch type

      } // case FIFTH

      // 6th-order Szabo-Babuska.
    case SIXTH:
      {
        switch (type)
          {
            // Szabo-Babuska shape functions on the triangle.
          case TRI6:
            {
              Real l1 = 1-p(0)-p(1);
              Real l2 = p(0);
              Real l3 = p(1);

              const Real x=l2-l1;
              const Real y=2.*l3-1;

              Real f=1;

              libmesh_assert_less (i, 28);


              if ((i== 4||i== 6) && (elem->point(0) > elem->point(1)))f=-1;
              if ((i== 9||i==11) && (elem->point(1) > elem->point(2)))f=-1;
              if ((i==14||i==16) && (elem->point(2) > elem->point(0)))f=-1;


              switch (i)
                {
                  //nodal modes
                case  0: return l1;
                case  1: return l2;
                case  2: return l3;

                  //side modes
                case  3: return   l1*l2*(-4.*sqrt6);
                case  4: return f*l1*l2*(-4.*sqrt10)*(l2-l1);
                case  5: return   -sqrt14*l1*l2*(5.0*l1*l1-1.0+(-10.0*l1+5.0*l2)*l2);
                case  6: return f*(-sqrt2)*l1*l2*((9.0-21.0*l1*l1)*l1+(-9.0+63.0*l1*l1+(-63.0*l1+21.0*l2)*l2)*l2);
                case  7: return   -sqrt22*l1*l2*(0.5+(-7.0+0.105E2*l1*l1)*l1*l1+((14.0-0.42E2*l1*l1)*l1+(-7.0+0.63E2*l1*l1+(-0.42E2*l1+0.105E2*l2)*l2)*l2)*l2);

                case  8: return   l2*l3*(-4.*sqrt6);
                case  9: return f*l2*l3*(-4.*sqrt10)*(l3-l2);
                case 10: return   -sqrt14*l2*l3*(5.0*l3*l3-1.0+(-10.0*l3+5.0*l2)*l2);
                case 11: return f*(-sqrt2)*l2*l3*((-9.0+21.0*l3*l3)*l3+(-63.0*l3*l3+9.0+(63.0*l3-21.0*l2)*l2)*l2);
                case 12: return   -sqrt22*l2*l3*(0.5+(-7.0+0.105E2*l3*l3)*l3*l3+((14.0-0.42E2*l3*l3)*l3+(-7.0+0.63E2*l3*l3+(-0.42E2*l3+0.105E2*l2)*l2)*l2)*l2);

                case 13: return   l3*l1*(-4.*sqrt6);
                case 14: return f*l3*l1*(-4.*sqrt10)*(l1-l3);
                case 15: return   -sqrt14*l3*l1*(5.0*l3*l3-1.0+(-10.0*l3+5.0*l1)*l1);
                case 16: return f*(-sqrt2)*l3*l1*((9.0-21.0*l3*l3)*l3+(-9.0+63.0*l3*l3+(-63.0*l3+21.0*l1)*l1)*l1);
                case 17: return   -sqrt22*l3*l1*(0.5+(-7.0+0.105E2*l3*l3)*l3*l3+((14.0-0.42E2*l3*l3)*l3+(-7.0+0.63E2*l3*l3+(-0.42E2*l3+0.105E2*l1)*l1)*l1)*l1);



                  //internal modes
                case 18: return l1*l2*l3;

                case 19: return l1*l2*l3*x;
                case 20: return l1*l2*l3*y;

                case 21: return 0.5*l1*l2*l3*(3.0*l1*l1-1.0+(-6.0*l1+3.0*l2)*l2);
                case 22: return l1*l2*l3*(l2-l1)*(2.0*l3-1.0);
                case 23: return 0.5*l1*l2*l3*(2.0+(-12.0+12.0*l3)*l3);
                case 24: return 0.5*l1*l2*l3*((3.0-5.0*l1*l1)*l1+(-3.0+15.0*l1*l1+(-15.0*l1+5.0*l2)*l2)*l2);
                case 25: return 0.5*l1*l2*l3*(3.0*l1*l1-1.0+(-6.0*l1+3.0*l2)*l2)*(2.0*l3-1.0);
                case 26: return 0.5*l1*l2*l3*(2.0+(-12.0+12.0*l3)*l3)*(l2-l1);
                case 27: return 0.5*l1*l2*l3*(-2.0+(24.0+(-60.0+40.0*l3)*l3)*l3);


                default:
                  libmesh_error_msg("Invalid i = " << i);
                }
            } // case TRI6

            // Szabo-Babuska shape functions on the quadrilateral.
          case QUAD8:
          case QUAD9:
            {
              // Compute quad shape functions as a tensor-product
              const Real xi  = p(0);
              const Real eta = p(1);

              libmesh_assert_less (i, 49);

              //                                0  1  2  3  4  5  6  7  8  9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48
              static const unsigned int i0[] = {0, 1, 1, 0, 2, 3, 4, 5, 6, 1, 1, 1, 1, 1, 2, 3, 4, 5, 6, 0, 0, 0, 0, 0, 2, 3, 4, 5, 6, 2, 3, 4, 5, 6, 2, 3, 4, 5, 6, 2, 3, 4, 5, 6, 2, 3, 4, 5, 6};
              static const unsigned int i1[] = {0, 0, 1, 1, 0, 0, 0, 0, 0, 2, 3, 4, 5, 6, 1, 1, 1, 1, 1, 2, 3, 4, 5, 6, 2, 2, 2, 2, 2, 3, 3, 3, 3, 3, 4, 4, 4, 4, 4, 5, 5, 5, 5, 5, 6, 6, 6, 6, 6};

              Real f=1.;

              switch(i)
                {
                case  5: // edge 0 points
                case  7:
                  if (elem->point(0) > elem->point(1))f = -1.;
                  break;
                case 10: // edge 1 points
                case 12:
                  if (elem->point(1) > elem->point(2))f = -1.;
                  break;
                case 15: // edge 2 points
                case 17:
                  if (elem->point(3) > elem->point(2))f = -1.;
                  break;
                case 20: // edge 3 points
                case 22:
                  if (elem->point(0) > elem->point(3))f = -1.;
                  break;

                default:
                  // Everything else keeps f=1
                  break;
                }

              return f*(FE<1,SZABAB>::shape(EDGE3, totalorder, i0[i], xi)*
                        FE<1,SZABAB>::shape(EDGE3, totalorder, i1[i], eta));

            } // case QUAD8/QUAD9

          default:
            libmesh_error_msg("Invalid element type = " << type);

          } // switch type

      } // case SIXTH


      // 7th-order Szabo-Babuska.
    case SEVENTH:
      {
        switch (type)
          {
            // Szabo-Babuska shape functions on the triangle.
          case TRI6:
            {

              Real l1 = 1-p(0)-p(1);
              Real l2 = p(0);
              Real l3 = p(1);

              const Real x=l2-l1;
              const Real y=2.*l3-1.;

              Real f=1;

              libmesh_assert_less (i, 36);


              if ((i>= 4&&i<= 8) && (elem->point(0) > elem->point(1)))f=-1;
              if ((i>=10&&i<=14) && (elem->point(1) > elem->point(2)))f=-1;
              if ((i>=16&&i<=20) && (elem->point(2) > elem->point(0)))f=-1;


              switch (i)
                {
                  //nodal modes
                case  0: return l1;
                case  1: return l2;
                case  2: return l3;

                  //side modes
                case  3: return   l1*l2*(-4.*sqrt6);
                case  4: return f*l1*l2*(-4.*sqrt10)*(l2-l1);

                case  5: return   -sqrt14*l1*l2*(5.0*l1*l1-1.0+(-10.0*l1+5.0*l2)*l2);
                case  6: return f*-sqrt2*l1*l2*((9.0-21.0*l1*l1)*l1+(-9.0+63.0*l1*l1+(-63.0*l1+21.0*l2)*l2)*l2);
                case  7: return   -sqrt22*l1*l2*(0.5+(-7.0+0.105E2*l1*l1)*l1*l1+((14.0-0.42E2*l1*l1)*l1+(-7.0+0.63E2*l1*l1+(-0.42E2*l1+0.105E2*l2)*l2)*l2)*l2);
                case  8: return f*-sqrt26*l1*l2*((-0.25E1+(15.0-0.165E2*l1*l1)*l1*l1)*l1+(0.25E1+(-45.0+0.825E2*l1*l1)*l1*l1+((45.0-0.165E3*l1*l1)*l1+(-15.0+0.165E3*l1*l1+(-0.825E2*l1+0.165E2*l2)*l2)*l2)*l2)*l2);

                case  9: return   l2*l3*(-4.*sqrt6);
                case 10: return f*l2*l3*(-4.*sqrt10)*(l3-l2);

                case 11: return   -sqrt14*l2*l3*(5.0*l3*l3-1.0+(-10.0*l3+5.0*l2)*l2);
                case 12: return f*-sqrt2*l2*l3*((-9.0+21.0*l3*l3)*l3+(-63.0*l3*l3+9.0+(63.0*l3-21.0*l2)*l2)*l2);
                case 13: return   -sqrt22*l2*l3*(0.5+(-7.0+0.105E2*l3*l3)*l3*l3+((14.0-0.42E2*l3*l3)*l3+(-7.0+0.63E2*l3*l3+(-0.42E2*l3+0.105E2*l2)*l2)*l2)*l2);
                case 14: return f*-sqrt26*l2*l3*((0.25E1+(-15.0+0.165E2*l3*l3)*l3*l3)*l3+(-0.25E1+(45.0-0.825E2*l3*l3)*l3*l3+((-45.0+0.165E3*l3*l3)*l3+(15.0-0.165E3*l3*l3+(0.825E2*l3-0.165E2*l2)*l2)*l2)*l2)*l2);

                case 15: return   l3*l1*(-4.*sqrt6);
                case 16: return f*l3*l1*(-4.*sqrt10)*(l1-l3);

                case 17: return   -sqrt14*l3*l1*(5.0*l3*l3-1.0+(-10.0*l3+5.0*l1)*l1);
                case 18: return -f*sqrt2*l3*l1*((9.-21.*l3*l3)*l3+(-9.+63.*l3*l3+(-63.*l3+21.*l1)*l1)*l1);
                case 19: return   -sqrt22*l3*l1*(0.5+(-7.0+0.105E2*l3*l3)*l3*l3+((14.0-0.42E2*l3*l3)*l3+(-7.0+0.63E2*l3*l3+(-0.42E2*l3+0.105E2*l1)*l1)*l1)*l1);
                case 20: return f*-sqrt26*l3*l1*((-0.25E1+(15.0-0.165E2*l3*l3)*l3*l3)*l3+(0.25E1+(-45.0+0.825E2*l3*l3)*l3*l3+((45.0-0.165E3*l3*l3)*l3+(-15.0+0.165E3*l3*l3+(-0.825E2*l3+0.165E2*l1)*l1)*l1)*l1)*l1);


                  //internal modes
                case 21: return l1*l2*l3;

                case 22: return l1*l2*l3*x;
                case 23: return l1*l2*l3*y;

                case 24: return l1*l2*l3*0.5*(3.*pow<2>(x)-1.);
                case 25: return l1*l2*l3*x*y;
                case 26: return l1*l2*l3*0.5*(3.*pow<2>(y)-1.);

                case 27: return 0.5*l1*l2*l3*((3.0-5.0*l1*l1)*l1+(-3.0+15.0*l1*l1+(-15.0*l1+5.0*l2)*l2)*l2);
                case 28: return 0.5*l1*l2*l3*(3.0*l1*l1-1.0+(-6.0*l1+3.0*l2)*l2)*(2.0*l3-1.0);
                case 29: return 0.5*l1*l2*l3*(2.0+(-12.0+12.0*l3)*l3)*(l2-l1);
                case 30: return 0.5*l1*l2*l3*(-2.0+(24.0+(-60.0+40.0*l3)*l3)*l3);
                case 31: return 0.125*l1*l2*l3*((-15.0+(70.0-63.0*l1*l1)*l1*l1)*l1+(15.0+(-210.0+315.0*l1*l1)*l1*l1+((210.0-630.0*l1*l1)*l1+(-70.0+630.0*l1*l1+(-315.0*l1+63.0*l2)*l2)*l2)*l2)*l2);
                case 32: return 0.5*l1*l2*l3*((3.0-5.0*l1*l1)*l1+(-3.0+15.0*l1*l1+(-15.0*l1+5.0*l2)*l2)*l2)*(2.0*l3-1.0);
                case 33: return 0.25*l1*l2*l3*(3.0*l1*l1-1.0+(-6.0*l1+3.0*l2)*l2)*(2.0+(-12.0+12.0*l3)*l3);
                case 34: return 0.5*l1*l2*l3*(-2.0+(24.0+(-60.0+40.0*l3)*l3)*l3)*(l2-l1);
                case 35: return 0.125*l1*l2*l3*(-8.0+(240.0+(-1680.0+(4480.0+(-5040.0+2016.0*l3)*l3)*l3)*l3)*l3);

                default:
                  libmesh_error_msg("Invalid i = " << i);
                }
            } // case TRI6

            // Szabo-Babuska shape functions on the quadrilateral.
          case QUAD8:
          case QUAD9:
            {
              // Compute quad shape functions as a tensor-product
              const Real xi  = p(0);
              const Real eta = p(1);

              libmesh_assert_less (i, 64);

              //                                0  1  2  3  4  5  6  7  8  9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63
              static const unsigned int i0[] = {0, 1, 1, 0, 2, 3, 4, 5, 6, 7, 1, 1, 1, 1, 1, 1, 2, 3, 4, 5, 6, 7, 0, 0, 0, 0, 0, 0, 2, 3, 4, 5, 6, 7, 2, 3, 4, 5, 6, 7, 2, 3, 4, 5, 6, 7, 2, 3, 4, 5, 6, 7, 2, 3, 4, 5, 6, 7, 2, 3, 4, 5, 6, 7};
              static const unsigned int i1[] = {0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 2, 3, 4, 5, 6, 7, 1, 1, 1, 1, 1, 1, 2, 3, 4, 5, 6, 7, 2, 2, 2, 2, 2, 2, 3, 3, 3, 3, 3, 3, 4, 4, 4, 4, 4, 4, 5, 5, 5, 5, 5, 5, 6, 6, 6, 6, 6, 6, 7, 7, 7, 7, 7, 7};

              Real f=1.;

              switch(i)
                {
                case  5: // edge 0 points
                case  7:
                case  9:
                  if (elem->point(0) > elem->point(1))f = -1.;
                  break;
                case 11: // edge 1 points
                case 13:
                case 15:
                  if (elem->point(1) > elem->point(2))f = -1.;
                  break;
                case 17: // edge 2 points
                case 19:
                case 21:
                  if (elem->point(3) > elem->point(2))f = -1.;
                  break;
                case 23: // edge 3 points
                case 25:
                case 27:
                  if (elem->point(0) > elem->point(3))f = -1.;
                  break;

                default:
                  // Everything else keeps f=1
                  break;
                }

              return f*(FE<1,SZABAB>::shape(EDGE3, totalorder, i0[i], xi)*
                        FE<1,SZABAB>::shape(EDGE3, totalorder, i1[i], eta));

            } // case QUAD8/QUAD9

          default:
            libmesh_error_msg("Invalid element type = " << type);

          } // switch type

      } // case SEVENTH


      // by default throw an error
    default:
      libmesh_error_msg("ERROR: Unsupported polynomial order!");
    } // switch order
}

shape() [66/109]

Real libMesh::FE< 1, MONOMIAL >::shape ( const Elem *  elem, const Order  order, const unsigned int  i, const Point &  p  )

inherited

Definition at line 70 of file fe_monomial_shape_1D.C.

{
  libmesh_assert(elem);

  return FE<1,MONOMIAL>::shape(elem->type(), static_cast<Order>(order + elem->p_level()), i, p);
}

shape() [67/109]

Real libMesh::FE< 1, SZABAB >::shape ( const Elem *  elem, const Order  order, const unsigned int  i, const Point &  p  )

inherited

Definition at line 79 of file fe_szabab_shape_1D.C.

{
  libmesh_assert(elem);

  return FE<1,SZABAB>::shape(elem->type(), static_cast<Order>(order + elem->p_level()), i, p);
}

shape() [68/109]

Real libMesh::FE< 1, L2_HIERARCHIC >::shape ( const Elem *  elem, const Order  order, const unsigned int  i, const Point &  p  )

inherited

Definition at line 101 of file fe_l2_hierarchic_shape_1D.C.

{
  libmesh_assert(elem);

  return FE<1,L2_HIERARCHIC>::shape(elem->type(), static_cast<Order>(order + elem->p_level()), i, p);
}

shape() [69/109]

Real libMesh::FE< 1, HIERARCHIC >::shape ( const Elem *  elem, const Order  order, const unsigned int  i, const Point &  p  )

inherited

Definition at line 101 of file fe_hierarchic_shape_1D.C.

{
  libmesh_assert(elem);

  return FE<1,HIERARCHIC>::shape(elem->type(), static_cast<Order>(order + elem->p_level()), i, p);
}

shape() [70/109]

static OutputShape libMesh::FE< Dim, T >::shape ( const ElemType  t, const Order  o, const unsigned int  i, const Point &  p  )

staticinherited

Returns

The value of the shape function at point p. This method allows you to specify the dimension, element type, and order directly. This allows the method to be static.

On a p-refined element, o should be the total order of the element.

shape() [71/109]

Real libMesh::FE< 1, L2_LAGRANGE >::shape ( const Elem *  elem, const Order  order, const unsigned int  i, const Point &  p  )

inherited

Definition at line 117 of file fe_l2_lagrange_shape_1D.C.

{
  libmesh_assert(elem);

  return FE<1,L2_LAGRANGE>::shape(elem->type(), static_cast<Order>(order + elem->p_level()), i, p);
}

shape() [72/109]

Real libMesh::FE< 1, LAGRANGE >::shape ( const Elem *  elem, const Order  order, const unsigned int  i, const Point &  p  )

inherited

Definition at line 117 of file fe_lagrange_shape_1D.C.

{
  libmesh_assert(elem);

  return FE<1,LAGRANGE>::shape(elem->type(), static_cast<Order>(order + elem->p_level()), i, p);
}

shape() [73/109]

Real libMesh::FE< 2, MONOMIAL >::shape ( const Elem *  elem, const Order  order, const unsigned int  i, const Point &  p  )

inherited

Definition at line 118 of file fe_monomial_shape_2D.C.

{
  libmesh_assert(elem);

  // by default call the orientation-independent shape functions
  return FE<2,MONOMIAL>::shape(elem->type(), static_cast<Order>(order + elem->p_level()), i, p);
}

shape() [74/109]

static OutputShape libMesh::FE< Dim, T >::shape ( const Elem *  elem, const Order  o, const unsigned int  i, const Point &  p  )

staticinherited

Returns

The value of the shape function at point p. This method allows you to specify the dimension, element type, and order directly. This allows the method to be static.

On a p-refined element, o should be the base order of the element.

shape() [75/109]

Real libMesh::FE< 1, HERMITE >::shape ( const ElemType  , const Order  , const unsigned  int, const Point &    )

inherited

Definition at line 175 of file fe_hermite_shape_1D.C.

{
  libmesh_error_msg("Hermite elements require the real element \nto construct gradient-based degrees of freedom.");
  return 0.;
}

shape() [76/109]

Real libMesh::FE< 3, MONOMIAL >::shape ( const Elem *  elem, const Order  order, const unsigned int  i, const Point &  p  )

inherited

Definition at line 186 of file fe_monomial_shape_3D.C.

{
  libmesh_assert(elem);

  // call the orientation-independent shape functions
  return FE<3,MONOMIAL>::shape(elem->type(), static_cast<Order>(order + elem->p_level()), i, p);
}

shape() [77/109]

Real libMesh::FE< 1, HERMITE >::shape ( const Elem *  elem, const Order  libmesh_dbg_varorder, const unsigned int  i, const Point &  p  )

inherited

Definition at line 187 of file fe_hermite_shape_1D.C.

{
  libmesh_assert(elem);

  // Coefficient naming: d(1)d(2n) is the coefficient of the
  // global shape function corresponding to value 1 in terms of the
  // local shape function corresponding to normal derivative 2
  Real d1xd1x, d2xd2x;

  hermite_compute_coefs(elem, d1xd1x, d2xd2x);

  const ElemType type = elem->type();

#ifndef NDEBUG
  const unsigned int totalorder = order + elem->p_level();
#endif

  switch (type)
    {
      // C1 functions on the C1 cubic edge
    case EDGE2:
    case EDGE3:
      {
        libmesh_assert_less (i, totalorder+1);

        switch (i)
          {
          case 0:
            return FEHermite<1>::hermite_raw_shape(0, p(0));
          case 1:
            return d1xd1x * FEHermite<1>::hermite_raw_shape(2, p(0));
          case 2:
            return FEHermite<1>::hermite_raw_shape(1, p(0));
          case 3:
            return d2xd2x * FEHermite<1>::hermite_raw_shape(3, p(0));
          default:
            return FEHermite<1>::hermite_raw_shape(i, p(0));
          }
      }
    default:
      libmesh_error_msg("ERROR: Unsupported element type = " << type);
    }
}

shape() [78/109]

Real libMesh::FE< 2, HERMITE >::shape ( const ElemType  , const Order  , const unsigned  int, const Point &    )

inherited

Definition at line 189 of file fe_hermite_shape_2D.C.

{
  libmesh_error_msg("Hermite elements require the real element \nto construct gradient-based degrees of freedom.");
  return 0.;
}

shape() [79/109]

Real libMesh::FE< 1, BERNSTEIN >::shape ( const Elem *  elem, const Order  order, const unsigned int  i, const Point &  p  )

inherited

Definition at line 190 of file fe_bernstein_shape_1D.C.

{
  libmesh_assert(elem);

  return FE<1,BERNSTEIN>::shape(elem->type(), static_cast<Order>(order + elem->p_level()), i, p);
}

shape() [80/109]

Real libMesh::FE< 2, HERMITE >::shape ( const Elem *  elem, const Order  order, const unsigned int  i, const Point &  p  )

inherited

Definition at line 201 of file fe_hermite_shape_2D.C.

{
  libmesh_assert(elem);

  std::vector<std::vector<Real>> dxdxi(2, std::vector<Real>(2, 0));

#ifdef DEBUG
  std::vector<Real> dxdeta(2), dydxi(2);
#endif

  hermite_compute_coefs(elem,dxdxi
#ifdef DEBUG
                        ,dxdeta,dydxi
#endif
                        );

  const ElemType type = elem->type();

  const Order totalorder = static_cast<Order>(order + elem->p_level());

  switch (type)
    {
    case QUAD4:
    case QUADSHELL4:
      libmesh_assert_less (totalorder, 4);
      libmesh_fallthrough();
    case QUAD8:
    case QUADSHELL8:
    case QUAD9:
      {
        libmesh_assert_less (i, (totalorder+1u)*(totalorder+1u));

        std::vector<unsigned int> bases1D;

        Real coef = hermite_bases_2D(bases1D, dxdxi, totalorder, i);

        return coef * FEHermite<1>::hermite_raw_shape(bases1D[0],p(0)) *
          FEHermite<1>::hermite_raw_shape(bases1D[1],p(1));
      }
    default:
      libmesh_error_msg("ERROR: Unsupported element type = " << type);
    }
}

shape() [81/109]

Real libMesh::FE< 2, L2_LAGRANGE >::shape ( const Elem *  elem, const Order  order, const unsigned int  i, const Point &  p  )

inherited

Definition at line 208 of file fe_l2_lagrange_shape_2D.C.

{
  libmesh_assert(elem);

  // call the orientation-independent shape functions
  return FE<2,L2_LAGRANGE>::shape(elem->type(), static_cast<Order>(order + elem->p_level()), i, p);
}

shape() [82/109]

Real libMesh::FE< 2, LAGRANGE >::shape ( const Elem *  elem, const Order  order, const unsigned int  i, const Point &  p  )

inherited

Definition at line 210 of file fe_lagrange_shape_2D.C.

{
  libmesh_assert(elem);

  // call the orientation-independent shape functions
  return FE<2,LAGRANGE>::shape(elem->type(), static_cast<Order>(order + elem->p_level()), i, p);
}

shape() [83/109]

Real libMesh::FE< 1, CLOUGH >::shape ( const ElemType  , const Order  , const unsigned  int, const Point &    )

inherited

Definition at line 211 of file fe_clough_shape_1D.C.

{
  libmesh_error_msg("Clough-Tocher elements require the real element \nto construct gradient-based degrees of freedom.");
  return 0.;
}

shape() [84/109]

Real libMesh::FE< 1, CLOUGH >::shape ( const Elem *  elem, const Order  order, const unsigned int  i, const Point &  p  )

inherited

Definition at line 223 of file fe_clough_shape_1D.C.

{
  libmesh_assert(elem);

  clough_compute_coefs(elem);

  const ElemType type = elem->type();

  const Order totalorder = static_cast<Order>(order + elem->p_level());

  switch (totalorder)
    {
      // 3rd-order C1 cubic element
    case THIRD:
      {
        switch (type)
          {
            // C1 functions on the C1 cubic edge
          case EDGE2:
          case EDGE3:
            {
              libmesh_assert_less (i, 4);

              switch (i)
                {
                case 0:
                  return clough_raw_shape(0, p);
                case 1:
                  return clough_raw_shape(1, p);
                case 2:
                  return d1xd1x * clough_raw_shape(2, p);
                case 3:
                  return d2xd2x * clough_raw_shape(3, p);
                default:
                  libmesh_error_msg("Invalid shape function index i = " << i);
                }
            }
          default:
            libmesh_error_msg("ERROR: Unsupported element type = " << type);
          }
      }
      // by default throw an error
    default:
      libmesh_error_msg("ERROR: Unsupported polynomial order = " << totalorder);
    }
}

shape() [85/109]

Real libMesh::FE< 3, L2_LAGRANGE >::shape ( const Elem *  elem, const Order  order, const unsigned int  i, const Point &  p  )

inherited

Definition at line 356 of file fe_l2_lagrange_shape_3D.C.

{
  libmesh_assert(elem);

  // call the orientation-independent shape functions
  return FE<3,L2_LAGRANGE>::shape(elem->type(), static_cast<Order>(order + elem->p_level()), i, p);
}

shape() [86/109]

Real libMesh::FE< 3, HERMITE >::shape ( const ElemType  , const Order  , const unsigned  int, const Point &    )

inherited

Definition at line 376 of file fe_hermite_shape_3D.C.

{
  libmesh_error_msg("Hermite elements require the real element \nto construct gradient-based degrees of freedom.");
  return 0.;
}

shape() [87/109]

Real libMesh::FE< 3, HERMITE >::shape ( const Elem *  elem, const Order  order, const unsigned int  i, const Point &  p  )

inherited

Definition at line 388 of file fe_hermite_shape_3D.C.

{
  libmesh_assert(elem);

  std::vector<std::vector<Real>> dxdxi(3, std::vector<Real>(2, 0));

#ifdef DEBUG
  std::vector<Real> dydxi(2), dzdeta(2), dxdzeta(2);
  std::vector<Real> dzdxi(2), dxdeta(2), dydzeta(2);
#endif //DEBUG

  hermite_compute_coefs(elem, dxdxi
#ifdef DEBUG
                        , dydxi, dzdeta, dxdzeta, dzdxi, dxdeta, dydzeta
#endif
                        );

  const ElemType type = elem->type();

  const Order totalorder = static_cast<Order>(order + elem->p_level());

  switch (totalorder)
    {
      // 3rd-order tricubic Hermite functions
    case THIRD:
      {
        switch (type)
          {
          case HEX8:
          case HEX20:
          case HEX27:
            {
              libmesh_assert_less (i, 64);

              std::vector<unsigned int> bases1D;

              Real coef = hermite_bases_3D(bases1D, dxdxi, totalorder, i);

              return coef *
                FEHermite<1>::hermite_raw_shape(bases1D[0],p(0)) *
                FEHermite<1>::hermite_raw_shape(bases1D[1],p(1)) *
                FEHermite<1>::hermite_raw_shape(bases1D[2],p(2));
            }
          default:
            libmesh_error_msg("ERROR: Unsupported element type " << type);
          }
      }
      // by default throw an error
    default:
      libmesh_error_msg("ERROR: Unsupported polynomial order " << totalorder);
    }
}

shape() [88/109]

RealGradient libMesh::FE< 0, LAGRANGE_VEC >::shape ( const ElemType  type, const Order  order, const unsigned int  i, const Point &  p  )

inherited

Definition at line 538 of file fe_lagrange_vec.C.

{
  Real value = FE<0,LAGRANGE>::shape( type, order, i, p );
  return libMesh::RealGradient( value );
}

shape() [89/109]

Real libMesh::FE< 2, SUBDIVISION >::shape ( const Elem *  elem, const Order  order, const unsigned int  i, const Point &  p  )

inherited

Subdivision finite elements.

Template specialization prototypes are needed for calling from inside FESubdivision::init_shape_functions

shape() [90/109]

Real libMesh::FE< 3, LAGRANGE >::shape ( const Elem *  elem, const Order  order, const unsigned int  i, const Point &  p  )

inherited

Definition at line 558 of file fe_lagrange_shape_3D.C.

{
  libmesh_assert(elem);

  // call the orientation-independent shape functions
  return FE<3,LAGRANGE>::shape(elem->type(), static_cast<Order>(order + elem->p_level()), i, p);
}

shape() [91/109]

RealGradient libMesh::FE< 1, LAGRANGE_VEC >::shape ( const ElemType  type, const Order  order, const unsigned int  i, const Point &  p  )

inherited

Definition at line 560 of file fe_lagrange_vec.C.

{
  Real value = FE<1,LAGRANGE>::shape( type, order, i, p );
  return libMesh::RealGradient( value );
}

shape() [92/109]

RealGradient libMesh::FE< 2, LAGRANGE_VEC >::shape ( const ElemType  type, const Order  order, const unsigned int  i, const Point &  p  )

inherited

Definition at line 582 of file fe_lagrange_vec.C.

{
  Real value = FE<2,LAGRANGE>::shape( type, order, i/2, p );

  switch( i%2 )
    {
    case 0:
      return libMesh::RealGradient( value );

    case 1:
      return libMesh::RealGradient( Real(0), value );

    default:
      libmesh_error_msg("i%2 must be either 0 or 1!");
    }

  //dummy
  return libMesh::RealGradient();
}

shape() [93/109]

RealGradient libMesh::FE< 0, NEDELEC_ONE >::shape ( const ElemType  , const Order  , const unsigned  int, const Point &    )

inherited

Definition at line 593 of file fe_nedelec_one.C.

{ NEDELEC_LOW_D_ERROR_MESSAGE }

shape() [94/109]

RealGradient libMesh::FE< 0, NEDELEC_ONE >::shape ( const Elem *  , const Order  , const unsigned  int, const Point &    )

inherited

Definition at line 596 of file fe_nedelec_one.C.

{ NEDELEC_LOW_D_ERROR_MESSAGE }

shape() [95/109]

RealGradient libMesh::FE< 1, NEDELEC_ONE >::shape ( const ElemType  , const Order  , const unsigned  int, const Point &    )

inherited

Definition at line 616 of file fe_nedelec_one.C.

{ NEDELEC_LOW_D_ERROR_MESSAGE }

shape() [96/109]

RealGradient libMesh::FE< 1, NEDELEC_ONE >::shape ( const Elem *  , const Order  , const unsigned  int, const Point &    )

inherited

Definition at line 619 of file fe_nedelec_one.C.

{ NEDELEC_LOW_D_ERROR_MESSAGE }

shape() [97/109]

Real libMesh::FE< 3, L2_HIERARCHIC >::shape ( const ElemType  , const Order  , const unsigned  int, const Point &    )

inherited

Definition at line 645 of file fe_l2_hierarchic_shape_3D.C.

{
  libmesh_error_msg("Hierarchic polynomials require the element type \nbecause edge and face orientation is needed.");
  return 0.;
}

shape() [98/109]

Real libMesh::FE< 3, HIERARCHIC >::shape ( const ElemType  , const Order  , const unsigned  int, const Point &    )

inherited

Definition at line 645 of file fe_hierarchic_shape_3D.C.

{
  libmesh_error_msg("Hierarchic polynomials require the element type \nbecause edge and face orientation is needed.");
  return 0.;
}

shape() [99/109]

RealGradient libMesh::FE< 3, LAGRANGE_VEC >::shape ( const ElemType  type, const Order  order, const unsigned int  i, const Point &  p  )

inherited

Definition at line 647 of file fe_lagrange_vec.C.

{
  Real value = FE<3,LAGRANGE>::shape( type, order, i/3, p );

  switch( i%3 )
    {
    case 0:
      return libMesh::RealGradient( value );

    case 1:
      return libMesh::RealGradient( Real(0), value );

    case 2:
      return libMesh::RealGradient( Real(0), Real(0), value );

    default:
      libmesh_error_msg("i%3 must be 0, 1, or 2!");
    }

  //dummy
  return libMesh::RealGradient();
}

shape() [100/109]

Real libMesh::FE< 3, HIERARCHIC >::shape ( const Elem *  elem, const Order  order, const unsigned int  i, const Point &  p  )

inherited

Definition at line 657 of file fe_hierarchic_shape_3D.C.

{
#if LIBMESH_DIM == 3

  libmesh_assert(elem);
  const ElemType type = elem->type();

  const Order totalorder = static_cast<Order>(order+elem->p_level());

  switch (type)
    {
    case HEX8:
    case HEX20:
      libmesh_assert_less (totalorder, 2);
      libmesh_fallthrough();
    case HEX27:
      {
        libmesh_assert_less (i, (totalorder+1u)*(totalorder+1u)*(totalorder+1u));

        // Compute hex shape functions as a tensor-product
        Real xi   = p(0);
        Real eta  = p(1);
        Real zeta = p(2);

        unsigned int i0, i1, i2;

        cube_indices(elem, totalorder, i, xi, eta, zeta, i0, i1, i2);

        return (FE<1,HIERARCHIC>::shape(EDGE3, totalorder, i0, xi)*
                FE<1,HIERARCHIC>::shape(EDGE3, totalorder, i1, eta)*
                FE<1,HIERARCHIC>::shape(EDGE3, totalorder, i2, zeta));
      }

    default:
      libmesh_error_msg("Invalid element type = " << type);
    }

#endif
}

shape() [101/109]

Real libMesh::FE< 3, L2_HIERARCHIC >::shape ( const Elem *  elem, const Order  order, const unsigned int  i, const Point &  p  )

inherited

Definition at line 657 of file fe_l2_hierarchic_shape_3D.C.

{
#if LIBMESH_DIM == 3

  libmesh_assert(elem);
  const ElemType type = elem->type();

  const Order totalorder = static_cast<Order>(order+elem->p_level());

  switch (type)
    {
    case HEX8:
    case HEX20:
      libmesh_assert_less (totalorder, 2);
      libmesh_fallthrough();
    case HEX27:
      {
        libmesh_assert_less (i, (totalorder+1u)*(totalorder+1u)*(totalorder+1u));

        // Compute hex shape functions as a tensor-product
        Real xi   = p(0);
        Real eta  = p(1);
        Real zeta = p(2);

        unsigned int i0, i1, i2;

        cube_indices(elem, totalorder, i, xi, eta, zeta, i0, i1, i2);

        return (FE<1,L2_HIERARCHIC>::shape(EDGE3, totalorder, i0, xi)*
                FE<1,L2_HIERARCHIC>::shape(EDGE3, totalorder, i1, eta)*
                FE<1,L2_HIERARCHIC>::shape(EDGE3, totalorder, i2, zeta));
      }

    default:
      libmesh_error_msg("Invalid element type = " << type);
    }

#endif
}

shape() [102/109]

Real libMesh::FE< 2, SUBDIVISION >::shape ( const ElemType  type, const Order  order, const unsigned int  i, const Point &  p  )

inherited

Definition at line 700 of file fe_subdivision_2D.C.

{
  switch (order)
    {
    case FOURTH:
      {
        switch (type)
          {
          case TRI3SUBDIVISION:
            libmesh_assert_less(i, 12);
            return FESubdivision::regular_shape(i,p(0),p(1));
          default:
            libmesh_error_msg("ERROR: Unsupported element type!");
          }
      }
    default:
      libmesh_error_msg("ERROR: Unsupported polynomial order!");
    }
}

shape() [103/109]

RealGradient libMesh::FE< 0, LAGRANGE_VEC >::shape ( const Elem *  elem, const Order  order, const unsigned int  i, const Point &  p  )

inherited

Definition at line 722 of file fe_lagrange_vec.C.

{
  Real value = FE<0,LAGRANGE>::shape( elem->type(), static_cast<Order>(order + elem->p_level()), i, p);
  return libMesh::RealGradient( value );
}

shape() [104/109]

Real libMesh::FE< 2, SUBDIVISION >::shape ( const Elem *  elem, const Order  order, const unsigned int  i, const Point &  p  )

inherited

Definition at line 726 of file fe_subdivision_2D.C.

{
  libmesh_assert(elem);
  return FE<2,SUBDIVISION>::shape(elem->type(), order, i, p);
}

shape() [105/109]

RealGradient libMesh::FE< 1, LAGRANGE_VEC >::shape ( const Elem *  elem, const Order  order, const unsigned int  i, const Point &  p  )

inherited

Definition at line 744 of file fe_lagrange_vec.C.

{
  Real value = FE<1,LAGRANGE>::shape( elem->type(), static_cast<Order>(order + elem->p_level()), i, p);
  return libMesh::RealGradient( value );
}

shape() [106/109]

RealGradient libMesh::FE< 2, LAGRANGE_VEC >::shape ( const Elem *  elem, const Order  order, const unsigned int  i, const Point &  p  )

inherited

Definition at line 766 of file fe_lagrange_vec.C.

{
  Real value = FE<2,LAGRANGE>::shape( elem->type(), static_cast<Order>(order + elem->p_level()), i/2, p );

  switch( i%2 )
    {
    case 0:
      return libMesh::RealGradient( value );

    case 1:
      return libMesh::RealGradient( Real(0), value );

    default:
      libmesh_error_msg("i%2 must be either 0 or 1!");
    }

  //dummy
  return libMesh::RealGradient();
}

shape() [107/109]

RealGradient libMesh::FE< 3, LAGRANGE_VEC >::shape ( const Elem *  elem, const Order  order, const unsigned int  i, const Point &  p  )

inherited

Definition at line 830 of file fe_lagrange_vec.C.

{
  Real value = FE<3,LAGRANGE>::shape( elem->type(), static_cast<Order>(order + elem->p_level()), i/3, p );

  switch( i%3 )
    {
    case 0:
      return libMesh::RealGradient( value );

    case 1:
      return libMesh::RealGradient( Real(0), value );

    case 2:
      return libMesh::RealGradient( Real(0), Real(0), value );

    default:
      libmesh_error_msg("i%3 must be 0, 1, or 2!");
    }

  //dummy
  return libMesh::RealGradient();
}

shape() [108/109]

Real libMesh::FE< 2, CLOUGH >::shape ( const ElemType  , const Order  , const unsigned  int, const Point &    )

inherited

Definition at line 1802 of file fe_clough_shape_2D.C.

{
  libmesh_error_msg("Clough-Tocher elements require the real element \nto construct gradient-based degrees of freedom.");
  return 0.;
}

shape() [109/109]

Real libMesh::FE< 2, CLOUGH >::shape ( const Elem *  elem, const Order  order, const unsigned int  i, const Point &  p  )

inherited

Definition at line 1814 of file fe_clough_shape_2D.C.

{
  libmesh_assert(elem);

  clough_compute_coefs(elem);

  const ElemType type = elem->type();

  const Order totalorder = static_cast<Order>(order + elem->p_level());

  switch (totalorder)
    {
      // 2nd-order restricted Clough-Tocher element
    case SECOND:
      {
        // There may be a bug in the 2nd order case; the 3rd order
        // Clough-Tocher elements are pretty uniformly better anyways
        // so use those instead.
        libmesh_experimental();
        switch (type)
          {
            // C1 functions on the Clough-Tocher triangle.
          case TRI6:
            {
              libmesh_assert_less (i, 9);
              // FIXME: it would be nice to calculate (and cache)
              // clough_raw_shape(j,p) only once per triangle, not 1-7
              // times
              switch (i)
                {
                  // Note: these DoF numbers are "scrambled" because my
                  // initial numbering conventions didn't match libMesh
                case 0:
                  return clough_raw_shape(0, p)
                    + d1d2n * clough_raw_shape(10, p)
                    + d1d3n * clough_raw_shape(11, p);
                case 3:
                  return clough_raw_shape(1, p)
                    + d2d3n * clough_raw_shape(11, p)
                    + d2d1n * clough_raw_shape(9, p);
                case 6:
                  return clough_raw_shape(2, p)
                    + d3d1n * clough_raw_shape(9, p)
                    + d3d2n * clough_raw_shape(10, p);
                case 1:
                  return d1xd1x * clough_raw_shape(3, p)
                    + d1xd1y * clough_raw_shape(4, p)
                    + d1xd2n * clough_raw_shape(10, p)
                    + d1xd3n * clough_raw_shape(11, p)
                    + 0.5 * N01x * d3nd3n * clough_raw_shape(11, p)
                    + 0.5 * N02x * d2nd2n * clough_raw_shape(10, p);
                case 2:
                  return d1yd1y * clough_raw_shape(4, p)
                    + d1yd1x * clough_raw_shape(3, p)
                    + d1yd2n * clough_raw_shape(10, p)
                    + d1yd3n * clough_raw_shape(11, p)
                    + 0.5 * N01y * d3nd3n * clough_raw_shape(11, p)
                    + 0.5 * N02y * d2nd2n * clough_raw_shape(10, p);
                case 4:
                  return d2xd2x * clough_raw_shape(5, p)
                    + d2xd2y * clough_raw_shape(6, p)
                    + d2xd3n * clough_raw_shape(11, p)
                    + d2xd1n * clough_raw_shape(9, p)
                    + 0.5 * N10x * d3nd3n * clough_raw_shape(11, p)
                    + 0.5 * N12x * d1nd1n * clough_raw_shape(9, p);
                case 5:
                  return d2yd2y * clough_raw_shape(6, p)
                    + d2yd2x * clough_raw_shape(5, p)
                    + d2yd3n * clough_raw_shape(11, p)
                    + d2yd1n * clough_raw_shape(9, p)
                    + 0.5 * N10y * d3nd3n * clough_raw_shape(11, p)
                    + 0.5 * N12y * d1nd1n * clough_raw_shape(9, p);
                case 7:
                  return d3xd3x * clough_raw_shape(7, p)
                    + d3xd3y * clough_raw_shape(8, p)
                    + d3xd1n * clough_raw_shape(9, p)
                    + d3xd2n * clough_raw_shape(10, p)
                    + 0.5 * N20x * d2nd2n * clough_raw_shape(10, p)
                    + 0.5 * N21x * d1nd1n * clough_raw_shape(9, p);
                case 8:
                  return d3yd3y * clough_raw_shape(8, p)
                    + d3yd3x * clough_raw_shape(7, p)
                    + d3yd1n * clough_raw_shape(9, p)
                    + d3yd2n * clough_raw_shape(10, p)
                    + 0.5 * N20y * d2nd2n * clough_raw_shape(10, p)
                    + 0.5 * N21y * d1nd1n * clough_raw_shape(9, p);
                default:
                  libmesh_error_msg("Invalid shape function index i = " << i);
                }
            }
          default:
            libmesh_error_msg("ERROR: Unsupported element type = " << type);
          }
      }
      // 3rd-order Clough-Tocher element
    case THIRD:
      {
        switch (type)
          {
            // C1 functions on the Clough-Tocher triangle.
          case TRI6:
            {
              libmesh_assert_less (i, 12);

              // FIXME: it would be nice to calculate (and cache)
              // clough_raw_shape(j,p) only once per triangle, not 1-7
              // times
              switch (i)
                {
                  // Note: these DoF numbers are "scrambled" because my
                  // initial numbering conventions didn't match libMesh
                case 0:
                  return clough_raw_shape(0, p)
                    + d1d2n * clough_raw_shape(10, p)
                    + d1d3n * clough_raw_shape(11, p);
                case 3:
                  return clough_raw_shape(1, p)
                    + d2d3n * clough_raw_shape(11, p)
                    + d2d1n * clough_raw_shape(9, p);
                case 6:
                  return clough_raw_shape(2, p)
                    + d3d1n * clough_raw_shape(9, p)
                    + d3d2n * clough_raw_shape(10, p);
                case 1:
                  return d1xd1x * clough_raw_shape(3, p)
                    + d1xd1y * clough_raw_shape(4, p)
                    + d1xd2n * clough_raw_shape(10, p)
                    + d1xd3n * clough_raw_shape(11, p);
                case 2:
                  return d1yd1y * clough_raw_shape(4, p)
                    + d1yd1x * clough_raw_shape(3, p)
                    + d1yd2n * clough_raw_shape(10, p)
                    + d1yd3n * clough_raw_shape(11, p);
                case 4:
                  return d2xd2x * clough_raw_shape(5, p)
                    + d2xd2y * clough_raw_shape(6, p)
                    + d2xd3n * clough_raw_shape(11, p)
                    + d2xd1n * clough_raw_shape(9, p);
                case 5:
                  return d2yd2y * clough_raw_shape(6, p)
                    + d2yd2x * clough_raw_shape(5, p)
                    + d2yd3n * clough_raw_shape(11, p)
                    + d2yd1n * clough_raw_shape(9, p);
                case 7:
                  return d3xd3x * clough_raw_shape(7, p)
                    + d3xd3y * clough_raw_shape(8, p)
                    + d3xd1n * clough_raw_shape(9, p)
                    + d3xd2n * clough_raw_shape(10, p);
                case 8:
                  return d3yd3y * clough_raw_shape(8, p)
                    + d3yd3x * clough_raw_shape(7, p)
                    + d3yd1n * clough_raw_shape(9, p)
                    + d3yd2n * clough_raw_shape(10, p);
                case 10:
                  return d1nd1n * clough_raw_shape(9, p);
                case 11:
                  return d2nd2n * clough_raw_shape(10, p);
                case 9:
                  return d3nd3n * clough_raw_shape(11, p);

                default:
                  libmesh_error_msg("Invalid shape function index i = " << i);
                }
            }
          default:
            libmesh_error_msg("ERROR: Unsupported element type = " << type);
          }
      }
      // by default throw an error
    default:
      libmesh_error_msg("ERROR: Unsupported polynomial order = " << order);
    }
}

shape_deriv() [1/109]

Real libMesh::FE< 3, SCALAR >::shape_deriv ( const ElemType  , const Order  , const unsigned  int, const unsigned  int, const Point &    )

inherited

Definition at line 47 of file fe_scalar_shape_3D.C.

{
  return 0.;
}

shape_deriv() [2/109]

Real libMesh::FE< 0, SCALAR >::shape_deriv ( const ElemType  , const Order  , const unsigned  int, const unsigned  int, const Point &    )

inherited

Definition at line 47 of file fe_scalar_shape_0D.C.

{
  return 0.;
}

shape_deriv() [3/109]

Real libMesh::FE< 1, SCALAR >::shape_deriv ( const ElemType  , const Order  , const unsigned  int, const unsigned  int, const Point &    )

inherited

Definition at line 47 of file fe_scalar_shape_1D.C.

{
  return 0.;
}

shape_deriv() [4/109]

Real libMesh::FE< 2, SCALAR >::shape_deriv ( const ElemType  , const Order  , const unsigned  int, const unsigned  int, const Point &    )

inherited

Definition at line 48 of file fe_scalar_shape_2D.C.

{
  return 0.;
}

shape_deriv() [5/109]

Real libMesh::FE< 3, SZABAB >::shape_deriv ( const ElemType  , const Order  , const unsigned  int, const unsigned  int, const Point &    )

inherited

Definition at line 55 of file fe_szabab_shape_3D.C.

{
  libmesh_error_msg("Szabo-Babuska polynomials are not defined in 3D");
  return 0.;
}

shape_deriv() [6/109]

Real libMesh::FE< 0, HERMITE >::shape_deriv ( const ElemType  , const Order  , const unsigned  int, const unsigned  int, const Point &    )

inherited

Definition at line 56 of file fe_hermite_shape_0D.C.

{
  libmesh_error_msg("No spatial derivatives in 0D!");
  return 0.;
}

shape_deriv() [7/109]

Real libMesh::FE< 0, HIERARCHIC >::shape_deriv ( const ElemType  , const Order  , const unsigned  int, const unsigned  int, const Point &    )

inherited

Definition at line 56 of file fe_hierarchic_shape_0D.C.

{
  libmesh_error_msg("No spatial derivatives in 0D!");
  return 0.;
}

shape_deriv() [8/109]

Real libMesh::FE< 0, L2_HIERARCHIC >::shape_deriv ( const ElemType  , const Order  , const unsigned  int, const unsigned  int, const Point &    )

inherited

Definition at line 56 of file fe_l2_hierarchic_shape_0D.C.

{
  libmesh_error_msg("No spatial derivatives in 0D!");
  return 0.;
}

shape_deriv() [9/109]

Real libMesh::FE< 0, L2_LAGRANGE >::shape_deriv ( const ElemType  , const Order  , const unsigned  int, const unsigned  int, const Point &    )

inherited

Definition at line 56 of file fe_l2_lagrange_shape_0D.C.

{
  libmesh_error_msg("No spatial derivatives in 0D!");
  return 0.;
}

shape_deriv() [10/109]

Real libMesh::FE< 0, LAGRANGE >::shape_deriv ( const ElemType  , const Order  , const unsigned  int, const unsigned  int, const Point &    )

inherited

Definition at line 56 of file fe_lagrange_shape_0D.C.

{
  libmesh_error_msg("No spatial derivatives in 0D!");
  return 0.;
}

shape_deriv() [11/109]

Real libMesh::FE< 0, MONOMIAL >::shape_deriv ( const ElemType  , const Order  , const unsigned  int, const unsigned  int, const Point &    )

inherited

Definition at line 56 of file fe_monomial_shape_0D.C.

{
  libmesh_error_msg("No spatial derivatives in 0D!");
  return 0.;
}

shape_deriv() [12/109]

Real libMesh::FE< 0, SZABAB >::shape_deriv ( const ElemType  , const Order  , const unsigned  int, const unsigned  int, const Point &    )

inherited

Definition at line 56 of file fe_szabab_shape_0D.C.

{
  libmesh_error_msg("No spatial derivatives in 0D!");
  return 0.;
}

shape_deriv() [13/109]

Real libMesh::FE< 0, XYZ >::shape_deriv ( const ElemType  , const Order  , const unsigned  int, const unsigned  int, const Point &    )

inherited

Definition at line 56 of file fe_xyz_shape_0D.C.

{
  libmesh_error_msg("No spatial derivatives in 0D!");
  return 0.;
}

shape_deriv() [14/109]

Real libMesh::FE< 0, BERNSTEIN >::shape_deriv ( const ElemType  , const Order  , const unsigned  int, const unsigned  int, const Point &    )

inherited

Definition at line 56 of file fe_bernstein_shape_0D.C.

{
  libmesh_error_msg("No spatial derivatives in 0D!");
  return 0.;
}

shape_deriv() [15/109]

Real libMesh::FE< 0, CLOUGH >::shape_deriv ( const ElemType  , const Order  , const unsigned  int, const unsigned  int, const Point &    )

inherited

Definition at line 56 of file fe_clough_shape_0D.C.

{
  libmesh_error_msg("No spatial derivatives in 0D!");
  return 0.;
}

shape_deriv() [16/109]

Real libMesh::FE< 0, SCALAR >::shape_deriv ( const Elem *  , const Order  , const unsigned  int, const unsigned  int, const Point &    )

inherited

Definition at line 57 of file fe_scalar_shape_0D.C.

{
  return 0.;
}

shape_deriv() [17/109]

Real libMesh::FE< 3, SCALAR >::shape_deriv ( const Elem *  , const Order  , const unsigned  int, const unsigned  int, const Point &    )

inherited

Definition at line 57 of file fe_scalar_shape_3D.C.

{
  return 0.;
}

shape_deriv() [18/109]

Real libMesh::FE< 1, SCALAR >::shape_deriv ( const Elem *  , const Order  , const unsigned  int, const unsigned  int, const Point &    )

inherited

Definition at line 57 of file fe_scalar_shape_1D.C.

{
  return 0.;
}

shape_deriv() [19/109]

Real libMesh::FE< 2, SCALAR >::shape_deriv ( const Elem *  , const Order  , const unsigned  int, const unsigned  int, const Point &    )

inherited

Definition at line 58 of file fe_scalar_shape_2D.C.

{
  return 0.;
}

shape_deriv() [20/109]

Real libMesh::FE< 3, CLOUGH >::shape_deriv ( const ElemType  , const Order  , const unsigned  int, const unsigned  int, const Point &    )

inherited

Definition at line 59 of file fe_clough_shape_3D.C.

{
  libmesh_error_msg("Clough-Tocher elements require the real element \nto construct gradient-based degrees of freedom.");
  return 0.;
}

shape_deriv() [21/109]

Real libMesh::FE< 3, SZABAB >::shape_deriv ( const Elem *  , const Order  , const unsigned  int, const unsigned  int, const Point &    )

inherited

Definition at line 68 of file fe_szabab_shape_3D.C.

{
  libmesh_error_msg("Szabo-Babuska polynomials are not defined in 3D");
  return 0.;
}

shape_deriv() [22/109]

Real libMesh::FE< 0, HIERARCHIC >::shape_deriv ( const Elem *  , const Order  , const unsigned  int, const unsigned  int, const Point &    )

inherited

Definition at line 69 of file fe_hierarchic_shape_0D.C.

{
  libmesh_error_msg("No spatial derivatives in 0D!");
  return 0.;
}

shape_deriv() [23/109]

Real libMesh::FE< 0, L2_LAGRANGE >::shape_deriv ( const Elem *  , const Order  , const unsigned  int, const unsigned  int, const Point &    )

inherited

Definition at line 69 of file fe_l2_lagrange_shape_0D.C.

{
  libmesh_error_msg("No spatial derivatives in 0D!");
  return 0.;
}

shape_deriv() [24/109]

Real libMesh::FE< 0, L2_HIERARCHIC >::shape_deriv ( const Elem *  , const Order  , const unsigned  int, const unsigned  int, const Point &    )

inherited

Definition at line 69 of file fe_l2_hierarchic_shape_0D.C.

{
  libmesh_error_msg("No spatial derivatives in 0D!");
  return 0.;
}

shape_deriv() [25/109]

Real libMesh::FE< 0, MONOMIAL >::shape_deriv ( const Elem *  , const Order  , const unsigned  int, const unsigned  int, const Point &    )

inherited

Definition at line 69 of file fe_monomial_shape_0D.C.

{
  libmesh_error_msg("No spatial derivatives in 0D!");
  return 0.;
}

shape_deriv() [26/109]

Real libMesh::FE< 0, SZABAB >::shape_deriv ( const Elem *  , const Order  , const unsigned  int, const unsigned  int, const Point &    )

inherited

Definition at line 69 of file fe_szabab_shape_0D.C.

{
  libmesh_error_msg("No spatial derivatives in 0D!");
  return 0.;
}

shape_deriv() [27/109]

Real libMesh::FE< 0, LAGRANGE >::shape_deriv ( const Elem *  , const Order  , const unsigned  int, const unsigned  int, const Point &    )

inherited

Definition at line 69 of file fe_lagrange_shape_0D.C.

{
  libmesh_error_msg("No spatial derivatives in 0D!");
  return 0.;
}

shape_deriv() [28/109]

Real libMesh::FE< 0, XYZ >::shape_deriv ( const Elem *  , const Order  , const unsigned  int, const unsigned  int, const Point &    )

inherited

Definition at line 69 of file fe_xyz_shape_0D.C.

{
  libmesh_error_msg("No spatial derivatives in 0D!");
  return 0.;
}

shape_deriv() [29/109]

Real libMesh::FE< 0, HERMITE >::shape_deriv ( const Elem *  , const Order  , const unsigned  int, const unsigned  int, const Point &    )

inherited

Definition at line 69 of file fe_hermite_shape_0D.C.

{
  libmesh_error_msg("No spatial derivatives in 0D!");
  return 0.;
}

shape_deriv() [30/109]

Real libMesh::FE< 0, CLOUGH >::shape_deriv ( const Elem *  , const Order  , const unsigned  int, const unsigned  int, const Point &    )

inherited

Definition at line 69 of file fe_clough_shape_0D.C.

{
  libmesh_error_msg("No spatial derivatives in 0D!");
  return 0.;
}

shape_deriv() [31/109]

Real libMesh::FE< 0, BERNSTEIN >::shape_deriv ( const Elem *  , const Order  , const unsigned  int, const unsigned  int, const Point &    )

inherited

Definition at line 69 of file fe_bernstein_shape_0D.C.

{
  libmesh_error_msg("No spatial derivatives in 0D!");
  return 0.;
}

shape_deriv() [32/109]

Real libMesh::FE< 3, CLOUGH >::shape_deriv ( const Elem *  libmesh_dbg_varelem, const Order  , const unsigned  int, const unsigned  int, const Point &    )

inherited

Definition at line 72 of file fe_clough_shape_3D.C.

{
  libmesh_assert(elem);
  libmesh_not_implemented();
  return 0.;
}

shape_deriv() [33/109]

Real libMesh::FE< 1, MONOMIAL >::shape_deriv ( const ElemType  , const Order  libmesh_dbg_varorder, const unsigned int  i, const unsigned int  libmesh_dbg_varj, const Point &  p  )

inherited

Definition at line 83 of file fe_monomial_shape_1D.C.

{
  // only d()/dxi in 1D!

  libmesh_assert_equal_to (j, 0);

  const Real xi = p(0);

  libmesh_assert_less_equal (i, static_cast<unsigned int>(order));

  // monomials. since they are hierarchic we only need one case block.
  switch (i)
    {
    case 0:
      return 0.;

    case 1:
      return 1.;

    case 2:
      return 2.*xi;

    case 3:
      return 3.*xi*xi;

    case 4:
      return 4.*xi*xi*xi;

    default:
      Real val = i;
      for (unsigned int index = 1; index != i; ++index)
        val *= xi;
      return val;
    }
}

shape_deriv() [34/109]

Real libMesh::FE< 1, SZABAB >::shape_deriv ( const ElemType  , const Order  libmesh_dbg_varorder, const unsigned int  i, const unsigned int  libmesh_dbg_varj, const Point &  p  )

inherited

Definition at line 92 of file fe_szabab_shape_1D.C.

{
  // only d()/dxi in 1D!
  libmesh_assert_equal_to (j, 0);

  const Real xi  = p(0);
  const Real xi2 = xi*xi;

  // Use this libmesh_assert rather than a switch with a single entry...
  // It will go away in optimized mode, essentially has the same effect.
  libmesh_assert_less_equal (order, SEVENTH);

  //   switch (order)
  //     {
  //     case FIRST:
  //     case SECOND:
  //     case THIRD:
  //     case FOURTH:
  //     case FIFTH:
  //     case SIXTH:
  //     case SEVENTH:

  switch(i)
    {
    case 0:return -1./2.;
    case 1:return 1./2.;
    case 2:return 1./2.*2.4494897427831780982*xi;
    case 3:return -1./4.*3.1622776601683793320+3./4.*3.1622776601683793320*xi2;
    case 4:return 1./16.*3.7416573867739413856*(-12.+20*xi2)*xi;
    case 5:return 9./16.*1.4142135623730950488+(-45./8.*1.4142135623730950488+105./16.*1.4142135623730950488*xi2)*xi2;
    case 6:return 1./32.*4.6904157598234295546*(30.+(-140.+126.*xi2)*xi2)*xi;
    case 7:return -5./32.*5.0990195135927848300+(105./32.*5.0990195135927848300+(-315./32.*5.0990195135927848300+231./32.*5.0990195135927848300*xi2)*xi2)*xi2;
    case 8:return 1./256.*5.4772255750516611346*(-280.+(2520.+(-5544.+3432.*xi2)*xi2)*xi2)*xi;

    default:
      libmesh_error_msg("Invalid shape function index!");
    }
}

shape_deriv() [35/109]

Real libMesh::FE< 1, XYZ >::shape_deriv ( const ElemType  , const Order  , const unsigned  int, const unsigned  int, const Point &    )

inherited

Definition at line 93 of file fe_xyz_shape_1D.C.

{
  libmesh_error_msg("XYZ polynomials require the element \nbecause the centroid is needed.");
  return 0.;
}

shape_deriv() [36/109]

Real libMesh::FE< 1, XYZ >::shape_deriv ( const Elem *  elem, const Order  libmesh_dbg_varorder, const unsigned int  i, const unsigned int  libmesh_dbg_varj, const Point &  point_in  )

inherited

Definition at line 106 of file fe_xyz_shape_1D.C.

{
  libmesh_assert(elem);
  libmesh_assert_less_equal (i, order + elem->p_level());

  // only d()/dxi in 1D!

  libmesh_assert_equal_to (j, 0);

  Point centroid = elem->centroid();
  Real max_distance = 0.;
  for (unsigned int p = 0; p < elem->n_nodes(); p++)
    {
      const Real distance = std::abs(centroid(0) - elem->point(p)(0));
      max_distance = std::max(distance, max_distance);
    }

  const Real x  = point_in(0);
  const Real xc = centroid(0);
  const Real dx = (x - xc)/max_distance;

  // monomials. since they are hierarchic we only need one case block.
  switch (i)
    {
    case 0:
      return 0.;

    case 1:
      return 1.;

    case 2:
      return 2.*dx/max_distance;

    case 3:
      return 3.*dx*dx/max_distance;

    case 4:
      return 4.*dx*dx*dx/max_distance;

    default:
      Real val = i;
      for (unsigned int index = 1; index != i; ++index)
        val *= dx;
      return val/max_distance;
    }
}

shape_deriv() [37/109]

Real libMesh::FE< 1, HIERARCHIC >::shape_deriv ( const ElemType  , const Order  libmesh_dbg_varorder, const unsigned int  i, const unsigned int  libmesh_dbg_varj, const Point &  p  )

inherited

Definition at line 114 of file fe_hierarchic_shape_1D.C.

{
  // only d()/dxi in 1D!

  libmesh_assert_equal_to (j, 0);
  libmesh_assert_less (i, order+1u);

  // Declare that we are using our own special power function
  // from the Utility namespace.  This saves typing later.
  using Utility::pow;

  const Real xi = p(0);

  Real returnval = 1.;

  switch (i)
    {
    case 0:
      returnval = -.5;
      break;
    case 1:
      returnval =  .5;
      break;
      // All even-terms have the same form.
      // xi^(p-1)/(p-1)!
    case 2:
      returnval = xi;
      break;
    case 4:
      returnval = pow<3>(xi)/6.;
      break;
    case 6:
      returnval = pow<5>(xi)/120.;
      break;
      // All odd-terms have the same form.
      // (p*xi^(p-1) - 1.)/p!
    case 3:
      returnval = (3*xi*xi - 1.)/6.;
      break;
    case 5:
      returnval = (5.*pow<4>(xi) - 1.)/120.;
      break;
    case 7:
      returnval = (7.*pow<6>(xi) - 1.)/5040.;
      break;
    default:
      Real denominator = 1.;
      for (unsigned int n=1; n != i; ++n)
        {
          returnval *= xi;
          denominator *= n;
        }
      // Odd:
      if (i % 2)
        returnval = (i * returnval - 1.)/denominator/i;
      // Even:
      else
        returnval = returnval/denominator;
      break;
    }

  return returnval;
}

shape_deriv() [38/109]

Real libMesh::FE< 1, L2_HIERARCHIC >::shape_deriv ( const ElemType  , const Order  libmesh_dbg_varorder, const unsigned int  i, const unsigned int  libmesh_dbg_varj, const Point &  p  )

inherited

Definition at line 114 of file fe_l2_hierarchic_shape_1D.C.

{
  // only d()/dxi in 1D!

  libmesh_assert_equal_to (j, 0);
  libmesh_assert_less (i, order+1u);

  // Declare that we are using our own special power function
  // from the Utility namespace.  This saves typing later.
  using Utility::pow;

  const Real xi = p(0);

  Real returnval = 1.;

  switch (i)
    {
    case 0:
      returnval = -.5;
      break;
    case 1:
      returnval =  .5;
      break;
      // All even-terms have the same form.
      // xi^(p-1)/(p-1)!
    case 2:
      returnval = xi;
      break;
    case 4:
      returnval = pow<3>(xi)/6.;
      break;
    case 6:
      returnval = pow<5>(xi)/120.;
      break;
      // All odd-terms have the same form.
      // (p*xi^(p-1) - 1.)/p!
    case 3:
      returnval = (3*xi*xi - 1.)/6.;
      break;
    case 5:
      returnval = (5.*pow<4>(xi) - 1.)/120.;
      break;
    case 7:
      returnval = (7.*pow<6>(xi) - 1.)/5040.;
      break;
    default:
      Real denominator = 1.;
      for (unsigned int n=1; n != i; ++n)
        {
          returnval *= xi;
          denominator *= n;
        }
      // Odd:
      if (i % 2)
        returnval = (i * returnval - 1.)/denominator/i;
      // Even:
      else
        returnval = returnval/denominator;
      break;
    }

  return returnval;
}

shape_deriv() [39/109]

Real libMesh::FE< 1, MONOMIAL >::shape_deriv ( const Elem *  elem, const Order  order, const unsigned int  i, const unsigned int  j, const Point &  p  )

inherited

Definition at line 126 of file fe_monomial_shape_1D.C.

{
  libmesh_assert(elem);

  return FE<1,MONOMIAL>::shape_deriv(elem->type(),
                                     static_cast<Order>(order + elem->p_level()), i, j, p);
}

shape_deriv() [40/109]

Real libMesh::FE< 1, L2_LAGRANGE >::shape_deriv ( const ElemType  , const Order  order, const unsigned int  i, const unsigned int  libmesh_dbg_varj, const Point &  p  )

inherited

Definition at line 130 of file fe_l2_lagrange_shape_1D.C.

{
  // only d()/dxi in 1D!

  libmesh_assert_equal_to (j, 0);

  const Real xi = p(0);


  switch (order)
    {
      // Lagrange linear shape function derivatives
    case FIRST:
      {
        libmesh_assert_less (i, 2);

        switch (i)
          {
          case 0:
            return -.5;

          case 1:
            return .5;

          default:
            libmesh_error_msg("Invalid shape function index i = " << i);
          }
      }


      // Lagrange quadratic shape function derivatives
    case SECOND:
      {
        libmesh_assert_less (i, 3);

        switch (i)
          {
          case 0:
            return xi-.5;

          case 1:
            return xi+.5;

          case 2:
            return -2.*xi;

          default:
            libmesh_error_msg("Invalid shape function index i = " << i);
          }
      }


      // Lagrange cubic shape function derivatives
    case THIRD:
      {
        libmesh_assert_less (i, 4);

        switch (i)
          {
          case 0:
            return -9./16.*(3.*xi*xi-2.*xi-1./9.);

          case 1:
            return -9./16.*(-3.*xi*xi-2.*xi+1./9.);

          case 2:
            return 27./16.*(3.*xi*xi-2./3.*xi-1.);

          case 3:
            return 27./16.*(-3.*xi*xi-2./3.*xi+1.);

          default:
            libmesh_error_msg("Invalid shape function index i = " << i);
          }
      }


    default:
      libmesh_error_msg("ERROR: Unsupported polynomial order = " << order);
    }
}

shape_deriv() [41/109]

Real libMesh::FE< 1, LAGRANGE >::shape_deriv ( const ElemType  , const Order  order, const unsigned int  i, const unsigned int  libmesh_dbg_varj, const Point &  p  )

inherited

Definition at line 130 of file fe_lagrange_shape_1D.C.

{
  // only d()/dxi in 1D!

  libmesh_assert_equal_to (j, 0);

  const Real xi = p(0);


  switch (order)
    {
      // Lagrange linear shape function derivatives
    case FIRST:
      {
        libmesh_assert_less (i, 2);

        switch (i)
          {
          case 0:
            return -.5;

          case 1:
            return .5;

          default:
            libmesh_error_msg("Invalid shape function index i = " << i);
          }
      }


      // Lagrange quadratic shape function derivatives
    case SECOND:
      {
        libmesh_assert_less (i, 3);

        switch (i)
          {
          case 0:
            return xi-.5;

          case 1:
            return xi+.5;

          case 2:
            return -2.*xi;

          default:
            libmesh_error_msg("Invalid shape function index i = " << i);
          }
      }


      // Lagrange cubic shape function derivatives
    case THIRD:
      {
        libmesh_assert_less (i, 4);

        switch (i)
          {
          case 0:
            return -9./16.*(3.*xi*xi-2.*xi-1./9.);

          case 1:
            return -9./16.*(-3.*xi*xi-2.*xi+1./9.);

          case 2:
            return 27./16.*(3.*xi*xi-2./3.*xi-1.);

          case 3:
            return 27./16.*(-3.*xi*xi-2./3.*xi+1.);

          default:
            libmesh_error_msg("Invalid shape function index i = " << i);
          }
      }


    default:
      libmesh_error_msg("ERROR: Unsupported polynomial order = " << order);
    }
}

shape_deriv() [42/109]

Real libMesh::FE< 2, MONOMIAL >::shape_deriv ( const ElemType  , const Order  libmesh_dbg_varorder, const unsigned int  i, const unsigned int  j, const Point &  p  )

inherited

Definition at line 132 of file fe_monomial_shape_2D.C.

{
#if LIBMESH_DIM > 1


  libmesh_assert_less (j, 2);

  libmesh_assert_less (i, (static_cast<unsigned int>(order)+1)*
                       (static_cast<unsigned int>(order)+2)/2);

  const Real xi  = p(0);
  const Real eta = p(1);

  // monomials. since they are hierarchic we only need one case block.

  switch (j)
    {
      // d()/dxi
    case 0:
      {
        switch (i)
          {
            // constants
          case 0:
            return 0.;

            // linears
          case 1:
            return 1.;

          case 2:
            return 0.;

            // quadratics
          case 3:
            return 2.*xi;

          case 4:
            return eta;

          case 5:
            return 0.;

            // cubics
          case 6:
            return 3.*xi*xi;

          case 7:
            return 2.*xi*eta;

          case 8:
            return eta*eta;

          case 9:
            return 0.;

            // quartics
          case 10:
            return 4.*xi*xi*xi;

          case 11:
            return 3.*xi*xi*eta;

          case 12:
            return 2.*xi*eta*eta;

          case 13:
            return eta*eta*eta;

          case 14:
            return 0.;

          default:
            unsigned int o = 0;
            for (; i >= (o+1)*(o+2)/2; o++) { }
            unsigned int ny = i - (o*(o+1)/2);
            unsigned int nx = o - ny;
            Real val = nx;
            for (unsigned int index=1; index < nx; index++)
              val *= xi;
            for (unsigned int index=0; index != ny; index++)
              val *= eta;
            return val;
          }
      }


      // d()/deta
    case 1:
      {
        switch (i)
          {
            // constants
          case 0:
            return 0.;

            // linears
          case 1:
            return 0.;

          case 2:
            return 1.;

            // quadratics
          case 3:
            return 0.;

          case 4:
            return xi;

          case 5:
            return 2.*eta;

            // cubics
          case 6:
            return 0.;

          case 7:
            return xi*xi;

          case 8:
            return 2.*xi*eta;

          case 9:
            return 3.*eta*eta;

            // quartics
          case 10:
            return 0.;

          case 11:
            return xi*xi*xi;

          case 12:
            return 2.*xi*xi*eta;

          case 13:
            return 3.*xi*eta*eta;

          case 14:
            return 4.*eta*eta*eta;

          default:
            unsigned int o = 0;
            for (; i >= (o+1)*(o+2)/2; o++) { }
            unsigned int ny = i - (o*(o+1)/2);
            unsigned int nx = o - ny;
            Real val = ny;
            for (unsigned int index=0; index != nx; index++)
              val *= xi;
            for (unsigned int index=1; index < ny; index++)
              val *= eta;
            return val;
          }
      }

    default:
      libmesh_error_msg("Invalid shape function derivative j = " << j);
    }

#else
  return 0.;
#endif
}

shape_deriv() [43/109]

static OutputShape libMesh::FE< Dim, T >::shape_deriv ( const ElemType  t, const Order  o, const unsigned int  i, const unsigned int  j, const Point &  p  )

staticinherited

Returns

The derivative of the shape function at point p. This method allows you to specify the dimension, element type, and order directly.

On a p-refined element, o should be the total order of the element.

shape_deriv() [44/109]

Real libMesh::FE< 1, SZABAB >::shape_deriv ( const Elem *  elem, const Order  order, const unsigned int  i, const unsigned int  j, const Point &  p  )

inherited

Definition at line 138 of file fe_szabab_shape_1D.C.

{
  libmesh_assert(elem);

  return FE<1,SZABAB>::shape_deriv(elem->type(),
                                   static_cast<Order>(order + elem->p_level()), i, j, p);
}

shape_deriv() [45/109]

static OutputShape libMesh::FE< Dim, T >::shape_deriv ( const Elem *  elem, const Order  o, const unsigned int  i, const unsigned int  j, const Point &  p  )

staticinherited

Returns

The derivative of the shape function. You must specify element type, and order directly.

On a p-refined element, o should be the base order of the element.

shape_deriv() [46/109]

Real libMesh::FE< 2, XYZ >::shape_deriv ( const ElemType  , const Order  , const unsigned  int, const unsigned  int, const Point &    )

inherited

Definition at line 152 of file fe_xyz_shape_2D.C.

{
  libmesh_error_msg("XYZ polynomials require the element \nbecause the centroid is needed.");
  return 0.;
}

shape_deriv() [47/109]

RealGradient libMesh::FE< 2, NEDELEC_ONE >::shape_deriv ( const ElemType  , const Order  , const unsigned  int, const unsigned  int, const Point &    )

inherited

Definition at line 165 of file fe_nedelec_one_shape_2D.C.

{
  libmesh_error_msg("Nedelec elements require the element type \nbecause edge orientation is needed.");
  return RealGradient();
}

shape_deriv() [48/109]

Real libMesh::FE< 2, XYZ >::shape_deriv ( const Elem *  elem, const Order  libmesh_dbg_varorder, const unsigned int  i, const unsigned int  j, const Point &  point_in  )

inherited

Definition at line 165 of file fe_xyz_shape_2D.C.

{
#if LIBMESH_DIM > 1


  libmesh_assert_less (j, 2);
  libmesh_assert(elem);

  Point centroid = elem->centroid();
  Point max_distance = Point(0.,0.,0.);
  for (unsigned int p = 0; p < elem->n_nodes(); p++)
    for (unsigned int d = 0; d < 2; d++)
      {
        const Real distance = std::abs(centroid(d) - elem->point(p)(d));
        max_distance(d) = std::max(distance, max_distance(d));
      }

  const Real x  = point_in(0);
  const Real y  = point_in(1);
  const Real xc = centroid(0);
  const Real yc = centroid(1);
  const Real distx = max_distance(0);
  const Real disty = max_distance(1);
  const Real dx = (x - xc)/distx;
  const Real dy = (y - yc)/disty;

#ifndef NDEBUG
  // totalorder is only used in the assertion below, so
  // we avoid declaring it when asserts are not active.
  const unsigned int totalorder = order + elem->p_level();
#endif
  libmesh_assert_less (i, (totalorder+1)*(totalorder+2)/2);

  // monomials. since they are hierarchic we only need one case block.

  switch (j)
    {
      // d()/dx
    case 0:
      {
        switch (i)
          {
            // constants
          case 0:
            return 0.;

            // linears
          case 1:
            return 1./distx;

          case 2:
            return 0.;

            // quadratics
          case 3:
            return 2.*dx/distx;

          case 4:
            return dy/distx;

          case 5:
            return 0.;

            // cubics
          case 6:
            return 3.*dx*dx/distx;

          case 7:
            return 2.*dx*dy/distx;

          case 8:
            return dy*dy/distx;

          case 9:
            return 0.;

            // quartics
          case 10:
            return 4.*dx*dx*dx/distx;

          case 11:
            return 3.*dx*dx*dy/distx;

          case 12:
            return 2.*dx*dy*dy/distx;

          case 13:
            return dy*dy*dy/distx;

          case 14:
            return 0.;

          default:
            unsigned int o = 0;
            for (; i >= (o+1)*(o+2)/2; o++) { }
            unsigned int i2 = i - (o*(o+1)/2);
            Real val = o - i2;
            for (unsigned int index=i2+1; index < o; index++)
              val *= dx;
            for (unsigned int index=0; index != i2; index++)
              val *= dy;
            return val/distx;
          }
      }


      // d()/dy
    case 1:
      {
        switch (i)
          {
            // constants
          case 0:
            return 0.;

            // linears
          case 1:
            return 0.;

          case 2:
            return 1./disty;

            // quadratics
          case 3:
            return 0.;

          case 4:
            return dx/disty;

          case 5:
            return 2.*dy/disty;

            // cubics
          case 6:
            return 0.;

          case 7:
            return dx*dx/disty;

          case 8:
            return 2.*dx*dy/disty;

          case 9:
            return 3.*dy*dy/disty;

            // quartics
          case 10:
            return 0.;

          case 11:
            return dx*dx*dx/disty;

          case 12:
            return 2.*dx*dx*dy/disty;

          case 13:
            return 3.*dx*dy*dy/disty;

          case 14:
            return 4.*dy*dy*dy/disty;

          default:
            unsigned int o = 0;
            for (; i >= (o+1)*(o+2)/2; o++) { }
            unsigned int i2 = i - (o*(o+1)/2);
            Real val = i2;
            for (unsigned int index=i2; index != o; index++)
              val *= dx;
            for (unsigned int index=1; index <= i2; index++)
              val *= dy;
            return val/disty;
          }
      }


    default:
      libmesh_error_msg("Invalid j = " << j);
    }

#else
  return 0.;
#endif
}

shape_deriv() [49/109]

RealGradient libMesh::FE< 2, NEDELEC_ONE >::shape_deriv ( const Elem *  elem, const Order  order, const unsigned int  i, const unsigned int  j, const Point &    )

inherited

Definition at line 178 of file fe_nedelec_one_shape_2D.C.

{
#if LIBMESH_DIM > 1
  libmesh_assert(elem);
  libmesh_assert_less (j, 2);

  const Order total_order = static_cast<Order>(order + elem->p_level());

  switch (total_order)
    {
      // linear Lagrange shape functions
    case FIRST:
      {
        switch (elem->type())
          {
          case QUAD8:
          case QUAD9:
            {
              libmesh_assert_less (i, 4);

              switch (j)
                {
                  // d()/dxi
                case 0:
                  {
                    switch(i)
                      {
                      case 0:
                      case 2:
                        return RealGradient();
                      case 1:
                        {
                          if (elem->point(1) > elem->point(2))
                            return RealGradient( 0.0, -0.25 );
                          else
                            return RealGradient( 0.0, 0.25 );
                        }
                      case 3:
                        {
                          if (elem->point(3) > elem->point(0))
                            return RealGradient( 0.0, -0.25 );
                          else
                            return RealGradient( 0.0, 0.25 );
                        }
                      default:
                        libmesh_error_msg("Invalid i = " << i);
                      }
                  } // j=0

                  // d()/deta
                case 1:
                  {
                    switch(i)
                      {
                      case 1:
                      case 3:
                        return RealGradient();
                      case 0:
                        {
                          if (elem->point(0) > elem->point(1))
                            return RealGradient( 0.25 );
                          else
                            return RealGradient( -0.25 );
                        }
                      case 2:
                        {
                          if (elem->point(2) > elem->point(3))
                            return RealGradient( 0.25 );
                          else
                            return RealGradient( -0.25 );
                        }
                      default:
                        libmesh_error_msg("Invalid i = " << i);
                      }
                  } // j=1

                default:
                  libmesh_error_msg("Invalid j = " << j);
                }

              return RealGradient();
            }

          case TRI6:
            {
              libmesh_assert_less (i, 3);

              // Account for edge flipping
              Real f = 1.0;

              switch(i)
                {
                case 0:
                  {
                    if (elem->point(0) > elem->point(1))
                      f = -1.0;
                    break;
                  }
                case 1:
                  {
                    if (elem->point(1) > elem->point(2))
                      f = -1.0;
                    break;
                  }
                case 2:
                  {
                    if (elem->point(2) > elem->point(0))
                      f = -1.0;
                    break;
                  }
                default:
                  libmesh_error_msg("Invalid i = " << i);
                }

              switch (j)
                {
                  // d()/dxi
                case 0:
                  {
                    return RealGradient( 0.0, f*1.0);
                  }
                  // d()/deta
                case 1:
                  {
                    return RealGradient( f*(-1.0) );
                  }
                default:
                  libmesh_error_msg("Invalid j = " << j);
                }
            }

          default:
            libmesh_error_msg("ERROR: Unsupported 2D element type!: " << elem->type());
          }
      }
      // unsupported order
    default:
      libmesh_error_msg("ERROR: Unsupported 2D FE order!: " << total_order);
    }
#else // LIBMESH_DIM > 1
  return RealGradient();
#endif
}

shape_deriv() [50/109]

Real libMesh::FE< 1, HIERARCHIC >::shape_deriv ( const Elem *  elem, const Order  order, const unsigned int  i, const unsigned int  j, const Point &  p  )

inherited

Definition at line 185 of file fe_hierarchic_shape_1D.C.

{
  libmesh_assert(elem);

  return FE<1,HIERARCHIC>::shape_deriv(elem->type(),
                                       static_cast<Order>(order + elem->p_level()), i, j, p);
}

shape_deriv() [51/109]

Real libMesh::FE< 1, L2_HIERARCHIC >::shape_deriv ( const Elem *  elem, const Order  order, const unsigned int  i, const unsigned int  j, const Point &  p  )

inherited

Definition at line 185 of file fe_l2_hierarchic_shape_1D.C.

{
  libmesh_assert(elem);

  return FE<1,L2_HIERARCHIC>::shape_deriv(elem->type(),
                                          static_cast<Order>(order + elem->p_level()), i, j, p);
}

shape_deriv() [52/109]

RealGradient libMesh::FE< 3, NEDELEC_ONE >::shape_deriv ( const ElemType  , const Order  , const unsigned  int, const unsigned  int, const Point &    )

inherited

Definition at line 194 of file fe_nedelec_one_shape_3D.C.

{
  libmesh_error_msg("Nedelec elements require the element type \nbecause edge orientation is needed.");
  return RealGradient();
}

shape_deriv() [53/109]

Real libMesh::FE< 3, MONOMIAL >::shape_deriv ( const ElemType  , const Order  libmesh_dbg_varorder, const unsigned int  i, const unsigned int  j, const Point &  p  )

inherited

Definition at line 200 of file fe_monomial_shape_3D.C.

{
#if LIBMESH_DIM == 3

  libmesh_assert_less (j, 3);

  libmesh_assert_less (i, (static_cast<unsigned int>(order)+1)*
                       (static_cast<unsigned int>(order)+2)*
                       (static_cast<unsigned int>(order)+3)/6);


  const Real xi   = p(0);
  const Real eta  = p(1);
  const Real zeta = p(2);

  // monomials. since they are hierarchic we only need one case block.
  switch (j)
    {
      // d()/dxi
    case 0:
      {
        switch (i)
          {
            // constant
          case 0:
            return 0.;

            // linear
          case 1:
            return 1.;

          case 2:
            return 0.;

          case 3:
            return 0.;

            // quadratic
          case 4:
            return 2.*xi;

          case 5:
            return eta;

          case 6:
            return 0.;

          case 7:
            return zeta;

          case 8:
            return 0.;

          case 9:
            return 0.;

            // cubic
          case 10:
            return 3.*xi*xi;

          case 11:
            return 2.*xi*eta;

          case 12:
            return eta*eta;

          case 13:
            return 0.;

          case 14:
            return 2.*xi*zeta;

          case 15:
            return eta*zeta;

          case 16:
            return 0.;

          case 17:
            return zeta*zeta;

          case 18:
            return 0.;

          case 19:
            return 0.;

            // quartics
          case 20:
            return 4.*xi*xi*xi;

          case 21:
            return 3.*xi*xi*eta;

          case 22:
            return 2.*xi*eta*eta;

          case 23:
            return eta*eta*eta;

          case 24:
            return 0.;

          case 25:
            return 3.*xi*xi*zeta;

          case 26:
            return 2.*xi*eta*zeta;

          case 27:
            return eta*eta*zeta;

          case 28:
            return 0.;

          case 29:
            return 2.*xi*zeta*zeta;

          case 30:
            return eta*zeta*zeta;

          case 31:
            return 0.;

          case 32:
            return zeta*zeta*zeta;

          case 33:
            return 0.;

          case 34:
            return 0.;

          default:
            unsigned int o = 0;
            for (; i >= (o+1)*(o+2)*(o+3)/6; o++) { }
            unsigned int i2 = i - (o*(o+1)*(o+2)/6);
            unsigned int block=o, nz = 0;
            for (; block < i2; block += (o-nz+1)) { nz++; }
            const unsigned int nx = block - i2;
            const unsigned int ny = o - nx - nz;
            Real val = nx;
            for (unsigned int index=1; index < nx; index++)
              val *= xi;
            for (unsigned int index=0; index != ny; index++)
              val *= eta;
            for (unsigned int index=0; index != nz; index++)
              val *= zeta;
            return val;
          }
      }


      // d()/deta
    case 1:
      {
        switch (i)
          {
            // constant
          case 0:
            return 0.;

            // linear
          case 1:
            return 0.;

          case 2:
            return 1.;

          case 3:
            return 0.;

            // quadratic
          case 4:
            return 0.;

          case 5:
            return xi;

          case 6:
            return 2.*eta;

          case 7:
            return 0.;

          case 8:
            return zeta;

          case 9:
            return 0.;

            // cubic
          case 10:
            return 0.;

          case 11:
            return xi*xi;

          case 12:
            return 2.*xi*eta;

          case 13:
            return 3.*eta*eta;

          case 14:
            return 0.;

          case 15:
            return xi*zeta;

          case 16:
            return 2.*eta*zeta;

          case 17:
            return 0.;

          case 18:
            return zeta*zeta;

          case 19:
            return 0.;

            // quartics
          case 20:
            return 0.;

          case 21:
            return xi*xi*xi;

          case 22:
            return 2.*xi*xi*eta;

          case 23:
            return 3.*xi*eta*eta;

          case 24:
            return 4.*eta*eta*eta;

          case 25:
            return 0.;

          case 26:
            return xi*xi*zeta;

          case 27:
            return 2.*xi*eta*zeta;

          case 28:
            return 3.*eta*eta*zeta;

          case 29:
            return 0.;

          case 30:
            return xi*zeta*zeta;

          case 31:
            return 2.*eta*zeta*zeta;

          case 32:
            return 0.;

          case 33:
            return zeta*zeta*zeta;

          case 34:
            return 0.;

          default:
            unsigned int o = 0;
            for (; i >= (o+1)*(o+2)*(o+3)/6; o++) { }
            unsigned int i2 = i - (o*(o+1)*(o+2)/6);
            unsigned int block=o, nz = 0;
            for (; block < i2; block += (o-nz+1)) { nz++; }
            const unsigned int nx = block - i2;
            const unsigned int ny = o - nx - nz;
            Real val = ny;
            for (unsigned int index=0; index != nx; index++)
              val *= xi;
            for (unsigned int index=1; index < ny; index++)
              val *= eta;
            for (unsigned int index=0; index != nz; index++)
              val *= zeta;
            return val;
          }
      }


      // d()/dzeta
    case 2:
      {
        switch (i)
          {
            // constant
          case 0:
            return 0.;

            // linear
          case 1:
            return 0.;

          case 2:
            return 0.;

          case 3:
            return 1.;

            // quadratic
          case 4:
            return 0.;

          case 5:
            return 0.;

          case 6:
            return 0.;

          case 7:
            return xi;

          case 8:
            return eta;

          case 9:
            return 2.*zeta;

            // cubic
          case 10:
            return 0.;

          case 11:
            return 0.;

          case 12:
            return 0.;

          case 13:
            return 0.;

          case 14:
            return xi*xi;

          case 15:
            return xi*eta;

          case 16:
            return eta*eta;

          case 17:
            return 2.*xi*zeta;

          case 18:
            return 2.*eta*zeta;

          case 19:
            return 3.*zeta*zeta;

            // quartics
          case 20:
            return 0.;

          case 21:
            return 0.;

          case 22:
            return 0.;

          case 23:
            return 0.;

          case 24:
            return 0.;

          case 25:
            return xi*xi*xi;

          case 26:
            return xi*xi*eta;

          case 27:
            return xi*eta*eta;

          case 28:
            return eta*eta*eta;

          case 29:
            return 2.*xi*xi*zeta;

          case 30:
            return 2.*xi*eta*zeta;

          case 31:
            return 2.*eta*eta*zeta;

          case 32:
            return 3.*xi*zeta*zeta;

          case 33:
            return 3.*eta*zeta*zeta;

          case 34:
            return 4.*zeta*zeta*zeta;

          default:
            unsigned int o = 0;
            for (; i >= (o+1)*(o+2)*(o+3)/6; o++) { }
            unsigned int i2 = i - (o*(o+1)*(o+2)/6);
            unsigned int block=o, nz = 0;
            for (; block < i2; block += (o-nz+1)) { nz++; }
            const unsigned int nx = block - i2;
            const unsigned int ny = o - nx - nz;
            Real val = nz;
            for (unsigned int index=0; index != nx; index++)
              val *= xi;
            for (unsigned int index=0; index != ny; index++)
              val *= eta;
            for (unsigned int index=1; index < nz; index++)
              val *= zeta;
            return val;
          }
      }

    default:
      libmesh_error_msg("Invalid shape function derivative j = " << j);
    }

#else
  return 0.;
#endif
}

shape_deriv() [54/109]

Real libMesh::FE< 1, BERNSTEIN >::shape_deriv ( const ElemType  , const Order  order, const unsigned int  i, const unsigned int  libmesh_dbg_varj, const Point &  p  )

inherited

Definition at line 203 of file fe_bernstein_shape_1D.C.

{
  // only d()/dxi in 1D!

  libmesh_assert_equal_to (j, 0);

  const Real xi = p(0);

  using Utility::pow;

  switch (order)
    {
    case FIRST:

      switch(i)
        {
        case 0:
          return -.5;
        case 1:
          return .5;
        default:
          libmesh_error_msg("Invalid shape function index i = " << i);
        }

    case SECOND:

      switch(i)
        {
        case 0:
          return (xi-1.)*.5;
        case 1:
          return (xi+1.)*.5;
        case 2:
          return -xi;
        default:
          libmesh_error_msg("Invalid shape function index i = " << i);
        }

    case THIRD:

      switch(i)
        {
        case 0:
          return -0.375*pow<2>(1.-xi);
        case 1:
          return  0.375*pow<2>(1.+xi);
        case 2:
          return -0.375 -.75*xi +1.125*pow<2>(xi);
        case 3:
          return  0.375 -.75*xi -1.125*pow<2>(xi);
        default:
          libmesh_error_msg("Invalid shape function index i = " << i);
        }

    case FOURTH:

      switch(i)
        {
        case 0:
          return -0.25*pow<3>(1.-xi);
        case 1:
          return  0.25*pow<3>(1.+xi);
        case 2:
          return -0.5 +1.5*pow<2>(xi)-pow<3>(xi);
        case 3:
          return  1.5*(pow<3>(xi)-xi);
        case 4:
          return  0.5 -1.5*pow<2>(xi)-pow<3>(xi);
        default:
          libmesh_error_msg("Invalid shape function index i = " << i);
        }

    case FIFTH:

      switch(i)
        {
        case 0:
          return -(5./32.)*pow<4>(xi-1.);
        case 1:
          return  (5./32.)*pow<4>(xi+1.);
        case 2:
          return  (5./32.)*pow<4>(1.-xi)         -(5./8.)*(1.+xi)*pow<3>(1.-xi);
        case 3:
          return  (5./ 8.)*(1.+xi)*pow<3>(1.-xi) -(15./16.)*pow<2>(1.+xi)*pow<2>(1.-xi);
        case 4:
          return -(5./ 8.)*pow<3>(1.+xi)*(1.-xi) +(15./16.)*pow<2>(1.+xi)*pow<2>(1.-xi);
        case 5:
          return  (5./ 8.)*pow<3>(1.+xi)*(1.-xi) -(5./32.)*pow<4>(1.+xi);
        default:
          libmesh_error_msg("Invalid shape function index i = " << i);
        }

    case SIXTH:

      switch(i)
        {
        case 0:
          return -( 3./32.)*pow<5>(1.-xi);
        case 1:
          return  ( 3./32.)*pow<5>(1.+xi);
        case 2:
          return  ( 3./32.)*pow<5>(1.-xi)-(15./32.)*(1.+xi)*pow<4>(1.-xi);
        case 3:
          return  (15./32.)*(1.+xi)*pow<4>(1.-xi)-(15./16.)*pow<2>(1.+xi)*pow<3>(1.-xi);
        case 4:
          return -(15./ 8.)*xi +(15./4.)*pow<3>(xi)-(15./8.)*pow<5>(xi);
        case 5:
          return -(15./32.)*(1.-xi)*pow<4>(1.+xi)+(15./16.)*pow<2>(1.-xi)*pow<3>(1.+xi);
        case 6:
          return  (15./32.)*pow<4>(1.+xi)*(1.-xi)-(3./32.)*pow<5>(1.+xi);
        default:
          libmesh_error_msg("Invalid shape function index i = " << i);
        }


    default:
      {
        libmesh_assert (order>6);

        // Use this for arbitrary orders
        const int p_order = static_cast<int>(order);
        const int m       = p_order-(i-1);
        const int n       = (i-1);

        Real binomial_p_i = 1;

        // the binomial coefficient (p choose n)
        // Using an unsigned long here will work for any of the orders we support.
        // Explicitly construct a Real to prevent conversion warnings
        if (i>1)
          binomial_p_i = Real(Utility::binomial(static_cast<unsigned long>(p_order),
                                                static_cast<unsigned long>(n)));

        switch(i)
          {
          case 0:
            return binomial_p_i * (-1./2.) * p_order * std::pow((1-xi)/2, p_order-1);
          case 1:
            return binomial_p_i * ( 1./2.) * p_order * std::pow((1+xi)/2, p_order-1);

          default:
            {
              return binomial_p_i * (1./2. * n * std::pow((1+xi)/2,n-1) * std::pow((1-xi)/2,m)
                                     - 1./2. * m * std::pow((1+xi)/2,n)   * std::pow((1-xi)/2,m-1));
            }
          }
      }

    }
}

shape_deriv() [55/109]

RealGradient libMesh::FE< 3, NEDELEC_ONE >::shape_deriv ( const Elem *  elem, const Order  order, const unsigned int  i, const unsigned int  j, const Point &  p  )

inherited

Definition at line 205 of file fe_nedelec_one_shape_3D.C.

{
#if LIBMESH_DIM == 3
  libmesh_assert(elem);
  libmesh_assert_less (j, 3);

  const Order totalorder = static_cast<Order>(order + elem->p_level());

  switch (totalorder)
    {
    case FIRST:
      {
        switch (elem->type())
          {
          case HEX20:
          case HEX27:
            {
              libmesh_assert_less (i, 12);

              const Real xi   = p(0);
              const Real eta  = p(1);
              const Real zeta = p(2);

              // Even with a loose inverse_map tolerance we ought to
              // be nearly on the element interior in master
              // coordinates
              libmesh_assert_less_equal ( std::fabs(xi),   1.0+TOLERANCE );
              libmesh_assert_less_equal ( std::fabs(eta),  1.0+TOLERANCE );
              libmesh_assert_less_equal ( std::fabs(zeta), 1.0+TOLERANCE );

              switch (j)
                {
                  // d()/dxi
                case 0:
                  {
                    switch(i)
                      {
                      case 0:
                      case 2:
                      case 8:
                      case 10:
                        return RealGradient();
                      case 1:
                        {
                          if (elem->point(1) > elem->point(2))
                            return RealGradient( 0.0, -0.125*(1.0-zeta) );
                          else
                            return RealGradient( 0.0, 0.125*(1.0-zeta) );
                        }
                      case 3:
                        {
                          if (elem->point(3) > elem->point(0))
                            return RealGradient( 0.0, 0.125*(-1.0+zeta) );
                          else
                            return RealGradient( 0.0, -0.125*(-1.0+zeta) );
                        }
                      case 4:
                        {
                          if (elem->point(0) > elem->point(4))
                            return RealGradient( 0.0, 0.0, -0.125*(-1.0+eta) );
                          else
                            return RealGradient( 0.0, 0.0,  0.125*(-1.0+eta) );
                        }
                      case 5:
                        {
                          if (elem->point(1) > elem->point(5))
                            return RealGradient( 0.0, 0.0, -0.125*(1.0-eta) );
                          else
                            return RealGradient( 0.0, 0.0,  0.125*(1.0-eta) );
                        }
                      case 6:
                        {
                          if (elem->point(2) > elem->point(6))
                            return RealGradient( 0.0, 0.0, -0.125*(1.0+eta) );
                          else
                            return RealGradient( 0.0, 0.0,  0.125*(1.0+eta) );
                        }
                      case 7:
                        {
                          if (elem->point(3) > elem->point(7))
                            return RealGradient( 0.0, 0.0, -0.125*(-1.0-eta) );
                          else
                            return RealGradient( 0.0, 0.0,  0.125*(-1.0-eta) );
                        }
                      case 9:
                        {
                          if (elem->point(5) > elem->point(6))
                            return RealGradient( 0.0, -0.125*(1.0+zeta), 0.0 );
                          else
                            return RealGradient( 0.0,  0.125*(1.0+zeta), 0.0 );
                        }
                      case 11:
                        {
                          if (elem->point(4) > elem->point(7))
                            return RealGradient( 0.0, -0.125*(-1.0-zeta), 0.0 );
                          else
                            return RealGradient( 0.0,  0.125*(-1.0-zeta), 0.0 );
                        }
                      default:
                        libmesh_error_msg("Invalid i = " << i);
                      } // switch(i)

                  } // j=0

                  // d()/deta
                case 1:
                  {
                    switch(i)
                      {
                      case 1:
                      case 3:
                      case 9:
                      case 11:
                        return RealGradient();
                      case 0:
                        {
                          if (elem->point(0) > elem->point(1))
                            return RealGradient( -0.125*(-1.0+zeta), 0.0, 0.0 );
                          else
                            return RealGradient(  0.125*(-1.0+zeta), 0.0, 0.0 );
                        }
                      case 2:
                        {
                          if (elem->point(2) > elem->point(3))
                            return RealGradient( 0.125*(1.0-zeta), 0.0, 0.0 );
                          else
                            return RealGradient( -0.125*(1.0-zeta), 0.0, 0.0 );
                        }
                      case 4:
                        {
                          if (elem->point(0) > elem->point(4))
                            return RealGradient( 0.0, 0.0, -0.125*(-1.0+xi) );
                          else
                            return RealGradient( 0.0, 0.0,  0.125*(-1.0+xi) );
                        }
                      case 5:
                        {
                          if (elem->point(1) > elem->point(5))
                            return RealGradient( 0.0, 0.0, -0.125*(-1.0-xi) );
                          else
                            return RealGradient( 0.0, 0.0,  0.125*(-1.0-xi) );
                        }
                      case 6:
                        {
                          if (elem->point(2) > elem->point(6))
                            return RealGradient( 0.0, 0.0, -0.125*(1.0+xi) );
                          else
                            return RealGradient( 0.0, 0.0,  0.125*(1.0+xi) );
                        }
                      case 7:
                        {
                          if (elem->point(3) > elem->point(7))
                            return RealGradient( 0.0, 0.0, -0.125*(1.0-xi) );
                          else
                            return RealGradient( 0.0, 0.0,  0.125*(1.0-xi) );
                        }
                      case 8:
                        {
                          if (elem->point(4) > elem->point(5))
                            return RealGradient( -0.125*(-1.0-zeta), 0.0, 0.0 );
                          else
                            return RealGradient(  0.125*(-1.0-zeta), 0.0, 0.0 );
                        }
                      case 10:
                        {
                          if (elem->point(7) > elem->point(6))
                            return RealGradient( -0.125*(1.0+zeta), 0.0, 0.0 );
                          else
                            return RealGradient(  0.125*(1.0+zeta), 0.0, 0.0 );
                        }
                      default:
                        libmesh_error_msg("Invalid i = " << i);
                      } // switch(i)

                  } // j=1

                  // d()/dzeta
                case 2:
                  {
                    switch(i)
                      {
                      case 4:
                      case 5:
                      case 6:
                      case 7:
                        return RealGradient();

                      case 0:
                        {
                          if (elem->point(0) > elem->point(1))
                            return RealGradient( -0.125*(-1.0+eta), 0.0, 0.0 );
                          else
                            return RealGradient(  0.125*(-1.0+eta), 0.0, 0.0 );
                        }
                      case 1:
                        {
                          if (elem->point(1) > elem->point(2))
                            return RealGradient( 0.0, -0.125*(-1.0-xi), 0.0 );
                          else
                            return RealGradient( 0.0,  0.125*(-1.0-xi), 0.0 );
                        }
                      case 2:
                        {
                          if (elem->point(2) > elem->point(3))
                            return RealGradient( 0.125*(-1.0-eta), 0.0, 0.0 );
                          else
                            return RealGradient( -0.125*(-1.0-eta), 0.0, 0.0 );
                        }
                      case 3:
                        {
                          if (elem->point(3) > elem->point(0))
                            return RealGradient( 0.0, 0.125*(-1.0+xi), 0.0 );
                          else
                            return RealGradient( 0.0,  -0.125*(-1.0+xi), 0.0 );
                        }
                      case 8:
                        {
                          if (elem->point(4) > elem->point(5))
                            return RealGradient( -0.125*(1.0-eta), 0.0, 0.0 );
                          else
                            return RealGradient(  0.125*(1.0-eta), 0.0, 0.0 );
                        }
                      case 9:
                        {
                          if (elem->point(5) > elem->point(6))
                            return RealGradient( 0.0, -0.125*(1.0+xi), 0.0 );
                          else
                            return RealGradient( 0.0,  0.125*(1.0+xi), 0.0 );
                        }
                      case 10:
                        {
                          if (elem->point(7) > elem->point(6))
                            return RealGradient( -0.125*(1.0+eta), 0.0, 0.0 );
                          else
                            return RealGradient(  0.125*(1.0+eta), 0.0, 0.0 );
                        }
                      case 11:
                        {
                          if (elem->point(4) > elem->point(7))
                            return RealGradient( 0.0, -0.125*(1.0-xi), 0.0 );
                          else
                            return RealGradient( 0.0,  0.125*(1.0-xi), 0.0 );
                        }
                      default:
                        libmesh_error_msg("Invalid i = " << i);
                      } // switch(i)

                  } // j = 2

                default:
                  libmesh_error_msg("Invalid j = " << j);
                }

              return RealGradient();
            }

          case TET10:
            {
              libmesh_assert_less (i, 6);

              libmesh_not_implemented();
              return RealGradient();
            }

          default:
            libmesh_error_msg("ERROR: Unsupported 3D element type!: " << elem->type());
          }
      }

      // unsupported order
    default:
      libmesh_error_msg("ERROR: Unsupported 3D FE order!: " << totalorder);
    }

#else
  return RealGradient();
#endif
}

shape_deriv() [56/109]

Real libMesh::FE< 1, L2_LAGRANGE >::shape_deriv ( const Elem *  elem, const Order  order, const unsigned int  i, const unsigned int  j, const Point &  p  )

inherited

Definition at line 219 of file fe_l2_lagrange_shape_1D.C.

{
  libmesh_assert(elem);

  return FE<1,L2_LAGRANGE>::shape_deriv(elem->type(),
                                        static_cast<Order>(order + elem->p_level()), i, j, p);
}

shape_deriv() [57/109]

Real libMesh::FE< 1, LAGRANGE >::shape_deriv ( const Elem *  elem, const Order  order, const unsigned int  i, const unsigned int  j, const Point &  p  )

inherited

Definition at line 219 of file fe_lagrange_shape_1D.C.

{
  libmesh_assert(elem);

  return FE<1,LAGRANGE>::shape_deriv(elem->type(),
                                     static_cast<Order>(order + elem->p_level()), i, j, p);
}

shape_deriv() [58/109]

Real libMesh::FE< 2, L2_LAGRANGE >::shape_deriv ( const ElemType  type, const Order  order, const unsigned int  i, const unsigned int  j, const Point &  p  )

inherited

Definition at line 222 of file fe_l2_lagrange_shape_2D.C.

{
#if LIBMESH_DIM > 1


  libmesh_assert_less (j, 2);

  switch (order)
    {
      // linear Lagrange shape functions
    case FIRST:
      {
        switch (type)
          {
          case QUAD4:
          case QUADSHELL4:
          case QUAD8:
          case QUADSHELL8:
          case QUAD9:
            {
              // Compute quad shape functions as a tensor-product
              const Real xi  = p(0);
              const Real eta = p(1);

              libmesh_assert_less (i, 4);

              //                                0  1  2  3
              static const unsigned int i0[] = {0, 1, 1, 0};
              static const unsigned int i1[] = {0, 0, 1, 1};

              switch (j)
                {
                  // d()/dxi
                case 0:
                  return (FE<1,L2_LAGRANGE>::shape_deriv(EDGE2, FIRST, i0[i], 0, xi)*
                          FE<1,L2_LAGRANGE>::shape      (EDGE2, FIRST, i1[i], eta));

                  // d()/deta
                case 1:
                  return (FE<1,L2_LAGRANGE>::shape      (EDGE2, FIRST, i0[i], xi)*
                          FE<1,L2_LAGRANGE>::shape_deriv(EDGE2, FIRST, i1[i], 0, eta));

                default:
                  libmesh_error_msg("ERROR: Invalid derivative index j = " << j);
                }
            }

          case TRI3:
          case TRISHELL3:
          case TRI6:
            {
              libmesh_assert_less (i, 3);

              const Real dzeta0dxi  = -1.;
              const Real dzeta1dxi  = 1.;
              const Real dzeta2dxi  = 0.;

              const Real dzeta0deta = -1.;
              const Real dzeta1deta = 0.;
              const Real dzeta2deta = 1.;

              switch (j)
                {
                  // d()/dxi
                case 0:
                  {
                    switch(i)
                      {
                      case 0:
                        return dzeta0dxi;

                      case 1:
                        return dzeta1dxi;

                      case 2:
                        return dzeta2dxi;

                      default:
                        libmesh_error_msg("Invalid shape function index i = " << i);
                      }
                  }
                  // d()/deta
                case 1:
                  {
                    switch(i)
                      {
                      case 0:
                        return dzeta0deta;

                      case 1:
                        return dzeta1deta;

                      case 2:
                        return dzeta2deta;

                      default:
                        libmesh_error_msg("Invalid shape function index i = " << i);
                      }
                  }
                default:
                  libmesh_error_msg("ERROR: Invalid derivative index j = " << j);
                }
            }

          default:
            libmesh_error_msg("ERROR: Unsupported 2D element type: " << type);
          }
      }


      // quadratic Lagrange shape functions
    case SECOND:
      {
        switch (type)
          {
          case QUAD8:
          case QUADSHELL8:
            {
              const Real xi  = p(0);
              const Real eta = p(1);

              libmesh_assert_less (i, 8);

              switch (j)
                {
                  // d/dxi
                case 0:
                  switch (i)
                    {
                    case 0:
                      return .25*(1. - eta)*((1. - xi)*(-1.) +
                                             (-1.)*(-1. - xi - eta));

                    case 1:
                      return .25*(1. - eta)*((1. + xi)*(1.) +
                                             (1.)*(-1. + xi - eta));

                    case 2:
                      return .25*(1. + eta)*((1. + xi)*(1.) +
                                             (1.)*(-1. + xi + eta));

                    case 3:
                      return .25*(1. + eta)*((1. - xi)*(-1.) +
                                             (-1.)*(-1. - xi + eta));

                    case 4:
                      return .5*(-2.*xi)*(1. - eta);

                    case 5:
                      return .5*(1.)*(1. - eta*eta);

                    case 6:
                      return .5*(-2.*xi)*(1. + eta);

                    case 7:
                      return .5*(-1.)*(1. - eta*eta);

                    default:
                      libmesh_error_msg("Invalid shape function index i = " << i);
                    }

                  // d/deta
                case 1:
                  switch (i)
                    {
                    case 0:
                      return .25*(1. - xi)*((1. - eta)*(-1.) +
                                            (-1.)*(-1. - xi - eta));

                    case 1:
                      return .25*(1. + xi)*((1. - eta)*(-1.) +
                                            (-1.)*(-1. + xi - eta));

                    case 2:
                      return .25*(1. + xi)*((1. + eta)*(1.) +
                                            (1.)*(-1. + xi + eta));

                    case 3:
                      return .25*(1. - xi)*((1. + eta)*(1.) +
                                            (1.)*(-1. - xi + eta));

                    case 4:
                      return .5*(1. - xi*xi)*(-1.);

                    case 5:
                      return .5*(1. + xi)*(-2.*eta);

                    case 6:
                      return .5*(1. - xi*xi)*(1.);

                    case 7:
                      return .5*(1. - xi)*(-2.*eta);

                    default:
                      libmesh_error_msg("Invalid shape function index i = " << i);
                    }

                default:
                  libmesh_error_msg("ERROR: Invalid derivative index j = " << j);
                }
            }

          case QUAD9:
            {
              // Compute quad shape functions as a tensor-product
              const Real xi  = p(0);
              const Real eta = p(1);

              libmesh_assert_less (i, 9);

              //                                0  1  2  3  4  5  6  7  8
              static const unsigned int i0[] = {0, 1, 1, 0, 2, 1, 2, 0, 2};
              static const unsigned int i1[] = {0, 0, 1, 1, 0, 2, 1, 2, 2};

              switch (j)
                {
                  // d()/dxi
                case 0:
                  return (FE<1,L2_LAGRANGE>::shape_deriv(EDGE3, SECOND, i0[i], 0, xi)*
                          FE<1,L2_LAGRANGE>::shape      (EDGE3, SECOND, i1[i], eta));

                  // d()/deta
                case 1:
                  return (FE<1,L2_LAGRANGE>::shape      (EDGE3, SECOND, i0[i], xi)*
                          FE<1,L2_LAGRANGE>::shape_deriv(EDGE3, SECOND, i1[i], 0, eta));

                default:
                  libmesh_error_msg("ERROR: Invalid derivative index j = " << j);
                }
            }

          case TRI6:
            {
              libmesh_assert_less (i, 6);

              const Real zeta1 = p(0);
              const Real zeta2 = p(1);
              const Real zeta0 = 1. - zeta1 - zeta2;

              const Real dzeta0dxi  = -1.;
              const Real dzeta1dxi  = 1.;
              const Real dzeta2dxi  = 0.;

              const Real dzeta0deta = -1.;
              const Real dzeta1deta = 0.;
              const Real dzeta2deta = 1.;

              switch(j)
                {
                case 0:
                  {
                    switch(i)
                      {
                      case 0:
                        return (4.*zeta0-1.)*dzeta0dxi;

                      case 1:
                        return (4.*zeta1-1.)*dzeta1dxi;

                      case 2:
                        return (4.*zeta2-1.)*dzeta2dxi;

                      case 3:
                        return 4.*zeta1*dzeta0dxi + 4.*zeta0*dzeta1dxi;

                      case 4:
                        return 4.*zeta2*dzeta1dxi + 4.*zeta1*dzeta2dxi;

                      case 5:
                        return 4.*zeta2*dzeta0dxi + 4*zeta0*dzeta2dxi;

                      default:
                        libmesh_error_msg("Invalid shape function index i = " << i);
                      }
                  }

                case 1:
                  {
                    switch(i)
                      {
                      case 0:
                        return (4.*zeta0-1.)*dzeta0deta;

                      case 1:
                        return (4.*zeta1-1.)*dzeta1deta;

                      case 2:
                        return (4.*zeta2-1.)*dzeta2deta;

                      case 3:
                        return 4.*zeta1*dzeta0deta + 4.*zeta0*dzeta1deta;

                      case 4:
                        return 4.*zeta2*dzeta1deta + 4.*zeta1*dzeta2deta;

                      case 5:
                        return 4.*zeta2*dzeta0deta + 4*zeta0*dzeta2deta;

                      default:
                        libmesh_error_msg("Invalid shape function index i = " << i);
                      }
                  }
                default:
                  libmesh_error_msg("ERROR: Invalid derivative index j = " << j);
                }
            }

          default:
            libmesh_error_msg("ERROR: Unsupported 2D element type: " << type);
          }
      }



      // unsupported order
    default:
      libmesh_error_msg("ERROR: Unsupported 2D FE order: " << order);
    }
#endif
}

shape_deriv() [59/109]

Real libMesh::FE< 3, XYZ >::shape_deriv ( const ElemType  , const Order  , const unsigned  int, const unsigned  int, const Point &    )

inherited

Definition at line 222 of file fe_xyz_shape_3D.C.

{
  libmesh_error_msg("XYZ polynomials require the element \nbecause the centroid is needed.");
  return 0.;
}

shape_deriv() [60/109]

Real libMesh::FE< 2, LAGRANGE >::shape_deriv ( const ElemType  type, const Order  order, const unsigned int  i, const unsigned int  j, const Point &  p  )

inherited

Definition at line 224 of file fe_lagrange_shape_2D.C.

{
#if LIBMESH_DIM > 1


  libmesh_assert_less (j, 2);

  switch (order)
    {
      // linear Lagrange shape functions
    case FIRST:
      {
        switch (type)
          {
          case QUAD4:
          case QUADSHELL4:
          case QUAD8:
          case QUADSHELL8:
          case QUAD9:
            {
              // Compute quad shape functions as a tensor-product
              const Real xi  = p(0);
              const Real eta = p(1);

              libmesh_assert_less (i, 4);

              //                                0  1  2  3
              static const unsigned int i0[] = {0, 1, 1, 0};
              static const unsigned int i1[] = {0, 0, 1, 1};

              switch (j)
                {
                  // d()/dxi
                case 0:
                  return (FE<1,LAGRANGE>::shape_deriv(EDGE2, FIRST, i0[i], 0, xi)*
                          FE<1,LAGRANGE>::shape      (EDGE2, FIRST, i1[i], eta));

                  // d()/deta
                case 1:
                  return (FE<1,LAGRANGE>::shape      (EDGE2, FIRST, i0[i], xi)*
                          FE<1,LAGRANGE>::shape_deriv(EDGE2, FIRST, i1[i], 0, eta));

                default:
                  libmesh_error_msg("ERROR: Invalid derivative index j = " << j);
                }
            }

          case TRI3:
          case TRISHELL3:
          case TRI6:
            {
              libmesh_assert_less (i, 3);

              const Real dzeta0dxi  = -1.;
              const Real dzeta1dxi  = 1.;
              const Real dzeta2dxi  = 0.;

              const Real dzeta0deta = -1.;
              const Real dzeta1deta = 0.;
              const Real dzeta2deta = 1.;

              switch (j)
                {
                  // d()/dxi
                case 0:
                  {
                    switch(i)
                      {
                      case 0:
                        return dzeta0dxi;

                      case 1:
                        return dzeta1dxi;

                      case 2:
                        return dzeta2dxi;

                      default:
                        libmesh_error_msg("Invalid shape function index i = " << i);
                      }
                  }
                  // d()/deta
                case 1:
                  {
                    switch(i)
                      {
                      case 0:
                        return dzeta0deta;

                      case 1:
                        return dzeta1deta;

                      case 2:
                        return dzeta2deta;

                      default:
                        libmesh_error_msg("Invalid shape function index i = " << i);
                      }
                  }
                default:
                  libmesh_error_msg("ERROR: Invalid derivative index j = " << j);
                }
            }

          default:
            libmesh_error_msg("ERROR: Unsupported 2D element type: " << type);
          }
      }


      // quadratic Lagrange shape functions
    case SECOND:
      {
        switch (type)
          {
          case QUAD8:
          case QUADSHELL8:
            {
              const Real xi  = p(0);
              const Real eta = p(1);

              libmesh_assert_less (i, 8);

              switch (j)
                {
                  // d/dxi
                case 0:
                  switch (i)
                    {
                    case 0:
                      return .25*(1. - eta)*((1. - xi)*(-1.) +
                                             (-1.)*(-1. - xi - eta));

                    case 1:
                      return .25*(1. - eta)*((1. + xi)*(1.) +
                                             (1.)*(-1. + xi - eta));

                    case 2:
                      return .25*(1. + eta)*((1. + xi)*(1.) +
                                             (1.)*(-1. + xi + eta));

                    case 3:
                      return .25*(1. + eta)*((1. - xi)*(-1.) +
                                             (-1.)*(-1. - xi + eta));

                    case 4:
                      return .5*(-2.*xi)*(1. - eta);

                    case 5:
                      return .5*(1.)*(1. - eta*eta);

                    case 6:
                      return .5*(-2.*xi)*(1. + eta);

                    case 7:
                      return .5*(-1.)*(1. - eta*eta);

                    default:
                      libmesh_error_msg("Invalid shape function index i = " << i);
                    }

                  // d/deta
                case 1:
                  switch (i)
                    {
                    case 0:
                      return .25*(1. - xi)*((1. - eta)*(-1.) +
                                            (-1.)*(-1. - xi - eta));

                    case 1:
                      return .25*(1. + xi)*((1. - eta)*(-1.) +
                                            (-1.)*(-1. + xi - eta));

                    case 2:
                      return .25*(1. + xi)*((1. + eta)*(1.) +
                                            (1.)*(-1. + xi + eta));

                    case 3:
                      return .25*(1. - xi)*((1. + eta)*(1.) +
                                            (1.)*(-1. - xi + eta));

                    case 4:
                      return .5*(1. - xi*xi)*(-1.);

                    case 5:
                      return .5*(1. + xi)*(-2.*eta);

                    case 6:
                      return .5*(1. - xi*xi)*(1.);

                    case 7:
                      return .5*(1. - xi)*(-2.*eta);

                    default:
                      libmesh_error_msg("Invalid shape function index i = " << i);
                    }

                default:
                  libmesh_error_msg("ERROR: Invalid derivative index j = " << j);
                }
            }

          case QUAD9:
            {
              // Compute quad shape functions as a tensor-product
              const Real xi  = p(0);
              const Real eta = p(1);

              libmesh_assert_less (i, 9);

              //                                0  1  2  3  4  5  6  7  8
              static const unsigned int i0[] = {0, 1, 1, 0, 2, 1, 2, 0, 2};
              static const unsigned int i1[] = {0, 0, 1, 1, 0, 2, 1, 2, 2};

              switch (j)
                {
                  // d()/dxi
                case 0:
                  return (FE<1,LAGRANGE>::shape_deriv(EDGE3, SECOND, i0[i], 0, xi)*
                          FE<1,LAGRANGE>::shape      (EDGE3, SECOND, i1[i], eta));

                  // d()/deta
                case 1:
                  return (FE<1,LAGRANGE>::shape      (EDGE3, SECOND, i0[i], xi)*
                          FE<1,LAGRANGE>::shape_deriv(EDGE3, SECOND, i1[i], 0, eta));

                default:
                  libmesh_error_msg("ERROR: Invalid derivative index j = " << j);
                }
            }

          case TRI6:
            {
              libmesh_assert_less (i, 6);

              const Real zeta1 = p(0);
              const Real zeta2 = p(1);
              const Real zeta0 = 1. - zeta1 - zeta2;

              const Real dzeta0dxi  = -1.;
              const Real dzeta1dxi  = 1.;
              const Real dzeta2dxi  = 0.;

              const Real dzeta0deta = -1.;
              const Real dzeta1deta = 0.;
              const Real dzeta2deta = 1.;

              switch(j)
                {
                case 0:
                  {
                    switch(i)
                      {
                      case 0:
                        return (4.*zeta0-1.)*dzeta0dxi;

                      case 1:
                        return (4.*zeta1-1.)*dzeta1dxi;

                      case 2:
                        return (4.*zeta2-1.)*dzeta2dxi;

                      case 3:
                        return 4.*zeta1*dzeta0dxi + 4.*zeta0*dzeta1dxi;

                      case 4:
                        return 4.*zeta2*dzeta1dxi + 4.*zeta1*dzeta2dxi;

                      case 5:
                        return 4.*zeta2*dzeta0dxi + 4*zeta0*dzeta2dxi;

                      default:
                        libmesh_error_msg("Invalid shape function index i = " << i);
                      }
                  }

                case 1:
                  {
                    switch(i)
                      {
                      case 0:
                        return (4.*zeta0-1.)*dzeta0deta;

                      case 1:
                        return (4.*zeta1-1.)*dzeta1deta;

                      case 2:
                        return (4.*zeta2-1.)*dzeta2deta;

                      case 3:
                        return 4.*zeta1*dzeta0deta + 4.*zeta0*dzeta1deta;

                      case 4:
                        return 4.*zeta2*dzeta1deta + 4.*zeta1*dzeta2deta;

                      case 5:
                        return 4.*zeta2*dzeta0deta + 4*zeta0*dzeta2deta;

                      default:
                        libmesh_error_msg("Invalid shape function index i = " << i);
                      }
                  }
                default:
                  libmesh_error_msg("ERROR: Invalid derivative index j = " << j);
                }
            }

          default:
            libmesh_error_msg("ERROR: Unsupported 2D element type: " << type);
          }
      }

      // unsupported order
    default:
      libmesh_error_msg("ERROR: Unsupported 2D FE order: " << order);
    }
#else // LIBMESH_DIM > 1
  return 0.;
#endif
}

shape_deriv() [61/109]

Real libMesh::FE< 2, HIERARCHIC >::shape_deriv ( const ElemType  , const Order  , const unsigned  int, const unsigned  int, const Point &    )

inherited

Definition at line 231 of file fe_hierarchic_shape_2D.C.

{
  libmesh_error_msg("Hierarchic polynomials require the element type \nbecause edge orientation is needed.");
  return 0.;
}

shape_deriv() [62/109]

Real libMesh::FE< 2, L2_HIERARCHIC >::shape_deriv ( const ElemType  , const Order  , const unsigned  int, const unsigned  int, const Point &    )

inherited

Definition at line 231 of file fe_l2_hierarchic_shape_2D.C.

{
  libmesh_error_msg("Hierarchic polynomials require the element type \nbecause edge orientation is needed.");
  return 0.;
}

shape_deriv() [63/109]

Real libMesh::FE< 3, XYZ >::shape_deriv ( const Elem *  elem, const Order  libmesh_dbg_varorder, const unsigned int  i, const unsigned int  j, const Point &  point_in  )

inherited

Definition at line 235 of file fe_xyz_shape_3D.C.

{
#if LIBMESH_DIM == 3

  libmesh_assert(elem);
  libmesh_assert_less (j, 3);

  Point centroid = elem->centroid();
  Point max_distance = Point(0.,0.,0.);
  for (unsigned int p = 0; p < elem->n_nodes(); p++)
    for (unsigned int d = 0; d < 3; d++)
      {
        const Real distance = std::abs(centroid(d) - elem->point(p)(d));
        max_distance(d) = std::max(distance, max_distance(d));
      }

  const Real x  = point_in(0);
  const Real y  = point_in(1);
  const Real z  = point_in(2);
  const Real xc = centroid(0);
  const Real yc = centroid(1);
  const Real zc = centroid(2);
  const Real distx = max_distance(0);
  const Real disty = max_distance(1);
  const Real distz = max_distance(2);
  const Real dx = (x - xc)/distx;
  const Real dy = (y - yc)/disty;
  const Real dz = (z - zc)/distz;

#ifndef NDEBUG
  // totalorder is only used in the assertion below, so
  // we avoid declaring it when asserts are not active.
  const unsigned int totalorder = static_cast<Order>(order + elem->p_level());
#endif
  libmesh_assert_less (i, (static_cast<unsigned int>(totalorder)+1)*
                       (static_cast<unsigned int>(totalorder)+2)*
                       (static_cast<unsigned int>(totalorder)+3)/6);

  switch (j)
    {
      // d()/dx
    case 0:
      {
        switch (i)
          {
            // constant
          case 0:
            return 0.;

            // linear
          case 1:
            return 1./distx;

          case 2:
            return 0.;

          case 3:
            return 0.;

            // quadratic
          case 4:
            return 2.*dx/distx;

          case 5:
            return dy/distx;

          case 6:
            return 0.;

          case 7:
            return dz/distx;

          case 8:
            return 0.;

          case 9:
            return 0.;

            // cubic
          case 10:
            return 3.*dx*dx/distx;

          case 11:
            return 2.*dx*dy/distx;

          case 12:
            return dy*dy/distx;

          case 13:
            return 0.;

          case 14:
            return 2.*dx*dz/distx;

          case 15:
            return dy*dz/distx;

          case 16:
            return 0.;

          case 17:
            return dz*dz/distx;

          case 18:
            return 0.;

          case 19:
            return 0.;

            // quartics
          case 20:
            return 4.*dx*dx*dx/distx;

          case 21:
            return 3.*dx*dx*dy/distx;

          case 22:
            return 2.*dx*dy*dy/distx;

          case 23:
            return dy*dy*dy/distx;

          case 24:
            return 0.;

          case 25:
            return 3.*dx*dx*dz/distx;

          case 26:
            return 2.*dx*dy*dz/distx;

          case 27:
            return dy*dy*dz/distx;

          case 28:
            return 0.;

          case 29:
            return 2.*dx*dz*dz/distx;

          case 30:
            return dy*dz*dz/distx;

          case 31:
            return 0.;

          case 32:
            return dz*dz*dz/distx;

          case 33:
            return 0.;

          case 34:
            return 0.;

          default:
            unsigned int o = 0;
            for (; i >= (o+1)*(o+2)*(o+3)/6; o++) { }
            unsigned int i2 = i - (o*(o+1)*(o+2)/6);
            unsigned int block=o, nz = 0;
            for (; block < i2; block += (o-nz+1)) { nz++; }
            const unsigned int nx = block - i2;
            const unsigned int ny = o - nx - nz;
            Real val = nx;
            for (unsigned int index=1; index < nx; index++)
              val *= dx;
            for (unsigned int index=0; index != ny; index++)
              val *= dy;
            for (unsigned int index=0; index != nz; index++)
              val *= dz;
            return val/distx;
          }
      }


      // d()/dy
    case 1:
      {
        switch (i)
          {
            // constant
          case 0:
            return 0.;

            // linear
          case 1:
            return 0.;

          case 2:
            return 1./disty;

          case 3:
            return 0.;

            // quadratic
          case 4:
            return 0.;

          case 5:
            return dx/disty;

          case 6:
            return 2.*dy/disty;

          case 7:
            return 0.;

          case 8:
            return dz/disty;

          case 9:
            return 0.;

            // cubic
          case 10:
            return 0.;

          case 11:
            return dx*dx/disty;

          case 12:
            return 2.*dx*dy/disty;

          case 13:
            return 3.*dy*dy/disty;

          case 14:
            return 0.;

          case 15:
            return dx*dz/disty;

          case 16:
            return 2.*dy*dz/disty;

          case 17:
            return 0.;

          case 18:
            return dz*dz/disty;

          case 19:
            return 0.;

            // quartics
          case 20:
            return 0.;

          case 21:
            return dx*dx*dx/disty;

          case 22:
            return 2.*dx*dx*dy/disty;

          case 23:
            return 3.*dx*dy*dy/disty;

          case 24:
            return 4.*dy*dy*dy/disty;

          case 25:
            return 0.;

          case 26:
            return dx*dx*dz/disty;

          case 27:
            return 2.*dx*dy*dz/disty;

          case 28:
            return 3.*dy*dy*dz/disty;

          case 29:
            return 0.;

          case 30:
            return dx*dz*dz/disty;

          case 31:
            return 2.*dy*dz*dz/disty;

          case 32:
            return 0.;

          case 33:
            return dz*dz*dz/disty;

          case 34:
            return 0.;

          default:
            unsigned int o = 0;
            for (; i >= (o+1)*(o+2)*(o+3)/6; o++) { }
            unsigned int i2 = i - (o*(o+1)*(o+2)/6);
            unsigned int block=o, nz = 0;
            for (; block < i2; block += (o-nz+1)) { nz++; }
            const unsigned int nx = block - i2;
            const unsigned int ny = o - nx - nz;
            Real val = ny;
            for (unsigned int index=0; index != nx; index++)
              val *= dx;
            for (unsigned int index=1; index < ny; index++)
              val *= dy;
            for (unsigned int index=0; index != nz; index++)
              val *= dz;
            return val/disty;
          }
      }


      // d()/dz
    case 2:
      {
        switch (i)
          {
            // constant
          case 0:
            return 0.;

            // linear
          case 1:
            return 0.;

          case 2:
            return 0.;

          case 3:
            return 1./distz;

            // quadratic
          case 4:
            return 0.;

          case 5:
            return 0.;

          case 6:
            return 0.;

          case 7:
            return dx/distz;

          case 8:
            return dy/distz;

          case 9:
            return 2.*dz/distz;

            // cubic
          case 10:
            return 0.;

          case 11:
            return 0.;

          case 12:
            return 0.;

          case 13:
            return 0.;

          case 14:
            return dx*dx/distz;

          case 15:
            return dx*dy/distz;

          case 16:
            return dy*dy/distz;

          case 17:
            return 2.*dx*dz/distz;

          case 18:
            return 2.*dy*dz/distz;

          case 19:
            return 3.*dz*dz/distz;

            // quartics
          case 20:
            return 0.;

          case 21:
            return 0.;

          case 22:
            return 0.;

          case 23:
            return 0.;

          case 24:
            return 0.;

          case 25:
            return dx*dx*dx/distz;

          case 26:
            return dx*dx*dy/distz;

          case 27:
            return dx*dy*dy/distz;

          case 28:
            return dy*dy*dy/distz;

          case 29:
            return 2.*dx*dx*dz/distz;

          case 30:
            return 2.*dx*dy*dz/distz;

          case 31:
            return 2.*dy*dy*dz/distz;

          case 32:
            return 3.*dx*dz*dz/distz;

          case 33:
            return 3.*dy*dz*dz/distz;

          case 34:
            return 4.*dz*dz*dz/distz;

          default:
            unsigned int o = 0;
            for (; i >= (o+1)*(o+2)*(o+3)/6; o++) { }
            unsigned int i2 = i - (o*(o+1)*(o+2)/6);
            unsigned int block=o, nz = 0;
            for (; block < i2; block += (o-nz+1)) { nz++; }
            const unsigned int nx = block - i2;
            const unsigned int ny = o - nx - nz;
            Real val = nz;
            for (unsigned int index=0; index != nx; index++)
              val *= dx;
            for (unsigned int index=0; index != ny; index++)
              val *= dy;
            for (unsigned int index=1; index < nz; index++)
              val *= dz;
            return val/distz;
          }
      }


    default:
      libmesh_error_msg("Invalid j = " << j);
    }

#else
  return 0.;
#endif
}

shape_deriv() [64/109]

Real libMesh::FE< 1, HERMITE >::shape_deriv ( const ElemType  , const Order  , const unsigned  int, const unsigned  int, const Point &    )

inherited

Definition at line 237 of file fe_hermite_shape_1D.C.

{
  libmesh_error_msg("Hermite elements require the real element \nto construct gradient-based degrees of freedom.");
  return 0.;
}

shape_deriv() [65/109]

Real libMesh::FE< 2, L2_HIERARCHIC >::shape_deriv ( const Elem *  elem, const Order  order, const unsigned int  i, const unsigned int  j, const Point &  p  )

inherited

Definition at line 244 of file fe_l2_hierarchic_shape_2D.C.

{
  libmesh_assert(elem);

  const ElemType type = elem->type();

  const Order totalorder = static_cast<Order>(order+elem->p_level());

  libmesh_assert_greater (totalorder, 0);

  switch (type)
    {
      // 1st & 2nd-order Hierarchics.
    case TRI3:
    case TRISHELL3:
    case TRI6:
      {
        const Real eps = 1.e-6;

        libmesh_assert_less (j, 2);

        switch (j)
          {
            //  d()/dxi
          case 0:
            {
              const Point pp(p(0)+eps, p(1));
              const Point pm(p(0)-eps, p(1));

              return (FE<2,L2_HIERARCHIC>::shape(elem, order, i, pp) -
                      FE<2,L2_HIERARCHIC>::shape(elem, order, i, pm))/2./eps;
            }

            // d()/deta
          case 1:
            {
              const Point pp(p(0), p(1)+eps);
              const Point pm(p(0), p(1)-eps);

              return (FE<2,L2_HIERARCHIC>::shape(elem, order, i, pp) -
                      FE<2,L2_HIERARCHIC>::shape(elem, order, i, pm))/2./eps;
            }


          default:
            libmesh_error_msg("Invalid derivative index j = " << j);
          }
      }

    case QUAD4:
    case QUADSHELL4:
      libmesh_assert_less (totalorder, 2);
      libmesh_fallthrough();
    case QUAD8:
    case QUADSHELL8:
    case QUAD9:
      {
        // Compute quad shape functions as a tensor-product
        const Real xi  = p(0);
        const Real eta = p(1);

        libmesh_assert_less (i, (totalorder+1u)*(totalorder+1u));

        // Example i, i0, i1 values for totalorder = 5:
        //                                    0  1  2  3  4  5  6  7  8  9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35
        //  static const unsigned int i0[] = {0, 1, 1, 0, 2, 3, 4, 5, 1, 1, 1, 1, 2, 3, 4, 5, 0, 0, 0, 0, 2, 3, 4, 5, 2, 3, 4, 5, 2, 3, 4, 5, 2, 3, 4, 5};
        //  static const unsigned int i1[] = {0, 0, 1, 1, 0, 0, 0, 0, 2, 3, 4, 5, 1, 1, 1, 1, 2, 3, 4, 5, 2, 2, 2, 2, 3, 3, 3, 3, 4, 4, 4, 4, 5, 5, 5, 5};

        unsigned int i0, i1;

        // Vertex DoFs
        if (i == 0)
          { i0 = 0; i1 = 0; }
        else if (i == 1)
          { i0 = 1; i1 = 0; }
        else if (i == 2)
          { i0 = 1; i1 = 1; }
        else if (i == 3)
          { i0 = 0; i1 = 1; }
        // Edge DoFs
        else if (i < totalorder + 3u)
          { i0 = i - 2; i1 = 0; }
        else if (i < 2u*totalorder + 2)
          { i0 = 1; i1 = i - totalorder - 1; }
        else if (i < 3u*totalorder + 1u)
          { i0 = i - 2u*totalorder; i1 = 1; }
        else if (i < 4u*totalorder)
          { i0 = 0; i1 = i - 3u*totalorder + 1; }
        // Interior DoFs
        else
          {
            unsigned int basisnum = i - 4*totalorder;
            i0 = square_number_column[basisnum] + 2;
            i1 = square_number_row[basisnum] + 2;
          }

        // Flip odd degree of freedom values if necessary
        // to keep continuity on sides
        Real f = 1.;

        if ((i0%2) && (i0 > 2) && (i1 == 0))
          f = (elem->point(0) > elem->point(1))?-1.:1.;
        else if ((i0%2) && (i0>2) && (i1 == 1))
          f = (elem->point(3) > elem->point(2))?-1.:1.;
        else if ((i0 == 0) && (i1%2) && (i1>2))
          f = (elem->point(0) > elem->point(3))?-1.:1.;
        else if ((i0 == 1) && (i1%2) && (i1>2))
          f = (elem->point(1) > elem->point(2))?-1.:1.;

        switch (j)
          {
            // d()/dxi
          case 0:
            return f*(FE<1,L2_HIERARCHIC>::shape_deriv(EDGE3, totalorder, i0, 0, xi)*
                      FE<1,L2_HIERARCHIC>::shape      (EDGE3, totalorder, i1,    eta));

            // d()/deta
          case 1:
            return f*(FE<1,L2_HIERARCHIC>::shape      (EDGE3, totalorder, i0,    xi)*
                      FE<1,L2_HIERARCHIC>::shape_deriv(EDGE3, totalorder, i1, 0, eta));

          default:
            libmesh_error_msg("Invalid derivative index j = " << j);
          }
      }

    default:
      libmesh_error_msg("ERROR: Unsupported element type = " << type);
    }

  return 0.;
}

shape_deriv() [66/109]

Real libMesh::FE< 2, HIERARCHIC >::shape_deriv ( const Elem *  elem, const Order  order, const unsigned int  i, const unsigned int  j, const Point &  p  )

inherited

Definition at line 244 of file fe_hierarchic_shape_2D.C.

{
  libmesh_assert(elem);

  const ElemType type = elem->type();

  const Order totalorder = static_cast<Order>(order+elem->p_level());

  libmesh_assert_greater (totalorder, 0);

  switch (type)
    {
      // 1st & 2nd-order Hierarchics.
    case TRI3:
    case TRISHELL3:
    case TRI6:
      {
        const Real eps = 1.e-6;

        libmesh_assert_less (j, 2);

        switch (j)
          {
            //  d()/dxi
          case 0:
            {
              const Point pp(p(0)+eps, p(1));
              const Point pm(p(0)-eps, p(1));

              return (FE<2,HIERARCHIC>::shape(elem, order, i, pp) -
                      FE<2,HIERARCHIC>::shape(elem, order, i, pm))/2./eps;
            }

            // d()/deta
          case 1:
            {
              const Point pp(p(0), p(1)+eps);
              const Point pm(p(0), p(1)-eps);

              return (FE<2,HIERARCHIC>::shape(elem, order, i, pp) -
                      FE<2,HIERARCHIC>::shape(elem, order, i, pm))/2./eps;
            }

          default:
            libmesh_error_msg("Invalid derivative index j = " << j);
          }
      }

    case QUAD4:
    case QUADSHELL4:
      libmesh_assert_less (totalorder, 2);
      libmesh_fallthrough();
    case QUAD8:
    case QUADSHELL8:
    case QUAD9:
      {
        // Compute quad shape functions as a tensor-product
        const Real xi  = p(0);
        const Real eta = p(1);

        libmesh_assert_less (i, (totalorder+1u)*(totalorder+1u));

        // Example i, i0, i1 values for totalorder = 5:
        //                                    0  1  2  3  4  5  6  7  8  9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35
        //  static const unsigned int i0[] = {0, 1, 1, 0, 2, 3, 4, 5, 1, 1, 1, 1, 2, 3, 4, 5, 0, 0, 0, 0, 2, 3, 4, 5, 2, 3, 4, 5, 2, 3, 4, 5, 2, 3, 4, 5};
        //  static const unsigned int i1[] = {0, 0, 1, 1, 0, 0, 0, 0, 2, 3, 4, 5, 1, 1, 1, 1, 2, 3, 4, 5, 2, 2, 2, 2, 3, 3, 3, 3, 4, 4, 4, 4, 5, 5, 5, 5};

        unsigned int i0, i1;

        // Vertex DoFs
        if (i == 0)
          { i0 = 0; i1 = 0; }
        else if (i == 1)
          { i0 = 1; i1 = 0; }
        else if (i == 2)
          { i0 = 1; i1 = 1; }
        else if (i == 3)
          { i0 = 0; i1 = 1; }
        // Edge DoFs
        else if (i < totalorder + 3u)
          { i0 = i - 2; i1 = 0; }
        else if (i < 2u*totalorder + 2)
          { i0 = 1; i1 = i - totalorder - 1; }
        else if (i < 3u*totalorder + 1u)
          { i0 = i - 2u*totalorder; i1 = 1; }
        else if (i < 4u*totalorder)
          { i0 = 0; i1 = i - 3u*totalorder + 1; }
        // Interior DoFs
        else
          {
            unsigned int basisnum = i - 4*totalorder;
            i0 = square_number_column[basisnum] + 2;
            i1 = square_number_row[basisnum] + 2;
          }

        // Flip odd degree of freedom values if necessary
        // to keep continuity on sides
        Real f = 1.;

        if ((i0%2) && (i0 > 2) && (i1 == 0))
          f = (elem->point(0) > elem->point(1))?-1.:1.;
        else if ((i0%2) && (i0>2) && (i1 == 1))
          f = (elem->point(3) > elem->point(2))?-1.:1.;
        else if ((i0 == 0) && (i1%2) && (i1>2))
          f = (elem->point(0) > elem->point(3))?-1.:1.;
        else if ((i0 == 1) && (i1%2) && (i1>2))
          f = (elem->point(1) > elem->point(2))?-1.:1.;

        switch (j)
          {
            // d()/dxi
          case 0:
            return f*(FE<1,HIERARCHIC>::shape_deriv(EDGE3, totalorder, i0, 0, xi)*
                      FE<1,HIERARCHIC>::shape      (EDGE3, totalorder, i1,    eta));

            // d()/deta
          case 1:
            return f*(FE<1,HIERARCHIC>::shape      (EDGE3, totalorder, i0,    xi)*
                      FE<1,HIERARCHIC>::shape_deriv(EDGE3, totalorder, i1, 0, eta));

          default:
            libmesh_error_msg("Invalid derivative index j = " << j);
          }
      }

    default:
      libmesh_error_msg("ERROR: Unsupported element type = " << type);
    }

  return 0.;
}

shape_deriv() [67/109]

Real libMesh::FE< 1, HERMITE >::shape_deriv ( const Elem *  elem, const Order  libmesh_dbg_varorder, const unsigned int  i, const unsigned  int, const Point &  p  )

inherited

Definition at line 250 of file fe_hermite_shape_1D.C.

{
  libmesh_assert(elem);

  // Coefficient naming: d(1)d(2n) is the coefficient of the
  // global shape function corresponding to value 1 in terms of the
  // local shape function corresponding to normal derivative 2
  Real d1xd1x, d2xd2x;

  hermite_compute_coefs(elem, d1xd1x, d2xd2x);

  const ElemType type = elem->type();

#ifndef NDEBUG
  const unsigned int totalorder = order + elem->p_level();
#endif

  switch (type)
    {
      // C1 functions on the C1 cubic edge
    case EDGE2:
    case EDGE3:
      {
        libmesh_assert_less (i, totalorder+1);

        switch (i)
          {
          case 0:
            return FEHermite<1>::hermite_raw_shape_deriv(0, p(0));
          case 1:
            return d1xd1x * FEHermite<1>::hermite_raw_shape_deriv(2, p(0));
          case 2:
            return FEHermite<1>::hermite_raw_shape_deriv(1, p(0));
          case 3:
            return d2xd2x * FEHermite<1>::hermite_raw_shape_deriv(3, p(0));
          default:
            return FEHermite<1>::hermite_raw_shape_deriv(i, p(0));
          }
      }
    default:
      libmesh_error_msg("ERROR: Unsupported element type = " << type);
    }
}

shape_deriv() [68/109]

Real libMesh::FE< 2, HERMITE >::shape_deriv ( const ElemType  , const Order  , const unsigned  int, const unsigned  int, const Point &    )

inherited

Definition at line 251 of file fe_hermite_shape_2D.C.

{
  libmesh_error_msg("Hermite elements require the real element \nto construct gradient-based degrees of freedom.");
  return 0.;
}

shape_deriv() [69/109]

Real libMesh::FE< 2, HERMITE >::shape_deriv ( const Elem *  elem, const Order  order, const unsigned int  i, const unsigned int  j, const Point &  p  )

inherited

Definition at line 264 of file fe_hermite_shape_2D.C.

{
  libmesh_assert(elem);
  libmesh_assert (j == 0 || j == 1);

  std::vector<std::vector<Real>> dxdxi(2, std::vector<Real>(2, 0));

#ifdef DEBUG
  std::vector<Real> dxdeta(2), dydxi(2);
#endif

  hermite_compute_coefs(elem,dxdxi
#ifdef DEBUG
                        ,dxdeta,dydxi
#endif
                        );

  const ElemType type = elem->type();

  const Order totalorder = static_cast<Order>(order + elem->p_level());

  switch (type)
    {
    case QUAD4:
    case QUADSHELL4:
      libmesh_assert_less (totalorder, 4);
      libmesh_fallthrough();
    case QUAD8:
    case QUADSHELL8:
    case QUAD9:
      {
        libmesh_assert_less (i, (totalorder+1u)*(totalorder+1u));

        std::vector<unsigned int> bases1D;

        Real coef = hermite_bases_2D(bases1D, dxdxi, totalorder, i);

        switch (j)
          {
          case 0:
            return coef *
              FEHermite<1>::hermite_raw_shape_deriv(bases1D[0],p(0)) *
              FEHermite<1>::hermite_raw_shape(bases1D[1],p(1));
          case 1:
            return coef *
              FEHermite<1>::hermite_raw_shape(bases1D[0],p(0)) *
              FEHermite<1>::hermite_raw_shape_deriv(bases1D[1],p(1));
          default:
            libmesh_error_msg("Invalid derivative index j = " << j);
          }
      }
    default:
      libmesh_error_msg("ERROR: Unsupported element type = " << type);
    }
}

shape_deriv() [70/109]

Real libMesh::FE< 1, CLOUGH >::shape_deriv ( const ElemType  , const Order  , const unsigned  int, const unsigned  int, const Point &    )

inherited

Definition at line 276 of file fe_clough_shape_1D.C.

{
  libmesh_error_msg("Clough-Tocher elements require the real element \nto construct gradient-based degrees of freedom.");
  return 0.;
}

shape_deriv() [71/109]

Real libMesh::FE< 1, CLOUGH >::shape_deriv ( const Elem *  elem, const Order  order, const unsigned int  i, const unsigned int  j, const Point &  p  )

inherited

Definition at line 289 of file fe_clough_shape_1D.C.

{
  libmesh_assert(elem);

  clough_compute_coefs(elem);

  const ElemType type = elem->type();

  const Order totalorder = static_cast<Order>(order + elem->p_level());

  switch (totalorder)
    {
      // 3rd-order C1 cubic element
    case THIRD:
      {
        switch (type)
          {
            // C1 functions on the C1 cubic edge
          case EDGE2:
          case EDGE3:
            {
              switch (i)
                {
                case 0:
                  return clough_raw_shape_deriv(0, j, p);
                case 1:
                  return clough_raw_shape_deriv(1, j, p);
                case 2:
                  return d1xd1x * clough_raw_shape_deriv(2, j, p);
                case 3:
                  return d2xd2x * clough_raw_shape_deriv(3, j, p);
                default:
                  libmesh_error_msg("Invalid shape function index i = " << i);
                }
            }
          default:
            libmesh_error_msg("ERROR: Unsupported element type = " << type);
          }
      }
      // by default throw an error
    default:
      libmesh_error_msg("ERROR: Unsupported polynomial order = " << totalorder);
    }
}

shape_deriv() [72/109]

Real libMesh::FE< 2, MONOMIAL >::shape_deriv ( const Elem *  elem, const Order  order, const unsigned int  i, const unsigned int  j, const Point &  p  )

inherited

Definition at line 304 of file fe_monomial_shape_2D.C.

{
  libmesh_assert(elem);

  // by default call the orientation-independent shape functions
  return FE<2,MONOMIAL>::shape_deriv(elem->type(), static_cast<Order>(order + elem->p_level()), i, j, p);
}

shape_deriv() [73/109]

Real libMesh::FE< 1, BERNSTEIN >::shape_deriv ( const Elem *  elem, const Order  order, const unsigned int  i, const unsigned int  j, const Point &  p  )

inherited

Definition at line 361 of file fe_bernstein_shape_1D.C.

{
  libmesh_assert(elem);

  return FE<1,BERNSTEIN>::shape_deriv(elem->type(),
                                      static_cast<Order>(order + elem->p_level()), i, j, p);
}

shape_deriv() [74/109]

Real libMesh::FE< 3, L2_LAGRANGE >::shape_deriv ( const ElemType  type, const Order  order, const unsigned int  i, const unsigned int  j, const Point &  p  )

inherited

Definition at line 371 of file fe_l2_lagrange_shape_3D.C.

{
#if LIBMESH_DIM == 3

  libmesh_assert_less (j, 3);

  switch (order)
    {
      // linear Lagrange shape functions
    case FIRST:
      {
        switch (type)
          {
            // trilinear hexahedral shape functions
          case HEX8:
          case HEX20:
          case HEX27:
            {
              libmesh_assert_less (i, 8);

              // Compute hex shape functions as a tensor-product
              const Real xi   = p(0);
              const Real eta  = p(1);
              const Real zeta = p(2);

              static const unsigned int i0[] = {0, 1, 1, 0, 0, 1, 1, 0};
              static const unsigned int i1[] = {0, 0, 1, 1, 0, 0, 1, 1};
              static const unsigned int i2[] = {0, 0, 0, 0, 1, 1, 1, 1};

              switch(j)
                {
                case 0:
                  return (FE<1,L2_LAGRANGE>::shape_deriv(EDGE2, FIRST, i0[i], 0, xi)*
                          FE<1,L2_LAGRANGE>::shape      (EDGE2, FIRST, i1[i], eta)*
                          FE<1,L2_LAGRANGE>::shape      (EDGE2, FIRST, i2[i], zeta));

                case 1:
                  return (FE<1,L2_LAGRANGE>::shape      (EDGE2, FIRST, i0[i], xi)*
                          FE<1,L2_LAGRANGE>::shape_deriv(EDGE2, FIRST, i1[i], 0, eta)*
                          FE<1,L2_LAGRANGE>::shape      (EDGE2, FIRST, i2[i], zeta));

                case 2:
                  return (FE<1,L2_LAGRANGE>::shape      (EDGE2, FIRST, i0[i], xi)*
                          FE<1,L2_LAGRANGE>::shape      (EDGE2, FIRST, i1[i], eta)*
                          FE<1,L2_LAGRANGE>::shape_deriv(EDGE2, FIRST, i2[i], 0, zeta));

                default:
                  libmesh_error_msg("Invalid j = " << j);
                }
            }

            // linear tetrahedral shape functions
          case TET4:
          case TET10:
            {
              libmesh_assert_less (i, 4);

              // Area coordinates, pg. 205, Vol. I, Carey, Oden, Becker FEM
              const Real dzeta0dxi = -1.;
              const Real dzeta1dxi =  1.;
              const Real dzeta2dxi =  0.;
              const Real dzeta3dxi =  0.;

              const Real dzeta0deta = -1.;
              const Real dzeta1deta =  0.;
              const Real dzeta2deta =  1.;
              const Real dzeta3deta =  0.;

              const Real dzeta0dzeta = -1.;
              const Real dzeta1dzeta =  0.;
              const Real dzeta2dzeta =  0.;
              const Real dzeta3dzeta =  1.;

              switch (j)
                {
                  // d()/dxi
                case 0:
                  {
                    switch(i)
                      {
                      case 0:
                        return dzeta0dxi;

                      case 1:
                        return dzeta1dxi;

                      case 2:
                        return dzeta2dxi;

                      case 3:
                        return dzeta3dxi;

                      default:
                        libmesh_error_msg("Invalid i = " << i);
                      }
                  }

                  // d()/deta
                case 1:
                  {
                    switch(i)
                      {
                      case 0:
                        return dzeta0deta;

                      case 1:
                        return dzeta1deta;

                      case 2:
                        return dzeta2deta;

                      case 3:
                        return dzeta3deta;

                      default:
                        libmesh_error_msg("Invalid i = " << i);
                      }
                  }

                  // d()/dzeta
                case 2:
                  {
                    switch(i)
                      {
                      case 0:
                        return dzeta0dzeta;

                      case 1:
                        return dzeta1dzeta;

                      case 2:
                        return dzeta2dzeta;

                      case 3:
                        return dzeta3dzeta;

                      default:
                        libmesh_error_msg("Invalid i = " << i);
                      }
                  }

                default:
                  libmesh_error_msg("Invalid shape function derivative j = " << j);
                }
            }

            // linear prism shape functions
          case PRISM6:
          case PRISM15:
          case PRISM18:
            {
              libmesh_assert_less (i, 6);

              // Compute prism shape functions as a tensor-product
              // of a triangle and an edge

              Point p2d(p(0),p(1));
              Point p1d(p(2));

              //                                0  1  2  3  4  5
              static const unsigned int i0[] = {0, 0, 0, 1, 1, 1};
              static const unsigned int i1[] = {0, 1, 2, 0, 1, 2};

              switch (j)
                {
                  // d()/dxi
                case 0:
                  return (FE<2,L2_LAGRANGE>::shape_deriv(TRI3,  FIRST, i1[i], 0, p2d)*
                          FE<1,L2_LAGRANGE>::shape(EDGE2, FIRST, i0[i], p1d));

                  // d()/deta
                case 1:
                  return (FE<2,L2_LAGRANGE>::shape_deriv(TRI3,  FIRST, i1[i], 1, p2d)*
                          FE<1,L2_LAGRANGE>::shape(EDGE2, FIRST, i0[i], p1d));

                  // d()/dzeta
                case 2:
                  return (FE<2,L2_LAGRANGE>::shape(TRI3,  FIRST, i1[i], p2d)*
                          FE<1,L2_LAGRANGE>::shape_deriv(EDGE2, FIRST, i0[i], 0, p1d));

                default:
                  libmesh_error_msg("Invalid shape function derivative j = " << j);
                }
            }

            // linear pyramid shape functions
          case PYRAMID5:
            {
              libmesh_assert_less (i, 5);

              const Real xi   = p(0);
              const Real eta  = p(1);
              const Real zeta = p(2);
              const Real eps  = 1.e-35;

              switch (j)
                {
                  // d/dxi
                case 0:
                  switch(i)
                    {
                    case 0:
                      return  .25*(zeta + eta - 1.)/((1. - zeta) + eps);

                    case 1:
                      return -.25*(zeta + eta - 1.)/((1. - zeta) + eps);

                    case 2:
                      return -.25*(zeta - eta - 1.)/((1. - zeta) + eps);

                    case 3:
                      return  .25*(zeta - eta - 1.)/((1. - zeta) + eps);

                    case 4:
                      return 0;

                    default:
                      libmesh_error_msg("Invalid i = " << i);
                    }


                  // d/deta
                case 1:
                  switch(i)
                    {
                    case 0:
                      return  .25*(zeta + xi - 1.)/((1. - zeta) + eps);

                    case 1:
                      return  .25*(zeta - xi - 1.)/((1. - zeta) + eps);

                    case 2:
                      return -.25*(zeta - xi - 1.)/((1. - zeta) + eps);

                    case 3:
                      return -.25*(zeta + xi - 1.)/((1. - zeta) + eps);

                    case 4:
                      return 0;

                    default:
                      libmesh_error_msg("Invalid i = " << i);
                    }


                  // d/dzeta
                case 2:
                  switch(i)
                    {
                    case 0:
                      {
                        const Real a=1.;
                        const Real b=1.;

                        return .25*(((zeta + a*xi - 1.)*(zeta + b*eta - 1.) +
                                     (1. - zeta)*((zeta + a*xi -1.) + (zeta + b*eta - 1.)))/
                                    ((1. - zeta)*(1. - zeta) + eps));
                      }

                    case 1:
                      {
                        const Real a=-1.;
                        const Real b=1.;

                        return .25*(((zeta + a*xi - 1.)*(zeta + b*eta - 1.) +
                                     (1. - zeta)*((zeta + a*xi -1.) + (zeta + b*eta - 1.)))/
                                    ((1. - zeta)*(1. - zeta) + eps));
                      }

                    case 2:
                      {
                        const Real a=-1.;
                        const Real b=-1.;

                        return .25*(((zeta + a*xi - 1.)*(zeta + b*eta - 1.) +
                                     (1. - zeta)*((zeta + a*xi -1.) + (zeta + b*eta - 1.)))/
                                    ((1. - zeta)*(1. - zeta) + eps));
                      }

                    case 3:
                      {
                        const Real a=1.;
                        const Real b=-1.;

                        return .25*(((zeta + a*xi - 1.)*(zeta + b*eta - 1.) +
                                     (1. - zeta)*((zeta + a*xi -1.) + (zeta + b*eta - 1.)))/
                                    ((1. - zeta)*(1. - zeta) + eps));
                      }

                    case 4:
                      return 1.;

                    default:
                      libmesh_error_msg("Invalid i = " << i);
                    }


                default:
                  libmesh_error_msg("Invalid j = " << j);
                }
            }


          default:
            libmesh_error_msg("ERROR: Unsupported 3D element type!: " << type);
          }
      }


      // quadratic Lagrange shape functions
    case SECOND:
      {
        switch (type)
          {

            // serendipity hexahedral quadratic shape functions
          case HEX20:
            {
              libmesh_assert_less (i, 20);

              const Real xi   = p(0);
              const Real eta  = p(1);
              const Real zeta = p(2);

              // these functions are defined for (x,y,z) in [0,1]^3
              // so transform the locations
              const Real x = .5*(xi   + 1.);
              const Real y = .5*(eta  + 1.);
              const Real z = .5*(zeta + 1.);

              // and don't forget the chain rule!

              switch (j)
                {

                  // d/dx*dx/dxi
                case 0:
                  switch (i)
                    {
                    case 0:
                      return .5*(1. - y)*(1. - z)*((1. - x)*(-2.) +
                                                   (-1.)*(1. - 2.*x - 2.*y - 2.*z));

                    case 1:
                      return .5*(1. - y)*(1. - z)*(x*(2.) +
                                                   (1.)*(2.*x - 2.*y - 2.*z - 1.));

                    case 2:
                      return .5*y*(1. - z)*(x*(2.) +
                                            (1.)*(2.*x + 2.*y - 2.*z - 3.));

                    case 3:
                      return .5*y*(1. - z)*((1. - x)*(-2.) +
                                            (-1.)*(2.*y - 2.*x - 2.*z - 1.));

                    case 4:
                      return .5*(1. - y)*z*((1. - x)*(-2.) +
                                            (-1.)*(2.*z - 2.*x - 2.*y - 1.));

                    case 5:
                      return .5*(1. - y)*z*(x*(2.) +
                                            (1.)*(2.*x - 2.*y + 2.*z - 3.));

                    case 6:
                      return .5*y*z*(x*(2.) +
                                     (1.)*(2.*x + 2.*y + 2.*z - 5.));

                    case 7:
                      return .5*y*z*((1. - x)*(-2.) +
                                     (-1.)*(2.*y - 2.*x + 2.*z - 3.));

                    case 8:
                      return 2.*(1. - y)*(1. - z)*(1. - 2.*x);

                    case 9:
                      return 2.*y*(1. - y)*(1. - z);

                    case 10:
                      return 2.*y*(1. - z)*(1. - 2.*x);

                    case 11:
                      return 2.*y*(1. - y)*(1. - z)*(-1.);

                    case 12:
                      return 2.*(1. - y)*z*(1. - z)*(-1.);

                    case 13:
                      return 2.*(1. - y)*z*(1. - z);

                    case 14:
                      return 2.*y*z*(1. - z);

                    case 15:
                      return 2.*y*z*(1. - z)*(-1.);

                    case 16:
                      return 2.*(1. - y)*z*(1. - 2.*x);

                    case 17:
                      return 2.*y*(1. - y)*z;

                    case 18:
                      return 2.*y*z*(1. - 2.*x);

                    case 19:
                      return 2.*y*(1. - y)*z*(-1.);

                    default:
                      libmesh_error_msg("Invalid i = " << i);
                    }


                  // d/dy*dy/deta
                case 1:
                  switch (i)
                    {
                    case 0:
                      return .5*(1. - x)*(1. - z)*((1. - y)*(-2.) +
                                                   (-1.)*(1. - 2.*x - 2.*y - 2.*z));

                    case 1:
                      return .5*x*(1. - z)*((1. - y)*(-2.) +
                                            (-1.)*(2.*x - 2.*y - 2.*z - 1.));

                    case 2:
                      return .5*x*(1. - z)*(y*(2.) +
                                            (1.)*(2.*x + 2.*y - 2.*z - 3.));

                    case 3:
                      return .5*(1. - x)*(1. - z)*(y*(2.) +
                                                   (1.)*(2.*y - 2.*x - 2.*z - 1.));

                    case 4:
                      return .5*(1. - x)*z*((1. - y)*(-2.) +
                                            (-1.)*(2.*z - 2.*x - 2.*y - 1.));

                    case 5:
                      return .5*x*z*((1. - y)*(-2.) +
                                     (-1.)*(2.*x - 2.*y + 2.*z - 3.));

                    case 6:
                      return .5*x*z*(y*(2.) +
                                     (1.)*(2.*x + 2.*y + 2.*z - 5.));

                    case 7:
                      return .5*(1. - x)*z*(y*(2.) +
                                            (1.)*(2.*y - 2.*x + 2.*z - 3.));

                    case 8:
                      return 2.*x*(1. - x)*(1. - z)*(-1.);

                    case 9:
                      return 2.*x*(1. - z)*(1. - 2.*y);

                    case 10:
                      return 2.*x*(1. - x)*(1. - z);

                    case 11:
                      return 2.*(1. - x)*(1. - z)*(1. - 2.*y);

                    case 12:
                      return 2.*(1. - x)*z*(1. - z)*(-1.);

                    case 13:
                      return 2.*x*z*(1. - z)*(-1.);

                    case 14:
                      return 2.*x*z*(1. - z);

                    case 15:
                      return 2.*(1. - x)*z*(1. - z);

                    case 16:
                      return 2.*x*(1. - x)*z*(-1.);

                    case 17:
                      return 2.*x*z*(1. - 2.*y);

                    case 18:
                      return 2.*x*(1. - x)*z;

                    case 19:
                      return 2.*(1. - x)*z*(1. - 2.*y);

                    default:
                      libmesh_error_msg("Invalid i = " << i);
                    }


                  // d/dz*dz/dzeta
                case 2:
                  switch (i)
                    {
                    case 0:
                      return .5*(1. - x)*(1. - y)*((1. - z)*(-2.) +
                                                   (-1.)*(1. - 2.*x - 2.*y - 2.*z));

                    case 1:
                      return .5*x*(1. - y)*((1. - z)*(-2.) +
                                            (-1.)*(2.*x - 2.*y - 2.*z - 1.));

                    case 2:
                      return .5*x*y*((1. - z)*(-2.) +
                                     (-1.)*(2.*x + 2.*y - 2.*z - 3.));

                    case 3:
                      return .5*(1. - x)*y*((1. - z)*(-2.) +
                                            (-1.)*(2.*y - 2.*x - 2.*z - 1.));

                    case 4:
                      return .5*(1. - x)*(1. - y)*(z*(2.) +
                                                   (1.)*(2.*z - 2.*x - 2.*y - 1.));

                    case 5:
                      return .5*x*(1. - y)*(z*(2.) +
                                            (1.)*(2.*x - 2.*y + 2.*z - 3.));

                    case 6:
                      return .5*x*y*(z*(2.) +
                                     (1.)*(2.*x + 2.*y + 2.*z - 5.));

                    case 7:
                      return .5*(1. - x)*y*(z*(2.) +
                                            (1.)*(2.*y - 2.*x + 2.*z - 3.));

                    case 8:
                      return 2.*x*(1. - x)*(1. - y)*(-1.);

                    case 9:
                      return 2.*x*y*(1. - y)*(-1.);

                    case 10:
                      return 2.*x*(1. - x)*y*(-1.);

                    case 11:
                      return 2.*(1. - x)*y*(1. - y)*(-1.);

                    case 12:
                      return 2.*(1. - x)*(1. - y)*(1. - 2.*z);

                    case 13:
                      return 2.*x*(1. - y)*(1. - 2.*z);

                    case 14:
                      return 2.*x*y*(1. - 2.*z);

                    case 15:
                      return 2.*(1. - x)*y*(1. - 2.*z);

                    case 16:
                      return 2.*x*(1. - x)*(1. - y);

                    case 17:
                      return 2.*x*y*(1. - y);

                    case 18:
                      return 2.*x*(1. - x)*y;

                    case 19:
                      return 2.*(1. - x)*y*(1. - y);

                    default:
                      libmesh_error_msg("Invalid i = " << i);
                    }

                default:
                  libmesh_error_msg("Invalid shape function derivative j = " << j);
                }
            }

            // triquadratic hexahedral shape functions
          case HEX27:
            {
              libmesh_assert_less (i, 27);

              // Compute hex shape functions as a tensor-product
              const Real xi   = p(0);
              const Real eta  = p(1);
              const Real zeta = p(2);

              // The only way to make any sense of this
              // is to look at the mgflo/mg2/mgf documentation
              // and make the cut-out cube!
              //                                0  1  2  3  4  5  6  7  8  9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26
              static const unsigned int i0[] = {0, 1, 1, 0, 0, 1, 1, 0, 2, 1, 2, 0, 0, 1, 1, 0, 2, 1, 2, 0, 2, 2, 1, 2, 0, 2, 2};
              static const unsigned int i1[] = {0, 0, 1, 1, 0, 0, 1, 1, 0, 2, 1, 2, 0, 0, 1, 1, 0, 2, 1, 2, 2, 0, 2, 1, 2, 2, 2};
              static const unsigned int i2[] = {0, 0, 0, 0, 1, 1, 1, 1, 0, 0, 0, 0, 2, 2, 2, 2, 1, 1, 1, 1, 0, 2, 2, 2, 2, 1, 2};

              switch(j)
                {
                case 0:
                  return (FE<1,L2_LAGRANGE>::shape_deriv(EDGE3, SECOND, i0[i], 0, xi)*
                          FE<1,L2_LAGRANGE>::shape      (EDGE3, SECOND, i1[i], eta)*
                          FE<1,L2_LAGRANGE>::shape      (EDGE3, SECOND, i2[i], zeta));

                case 1:
                  return (FE<1,L2_LAGRANGE>::shape      (EDGE3, SECOND, i0[i], xi)*
                          FE<1,L2_LAGRANGE>::shape_deriv(EDGE3, SECOND, i1[i], 0, eta)*
                          FE<1,L2_LAGRANGE>::shape      (EDGE3, SECOND, i2[i], zeta));

                case 2:
                  return (FE<1,L2_LAGRANGE>::shape      (EDGE3, SECOND, i0[i], xi)*
                          FE<1,L2_LAGRANGE>::shape      (EDGE3, SECOND, i1[i], eta)*
                          FE<1,L2_LAGRANGE>::shape_deriv(EDGE3, SECOND, i2[i], 0, zeta));

                default:
                  libmesh_error_msg("Invalid j = " << j);
                }
            }

            // quadratic tetrahedral shape functions
          case TET10:
            {
              libmesh_assert_less (i, 10);

              // Area coordinates, pg. 205, Vol. I, Carey, Oden, Becker FEM
              const Real zeta1 = p(0);
              const Real zeta2 = p(1);
              const Real zeta3 = p(2);
              const Real zeta0 = 1. - zeta1 - zeta2 - zeta3;

              const Real dzeta0dxi = -1.;
              const Real dzeta1dxi =  1.;
              const Real dzeta2dxi =  0.;
              const Real dzeta3dxi =  0.;

              const Real dzeta0deta = -1.;
              const Real dzeta1deta =  0.;
              const Real dzeta2deta =  1.;
              const Real dzeta3deta =  0.;

              const Real dzeta0dzeta = -1.;
              const Real dzeta1dzeta =  0.;
              const Real dzeta2dzeta =  0.;
              const Real dzeta3dzeta =  1.;

              switch (j)
                {
                  // d()/dxi
                case 0:
                  {
                    switch(i)
                      {
                      case 0:
                        return (4.*zeta0 - 1.)*dzeta0dxi;

                      case 1:
                        return (4.*zeta1 - 1.)*dzeta1dxi;

                      case 2:
                        return (4.*zeta2 - 1.)*dzeta2dxi;

                      case 3:
                        return (4.*zeta3 - 1.)*dzeta3dxi;

                      case 4:
                        return 4.*(zeta0*dzeta1dxi + dzeta0dxi*zeta1);

                      case 5:
                        return 4.*(zeta1*dzeta2dxi + dzeta1dxi*zeta2);

                      case 6:
                        return 4.*(zeta0*dzeta2dxi + dzeta0dxi*zeta2);

                      case 7:
                        return 4.*(zeta0*dzeta3dxi + dzeta0dxi*zeta3);

                      case 8:
                        return 4.*(zeta1*dzeta3dxi + dzeta1dxi*zeta3);

                      case 9:
                        return 4.*(zeta2*dzeta3dxi + dzeta2dxi*zeta3);

                      default:
                        libmesh_error_msg("Invalid i = " << i);
                      }
                  }

                  // d()/deta
                case 1:
                  {
                    switch(i)
                      {
                      case 0:
                        return (4.*zeta0 - 1.)*dzeta0deta;

                      case 1:
                        return (4.*zeta1 - 1.)*dzeta1deta;

                      case 2:
                        return (4.*zeta2 - 1.)*dzeta2deta;

                      case 3:
                        return (4.*zeta3 - 1.)*dzeta3deta;

                      case 4:
                        return 4.*(zeta0*dzeta1deta + dzeta0deta*zeta1);

                      case 5:
                        return 4.*(zeta1*dzeta2deta + dzeta1deta*zeta2);

                      case 6:
                        return 4.*(zeta0*dzeta2deta + dzeta0deta*zeta2);

                      case 7:
                        return 4.*(zeta0*dzeta3deta + dzeta0deta*zeta3);

                      case 8:
                        return 4.*(zeta1*dzeta3deta + dzeta1deta*zeta3);

                      case 9:
                        return 4.*(zeta2*dzeta3deta + dzeta2deta*zeta3);

                      default:
                        libmesh_error_msg("Invalid i = " << i);
                      }
                  }

                  // d()/dzeta
                case 2:
                  {
                    switch(i)
                      {
                      case 0:
                        return (4.*zeta0 - 1.)*dzeta0dzeta;

                      case 1:
                        return (4.*zeta1 - 1.)*dzeta1dzeta;

                      case 2:
                        return (4.*zeta2 - 1.)*dzeta2dzeta;

                      case 3:
                        return (4.*zeta3 - 1.)*dzeta3dzeta;

                      case 4:
                        return 4.*(zeta0*dzeta1dzeta + dzeta0dzeta*zeta1);

                      case 5:
                        return 4.*(zeta1*dzeta2dzeta + dzeta1dzeta*zeta2);

                      case 6:
                        return 4.*(zeta0*dzeta2dzeta + dzeta0dzeta*zeta2);

                      case 7:
                        return 4.*(zeta0*dzeta3dzeta + dzeta0dzeta*zeta3);

                      case 8:
                        return 4.*(zeta1*dzeta3dzeta + dzeta1dzeta*zeta3);

                      case 9:
                        return 4.*(zeta2*dzeta3dzeta + dzeta2dzeta*zeta3);

                      default:
                        libmesh_error_msg("Invalid i = " << i);
                      }
                  }

                default:
                  libmesh_error_msg("Invalid j = " << j);
                }
            }



            // quadratic prism shape functions
          case PRISM18:
            {
              libmesh_assert_less (i, 18);

              // Compute prism shape functions as a tensor-product
              // of a triangle and an edge

              Point p2d(p(0),p(1));
              Point p1d(p(2));

              //                                0  1  2  3  4  5  6  7  8  9 10 11 12 13 14 15 16 17
              static const unsigned int i0[] = {0, 0, 0, 1, 1, 1, 0, 0, 0, 2, 2, 2, 1, 1, 1, 2, 2, 2};
              static const unsigned int i1[] = {0, 1, 2, 0, 1, 2, 3, 4, 5, 0, 1, 2, 3, 4, 5, 3, 4, 5};

              switch (j)
                {
                  // d()/dxi
                case 0:
                  return (FE<2,L2_LAGRANGE>::shape_deriv(TRI6,  SECOND, i1[i], 0, p2d)*
                          FE<1,L2_LAGRANGE>::shape(EDGE3, SECOND, i0[i], p1d));

                  // d()/deta
                case 1:
                  return (FE<2,L2_LAGRANGE>::shape_deriv(TRI6,  SECOND, i1[i], 1, p2d)*
                          FE<1,L2_LAGRANGE>::shape(EDGE3, SECOND, i0[i], p1d));

                  // d()/dzeta
                case 2:
                  return (FE<2,L2_LAGRANGE>::shape(TRI6,  SECOND, i1[i], p2d)*
                          FE<1,L2_LAGRANGE>::shape_deriv(EDGE3, SECOND, i0[i], 0, p1d));

                default:
                  libmesh_error_msg("Invalid shape function derivative j = " << j);
                }
            }


          default:
            libmesh_error_msg("ERROR: Unsupported 3D element type!: " << type);
          }
      }


      // unsupported order
    default:
      libmesh_error_msg("ERROR: Unsupported 3D FE order!: " << order);
    }

#endif
}

shape_deriv() [75/109]

Real libMesh::FE< 2, BERNSTEIN >::shape_deriv ( const ElemType  , const Order  , const unsigned  int, const unsigned  int, const Point &    )

inherited

Definition at line 386 of file fe_bernstein_shape_2D.C.

{
  libmesh_error_msg("Bernstein polynomials require the element type \nbecause edge orientation is needed.");
  return 0.;
}

shape_deriv() [76/109]

Real libMesh::FE< 2, BERNSTEIN >::shape_deriv ( const Elem *  elem, const Order  order, const unsigned int  i, const unsigned int  j, const Point &  p  )

inherited

Definition at line 399 of file fe_bernstein_shape_2D.C.

{
  libmesh_assert(elem);

  const ElemType type = elem->type();

  const Order totalorder = static_cast<Order>(order + elem->p_level());

  switch (type)
    {
      // Hierarchic shape functions on the quadrilateral.
    case QUAD4:
    case QUAD9:
      {
        // Compute quad shape functions as a tensor-product
        const Real xi  = p(0);
        const Real eta = p(1);

        libmesh_assert_less (i, (totalorder+1u)*(totalorder+1u));

        unsigned int i0, i1;

        // Vertex DoFs
        if (i == 0)
          { i0 = 0; i1 = 0; }
        else if (i == 1)
          { i0 = 1; i1 = 0; }
        else if (i == 2)
          { i0 = 1; i1 = 1; }
        else if (i == 3)
          { i0 = 0; i1 = 1; }


        // Edge DoFs
        else if (i < totalorder + 3u)
          { i0 = i - 2; i1 = 0; }
        else if (i < 2u*totalorder + 2)
          { i0 = 1; i1 = i - totalorder - 1; }
        else if (i < 3u*totalorder + 1)
          { i0 = i - 2u*totalorder; i1 = 1; }
        else if (i < 4u*totalorder)
          { i0 = 0; i1 = i - 3u*totalorder + 1; }
        // Interior DoFs
        else
          {
            unsigned int basisnum = i - 4*totalorder;
            i0 = square_number_column[basisnum] + 2;
            i1 = square_number_row[basisnum] + 2;
          }


        // Flip odd degree of freedom values if necessary
        // to keep continuity on sides
        if      ((i>= 4                 && i<= 4+  totalorder-2u) && elem->point(0) > elem->point(1)) i0=totalorder+2-i0;
        else if ((i>= 4+  totalorder-1u && i<= 4+2*totalorder-3u) && elem->point(1) > elem->point(2)) i1=totalorder+2-i1;
        else if ((i>= 4+2*totalorder-2u && i<= 4+3*totalorder-4u) && elem->point(3) > elem->point(2)) i0=totalorder+2-i0;
        else if ((i>= 4+3*totalorder-3u && i<= 4+4*totalorder-5u) && elem->point(0) > elem->point(3)) i1=totalorder+2-i1;

        switch (j)
          {
            // d()/dxi
          case 0:
            return (FE<1,BERNSTEIN>::shape_deriv(EDGE3, totalorder, i0, 0, xi)*
                    FE<1,BERNSTEIN>::shape      (EDGE3, totalorder, i1,    eta));

            // d()/deta
          case 1:
            return (FE<1,BERNSTEIN>::shape      (EDGE3, totalorder, i0,    xi)*
                    FE<1,BERNSTEIN>::shape_deriv(EDGE3, totalorder, i1, 0, eta));

          default:
            libmesh_error_msg("Invalid shape function derivative j = " << j);
          }
      }

      // Bernstein shape functions on the 8-noded quadrilateral
      // is handled separately.
    case QUAD8:
    case QUADSHELL8:
      {
        libmesh_assert_less (totalorder, 3);

        const Real xi  = p(0);
        const Real eta = p(1);

        libmesh_assert_less (i, 8);

        //                                0  1  2  3  4  5  6  7  8
        static const unsigned int i0[] = {0, 1, 1, 0, 2, 1, 2, 0, 2};
        static const unsigned int i1[] = {0, 0, 1, 1, 0, 2, 1, 2, 2};
        static const Real scal[] = {-0.25, -0.25, -0.25, -0.25, 0.5, 0.5, 0.5, 0.5};
        switch (j)
          {
            // d()/dxi
          case 0:
            return (FE<1,BERNSTEIN>::shape_deriv(EDGE3, totalorder, i0[i], 0, xi)*
                    FE<1,BERNSTEIN>::shape      (EDGE3, totalorder, i1[i],    eta)
                    +scal[i]*
                    FE<1,BERNSTEIN>::shape_deriv(EDGE3, totalorder, i0[8], 0, xi)*
                    FE<1,BERNSTEIN>::shape      (EDGE3, totalorder, i1[8],    eta));

            // d()/deta
          case 1:
            return (FE<1,BERNSTEIN>::shape      (EDGE3, totalorder, i0[i],    xi)*
                    FE<1,BERNSTEIN>::shape_deriv(EDGE3, totalorder, i1[i], 0, eta)
                    +scal[i]*
                    FE<1,BERNSTEIN>::shape      (EDGE3, totalorder, i0[8],    xi)*
                    FE<1,BERNSTEIN>::shape_deriv(EDGE3, totalorder, i1[8], 0, eta));

          default:
            libmesh_error_msg("Invalid shape function derivative j = " << j);
          }
      }

    case TRI3:
    case TRISHELL3:
      libmesh_assert_less (totalorder, 2);
      libmesh_fallthrough();
    case TRI6:
      {
        // I have been lazy here and am using finite differences
        // to compute the derivatives!
        const Real eps = 1.e-6;

        switch (j)
          {
            //  d()/dxi
          case 0:
            {
              const Point pp(p(0)+eps, p(1));
              const Point pm(p(0)-eps, p(1));

              return (FE<2,BERNSTEIN>::shape(elem, totalorder, i, pp) -
                      FE<2,BERNSTEIN>::shape(elem, totalorder, i, pm))/2./eps;
            }

            // d()/deta
          case 1:
            {
              const Point pp(p(0), p(1)+eps);
              const Point pm(p(0), p(1)-eps);

              return (FE<2,BERNSTEIN>::shape(elem, totalorder, i, pp) -
                      FE<2,BERNSTEIN>::shape(elem, totalorder, i, pm))/2./eps;
            }


          default:
            libmesh_error_msg("Invalid shape function derivative j = " << j);
          }
      }

    default:
      libmesh_error_msg("ERROR: Unsupported element type = " << type);
    }
}

shape_deriv() [77/109]

Real libMesh::FE< 3, HERMITE >::shape_deriv ( const ElemType  , const Order  , const unsigned  int, const unsigned  int, const Point &    )

inherited

Definition at line 447 of file fe_hermite_shape_3D.C.

{
  libmesh_error_msg("Hermite elements require the real element \nto construct gradient-based degrees of freedom.");
  return 0.;
}

shape_deriv() [78/109]

Real libMesh::FE< 3, HERMITE >::shape_deriv ( const Elem *  elem, const Order  order, const unsigned int  i, const unsigned int  j, const Point &  p  )

inherited

Definition at line 460 of file fe_hermite_shape_3D.C.

{
  libmesh_assert(elem);
  libmesh_assert (j == 0 || j == 1 || j == 2);

  std::vector<std::vector<Real>> dxdxi(3, std::vector<Real>(2, 0));

#ifdef DEBUG
  std::vector<Real> dydxi(2), dzdeta(2), dxdzeta(2);
  std::vector<Real> dzdxi(2), dxdeta(2), dydzeta(2);
#endif //DEBUG

  hermite_compute_coefs(elem, dxdxi
#ifdef DEBUG
                        , dydxi, dzdeta, dxdzeta, dzdxi, dxdeta, dydzeta
#endif
                        );

  const ElemType type = elem->type();

  const Order totalorder = static_cast<Order>(order + elem->p_level());

  switch (totalorder)
    {
      // 3rd-order tricubic Hermite functions
    case THIRD:
      {
        switch (type)
          {
          case HEX8:
          case HEX20:
          case HEX27:
            {
              libmesh_assert_less (i, 64);

              std::vector<unsigned int> bases1D;

              Real coef = hermite_bases_3D(bases1D, dxdxi, totalorder, i);

              switch (j) // Derivative type
                {
                case 0:
                  return coef *
                    FEHermite<1>::hermite_raw_shape_deriv(bases1D[0],p(0)) *
                    FEHermite<1>::hermite_raw_shape(bases1D[1],p(1)) *
                    FEHermite<1>::hermite_raw_shape(bases1D[2],p(2));
                  break;
                case 1:
                  return coef *
                    FEHermite<1>::hermite_raw_shape(bases1D[0],p(0)) *
                    FEHermite<1>::hermite_raw_shape_deriv(bases1D[1],p(1)) *
                    FEHermite<1>::hermite_raw_shape(bases1D[2],p(2));
                  break;
                case 2:
                  return coef *
                    FEHermite<1>::hermite_raw_shape(bases1D[0],p(0)) *
                    FEHermite<1>::hermite_raw_shape(bases1D[1],p(1)) *
                    FEHermite<1>::hermite_raw_shape_deriv(bases1D[2],p(2));
                  break;
                default:
                  libmesh_error_msg("Invalid shape function derivative j = " << j);
                }

            }
          default:
            libmesh_error_msg("ERROR: Unsupported element type " << type);
          }
      }
      // by default throw an error
    default:
      libmesh_error_msg("ERROR: Unsupported polynomial order " << totalorder);
    }
}

shape_deriv() [79/109]

RealGradient libMesh::FE< 0, LAGRANGE_VEC >::shape_deriv ( const ElemType  type, const Order  order, const unsigned int  i, const unsigned int  j, const Point &  p  )

inherited

Definition at line 544 of file fe_lagrange_vec.C.

{
  Real value = FE<0,LAGRANGE>::shape_deriv( type, order, i, j, p );
  return libMesh::RealGradient( value );
}

shape_deriv() [80/109]

Real libMesh::FE< 2, L2_LAGRANGE >::shape_deriv ( const Elem *  elem, const Order  order, const unsigned int  i, const unsigned int  j, const Point &  p  )

inherited

Definition at line 550 of file fe_l2_lagrange_shape_2D.C.

{
  libmesh_assert(elem);


  // call the orientation-independent shape functions
  return FE<2,L2_LAGRANGE>::shape_deriv(elem->type(), static_cast<Order>(order + elem->p_level()), i, j, p);
}

shape_deriv() [81/109]

Real libMesh::FE< 2, LAGRANGE >::shape_deriv ( const Elem *  elem, const Order  order, const unsigned int  i, const unsigned int  j, const Point &  p  )

inherited

Definition at line 552 of file fe_lagrange_shape_2D.C.

{
  libmesh_assert(elem);


  // call the orientation-independent shape functions
  return FE<2,LAGRANGE>::shape_deriv(elem->type(), static_cast<Order>(order + elem->p_level()), i, j, p);
}

shape_deriv() [82/109]

Real libMesh::FE< 2, SUBDIVISION >::shape_deriv ( const Elem *  elem, const Order  order, const unsigned int  i, const unsigned int  j, const Point &  p  )

inherited

shape_deriv() [83/109]

RealGradient libMesh::FE< 1, LAGRANGE_VEC >::shape_deriv ( const ElemType  type, const Order  order, const unsigned int  i, const unsigned int  j, const Point &  p  )

inherited

Definition at line 566 of file fe_lagrange_vec.C.

{
  Real value = FE<1,LAGRANGE>::shape_deriv( type, order, i, j, p );
  return libMesh::RealGradient( value );
}

shape_deriv() [84/109]

Real libMesh::FE< 3, LAGRANGE >::shape_deriv ( const ElemType  type, const Order  order, const unsigned int  i, const unsigned int  j, const Point &  p  )

inherited

Definition at line 573 of file fe_lagrange_shape_3D.C.

{
#if LIBMESH_DIM == 3

  libmesh_assert_less (j, 3);

  switch (order)
    {
      // linear Lagrange shape functions
    case FIRST:
      {
        switch (type)
          {
            // trilinear hexahedral shape functions
          case HEX8:
          case HEX20:
          case HEX27:
            {
              libmesh_assert_less (i, 8);

              // Compute hex shape functions as a tensor-product
              const Real xi   = p(0);
              const Real eta  = p(1);
              const Real zeta = p(2);

              static const unsigned int i0[] = {0, 1, 1, 0, 0, 1, 1, 0};
              static const unsigned int i1[] = {0, 0, 1, 1, 0, 0, 1, 1};
              static const unsigned int i2[] = {0, 0, 0, 0, 1, 1, 1, 1};

              switch(j)
                {
                case 0:
                  return (FE<1,LAGRANGE>::shape_deriv(EDGE2, FIRST, i0[i], 0, xi)*
                          FE<1,LAGRANGE>::shape      (EDGE2, FIRST, i1[i], eta)*
                          FE<1,LAGRANGE>::shape      (EDGE2, FIRST, i2[i], zeta));

                case 1:
                  return (FE<1,LAGRANGE>::shape      (EDGE2, FIRST, i0[i], xi)*
                          FE<1,LAGRANGE>::shape_deriv(EDGE2, FIRST, i1[i], 0, eta)*
                          FE<1,LAGRANGE>::shape      (EDGE2, FIRST, i2[i], zeta));

                case 2:
                  return (FE<1,LAGRANGE>::shape      (EDGE2, FIRST, i0[i], xi)*
                          FE<1,LAGRANGE>::shape      (EDGE2, FIRST, i1[i], eta)*
                          FE<1,LAGRANGE>::shape_deriv(EDGE2, FIRST, i2[i], 0, zeta));

                default:
                  libmesh_error_msg("Invalid j = " << j);
                }
            }

            // linear tetrahedral shape functions
          case TET4:
          case TET10:
            {
              libmesh_assert_less (i, 4);

              // Area coordinates, pg. 205, Vol. I, Carey, Oden, Becker FEM
              const Real dzeta0dxi = -1.;
              const Real dzeta1dxi =  1.;
              const Real dzeta2dxi =  0.;
              const Real dzeta3dxi =  0.;

              const Real dzeta0deta = -1.;
              const Real dzeta1deta =  0.;
              const Real dzeta2deta =  1.;
              const Real dzeta3deta =  0.;

              const Real dzeta0dzeta = -1.;
              const Real dzeta1dzeta =  0.;
              const Real dzeta2dzeta =  0.;
              const Real dzeta3dzeta =  1.;

              switch (j)
                {
                  // d()/dxi
                case 0:
                  {
                    switch(i)
                      {
                      case 0:
                        return dzeta0dxi;

                      case 1:
                        return dzeta1dxi;

                      case 2:
                        return dzeta2dxi;

                      case 3:
                        return dzeta3dxi;

                      default:
                        libmesh_error_msg("Invalid i = " << i);
                      }
                  }

                  // d()/deta
                case 1:
                  {
                    switch(i)
                      {
                      case 0:
                        return dzeta0deta;

                      case 1:
                        return dzeta1deta;

                      case 2:
                        return dzeta2deta;

                      case 3:
                        return dzeta3deta;

                      default:
                        libmesh_error_msg("Invalid i = " << i);
                      }
                  }

                  // d()/dzeta
                case 2:
                  {
                    switch(i)
                      {
                      case 0:
                        return dzeta0dzeta;

                      case 1:
                        return dzeta1dzeta;

                      case 2:
                        return dzeta2dzeta;

                      case 3:
                        return dzeta3dzeta;

                      default:
                        libmesh_error_msg("Invalid i = " << i);
                      }
                  }

                default:
                  libmesh_error_msg("Invalid shape function derivative j = " << j);
                }
            }

            // linear prism shape functions
          case PRISM6:
          case PRISM15:
          case PRISM18:
            {
              libmesh_assert_less (i, 6);

              // Compute prism shape functions as a tensor-product
              // of a triangle and an edge

              Point p2d(p(0),p(1));
              Point p1d(p(2));

              //                                0  1  2  3  4  5
              static const unsigned int i0[] = {0, 0, 0, 1, 1, 1};
              static const unsigned int i1[] = {0, 1, 2, 0, 1, 2};

              switch (j)
                {
                  // d()/dxi
                case 0:
                  return (FE<2,LAGRANGE>::shape_deriv(TRI3,  FIRST, i1[i], 0, p2d)*
                          FE<1,LAGRANGE>::shape(EDGE2, FIRST, i0[i], p1d));

                  // d()/deta
                case 1:
                  return (FE<2,LAGRANGE>::shape_deriv(TRI3,  FIRST, i1[i], 1, p2d)*
                          FE<1,LAGRANGE>::shape(EDGE2, FIRST, i0[i], p1d));

                  // d()/dzeta
                case 2:
                  return (FE<2,LAGRANGE>::shape(TRI3,  FIRST, i1[i], p2d)*
                          FE<1,LAGRANGE>::shape_deriv(EDGE2, FIRST, i0[i], 0, p1d));

                default:
                  libmesh_error_msg("Invalid shape function derivative j = " << j);
                }
            }

            // linear pyramid shape functions
          case PYRAMID5:
          case PYRAMID13:
          case PYRAMID14:
            {
              libmesh_assert_less (i, 5);

              const Real xi   = p(0);
              const Real eta  = p(1);
              const Real zeta = p(2);
              const Real eps  = 1.e-35;

              switch (j)
                {
                  // d/dxi
                case 0:
                  switch(i)
                    {
                    case 0:
                      return  .25*(zeta + eta - 1.)/((1. - zeta) + eps);

                    case 1:
                      return -.25*(zeta + eta - 1.)/((1. - zeta) + eps);

                    case 2:
                      return -.25*(zeta - eta - 1.)/((1. - zeta) + eps);

                    case 3:
                      return  .25*(zeta - eta - 1.)/((1. - zeta) + eps);

                    case 4:
                      return 0;

                    default:
                      libmesh_error_msg("Invalid i = " << i);
                    }


                  // d/deta
                case 1:
                  switch(i)
                    {
                    case 0:
                      return  .25*(zeta + xi - 1.)/((1. - zeta) + eps);

                    case 1:
                      return  .25*(zeta - xi - 1.)/((1. - zeta) + eps);

                    case 2:
                      return -.25*(zeta - xi - 1.)/((1. - zeta) + eps);

                    case 3:
                      return -.25*(zeta + xi - 1.)/((1. - zeta) + eps);

                    case 4:
                      return 0;

                    default:
                      libmesh_error_msg("Invalid i = " << i);
                    }


                  // d/dzeta
                case 2:
                  {
                    // We computed the derivatives with general eps and
                    // then let eps tend to zero in the numerators...
                    Real
                      num = zeta*(2. - zeta) - 1.,
                      den = (1. - zeta + eps)*(1. - zeta + eps);

                    switch(i)
                      {
                      case 0:
                      case 2:
                        return .25*(num + xi*eta)/den;

                      case 1:
                      case 3:
                        return .25*(num - xi*eta)/den;

                      case 4:
                        return 1.;

                      default:
                        libmesh_error_msg("Invalid i = " << i);
                      }
                  }

                default:
                  libmesh_error_msg("Invalid j = " << j);
                }
            }


          default:
            libmesh_error_msg("ERROR: Unsupported 3D element type!: " << type);
          }
      }


      // quadratic Lagrange shape functions
    case SECOND:
      {
        switch (type)
          {

            // serendipity hexahedral quadratic shape functions
          case HEX20:
            {
              libmesh_assert_less (i, 20);

              const Real xi   = p(0);
              const Real eta  = p(1);
              const Real zeta = p(2);

              // these functions are defined for (x,y,z) in [0,1]^3
              // so transform the locations
              const Real x = .5*(xi   + 1.);
              const Real y = .5*(eta  + 1.);
              const Real z = .5*(zeta + 1.);

              // and don't forget the chain rule!

              switch (j)
                {

                  // d/dx*dx/dxi
                case 0:
                  switch (i)
                    {
                    case 0:
                      return .5*(1. - y)*(1. - z)*((1. - x)*(-2.) +
                                                   (-1.)*(1. - 2.*x - 2.*y - 2.*z));

                    case 1:
                      return .5*(1. - y)*(1. - z)*(x*(2.) +
                                                   (1.)*(2.*x - 2.*y - 2.*z - 1.));

                    case 2:
                      return .5*y*(1. - z)*(x*(2.) +
                                            (1.)*(2.*x + 2.*y - 2.*z - 3.));

                    case 3:
                      return .5*y*(1. - z)*((1. - x)*(-2.) +
                                            (-1.)*(2.*y - 2.*x - 2.*z - 1.));

                    case 4:
                      return .5*(1. - y)*z*((1. - x)*(-2.) +
                                            (-1.)*(2.*z - 2.*x - 2.*y - 1.));

                    case 5:
                      return .5*(1. - y)*z*(x*(2.) +
                                            (1.)*(2.*x - 2.*y + 2.*z - 3.));

                    case 6:
                      return .5*y*z*(x*(2.) +
                                     (1.)*(2.*x + 2.*y + 2.*z - 5.));

                    case 7:
                      return .5*y*z*((1. - x)*(-2.) +
                                     (-1.)*(2.*y - 2.*x + 2.*z - 3.));

                    case 8:
                      return 2.*(1. - y)*(1. - z)*(1. - 2.*x);

                    case 9:
                      return 2.*y*(1. - y)*(1. - z);

                    case 10:
                      return 2.*y*(1. - z)*(1. - 2.*x);

                    case 11:
                      return 2.*y*(1. - y)*(1. - z)*(-1.);

                    case 12:
                      return 2.*(1. - y)*z*(1. - z)*(-1.);

                    case 13:
                      return 2.*(1. - y)*z*(1. - z);

                    case 14:
                      return 2.*y*z*(1. - z);

                    case 15:
                      return 2.*y*z*(1. - z)*(-1.);

                    case 16:
                      return 2.*(1. - y)*z*(1. - 2.*x);

                    case 17:
                      return 2.*y*(1. - y)*z;

                    case 18:
                      return 2.*y*z*(1. - 2.*x);

                    case 19:
                      return 2.*y*(1. - y)*z*(-1.);

                    default:
                      libmesh_error_msg("Invalid i = " << i);
                    }


                  // d/dy*dy/deta
                case 1:
                  switch (i)
                    {
                    case 0:
                      return .5*(1. - x)*(1. - z)*((1. - y)*(-2.) +
                                                   (-1.)*(1. - 2.*x - 2.*y - 2.*z));

                    case 1:
                      return .5*x*(1. - z)*((1. - y)*(-2.) +
                                            (-1.)*(2.*x - 2.*y - 2.*z - 1.));

                    case 2:
                      return .5*x*(1. - z)*(y*(2.) +
                                            (1.)*(2.*x + 2.*y - 2.*z - 3.));

                    case 3:
                      return .5*(1. - x)*(1. - z)*(y*(2.) +
                                                   (1.)*(2.*y - 2.*x - 2.*z - 1.));

                    case 4:
                      return .5*(1. - x)*z*((1. - y)*(-2.) +
                                            (-1.)*(2.*z - 2.*x - 2.*y - 1.));

                    case 5:
                      return .5*x*z*((1. - y)*(-2.) +
                                     (-1.)*(2.*x - 2.*y + 2.*z - 3.));

                    case 6:
                      return .5*x*z*(y*(2.) +
                                     (1.)*(2.*x + 2.*y + 2.*z - 5.));

                    case 7:
                      return .5*(1. - x)*z*(y*(2.) +
                                            (1.)*(2.*y - 2.*x + 2.*z - 3.));

                    case 8:
                      return 2.*x*(1. - x)*(1. - z)*(-1.);

                    case 9:
                      return 2.*x*(1. - z)*(1. - 2.*y);

                    case 10:
                      return 2.*x*(1. - x)*(1. - z);

                    case 11:
                      return 2.*(1. - x)*(1. - z)*(1. - 2.*y);

                    case 12:
                      return 2.*(1. - x)*z*(1. - z)*(-1.);

                    case 13:
                      return 2.*x*z*(1. - z)*(-1.);

                    case 14:
                      return 2.*x*z*(1. - z);

                    case 15:
                      return 2.*(1. - x)*z*(1. - z);

                    case 16:
                      return 2.*x*(1. - x)*z*(-1.);

                    case 17:
                      return 2.*x*z*(1. - 2.*y);

                    case 18:
                      return 2.*x*(1. - x)*z;

                    case 19:
                      return 2.*(1. - x)*z*(1. - 2.*y);

                    default:
                      libmesh_error_msg("Invalid i = " << i);
                    }


                  // d/dz*dz/dzeta
                case 2:
                  switch (i)
                    {
                    case 0:
                      return .5*(1. - x)*(1. - y)*((1. - z)*(-2.) +
                                                   (-1.)*(1. - 2.*x - 2.*y - 2.*z));

                    case 1:
                      return .5*x*(1. - y)*((1. - z)*(-2.) +
                                            (-1.)*(2.*x - 2.*y - 2.*z - 1.));

                    case 2:
                      return .5*x*y*((1. - z)*(-2.) +
                                     (-1.)*(2.*x + 2.*y - 2.*z - 3.));

                    case 3:
                      return .5*(1. - x)*y*((1. - z)*(-2.) +
                                            (-1.)*(2.*y - 2.*x - 2.*z - 1.));

                    case 4:
                      return .5*(1. - x)*(1. - y)*(z*(2.) +
                                                   (1.)*(2.*z - 2.*x - 2.*y - 1.));

                    case 5:
                      return .5*x*(1. - y)*(z*(2.) +
                                            (1.)*(2.*x - 2.*y + 2.*z - 3.));

                    case 6:
                      return .5*x*y*(z*(2.) +
                                     (1.)*(2.*x + 2.*y + 2.*z - 5.));

                    case 7:
                      return .5*(1. - x)*y*(z*(2.) +
                                            (1.)*(2.*y - 2.*x + 2.*z - 3.));

                    case 8:
                      return 2.*x*(1. - x)*(1. - y)*(-1.);

                    case 9:
                      return 2.*x*y*(1. - y)*(-1.);

                    case 10:
                      return 2.*x*(1. - x)*y*(-1.);

                    case 11:
                      return 2.*(1. - x)*y*(1. - y)*(-1.);

                    case 12:
                      return 2.*(1. - x)*(1. - y)*(1. - 2.*z);

                    case 13:
                      return 2.*x*(1. - y)*(1. - 2.*z);

                    case 14:
                      return 2.*x*y*(1. - 2.*z);

                    case 15:
                      return 2.*(1. - x)*y*(1. - 2.*z);

                    case 16:
                      return 2.*x*(1. - x)*(1. - y);

                    case 17:
                      return 2.*x*y*(1. - y);

                    case 18:
                      return 2.*x*(1. - x)*y;

                    case 19:
                      return 2.*(1. - x)*y*(1. - y);

                    default:
                      libmesh_error_msg("Invalid i = " << i);
                    }

                default:
                  libmesh_error_msg("Invalid shape function derivative j = " << j);
                }
            }

            // triquadratic hexahedral shape functions
          case HEX27:
            {
              libmesh_assert_less (i, 27);

              // Compute hex shape functions as a tensor-product
              const Real xi   = p(0);
              const Real eta  = p(1);
              const Real zeta = p(2);

              // The only way to make any sense of this
              // is to look at the mgflo/mg2/mgf documentation
              // and make the cut-out cube!
              //                                0  1  2  3  4  5  6  7  8  9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26
              static const unsigned int i0[] = {0, 1, 1, 0, 0, 1, 1, 0, 2, 1, 2, 0, 0, 1, 1, 0, 2, 1, 2, 0, 2, 2, 1, 2, 0, 2, 2};
              static const unsigned int i1[] = {0, 0, 1, 1, 0, 0, 1, 1, 0, 2, 1, 2, 0, 0, 1, 1, 0, 2, 1, 2, 2, 0, 2, 1, 2, 2, 2};
              static const unsigned int i2[] = {0, 0, 0, 0, 1, 1, 1, 1, 0, 0, 0, 0, 2, 2, 2, 2, 1, 1, 1, 1, 0, 2, 2, 2, 2, 1, 2};

              switch(j)
                {
                case 0:
                  return (FE<1,LAGRANGE>::shape_deriv(EDGE3, SECOND, i0[i], 0, xi)*
                          FE<1,LAGRANGE>::shape      (EDGE3, SECOND, i1[i], eta)*
                          FE<1,LAGRANGE>::shape      (EDGE3, SECOND, i2[i], zeta));

                case 1:
                  return (FE<1,LAGRANGE>::shape      (EDGE3, SECOND, i0[i], xi)*
                          FE<1,LAGRANGE>::shape_deriv(EDGE3, SECOND, i1[i], 0, eta)*
                          FE<1,LAGRANGE>::shape      (EDGE3, SECOND, i2[i], zeta));

                case 2:
                  return (FE<1,LAGRANGE>::shape      (EDGE3, SECOND, i0[i], xi)*
                          FE<1,LAGRANGE>::shape      (EDGE3, SECOND, i1[i], eta)*
                          FE<1,LAGRANGE>::shape_deriv(EDGE3, SECOND, i2[i], 0, zeta));

                default:
                  libmesh_error_msg("Invalid j = " << j);
                }
            }

            // quadratic tetrahedral shape functions
          case TET10:
            {
              libmesh_assert_less (i, 10);

              // Area coordinates, pg. 205, Vol. I, Carey, Oden, Becker FEM
              const Real zeta1 = p(0);
              const Real zeta2 = p(1);
              const Real zeta3 = p(2);
              const Real zeta0 = 1. - zeta1 - zeta2 - zeta3;

              const Real dzeta0dxi = -1.;
              const Real dzeta1dxi =  1.;
              const Real dzeta2dxi =  0.;
              const Real dzeta3dxi =  0.;

              const Real dzeta0deta = -1.;
              const Real dzeta1deta =  0.;
              const Real dzeta2deta =  1.;
              const Real dzeta3deta =  0.;

              const Real dzeta0dzeta = -1.;
              const Real dzeta1dzeta =  0.;
              const Real dzeta2dzeta =  0.;
              const Real dzeta3dzeta =  1.;

              switch (j)
                {
                  // d()/dxi
                case 0:
                  {
                    switch(i)
                      {
                      case 0:
                        return (4.*zeta0 - 1.)*dzeta0dxi;

                      case 1:
                        return (4.*zeta1 - 1.)*dzeta1dxi;

                      case 2:
                        return (4.*zeta2 - 1.)*dzeta2dxi;

                      case 3:
                        return (4.*zeta3 - 1.)*dzeta3dxi;

                      case 4:
                        return 4.*(zeta0*dzeta1dxi + dzeta0dxi*zeta1);

                      case 5:
                        return 4.*(zeta1*dzeta2dxi + dzeta1dxi*zeta2);

                      case 6:
                        return 4.*(zeta0*dzeta2dxi + dzeta0dxi*zeta2);

                      case 7:
                        return 4.*(zeta0*dzeta3dxi + dzeta0dxi*zeta3);

                      case 8:
                        return 4.*(zeta1*dzeta3dxi + dzeta1dxi*zeta3);

                      case 9:
                        return 4.*(zeta2*dzeta3dxi + dzeta2dxi*zeta3);

                      default:
                        libmesh_error_msg("Invalid i = " << i);
                      }
                  }

                  // d()/deta
                case 1:
                  {
                    switch(i)
                      {
                      case 0:
                        return (4.*zeta0 - 1.)*dzeta0deta;

                      case 1:
                        return (4.*zeta1 - 1.)*dzeta1deta;

                      case 2:
                        return (4.*zeta2 - 1.)*dzeta2deta;

                      case 3:
                        return (4.*zeta3 - 1.)*dzeta3deta;

                      case 4:
                        return 4.*(zeta0*dzeta1deta + dzeta0deta*zeta1);

                      case 5:
                        return 4.*(zeta1*dzeta2deta + dzeta1deta*zeta2);

                      case 6:
                        return 4.*(zeta0*dzeta2deta + dzeta0deta*zeta2);

                      case 7:
                        return 4.*(zeta0*dzeta3deta + dzeta0deta*zeta3);

                      case 8:
                        return 4.*(zeta1*dzeta3deta + dzeta1deta*zeta3);

                      case 9:
                        return 4.*(zeta2*dzeta3deta + dzeta2deta*zeta3);

                      default:
                        libmesh_error_msg("Invalid i = " << i);
                      }
                  }

                  // d()/dzeta
                case 2:
                  {
                    switch(i)
                      {
                      case 0:
                        return (4.*zeta0 - 1.)*dzeta0dzeta;

                      case 1:
                        return (4.*zeta1 - 1.)*dzeta1dzeta;

                      case 2:
                        return (4.*zeta2 - 1.)*dzeta2dzeta;

                      case 3:
                        return (4.*zeta3 - 1.)*dzeta3dzeta;

                      case 4:
                        return 4.*(zeta0*dzeta1dzeta + dzeta0dzeta*zeta1);

                      case 5:
                        return 4.*(zeta1*dzeta2dzeta + dzeta1dzeta*zeta2);

                      case 6:
                        return 4.*(zeta0*dzeta2dzeta + dzeta0dzeta*zeta2);

                      case 7:
                        return 4.*(zeta0*dzeta3dzeta + dzeta0dzeta*zeta3);

                      case 8:
                        return 4.*(zeta1*dzeta3dzeta + dzeta1dzeta*zeta3);

                      case 9:
                        return 4.*(zeta2*dzeta3dzeta + dzeta2dzeta*zeta3);

                      default:
                        libmesh_error_msg("Invalid i = " << i);
                      }
                  }

                default:
                  libmesh_error_msg("Invalid j = " << j);
                }
            }


            // "serendipity" prism
          case PRISM15:
            {
              libmesh_assert_less (i, 15);

              const Real xi   = p(0);
              const Real eta  = p(1);
              const Real zeta = p(2);

              switch (j)
                {
                  // d()/dxi
                case 0:
                  {
                    switch(i)
                      {
                      case 0:
                        return (2.*xi + 2.*eta + 0.5*zeta - 1.)*(1. - zeta);
                      case 1:
                        return (2.*xi - 1. - 0.5*zeta)*(1. - zeta);
                      case 2:
                        return 0.;
                      case 3:
                        return (2.*xi + 2.*eta - 0.5*zeta - 1.)*(1. + zeta);
                      case 4:
                        return (2.*xi - 1. + 0.5*zeta)*(1. + zeta);
                      case 5:
                        return 0.;
                      case 6:
                        return (4.*xi + 2.*eta - 2.)*(zeta - 1.);
                      case 7:
                        return -2.*(zeta - 1.)*eta;
                      case 8:
                        return 2.*(zeta - 1.)*eta;
                      case 9:
                        return (zeta - 1.)*(1. + zeta);
                      case 10:
                        return (1. - zeta)*(1. + zeta);
                      case 11:
                        return 0.;
                      case 12:
                        return (-4.*xi - 2.*eta + 2.)*(1. + zeta);
                      case 13:
                        return 2.*(1. + zeta)*eta;
                      case 14:
                        return -2.*(1. + zeta)*eta;
                      default:
                        libmesh_error_msg("Invalid i = " << i);
                      }
                  }

                  // d()/deta
                case 1:
                  {
                    switch(i)
                      {
                      case 0:
                        return (2.*xi + 2.*eta + 0.5*zeta - 1.)*(1. - zeta);
                      case 1:
                        return 0.;
                      case 2:
                        return (2.*eta - 1. - 0.5*zeta)*(1. - zeta);
                      case 3:
                        return (2.*xi + 2.*eta - 0.5*zeta - 1.)*(1. + zeta);
                      case 4:
                        return 0.;
                      case 5:
                        return (2.*eta - 1. + 0.5*zeta)*(1. + zeta);
                      case 6:
                        return 2.*(zeta - 1.)*xi;
                      case 7:
                        return 2.*(1. - zeta)*xi;
                      case 8:
                        return (2.*xi + 4.*eta - 2.)*(zeta - 1.);
                      case 9:
                        return (zeta - 1.)*(1. + zeta);
                      case 10:
                        return 0.;
                      case 11:
                        return (1. - zeta)*(1. + zeta);
                      case 12:
                        return -2.*(1. + zeta)*xi;
                      case 13:
                        return 2.*(1. + zeta)*xi;
                      case 14:
                        return (-2.*xi - 4.*eta + 2.)*(1. + zeta);

                      default:
                        libmesh_error_msg("Invalid i = " << i);
                      }
                  }

                  // d()/dzeta
                case 2:
                  {
                    switch(i)
                      {
                      case 0:
                        return (-xi - eta - zeta + 0.5)*(xi + eta - 1.);
                      case 1:
                        return -0.5*xi*(2.*xi - 1. - 2.*zeta);
                      case 2:
                        return -0.5*eta*(2.*eta - 1. - 2.*zeta);
                      case 3:
                        return (xi + eta - zeta - 0.5)*(xi + eta - 1.);
                      case 4:
                        return 0.5*xi*(2.*xi - 1. + 2.*zeta);
                      case 5:
                        return 0.5*eta*(2.*eta - 1. + 2.*zeta);
                      case 6:
                        return 2.*xi*(xi + eta - 1.);
                      case 7:
                        return -2.*xi*eta;
                      case 8:
                        return 2.*eta*(xi + eta - 1.);
                      case 9:
                        return 2.*zeta*(xi + eta - 1.);
                      case 10:
                        return -2.*xi*zeta;
                      case 11:
                        return -2.*eta*zeta;
                      case 12:
                        return 2.*xi*(1. - xi - eta);
                      case 13:
                        return 2.*xi*eta;
                      case 14:
                        return 2.*eta*(1. - xi - eta);

                      default:
                        libmesh_error_msg("Invalid i = " << i);
                      }
                  }

                default:
                  libmesh_error_msg("Invalid j = " << j);
                }
            }



            // quadratic prism shape functions
          case PRISM18:
            {
              libmesh_assert_less (i, 18);

              // Compute prism shape functions as a tensor-product
              // of a triangle and an edge

              Point p2d(p(0),p(1));
              Point p1d(p(2));

              //                                0  1  2  3  4  5  6  7  8  9 10 11 12 13 14 15 16 17
              static const unsigned int i0[] = {0, 0, 0, 1, 1, 1, 0, 0, 0, 2, 2, 2, 1, 1, 1, 2, 2, 2};
              static const unsigned int i1[] = {0, 1, 2, 0, 1, 2, 3, 4, 5, 0, 1, 2, 3, 4, 5, 3, 4, 5};

              switch (j)
                {
                  // d()/dxi
                case 0:
                  return (FE<2,LAGRANGE>::shape_deriv(TRI6,  SECOND, i1[i], 0, p2d)*
                          FE<1,LAGRANGE>::shape(EDGE3, SECOND, i0[i], p1d));

                  // d()/deta
                case 1:
                  return (FE<2,LAGRANGE>::shape_deriv(TRI6,  SECOND, i1[i], 1, p2d)*
                          FE<1,LAGRANGE>::shape(EDGE3, SECOND, i0[i], p1d));

                  // d()/dzeta
                case 2:
                  return (FE<2,LAGRANGE>::shape(TRI6,  SECOND, i1[i], p2d)*
                          FE<1,LAGRANGE>::shape_deriv(EDGE3, SECOND, i0[i], 0, p1d));

                default:
                  libmesh_error_msg("Invalid shape function derivative j = " << j);
                }
            }

            // G. Bedrosian, "Shape functions and integration formulas for
            // three-dimensional finite element analysis", Int. J. Numerical
            // Methods Engineering, vol 35, p. 95-108, 1992.
          case PYRAMID13:
            {
              libmesh_assert_less (i, 13);

              const Real xi   = p(0);
              const Real eta  = p(1);
              const Real zeta = p(2);
              const Real eps  = 1.e-35;

              // Denominators are perturbed by epsilon to avoid
              // divide-by-zero issues.
              Real
                den = (-1. + zeta + eps),
                den2 = den*den,
                xi2 = xi*xi,
                eta2 = eta*eta,
                zeta2 = zeta*zeta,
                zeta3 = zeta2*zeta;

              switch (j)
                {
                  // d/dxi
                case 0:
                  switch(i)
                    {
                    case 0:
                      return 0.25*(-zeta - eta + 2.*eta*zeta - 2.*xi + 2.*zeta*xi + 2.*eta*xi + zeta2 + eta2)/den;

                    case 1:
                      return -0.25*(-zeta - eta + 2.*eta*zeta + 2.*xi - 2.*zeta*xi - 2.*eta*xi + zeta2 + eta2)/den;

                    case 2:
                      return -0.25*(-zeta + eta - 2.*eta*zeta + 2.*xi - 2.*zeta*xi + 2.*eta*xi + zeta2 + eta2)/den;

                    case 3:
                      return 0.25*(-zeta + eta - 2.*eta*zeta - 2.*xi + 2.*zeta*xi - 2.*eta*xi + zeta2 + eta2)/den;

                    case 4:
                      return 0.;

                    case 5:
                      return -(-1. + eta + zeta)*xi/den;

                    case 6:
                      return 0.5*(-1. + eta + zeta)*(1. + eta - zeta)/den;

                    case 7:
                      return (1. + eta - zeta)*xi/den;

                    case 8:
                      return -0.5*(-1. + eta + zeta)*(1. + eta - zeta)/den;

                    case 9:
                      return -(-1. + eta + zeta)*zeta/den;

                    case 10:
                      return (-1. + eta + zeta)*zeta/den;

                    case 11:
                      return -(1. + eta - zeta)*zeta/den;

                    case 12:
                      return (1. + eta - zeta)*zeta/den;

                    default:
                      libmesh_error_msg("Invalid i = " << i);
                    }

                  // d/deta
                case 1:
                  switch(i)
                    {
                    case 0:
                      return 0.25*(-zeta - 2.*eta + 2.*eta*zeta - xi + 2.*zeta*xi + 2.*eta*xi + zeta2 + xi2)/den;

                    case 1:
                      return -0.25*(zeta + 2.*eta - 2.*eta*zeta - xi + 2.*zeta*xi + 2.*eta*xi - zeta2 - xi2)/den;

                    case 2:
                      return -0.25*(-zeta + 2.*eta - 2.*eta*zeta + xi - 2.*zeta*xi + 2.*eta*xi + zeta2 + xi2)/den;

                    case 3:
                      return 0.25*(zeta - 2.*eta + 2.*eta*zeta + xi - 2.*zeta*xi + 2.*eta*xi - zeta2 - xi2)/den;

                    case 4:
                      return 0.;

                    case 5:
                      return -0.5*(-1. + xi + zeta)*(1. + xi - zeta)/den;

                    case 6:
                      return (1. + xi - zeta)*eta/den;

                    case 7:
                      return 0.5*(-1. + xi + zeta)*(1. + xi - zeta)/den;

                    case 8:
                      return -(-1. + xi + zeta)*eta/den;

                    case 9:
                      return -(-1. + xi + zeta)*zeta/den;

                    case 10:
                      return (1. + xi - zeta)*zeta/den;

                    case 11:
                      return -(1. + xi - zeta)*zeta/den;

                    case 12:
                      return (-1. + xi + zeta)*zeta/den;

                    default:
                      libmesh_error_msg("Invalid i = " << i);
                    }

                  // d/dzeta
                case 2:
                  {
                    switch(i)
                      {
                      case 0:
                        return -0.25*(xi + eta + 1.)*(-1. + 2.*zeta - zeta2 + eta*xi)/den2;

                      case 1:
                        return 0.25*(eta - xi + 1.)*(1. - 2.*zeta + zeta2 + eta*xi)/den2;

                      case 2:
                        return 0.25*(xi + eta - 1.)*(-1. + 2.*zeta - zeta2 + eta*xi)/den2;

                      case 3:
                        return -0.25*(eta - xi - 1.)*(1. - 2.*zeta + zeta2 + eta*xi)/den2;

                      case 4:
                        return 4.*zeta - 1.;

                      case 5:
                        return 0.5*(-2 + eta + 6.*zeta + eta*xi2 + eta*zeta2 - 6.*zeta2 + 2.*zeta3 - 2.*eta*zeta)/den2;

                      case 6:
                        return -0.5*(2 - 6.*zeta + xi + xi*zeta2 + eta2*xi + 6.*zeta2 - 2.*zeta3 - 2.*zeta*xi)/den2;

                      case 7:
                        return -0.5*(2 + eta - 6.*zeta + eta*xi2 + eta*zeta2 + 6.*zeta2 - 2.*zeta3 - 2.*eta*zeta)/den2;

                      case 8:
                        return 0.5*(-2 + 6.*zeta + xi + xi*zeta2 + eta2*xi - 6.*zeta2 + 2.*zeta3 - 2.*zeta*xi)/den2;

                      case 9:
                        return (1. - eta - 4.*zeta - xi - xi*zeta2 - eta*zeta2 + eta*xi + 5.*zeta2 - 2.*zeta3 + 2.*eta*zeta + 2.*zeta*xi)/den2;

                      case 10:
                        return -(-1. + eta + 4.*zeta - xi - xi*zeta2 + eta*zeta2 + eta*xi - 5.*zeta2 + 2.*zeta3 - 2.*eta*zeta + 2.*zeta*xi)/den2;

                      case 11:
                        return (1. + eta - 4.*zeta + xi + xi*zeta2 + eta*zeta2 + eta*xi + 5.*zeta2 - 2.*zeta3 - 2.*eta*zeta - 2.*zeta*xi)/den2;

                      case 12:
                        return -(-1. - eta + 4.*zeta + xi + xi*zeta2 - eta*zeta2 + eta*xi - 5.*zeta2 + 2.*zeta3 + 2.*eta*zeta - 2.*zeta*xi)/den2;

                      default:
                        libmesh_error_msg("Invalid i = " << i);
                      }
                  }

                default:
                  libmesh_error_msg("Invalid j = " << j);
                }
            }

            // Quadratic shape functions, as defined in R. Graglia, "Higher order
            // bases on pyramidal elements", IEEE Trans Antennas and Propagation,
            // vol 47, no 5, May 1999.
          case PYRAMID14:
            {
              libmesh_assert_less (i, 14);

              const Real xi   = p(0);
              const Real eta  = p(1);
              const Real zeta = p(2);
              const Real eps  = 1.e-35;

              // The "normalized coordinates" defined by Graglia.  These are
              // the planes which define the faces of the pyramid.
              Real
                p1 = 0.5*(1. - eta - zeta), // back
                p2 = 0.5*(1. + xi  - zeta), // left
                p3 = 0.5*(1. + eta - zeta), // front
                p4 = 0.5*(1. - xi  - zeta); // right

              // Denominators are perturbed by epsilon to avoid
              // divide-by-zero issues.
              Real
                den = (-1. + zeta + eps),
                den2 = den*den,
                den3 = den2*den;

              switch (j)
                {
                  // d/dxi
                case 0:
                  switch(i)
                    {
                    case 0:
                      return 0.5*p1*(-xi*eta + zeta - zeta*zeta + 2.*p4*eta)/den2;

                    case 1:
                      return -0.5*p1*(xi*eta + zeta - zeta*zeta + 2.*p2*eta)/den2;

                    case 2:
                      return 0.5*p3*(xi*eta - zeta + zeta*zeta + 2.*p2*eta)/den2;

                    case 3:
                      return -0.5*p3*(-xi*eta - zeta + zeta*zeta + 2.*p4*eta)/den2;

                    case 4:
                      return 0.;

                    case 5:
                      return 2.*p1*eta*xi/den2;

                    case 6:
                      return 2.*p1*p3*(xi + 2.*p2)/den2;

                    case 7:
                      return -2.*p3*eta*xi/den2;

                    case 8:
                      return -2.*p1*p3*(-xi + 2.*p4)/den2;

                    case 9:
                      return 2.*p1*zeta/den;

                    case 10:
                      return -2.*p1*zeta/den;

                    case 11:
                      return -2.*p3*zeta/den;

                    case 12:
                      return 2.*p3*zeta/den;

                    case 13:
                      return -8.*p1*p3*xi/den2;

                    default:
                      libmesh_error_msg("Invalid i = " << i);
                    }

                  // d/deta
                case 1:
                  switch(i)
                    {
                    case 0:
                      return -0.5*p4*(xi*eta - zeta + zeta*zeta - 2.*p1*xi)/den2;

                    case 1:
                      return 0.5*p2*(xi*eta + zeta - zeta*zeta - 2.*p1*xi)/den2;

                    case 2:
                      return 0.5*p2*(xi*eta - zeta + zeta*zeta + 2.*p3*xi)/den2;

                    case 3:
                      return -0.5*p4*(xi*eta + zeta - zeta*zeta + 2.*p3*xi)/den2;

                    case 4:
                      return 0.;

                    case 5:
                      return 2.*p2*p4*(eta - 2.*p1)/den2;

                    case 6:
                      return -2.*p2*xi*eta/den2;

                    case 7:
                      return 2.*p2*p4*(eta + 2.*p3)/den2;

                    case 8:
                      return 2.*p4*xi*eta/den2;

                    case 9:
                      return 2.*p4*zeta/den;

                    case 10:
                      return 2.*p2*zeta/den;

                    case 11:
                      return -2.*p2*zeta/den;

                    case 12:
                      return -2.*p4*zeta/den;

                    case 13:
                      return -8.*p2*p4*eta/den2;

                    default:
                      libmesh_error_msg("Invalid i = " << i);
                    }


                  // d/dzeta
                case 2:
                  {
                    switch(i)
                      {
                      case 0:
                        return -0.5*p1*(xi*eta - zeta + zeta*zeta)/den2
                          - 0.5*p4*(xi*eta - zeta + zeta*zeta)/den2
                          + p4*p1*(2.*zeta - 1)/den2
                          - 2.*p4*p1*(xi*eta - zeta + zeta*zeta)/den3;

                      case 1:
                        return 0.5*p2*(xi*eta + zeta - zeta*zeta)/den2
                          + 0.5*p1*(xi*eta + zeta - zeta*zeta)/den2
                          - p1*p2*(1 - 2.*zeta)/den2
                          + 2.*p1*p2*(xi*eta + zeta - zeta*zeta)/den3;

                      case 2:
                        return -0.5*p3*(xi*eta - zeta + zeta*zeta)/den2
                          - 0.5*p2*(xi*eta - zeta + zeta*zeta)/den2
                          + p2*p3*(2.*zeta - 1)/den2
                          - 2.*p2*p3*(xi*eta - zeta + zeta*zeta)/den3;

                      case 3:
                        return 0.5*p4*(xi*eta + zeta - zeta*zeta)/den2
                          + 0.5*p3*(xi*eta + zeta - zeta*zeta)/den2
                          - p3*p4*(1 - 2.*zeta)/den2
                          + 2.*p3*p4*(xi*eta + zeta - zeta*zeta)/den3;

                      case 4:
                        return 4.*zeta - 1.;

                      case 5:
                        return 2.*p4*p1*eta/den2
                          + 2.*p2*p4*eta/den2
                          + 2.*p1*p2*eta/den2
                          + 8.*p2*p1*p4*eta/den3;

                      case 6:
                        return -2.*p2*p3*xi/den2
                          - 2.*p1*p3*xi/den2
                          - 2.*p1*p2*xi/den2
                          - 8.*p1*p2*p3*xi/den3;

                      case 7:
                        return -2.*p3*p4*eta/den2
                          - 2.*p2*p4*eta/den2
                          - 2.*p2*p3*eta/den2
                          - 8.*p2*p3*p4*eta/den3;

                      case 8:
                        return 2.*p4*p1*xi/den2
                          + 2.*p1*p3*xi/den2
                          + 2.*p3*p4*xi/den2
                          + 8.*p3*p4*p1*xi/den3;

                      case 9:
                        return 2.*p4*zeta/den
                          + 2.*p1*zeta/den
                          - 4.*p1*p4/den
                          + 4.*p1*p4*zeta/den2;

                      case 10:
                        return 2.*p1*zeta/den
                          + 2.*p2*zeta/den
                          - 4.*p2*p1/den
                          + 4.*p2*p1*zeta/den2;

                      case 11:
                        return 2.*p2*zeta/den
                          + 2.*p3*zeta/den
                          - 4.*p3*p2/den
                          + 4.*p3*p2*zeta/den2;

                      case 12:
                        return 2.*p3*zeta/den
                          + 2.*p4*zeta/den
                          - 4.*p4*p3/den
                          + 4.*p4*p3*zeta/den2;

                      case 13:
                        return -8.*p2*p3*p4/den2
                          - 8.*p3*p4*p1/den2
                          - 8.*p2*p1*p4/den2
                          - 8.*p1*p2*p3/den2
                          - 32.*p1*p2*p3*p4/den3;

                      default:
                        libmesh_error_msg("Invalid i = " << i);
                      }
                  }

                default:
                  libmesh_error_msg("Invalid j = " << j);
                }
            }


          default:
            libmesh_error_msg("ERROR: Unsupported 3D element type!: " << type);
          }
      }


      // unsupported order
    default:
      libmesh_error_msg("ERROR: Unsupported 3D FE order!: " << order);
    }

#else
  return 0.;
#endif
}

shape_deriv() [85/109]

RealGradient libMesh::FE< 0, NEDELEC_ONE >::shape_deriv ( const ElemType  , const Order  , const unsigned  int, const unsigned  int, const Point &    )

inherited

Definition at line 599 of file fe_nedelec_one.C.

{ NEDELEC_LOW_D_ERROR_MESSAGE }

shape_deriv() [86/109]

RealGradient libMesh::FE< 2, LAGRANGE_VEC >::shape_deriv ( const ElemType  type, const Order  order, const unsigned int  i, const unsigned int  j, const Point &  p  )

inherited

Definition at line 602 of file fe_lagrange_vec.C.

{
  Real value = FE<2,LAGRANGE>::shape_deriv( type, order, i/2, j, p );

  switch( i%2 )
    {
    case 0:
      return libMesh::RealGradient( value );

    case 1:
      return libMesh::RealGradient( Real(0), value );

    default:
      libmesh_error_msg("i%2 must be either 0 or 1!");
    }

  //dummy
  return libMesh::RealGradient();
}

shape_deriv() [87/109]

RealGradient libMesh::FE< 0, NEDELEC_ONE >::shape_deriv ( const Elem *  , const Order  , const unsigned  int, const unsigned  int, const Point &    )

inherited

Definition at line 603 of file fe_nedelec_one.C.

{ NEDELEC_LOW_D_ERROR_MESSAGE }

shape_deriv() [88/109]

RealGradient libMesh::FE< 1, NEDELEC_ONE >::shape_deriv ( const ElemType  , const Order  , const unsigned  int, const unsigned  int, const Point &    )

inherited

Definition at line 622 of file fe_nedelec_one.C.

{ NEDELEC_LOW_D_ERROR_MESSAGE }

shape_deriv() [89/109]

RealGradient libMesh::FE< 1, NEDELEC_ONE >::shape_deriv ( const Elem *  , const Order  , const unsigned  int, const unsigned  int, const Point &    )

inherited

Definition at line 626 of file fe_nedelec_one.C.

{ NEDELEC_LOW_D_ERROR_MESSAGE }

shape_deriv() [90/109]

Real libMesh::FE< 3, MONOMIAL >::shape_deriv ( const Elem *  elem, const Order  order, const unsigned int  i, const unsigned int  j, const Point &  p  )

inherited

Definition at line 638 of file fe_monomial_shape_3D.C.

{
  libmesh_assert(elem);

  // call the orientation-independent shape function derivatives
  return FE<3,MONOMIAL>::shape_deriv(elem->type(), static_cast<Order>(order + elem->p_level()), i, j, p);
}

shape_deriv() [91/109]

RealGradient libMesh::FE< 3, LAGRANGE_VEC >::shape_deriv ( const ElemType  type, const Order  order, const unsigned int  i, const unsigned int  j, const Point &  p  )

inherited

Definition at line 670 of file fe_lagrange_vec.C.

{
  Real value = FE<3,LAGRANGE>::shape_deriv( type, order, i/3, j, p );

  switch( i%3 )
    {
    case 0:
      return libMesh::RealGradient( value );

    case 1:
      return libMesh::RealGradient( Real(0), value );

    case 2:
      return libMesh::RealGradient( Real(0), Real(0), value );

    default:
      libmesh_error_msg("i%3 must be 0, 1, or 2!");
    }

  //dummy
  return libMesh::RealGradient();
}

shape_deriv() [92/109]

Real libMesh::FE< 3, L2_HIERARCHIC >::shape_deriv ( const ElemType  , const Order  , const unsigned  int, const unsigned  int, const Point &    )

inherited

Definition at line 704 of file fe_l2_hierarchic_shape_3D.C.

{
  libmesh_error_msg("Hierarchic polynomials require the element type \nbecause edge and face orientation is needed.");
  return 0.;
}

shape_deriv() [93/109]

Real libMesh::FE< 3, HIERARCHIC >::shape_deriv ( const ElemType  , const Order  , const unsigned  int, const unsigned  int, const Point &    )

inherited

Definition at line 704 of file fe_hierarchic_shape_3D.C.

{
  libmesh_error_msg("Hierarchic polynomials require the element type \nbecause edge and face orientation is needed.");
  return 0.;
}

shape_deriv() [94/109]

Real libMesh::FE< 3, L2_HIERARCHIC >::shape_deriv ( const Elem *  elem, const Order  order, const unsigned int  i, const unsigned int  j, const Point &  p  )

inherited

Definition at line 717 of file fe_l2_hierarchic_shape_3D.C.

{
#if LIBMESH_DIM == 3
  libmesh_assert(elem);

  libmesh_assert_less (j, 3);

  // cheat by using finite difference approximations:
  const Real eps = 1.e-6;
  Point pp, pm;

  switch (j)
    {
      // d()/dxi
    case 0:
      {
        pp = Point(p(0)+eps, p(1), p(2));
        pm = Point(p(0)-eps, p(1), p(2));
        break;
      }

      // d()/deta
    case 1:
      {
        pp = Point(p(0), p(1)+eps, p(2));
        pm = Point(p(0), p(1)-eps, p(2));
        break;
      }

      // d()/dzeta
    case 2:
      {
        pp = Point(p(0), p(1), p(2)+eps);
        pm = Point(p(0), p(1), p(2)-eps);
        break;
      }

    default:
      libmesh_error_msg("Invalid derivative index j = " << j);
    }

  return (FE<3,L2_HIERARCHIC>::shape(elem, order, i, pp) -
          FE<3,L2_HIERARCHIC>::shape(elem, order, i, pm))/2./eps;
#endif
}

shape_deriv() [95/109]

Real libMesh::FE< 3, HIERARCHIC >::shape_deriv ( const Elem *  elem, const Order  order, const unsigned int  i, const unsigned int  j, const Point &  p  )

inherited

Definition at line 717 of file fe_hierarchic_shape_3D.C.

{
#if LIBMESH_DIM == 3
  libmesh_assert(elem);

  libmesh_assert_less (j, 3);

  // cheat by using finite difference approximations:
  const Real eps = 1.e-6;
  Point pp, pm;

  switch (j)
    {
      // d()/dxi
    case 0:
      {
        pp = Point(p(0)+eps, p(1), p(2));
        pm = Point(p(0)-eps, p(1), p(2));
        break;
      }

      // d()/deta
    case 1:
      {
        pp = Point(p(0), p(1)+eps, p(2));
        pm = Point(p(0), p(1)-eps, p(2));
        break;
      }

      // d()/dzeta
    case 2:
      {
        pp = Point(p(0), p(1), p(2)+eps);
        pm = Point(p(0), p(1), p(2)-eps);
        break;
      }

    default:
      libmesh_error_msg("Invalid derivative index j = " << j);
    }

  return (FE<3,HIERARCHIC>::shape(elem, order, i, pp) -
          FE<3,HIERARCHIC>::shape(elem, order, i, pm))/2./eps;
#endif
}

shape_deriv() [96/109]

RealGradient libMesh::FE< 0, LAGRANGE_VEC >::shape_deriv ( const Elem *  elem, const Order  order, const unsigned int  i, const unsigned int  j, const Point &  p  )

inherited

Definition at line 728 of file fe_lagrange_vec.C.

{
  Real value = FE<0,LAGRANGE>::shape_deriv( elem->type(), static_cast<Order>(order + elem->p_level()), i, j, p);
  return libMesh::RealGradient( value );
}

shape_deriv() [97/109]

Real libMesh::FE< 2, SZABAB >::shape_deriv ( const ElemType  , const Order  , const unsigned  int, const unsigned  int, const Point &    )

inherited

Definition at line 737 of file fe_szabab_shape_2D.C.

{
  libmesh_error_msg("Szabo-Babuska polynomials require the element type \nbecause edge orientation is needed.");
  return 0.;
}

shape_deriv() [98/109]

Real libMesh::FE< 2, SUBDIVISION >::shape_deriv ( const ElemType  type, const Order  order, const unsigned int  i, const unsigned int  j, const Point &  p  )

inherited

Definition at line 738 of file fe_subdivision_2D.C.

{
  switch (order)
    {
    case FOURTH:
      {
        switch (type)
          {
          case TRI3SUBDIVISION:
            libmesh_assert_less(i, 12);
            return FESubdivision::regular_shape_deriv(i,j,p(0),p(1));
          default:
            libmesh_error_msg("ERROR: Unsupported element type!");
          }
      }
    default:
      libmesh_error_msg("ERROR: Unsupported polynomial order!");
    }
}

shape_deriv() [99/109]

Real libMesh::FE< 2, SZABAB >::shape_deriv ( const Elem *  elem, const Order  order, const unsigned int  i, const unsigned int  j, const Point &  p  )

inherited

Definition at line 750 of file fe_szabab_shape_2D.C.

{
  libmesh_assert(elem);

  const ElemType type = elem->type();

  const Order totalorder = static_cast<Order>(order + elem->p_level());

  switch (totalorder)
    {

      // 1st & 2nd-order Szabo-Babuska.
    case FIRST:
    case SECOND:
      {
        switch (type)
          {

            // Szabo-Babuska shape functions on the triangle.
          case TRI3:
          case TRI6:
            {
              // Here we use finite differences to compute the derivatives!
              const Real eps = 1.e-6;

              libmesh_assert_less (i, 6);
              libmesh_assert_less (j, 2);

              switch (j)
                {
                  //  d()/dxi
                case 0:
                  {
                    const Point pp(p(0)+eps, p(1));
                    const Point pm(p(0)-eps, p(1));

                    return (FE<2,SZABAB>::shape(elem, order, i, pp) -
                            FE<2,SZABAB>::shape(elem, order, i, pm))/2./eps;
                  }

                  // d()/deta
                case 1:
                  {
                    const Point pp(p(0), p(1)+eps);
                    const Point pm(p(0), p(1)-eps);

                    return (FE<2,SZABAB>::shape(elem, order, i, pp) -
                            FE<2,SZABAB>::shape(elem, order, i, pm))/2./eps;
                  }

                default:
                  libmesh_error_msg("Invalid j = " << j);
                }
            }



            // Szabo-Babuska shape functions on the quadrilateral.
          case QUAD4:
          case QUAD8:
          case QUAD9:
            {
              // Compute quad shape functions as a tensor-product
              const Real xi  = p(0);
              const Real eta = p(1);

              libmesh_assert_less (i, 9);

              //                                0  1  2  3  4  5  6  7  8
              static const unsigned int i0[] = {0, 1, 1, 0, 2, 1, 2, 0, 2};
              static const unsigned int i1[] = {0, 0, 1, 1, 0, 2, 1, 2, 2};

              switch (j)
                {
                  // d()/dxi
                case 0:
                  return (FE<1,SZABAB>::shape_deriv(EDGE3, totalorder, i0[i], 0, xi)*
                          FE<1,SZABAB>::shape      (EDGE3, totalorder, i1[i],    eta));

                  // d()/deta
                case 1:
                  return (FE<1,SZABAB>::shape      (EDGE3, totalorder, i0[i],    xi)*
                          FE<1,SZABAB>::shape_deriv(EDGE3, totalorder, i1[i], 0, eta));

                default:
                  libmesh_error_msg("Invalid j = " << j);
                }
            }

          default:
            libmesh_error_msg("Invalid element type = " << type);
          }
      }



      // 3rd-order Szabo-Babuska.
    case THIRD:
      {
        switch (type)
          {
            // Szabo-Babuska shape functions on the triangle.
          case TRI6:
            {
              // Here we use finite differences to compute the derivatives!
              const Real eps = 1.e-6;

              libmesh_assert_less (i, 10);
              libmesh_assert_less (j, 2);

              switch (j)
                {
                  //  d()/dxi
                case 0:
                  {
                    const Point pp(p(0)+eps, p(1));
                    const Point pm(p(0)-eps, p(1));

                    return (FE<2,SZABAB>::shape(elem, order, i, pp) -
                            FE<2,SZABAB>::shape(elem, order, i, pm))/2./eps;
                  }

                  // d()/deta
                case 1:
                  {
                    const Point pp(p(0), p(1)+eps);
                    const Point pm(p(0), p(1)-eps);

                    return (FE<2,SZABAB>::shape(elem, order, i, pp) -
                            FE<2,SZABAB>::shape(elem, order, i, pm))/2./eps;
                  }


                default:
                  libmesh_error_msg("Invalid j = " << j);
                }
            }


            // Szabo-Babuska shape functions on the quadrilateral.
          case QUAD8:
          case QUAD9:
            {
              // Compute quad shape functions as a tensor-product
              const Real xi  = p(0);
              const Real eta = p(1);

              libmesh_assert_less (i, 16);

              //                                0   1   2   3   4   5   6   7   8   9  10  11  12  13  14  15
              static const unsigned int i0[] = {0,  1,  1,  0,  2,  3,  1,  1,  2,  3,  0,  0,  2,  3,  2,  3};
              static const unsigned int i1[] = {0,  0,  1,  1,  0,  0,  2,  3,  1,  1,  2,  3,  2,  2,  3,  3};

              Real f=1.;

              switch(i)
                {
                case  5: // edge 0 points
                  if (elem->point(0) > elem->point(1))f = -1.;
                  break;
                case  7: // edge 1 points
                  if (elem->point(1) > elem->point(2))f = -1.;
                  break;
                case  9: // edge 2 points
                  if (elem->point(3) > elem->point(2))f = -1.;
                  break;
                case 11: // edge 3 points
                  if (elem->point(0) > elem->point(3))f = -1.;
                  break;

                default:
                  // Everything else keeps f=1
                  break;
                }


              switch (j)
                {
                  // d()/dxi
                case 0:
                  return f*(FE<1,SZABAB>::shape_deriv(EDGE3, totalorder, i0[i], 0, xi)*
                            FE<1,SZABAB>::shape      (EDGE3, totalorder, i1[i],    eta));

                  // d()/deta
                case 1:
                  return f*(FE<1,SZABAB>::shape      (EDGE3, totalorder, i0[i],    xi)*
                            FE<1,SZABAB>::shape_deriv(EDGE3, totalorder, i1[i], 0, eta));

                default:
                  libmesh_error_msg("Invalid j = " << j);
                }
            }

          default:
            libmesh_error_msg("Invalid element type = " << type);
          }
      }




      // 4th-order Szabo-Babuska.
    case FOURTH:
      {
        switch (type)
          {

            // Szabo-Babuska shape functions on the triangle.
          case TRI6:
            {
              // Here we use finite differences to compute the derivatives!
              const Real eps = 1.e-6;

              libmesh_assert_less (i, 15);
              libmesh_assert_less (j, 2);

              switch (j)
                {
                  //  d()/dxi
                case 0:
                  {
                    const Point pp(p(0)+eps, p(1));
                    const Point pm(p(0)-eps, p(1));

                    return (FE<2,SZABAB>::shape(elem, order, i, pp) -
                            FE<2,SZABAB>::shape(elem, order, i, pm))/2./eps;
                  }

                  // d()/deta
                case 1:
                  {
                    const Point pp(p(0), p(1)+eps);
                    const Point pm(p(0), p(1)-eps);

                    return (FE<2,SZABAB>::shape(elem, order, i, pp) -
                            FE<2,SZABAB>::shape(elem, order, i, pm))/2./eps;
                  }


                default:
                  libmesh_error_msg("Invalid j = " << j);
                }
            }



            // Szabo-Babuska shape functions on the quadrilateral.
          case QUAD8:
          case QUAD9:
            {
              // Compute quad shape functions as a tensor-product
              const Real xi  = p(0);
              const Real eta = p(1);

              libmesh_assert_less (i, 25);

              //                                0  1  2  3  4  5  6  7  8  9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24
              static const unsigned int i0[] = {0, 1, 1, 0, 2, 3, 4, 1, 1, 1, 2, 3, 4, 0, 0, 0, 2, 3, 4, 2, 3, 4, 2, 3, 4};
              static const unsigned int i1[] = {0, 0, 1, 1, 0, 0, 0, 2, 3, 4, 1, 1, 1, 2, 3, 4, 2, 2, 2, 3, 3, 3, 4, 4, 4};

              Real f=1.;

              switch(i)
                {
                case  5: // edge 0 points
                  if (elem->point(0) > elem->point(1))f = -1.;
                  break;
                case  8: // edge 1 points
                  if (elem->point(1) > elem->point(2))f = -1.;
                  break;
                case 11: // edge 2 points
                  if (elem->point(3) > elem->point(2))f = -1.;
                  break;
                case 14: // edge 3 points
                  if (elem->point(0) > elem->point(3))f = -1.;
                  break;

                default:
                  // Everything else keeps f=1
                  break;
                }


              switch (j)
                {
                  // d()/dxi
                case 0:
                  return f*(FE<1,SZABAB>::shape_deriv(EDGE3, totalorder, i0[i], 0, xi)*
                            FE<1,SZABAB>::shape      (EDGE3, totalorder, i1[i],    eta));

                  // d()/deta
                case 1:
                  return f*(FE<1,SZABAB>::shape      (EDGE3, totalorder, i0[i],    xi)*
                            FE<1,SZABAB>::shape_deriv(EDGE3, totalorder, i1[i], 0, eta));

                default:
                  libmesh_error_msg("Invalid j = " << j);
                }
            }

          default:
            libmesh_error_msg("Invalid element type = " << type);
          }
      }




      // 5th-order Szabo-Babuska.
    case FIFTH:
      {
        // Szabo-Babuska shape functions on the quadrilateral.
        switch (type)
          {

            // Szabo-Babuska shape functions on the triangle.
          case TRI6:
            {
              // Here we use finite differences to compute the derivatives!
              const Real eps = 1.e-6;

              libmesh_assert_less (i, 21);
              libmesh_assert_less (j, 2);

              switch (j)
                {
                  //  d()/dxi
                case 0:
                  {
                    const Point pp(p(0)+eps, p(1));
                    const Point pm(p(0)-eps, p(1));

                    return (FE<2,SZABAB>::shape(elem, order, i, pp) -
                            FE<2,SZABAB>::shape(elem, order, i, pm))/2./eps;
                  }

                  // d()/deta
                case 1:
                  {
                    const Point pp(p(0), p(1)+eps);
                    const Point pm(p(0), p(1)-eps);

                    return (FE<2,SZABAB>::shape(elem, order, i, pp) -
                            FE<2,SZABAB>::shape(elem, order, i, pm))/2./eps;
                  }

                default:
                  libmesh_error_msg("Invalid j = " << j);
                }
            }



          case QUAD8:
          case QUAD9:
            {
              // Compute quad shape functions as a tensor-product
              const Real xi  = p(0);
              const Real eta = p(1);

              libmesh_assert_less (i, 36);

              //                                0  1  2  3  4  5  6  7  8  9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35
              static const unsigned int i0[] = {0, 1, 1, 0, 2, 3, 4, 5, 1, 1, 1, 1, 2, 3, 4, 5, 0, 0, 0, 0, 2, 3, 4, 5, 2, 3, 4, 5, 2, 3, 4, 5, 2, 3, 4, 5};
              static const unsigned int i1[] = {0, 0, 1, 1, 0, 0, 0, 0, 2, 3, 4, 5, 1, 1, 1, 1, 2, 3, 4, 5, 2, 2, 2, 2, 3, 3, 3, 3, 4, 4, 4, 4, 5, 5, 5, 5};

              Real f=1.;

              switch(i)
                {
                case  5: // edge 0 points
                case  7:
                  if (elem->point(0) > elem->point(1))f = -1.;
                  break;
                case  9: // edge 1 points
                case 11:
                  if (elem->point(1) > elem->point(2))f = -1.;
                  break;
                case 13: // edge 2 points
                case 15:
                  if (elem->point(3) > elem->point(2))f = -1.;
                  break;
                case 14: // edge 3 points
                case 19:
                  if (elem->point(0) > elem->point(3))f = -1.;
                  break;

                default:
                  // Everything else keeps f=1
                  break;
                }


              switch (j)
                {
                  // d()/dxi
                case 0:
                  return f*(FE<1,SZABAB>::shape_deriv(EDGE3, totalorder, i0[i], 0, xi)*
                            FE<1,SZABAB>::shape      (EDGE3, totalorder, i1[i],    eta));

                  // d()/deta
                case 1:
                  return f*(FE<1,SZABAB>::shape      (EDGE3, totalorder, i0[i],    xi)*
                            FE<1,SZABAB>::shape_deriv(EDGE3, totalorder, i1[i], 0, eta));

                default:
                  libmesh_error_msg("Invalid j = " << j);
                }
            }

          default:
            libmesh_error_msg("Invalid element type = " << type);
          }
      }


      // 6th-order Szabo-Babuska.
    case SIXTH:
      {
        // Szabo-Babuska shape functions on the quadrilateral.
        switch (type)
          {

            // Szabo-Babuska shape functions on the triangle.
          case TRI6:
            {
              // Here we use finite differences to compute the derivatives!
              const Real eps = 1.e-6;

              libmesh_assert_less (i, 28);
              libmesh_assert_less (j, 2);

              switch (j)
                {
                  //  d()/dxi
                case 0:
                  {
                    const Point pp(p(0)+eps, p(1));
                    const Point pm(p(0)-eps, p(1));

                    return (FE<2,SZABAB>::shape(elem, order, i, pp) -
                            FE<2,SZABAB>::shape(elem, order, i, pm))/2./eps;
                  }

                  // d()/deta
                case 1:
                  {
                    const Point pp(p(0), p(1)+eps);
                    const Point pm(p(0), p(1)-eps);

                    return (FE<2,SZABAB>::shape(elem, order, i, pp) -
                            FE<2,SZABAB>::shape(elem, order, i, pm))/2./eps;
                  }

                default:
                  libmesh_error_msg("Invalid j = " << j);
                }
            }



          case QUAD8:
          case QUAD9:
            {
              // Compute quad shape functions as a tensor-product
              const Real xi  = p(0);
              const Real eta = p(1);

              libmesh_assert_less (i, 49);

              //                                0  1  2  3  4  5  6  7  8  9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48
              static const unsigned int i0[] = {0, 1, 1, 0, 2, 3, 4, 5, 6, 1, 1, 1, 1, 1, 2, 3, 4, 5, 6, 0, 0, 0, 0, 0, 2, 3, 4, 5, 6, 2, 3, 4, 5, 6, 2, 3, 4, 5, 6, 2, 3, 4, 5, 6, 2, 3, 4, 5, 6};
              static const unsigned int i1[] = {0, 0, 1, 1, 0, 0, 0, 0, 0, 2, 3, 4, 5, 6, 1, 1, 1, 1, 1, 2, 3, 4, 5, 6, 2, 2, 2, 2, 2, 3, 3, 3, 3, 3, 4, 4, 4, 4, 4, 5, 5, 5, 5, 5, 6, 6, 6, 6, 6};

              Real f=1.;

              switch(i)
                {
                case  5: // edge 0 points
                case  7:
                  if (elem->point(0) > elem->point(1))f = -1.;
                  break;
                case 10: // edge 1 points
                case 12:
                  if (elem->point(1) > elem->point(2))f = -1.;
                  break;
                case 15: // edge 2 points
                case 17:
                  if (elem->point(3) > elem->point(2))f = -1.;
                  break;
                case 20: // edge 3 points
                case 22:
                  if (elem->point(0) > elem->point(3))f = -1.;
                  break;

                default:
                  // Everything else keeps f=1
                  break;
                }


              switch (j)
                {
                  // d()/dxi
                case 0:
                  return f*(FE<1,SZABAB>::shape_deriv(EDGE3, totalorder, i0[i], 0, xi)*
                            FE<1,SZABAB>::shape      (EDGE3, totalorder, i1[i],    eta));

                  // d()/deta
                case 1:
                  return f*(FE<1,SZABAB>::shape      (EDGE3, totalorder, i0[i],    xi)*
                            FE<1,SZABAB>::shape_deriv(EDGE3, totalorder, i1[i], 0, eta));

                default:
                  libmesh_error_msg("Invalid j = " << j);
                }
            }

          default:
            libmesh_error_msg("Invalid element type = " << type);
          }
      }


      // 7th-order Szabo-Babuska.
    case SEVENTH:
      {
        // Szabo-Babuska shape functions on the quadrilateral.
        switch (type)
          {

            // Szabo-Babuska shape functions on the triangle.
          case TRI6:
            {
              // Here we use finite differences to compute the derivatives!
              const Real eps = 1.e-6;

              libmesh_assert_less (i, 36);
              libmesh_assert_less (j, 2);

              switch (j)
                {
                  //  d()/dxi
                case 0:
                  {
                    const Point pp(p(0)+eps, p(1));
                    const Point pm(p(0)-eps, p(1));

                    return (FE<2,SZABAB>::shape(elem, order, i, pp) -
                            FE<2,SZABAB>::shape(elem, order, i, pm))/2./eps;
                  }

                  // d()/deta
                case 1:
                  {
                    const Point pp(p(0), p(1)+eps);
                    const Point pm(p(0), p(1)-eps);

                    return (FE<2,SZABAB>::shape(elem, order, i, pp) -
                            FE<2,SZABAB>::shape(elem, order, i, pm))/2./eps;
                  }

                default:
                  libmesh_error_msg("Invalid j = " << j);
                }
            }



          case QUAD8:
          case QUAD9:
            {
              // Compute quad shape functions as a tensor-product
              const Real xi  = p(0);
              const Real eta = p(1);

              libmesh_assert_less (i, 64);

              //                                0  1  2  3  4  5  6  7  8  9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63
              static const unsigned int i0[] = {0, 1, 1, 0, 2, 3, 4, 5, 6, 7, 1, 1, 1, 1, 1, 1, 2, 3, 4, 5, 6, 7, 0, 0, 0, 0, 0, 0, 2, 3, 4, 5, 6, 7, 2, 3, 4, 5, 6, 7, 2, 3, 4, 5, 6, 7, 2, 3, 4, 5, 6, 7, 2, 3, 4, 5, 6, 7, 2, 3, 4, 5, 6, 7};
              static const unsigned int i1[] = {0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 2, 3, 4, 5, 6, 7, 1, 1, 1, 1, 1, 1, 2, 3, 4, 5, 6, 7, 2, 2, 2, 2, 2, 2, 3, 3, 3, 3, 3, 3, 4, 4, 4, 4, 4, 4, 5, 5, 5, 5, 5, 5, 6, 6, 6, 6, 6, 6, 7, 7, 7, 7, 7, 7};

              Real f=1.;

              switch(i)
                {
                case  5: // edge 0 points
                case  7:
                case  9:
                  if (elem->point(0) > elem->point(1))f = -1.;
                  break;
                case 11: // edge 1 points
                case 13:
                case 15:
                  if (elem->point(1) > elem->point(2))f = -1.;
                  break;
                case 17: // edge 2 points
                case 19:
                case 21:
                  if (elem->point(3) > elem->point(2))f = -1.;
                  break;
                case 23: // edge 3 points
                case 25:
                case 27:
                  if (elem->point(0) > elem->point(3))f = -1.;
                  break;

                default:
                  // Everything else keeps f=1
                  break;
                }


              switch (j)
                {
                  // d()/dxi
                case 0:
                  return f*(FE<1,SZABAB>::shape_deriv(EDGE3, totalorder, i0[i], 0, xi)*
                            FE<1,SZABAB>::shape      (EDGE3, totalorder, i1[i],    eta));

                  // d()/deta
                case 1:
                  return f*(FE<1,SZABAB>::shape      (EDGE3, totalorder, i0[i],    xi)*
                            FE<1,SZABAB>::shape_deriv(EDGE3, totalorder, i1[i], 0, eta));

                default:
                  libmesh_error_msg("Invalid j = " << j);
                }
            }

          default:
            libmesh_error_msg("Invalid element type = " << type);
          }
      }



      // by default throw an error;call the orientation-independent shape functions
    default:
      libmesh_error_msg("ERROR: Unsupported polynomial order!");
    }
}

shape_deriv() [100/109]

RealGradient libMesh::FE< 1, LAGRANGE_VEC >::shape_deriv ( const Elem *  elem, const Order  order, const unsigned int  i, const unsigned int  j, const Point &  p  )

inherited

Definition at line 750 of file fe_lagrange_vec.C.

{
  Real value = FE<1,LAGRANGE>::shape_deriv( elem->type(), static_cast<Order>(order + elem->p_level()), i, j, p);
  return libMesh::RealGradient( value );
}

shape_deriv() [101/109]

Real libMesh::FE< 2, SUBDIVISION >::shape_deriv ( const Elem *  elem, const Order  order, const unsigned int  i, const unsigned int  j, const Point &  p  )

inherited

Definition at line 765 of file fe_subdivision_2D.C.

{
  libmesh_assert(elem);
  return FE<2,SUBDIVISION>::shape_deriv(elem->type(), order, i, j, p);
}

shape_deriv() [102/109]

RealGradient libMesh::FE< 2, LAGRANGE_VEC >::shape_deriv ( const Elem *  elem, const Order  order, const unsigned int  i, const unsigned int  j, const Point &  p  )

inherited

Definition at line 786 of file fe_lagrange_vec.C.

{
  Real value = FE<2,LAGRANGE>::shape_deriv( elem->type(), static_cast<Order>(order + elem->p_level()), i/2, j, p );

  switch( i%2 )
    {
    case 0:
      return libMesh::RealGradient( value );

    case 1:
      return libMesh::RealGradient( Real(0), value );

    default:
      libmesh_error_msg("i%2 must be either 0 or 1!");
    }

  //dummy
  return libMesh::RealGradient();
}

shape_deriv() [103/109]

RealGradient libMesh::FE< 3, LAGRANGE_VEC >::shape_deriv ( const Elem *  elem, const Order  order, const unsigned int  i, const unsigned int  j, const Point &  p  )

inherited

Definition at line 853 of file fe_lagrange_vec.C.

{
  Real value = FE<3,LAGRANGE>::shape_deriv( elem->type(), static_cast<Order>(order + elem->p_level()), i/3, j, p );

  switch( i%3 )
    {
    case 0:
      return libMesh::RealGradient( value );

    case 1:
      return libMesh::RealGradient( Real(0), value );

    case 2:
      return libMesh::RealGradient( Real(0), Real(0), value );

    default:
      libmesh_error_msg("i%3 must be 0, 1, or 2!");
    }

  //dummy
  return libMesh::RealGradient();
}

shape_deriv() [104/109]

Real libMesh::FE< 3, L2_LAGRANGE >::shape_deriv ( const Elem *  elem, const Order  order, const unsigned int  i, const unsigned int  j, const Point &  p  )

inherited

Definition at line 1195 of file fe_l2_lagrange_shape_3D.C.

{
  libmesh_assert(elem);

  // call the orientation-independent shape function derivatives
  return FE<3,L2_LAGRANGE>::shape_deriv(elem->type(), static_cast<Order>(order + elem->p_level()), i, j, p);
}

shape_deriv() [105/109]

Real libMesh::FE< 3, BERNSTEIN >::shape_deriv ( const ElemType  , const Order  , const unsigned  int, const unsigned  int, const Point &    )

inherited

Definition at line 1382 of file fe_bernstein_shape_3D.C.

{
  libmesh_error_msg("Bernstein polynomials require the element type \nbecause edge and face orientation is needed.");
  return 0.;
}

shape_deriv() [106/109]

Real libMesh::FE< 3, BERNSTEIN >::shape_deriv ( const Elem *  elem, const Order  order, const unsigned int  i, const unsigned int  j, const Point &  p  )

inherited

Definition at line 1395 of file fe_bernstein_shape_3D.C.

{

#if LIBMESH_DIM == 3
  libmesh_assert(elem);
  const ElemType type = elem->type();

  const Order totalorder = static_cast<Order>(order + elem->p_level());

  libmesh_assert_less (j, 3);

  switch (totalorder)
    {
      // 1st order Bernstein.
    case FIRST:
      {
        switch (type)
          {
            // Bernstein shape functions on the tetrahedron.
          case TET4:
          case TET10:
            {
              // I have been lazy here and am using finite differences
              // to compute the derivatives!
              const Real eps = 1.e-6;

              libmesh_assert_less (i, 4);
              libmesh_assert_less (j, 3);


              switch (j)
                {
                  //  d()/dxi
                case 0:
                  {
                    const Point pp(p(0)+eps, p(1), p(2));
                    const Point pm(p(0)-eps, p(1), p(2));

                    return (FE<3,BERNSTEIN>::shape(elem, order, i, pp) -
                            FE<3,BERNSTEIN>::shape(elem, order, i, pm))/2./eps;
                  }

                  // d()/deta
                case 1:
                  {
                    const Point pp(p(0), p(1)+eps, p(2));
                    const Point pm(p(0), p(1)-eps, p(2));

                    return (FE<3,BERNSTEIN>::shape(elem, order, i, pp) -
                            FE<3,BERNSTEIN>::shape(elem, order, i, pm))/2./eps;
                  }
                  // d()/dzeta
                case 2:
                  {
                    const Point pp(p(0), p(1), p(2)+eps);
                    const Point pm(p(0), p(1), p(2)-eps);

                    return (FE<3,BERNSTEIN>::shape(elem, order, i, pp) -
                            FE<3,BERNSTEIN>::shape(elem, order, i, pm))/2./eps;
                  }
                default:
                  libmesh_error_msg("Invalid derivative index j = " << j);
                }
            }




            // Bernstein shape functions on the hexahedral.
          case HEX8:
          case HEX20:
          case HEX27:
            {
              libmesh_assert_less (i, 8);

              // Compute hex shape functions as a tensor-product
              const Real xi   = p(0);
              const Real eta  = p(1);
              const Real zeta = p(2);

              // The only way to make any sense of this
              // is to look at the mgflo/mg2/mgf documentation
              // and make the cut-out cube!
              //                                0  1  2  3  4  5  6  7
              static const unsigned int i0[] = {0, 1, 1, 0, 0, 1, 1, 0};
              static const unsigned int i1[] = {0, 0, 1, 1, 0, 0, 1, 1};
              static const unsigned int i2[] = {0, 0, 0, 0, 1, 1, 1, 1};

              switch (j)
                {
                  // d()/dxi
                case 0:
                  return (FE<1,BERNSTEIN>::shape_deriv(EDGE3, totalorder, i0[i], 0, xi)*
                          FE<1,BERNSTEIN>::shape      (EDGE3, totalorder, i1[i],    eta)*
                          FE<1,BERNSTEIN>::shape      (EDGE3, totalorder, i2[i],    zeta));

                  // d()/deta
                case 1:
                  return (FE<1,BERNSTEIN>::shape      (EDGE3, totalorder, i0[i],     xi)*
                          FE<1,BERNSTEIN>::shape_deriv(EDGE3, totalorder, i1[i], 0, eta)*
                          FE<1,BERNSTEIN>::shape      (EDGE3, totalorder, i2[i],    zeta));

                  // d()/dzeta
                case 2:
                  return (FE<1,BERNSTEIN>::shape      (EDGE3, totalorder, i0[i],    xi)*
                          FE<1,BERNSTEIN>::shape      (EDGE3, totalorder, i1[i],    eta)*
                          FE<1,BERNSTEIN>::shape_deriv(EDGE3, totalorder, i2[i], 0, zeta));

                default:
                  libmesh_error_msg("Invalid derivative index j = " << j);
                }
            }

          default:
            libmesh_error_msg("Invalid element type = " << type);
          }
      }




    case SECOND:
      {
        switch (type)
          {
            // Bernstein shape functions on the tetrahedron.
          case TET10:
            {
              // I have been lazy here and am using finite differences
              // to compute the derivatives!
              const Real eps = 1.e-6;

              libmesh_assert_less (i, 10);
              libmesh_assert_less (j, 3);


              switch (j)
                {
                  //  d()/dxi
                case 0:
                  {
                    const Point pp(p(0)+eps, p(1), p(2));
                    const Point pm(p(0)-eps, p(1), p(2));

                    return (FE<3,BERNSTEIN>::shape(elem, order, i, pp) -
                            FE<3,BERNSTEIN>::shape(elem, order, i, pm))/2./eps;
                  }

                  // d()/deta
                case 1:
                  {
                    const Point pp(p(0), p(1)+eps, p(2));
                    const Point pm(p(0), p(1)-eps, p(2));

                    return (FE<3,BERNSTEIN>::shape(elem, order, i, pp) -
                            FE<3,BERNSTEIN>::shape(elem, order, i, pm))/2./eps;
                  }
                  // d()/dzeta
                case 2:
                  {
                    const Point pp(p(0), p(1), p(2)+eps);
                    const Point pm(p(0), p(1), p(2)-eps);

                    return (FE<3,BERNSTEIN>::shape(elem, order, i, pp) -
                            FE<3,BERNSTEIN>::shape(elem, order, i, pm))/2./eps;
                  }
                default:
                  libmesh_error_msg("Invalid derivative index j = " << j);
                }
            }

            // Bernstein shape functions on the hexahedral.
          case HEX20:
            {
              libmesh_assert_less (i, 20);

              // Compute hex shape functions as a tensor-product
              const Real xi   = p(0);
              const Real eta  = p(1);
              const Real zeta = p(2);

              // The only way to make any sense of this
              // is to look at the mgflo/mg2/mgf documentation
              // and make the cut-out cube!
              //                                0  1  2  3  4  5  6  7  8  9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26
              static const unsigned int i0[] = {0, 1, 1, 0, 0, 1, 1, 0, 2, 1, 2, 0, 0, 1, 1, 0, 2, 1, 2, 0, 2, 2, 1, 2, 0, 2, 2};
              static const unsigned int i1[] = {0, 0, 1, 1, 0, 0, 1, 1, 0, 2, 1, 2, 0, 0, 1, 1, 0, 2, 1, 2, 2, 0, 2, 1, 2, 2, 2};
              static const unsigned int i2[] = {0, 0, 0, 0, 1, 1, 1, 1, 0, 0, 0, 0, 2, 2, 2, 2, 1, 1, 1, 1, 0, 2, 2, 2, 2, 1, 2};
              static const Real scal20[] =     {-0.25, -0.25, -0.25, -0.25, 0,     0,     0,     0,     0.5,   0.5,   0.5,   0.5,   0,     0,     0,     0,     0,     0,     0,     0};
              static const Real scal21[] =     {-0.25, -0.25, 0,     0,     -0.25, -0.25, 0,     0,     0.5,   0,     0,     0,     0.5,   0.5,   0,     0,     0.5,   0,     0,     0};
              static const Real scal22[] =     {0,     -0.25, -0.25, 0,     0,     -0.25, -0.25, 0,     0,     0.5,   0,     0,     0,     0.5,   0.5,   0,     0,     0.5,   0,     0};
              static const Real scal23[] =     {0,     0,     -0.25, -0.25, 0,     0,     -0.25, -0.25, 0,     0,     0.5,   0,     0,     0,     0.5,   0.5,   0,     0,     0.5,   0};
              static const Real scal24[] =     {-0.25, 0,     0,     -0.25, -0.25, 0,     0,     -0.25, 0,     0,     0,     0.5,   0.5,   0,     0,     0.5,   0,     0,     0,     0.5};
              static const Real scal25[] =     {0,     0,     0,     0,     -0.25, -0.25, -0.25, -0.25, 0,     0,     0,     0,     0,     0,     0,     0,     0.5,   0.5,   0.5,   0.5};
              static const Real scal26[] =     {-0.25, -0.25, -0.25, -0.25, -0.25, -0.25, -0.25, -0.25, 0.25,  0.25,  0.25,  0.25,  0.25,  0.25,  0.25,  0.25,  0.25,  0.25,  0.25,  0.25};

              switch (j)
                {
                  // d()/dxi
                case 0:
                  return (FE<1,BERNSTEIN>::shape_deriv(EDGE3, totalorder, i0[i], 0, xi)*
                          FE<1,BERNSTEIN>::shape      (EDGE3, totalorder, i1[i],    eta)*
                          FE<1,BERNSTEIN>::shape      (EDGE3, totalorder, i2[i],    zeta)
                          +scal20[i]*
                          FE<1,BERNSTEIN>::shape_deriv(EDGE3, totalorder, i0[20], 0, xi)*
                          FE<1,BERNSTEIN>::shape      (EDGE3, totalorder, i1[20],    eta)*
                          FE<1,BERNSTEIN>::shape      (EDGE3, totalorder, i2[20],    zeta)
                          +scal21[i]*
                          FE<1,BERNSTEIN>::shape_deriv(EDGE3, totalorder, i0[21], 0, xi)*
                          FE<1,BERNSTEIN>::shape      (EDGE3, totalorder, i1[21],    eta)*
                          FE<1,BERNSTEIN>::shape      (EDGE3, totalorder, i2[21],    zeta)
                          +scal22[i]*
                          FE<1,BERNSTEIN>::shape_deriv(EDGE3, totalorder, i0[22], 0, xi)*
                          FE<1,BERNSTEIN>::shape      (EDGE3, totalorder, i1[22],    eta)*
                          FE<1,BERNSTEIN>::shape      (EDGE3, totalorder, i2[22],    zeta)
                          +scal23[i]*
                          FE<1,BERNSTEIN>::shape_deriv(EDGE3, totalorder, i0[23], 0, xi)*
                          FE<1,BERNSTEIN>::shape      (EDGE3, totalorder, i1[23],    eta)*
                          FE<1,BERNSTEIN>::shape      (EDGE3, totalorder, i2[23],    zeta)
                          +scal24[i]*
                          FE<1,BERNSTEIN>::shape_deriv(EDGE3, totalorder, i0[24], 0, xi)*
                          FE<1,BERNSTEIN>::shape      (EDGE3, totalorder, i1[24],    eta)*
                          FE<1,BERNSTEIN>::shape      (EDGE3, totalorder, i2[24],    zeta)
                          +scal25[i]*
                          FE<1,BERNSTEIN>::shape_deriv(EDGE3, totalorder, i0[25], 0, xi)*
                          FE<1,BERNSTEIN>::shape      (EDGE3, totalorder, i1[25],    eta)*
                          FE<1,BERNSTEIN>::shape      (EDGE3, totalorder, i2[25],    zeta)
                          +scal26[i]*
                          FE<1,BERNSTEIN>::shape_deriv(EDGE3, totalorder, i0[26], 0, xi)*
                          FE<1,BERNSTEIN>::shape      (EDGE3, totalorder, i1[26],    eta)*
                          FE<1,BERNSTEIN>::shape      (EDGE3, totalorder, i2[26],    zeta));

                  // d()/deta
                case 1:
                  return (FE<1,BERNSTEIN>::shape      (EDGE3, totalorder, i0[i],     xi)*
                          FE<1,BERNSTEIN>::shape_deriv(EDGE3, totalorder, i1[i], 0, eta)*
                          FE<1,BERNSTEIN>::shape      (EDGE3, totalorder, i2[i],    zeta)
                          +scal20[i]*
                          FE<1,BERNSTEIN>::shape      (EDGE3, totalorder, i0[20],     xi)*
                          FE<1,BERNSTEIN>::shape_deriv(EDGE3, totalorder, i1[20], 0, eta)*
                          FE<1,BERNSTEIN>::shape      (EDGE3, totalorder, i2[20],    zeta)
                          +scal21[i]*
                          FE<1,BERNSTEIN>::shape      (EDGE3, totalorder, i0[21],     xi)*
                          FE<1,BERNSTEIN>::shape_deriv(EDGE3, totalorder, i1[21], 0, eta)*
                          FE<1,BERNSTEIN>::shape      (EDGE3, totalorder, i2[21],    zeta)
                          +scal22[i]*
                          FE<1,BERNSTEIN>::shape      (EDGE3, totalorder, i0[22],     xi)*
                          FE<1,BERNSTEIN>::shape_deriv(EDGE3, totalorder, i1[22], 0, eta)*
                          FE<1,BERNSTEIN>::shape      (EDGE3, totalorder, i2[22],    zeta)
                          +scal23[i]*
                          FE<1,BERNSTEIN>::shape      (EDGE3, totalorder, i0[23],     xi)*
                          FE<1,BERNSTEIN>::shape_deriv(EDGE3, totalorder, i1[23], 0, eta)*
                          FE<1,BERNSTEIN>::shape      (EDGE3, totalorder, i2[23],    zeta)
                          +scal24[i]*
                          FE<1,BERNSTEIN>::shape      (EDGE3, totalorder, i0[24],     xi)*
                          FE<1,BERNSTEIN>::shape_deriv(EDGE3, totalorder, i1[24], 0, eta)*
                          FE<1,BERNSTEIN>::shape      (EDGE3, totalorder, i2[24],    zeta)
                          +scal25[i]*
                          FE<1,BERNSTEIN>::shape      (EDGE3, totalorder, i0[25],     xi)*
                          FE<1,BERNSTEIN>::shape_deriv(EDGE3, totalorder, i1[25], 0, eta)*
                          FE<1,BERNSTEIN>::shape      (EDGE3, totalorder, i2[25],    zeta)
                          +scal26[i]*
                          FE<1,BERNSTEIN>::shape      (EDGE3, totalorder, i0[26],     xi)*
                          FE<1,BERNSTEIN>::shape_deriv(EDGE3, totalorder, i1[26], 0, eta)*
                          FE<1,BERNSTEIN>::shape      (EDGE3, totalorder, i2[26],    zeta));

                  // d()/dzeta
                case 2:
                  return (FE<1,BERNSTEIN>::shape      (EDGE3, totalorder, i0[i],    xi)*
                          FE<1,BERNSTEIN>::shape      (EDGE3, totalorder, i1[i],    eta)*
                          FE<1,BERNSTEIN>::shape_deriv(EDGE3, totalorder, i2[i], 0, zeta)
                          +scal20[i]*
                          FE<1,BERNSTEIN>::shape      (EDGE3, totalorder, i0[20],    xi)*
                          FE<1,BERNSTEIN>::shape      (EDGE3, totalorder, i1[20],    eta)*
                          FE<1,BERNSTEIN>::shape_deriv(EDGE3, totalorder, i2[20], 0, zeta)
                          +scal21[i]*
                          FE<1,BERNSTEIN>::shape      (EDGE3, totalorder, i0[21],    xi)*
                          FE<1,BERNSTEIN>::shape      (EDGE3, totalorder, i1[21],    eta)*
                          FE<1,BERNSTEIN>::shape_deriv(EDGE3, totalorder, i2[21], 0, zeta)
                          +scal22[i]*
                          FE<1,BERNSTEIN>::shape      (EDGE3, totalorder, i0[22],    xi)*
                          FE<1,BERNSTEIN>::shape      (EDGE3, totalorder, i1[22],    eta)*
                          FE<1,BERNSTEIN>::shape_deriv(EDGE3, totalorder, i2[22], 0, zeta)
                          +scal23[i]*
                          FE<1,BERNSTEIN>::shape      (EDGE3, totalorder, i0[23],    xi)*
                          FE<1,BERNSTEIN>::shape      (EDGE3, totalorder, i1[23],    eta)*
                          FE<1,BERNSTEIN>::shape_deriv(EDGE3, totalorder, i2[23], 0, zeta)
                          +scal24[i]*
                          FE<1,BERNSTEIN>::shape      (EDGE3, totalorder, i0[24],    xi)*
                          FE<1,BERNSTEIN>::shape      (EDGE3, totalorder, i1[24],    eta)*
                          FE<1,BERNSTEIN>::shape_deriv(EDGE3, totalorder, i2[24], 0, zeta)
                          +scal25[i]*
                          FE<1,BERNSTEIN>::shape      (EDGE3, totalorder, i0[25],    xi)*
                          FE<1,BERNSTEIN>::shape      (EDGE3, totalorder, i1[25],    eta)*
                          FE<1,BERNSTEIN>::shape_deriv(EDGE3, totalorder, i2[25], 0, zeta)
                          +scal26[i]*
                          FE<1,BERNSTEIN>::shape      (EDGE3, totalorder, i0[26],    xi)*
                          FE<1,BERNSTEIN>::shape      (EDGE3, totalorder, i1[26],    eta)*
                          FE<1,BERNSTEIN>::shape_deriv(EDGE3, totalorder, i2[26], 0, zeta));

                default:
                  libmesh_error_msg("Invalid derivative index j = " << j);
                }
            }

            // Bernstein shape functions on the hexahedral.
          case HEX27:
            {
              libmesh_assert_less (i, 27);

              // Compute hex shape functions as a tensor-product
              const Real xi   = p(0);
              const Real eta  = p(1);
              const Real zeta = p(2);

              // The only way to make any sense of this
              // is to look at the mgflo/mg2/mgf documentation
              // and make the cut-out cube!
              //                                0  1  2  3  4  5  6  7  8  9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26
              static const unsigned int i0[] = {0, 1, 1, 0, 0, 1, 1, 0, 2, 1, 2, 0, 0, 1, 1, 0, 2, 1, 2, 0, 2, 2, 1, 2, 0, 2, 2};
              static const unsigned int i1[] = {0, 0, 1, 1, 0, 0, 1, 1, 0, 2, 1, 2, 0, 0, 1, 1, 0, 2, 1, 2, 2, 0, 2, 1, 2, 2, 2};
              static const unsigned int i2[] = {0, 0, 0, 0, 1, 1, 1, 1, 0, 0, 0, 0, 2, 2, 2, 2, 1, 1, 1, 1, 0, 2, 2, 2, 2, 1, 2};

              switch (j)
                {
                  // d()/dxi
                case 0:
                  return (FE<1,BERNSTEIN>::shape_deriv(EDGE3, totalorder, i0[i], 0, xi)*
                          FE<1,BERNSTEIN>::shape      (EDGE3, totalorder, i1[i],    eta)*
                          FE<1,BERNSTEIN>::shape      (EDGE3, totalorder, i2[i],    zeta));

                  // d()/deta
                case 1:
                  return (FE<1,BERNSTEIN>::shape      (EDGE3, totalorder, i0[i],     xi)*
                          FE<1,BERNSTEIN>::shape_deriv(EDGE3, totalorder, i1[i], 0, eta)*
                          FE<1,BERNSTEIN>::shape      (EDGE3, totalorder, i2[i],    zeta));

                  // d()/dzeta
                case 2:
                  return (FE<1,BERNSTEIN>::shape      (EDGE3, totalorder, i0[i],    xi)*
                          FE<1,BERNSTEIN>::shape      (EDGE3, totalorder, i1[i],    eta)*
                          FE<1,BERNSTEIN>::shape_deriv(EDGE3, totalorder, i2[i], 0, zeta));

                default:
                  libmesh_error_msg("Invalid derivative index j = " << j);
                }
            }


          default:
            libmesh_error_msg("Invalid element type = " << type);
          }
      }



      // 3rd-order Bernstein.
    case THIRD:
      {
        switch (type)
          {

            //     // Bernstein shape functions derivatives.
            //   case TET10:
            //     {
            //       // I have been lazy here and am using finite differences
            //       // to compute the derivatives!
            //       const Real eps = 1.e-6;

            //       libmesh_assert_less (i, 20);
            //       libmesh_assert_less (j, 3);

            // switch (j)
            // {
            //   //  d()/dxi
            // case 0:
            //   {
            //     const Point pp(p(0)+eps, p(1), p(2));
            //     const Point pm(p(0)-eps, p(1), p(2));

            //     return (FE<3,BERNSTEIN>::shape(elem, order, i, pp) -
            //     FE<3,BERNSTEIN>::shape(elem, order, i, pm))/2./eps;
            //   }

            //   // d()/deta
            // case 1:
            //   {
            //     const Point pp(p(0), p(1)+eps, p(2));
            //     const Point pm(p(0), p(1)-eps, p(2));

            //     return (FE<3,BERNSTEIN>::shape(elem, order, i, pp) -
            //     FE<3,BERNSTEIN>::shape(elem, order, i, pm))/2./eps;
            //   }
            //   // d()/dzeta
            // case 2:
            //   {
            //     const Point pp(p(0), p(1), p(2)+eps);
            //     const Point pm(p(0), p(1), p(2)-eps);

            //     return (FE<3,BERNSTEIN>::shape(elem, order, i, pp) -
            //     FE<3,BERNSTEIN>::shape(elem, order, i, pm))/2./eps;
            //   }
            // default:
            // libmesh_error_msg("Invalid derivative index j = " << j);
            // }


            //     }


            // Bernstein shape functions on the hexahedral.
          case HEX27:
            {
              // I have been lazy here and am using finite differences
              // to compute the derivatives!
              const Real eps = 1.e-6;

              libmesh_assert_less (i, 64);
              libmesh_assert_less (j, 3);

              switch (j)
                {
                  //  d()/dxi
                case 0:
                  {
                    const Point pp(p(0)+eps, p(1), p(2));
                    const Point pm(p(0)-eps, p(1), p(2));

                    return (FE<3,BERNSTEIN>::shape(elem, order, i, pp) -
                            FE<3,BERNSTEIN>::shape(elem, order, i, pm))/2./eps;
                  }

                  // d()/deta
                case 1:
                  {
                    const Point pp(p(0), p(1)+eps, p(2));
                    const Point pm(p(0), p(1)-eps, p(2));

                    return (FE<3,BERNSTEIN>::shape(elem, order, i, pp) -
                            FE<3,BERNSTEIN>::shape(elem, order, i, pm))/2./eps;
                  }
                  // d()/dzeta
                case 2:
                  {
                    const Point pp(p(0), p(1), p(2)+eps);
                    const Point pm(p(0), p(1), p(2)-eps);

                    return (FE<3,BERNSTEIN>::shape(elem, order, i, pp) -
                            FE<3,BERNSTEIN>::shape(elem, order, i, pm))/2./eps;
                  }
                default:
                  libmesh_error_msg("Invalid derivative index j = " << j);
                }

            }

            //       // Compute hex shape functions as a tensor-product
            //       const Real xi    = p(0);
            //       const Real eta   = p(1);
            //       const Real zeta  = p(2);
            //       Real xi_mapped   = p(0);
            //       Real eta_mapped  = p(1);
            //       Real zeta_mapped = p(2);

            //       // The only way to make any sense of this
            //       // is to look at the mgflo/mg2/mgf documentation
            //       // and make the cut-out cube!
            //       //  Nodes                         0  1  2  3  4  5  6  7  8  8  9  9 10 10 11 11 12 12 13 13 14 14 15 15 16 16 17 17 18 18 19 19 20 20 20 20 21 21 21 21 22 22 22 22 23 23 23 23 24 24 24 24 25 25 25 25 26 26 26 26 26 26 26 26
            //       //  DOFS                          0  1  2  3  4  5  6  7  8  9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 18 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 60 62 63
            //       static const unsigned int i0[] = {0, 1, 1, 0, 0, 1, 1, 0, 2, 3, 1, 1, 2, 3, 0, 0, 0, 0, 1, 1, 1, 1, 0, 0, 2, 3, 1, 1, 2, 3, 0, 0, 2, 3, 2, 3, 2, 3, 2, 3, 1, 1, 1, 1, 2, 3, 2, 3, 0, 0, 0, 0, 2, 3, 2, 3, 2, 3, 2, 3, 2, 3, 2, 3};
            //       static const unsigned int i1[] = {0, 0, 1, 1, 0, 0, 1, 1, 0, 0, 2, 3, 1, 1, 2, 3, 0, 0, 0, 0, 1, 1, 1, 1, 0, 0, 2, 3, 1, 1, 2, 3, 2, 2, 3, 3, 0, 0, 0, 0, 2, 3, 2, 3, 1, 1, 1, 1, 2, 3, 2, 3, 2, 2, 3, 3, 2, 2, 3, 3, 2, 2, 3, 3};
            //       static const unsigned int i2[] = {0, 0, 0, 0, 1, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 2, 3, 2, 3, 2, 3, 2, 3, 1, 1, 1, 1, 1, 1, 1, 1, 0, 0, 0, 0, 2, 2, 3, 3, 2, 2, 3, 3, 2, 2, 3, 3, 2, 2, 3, 3, 1, 1, 1, 1, 2, 2, 2, 2, 3, 3, 3, 3};



            //       // handle the edge orientation
            //       {
            // // Edge 0
            // if ((i1[i] == 0) && (i2[i] == 0))
            //   {
            //     if (elem->node_id(0) != std::min(elem->node_id(0), elem->node_id(1)))
            //       xi_mapped = -xi;
            //   }
            // // Edge 1
            // else if ((i0[i] == 1) && (i2[i] == 0))
            //   {
            //     if (elem->node_id(1) != std::min(elem->node_id(1), elem->node_id(2)))
            //       eta_mapped = -eta;
            //   }
            // // Edge 2
            // else if ((i1[i] == 1) && (i2[i] == 0))
            //   {
            //     if (elem->node_id(3) != std::min(elem->node_id(3), elem->node_id(2)))
            //       xi_mapped = -xi;
            //   }
            // // Edge 3
            // else if ((i0[i] == 0) && (i2[i] == 0))
            //   {
            //     if (elem->node_id(0) != std::min(elem->node_id(0), elem->node_id(3)))
            //       eta_mapped = -eta;
            //   }
            // // Edge 4
            // else if ((i0[i] == 0) && (i1[i] == 0))
            //   {
            //     if (elem->node_id(0) != std::min(elem->node_id(0), elem->node_id(4)))
            //       zeta_mapped = -zeta;
            //   }
            // // Edge 5
            // else if ((i0[i] == 1) && (i1[i] == 0))
            //   {
            //     if (elem->node_id(1) != std::min(elem->node_id(1), elem->node_id(5)))
            //       zeta_mapped = -zeta;
            //   }
            // // Edge 6
            // else if ((i0[i] == 1) && (i1[i] == 1))
            //   {
            //     if (elem->node_id(2) != std::min(elem->node_id(2), elem->node_id(6)))
            //       zeta_mapped = -zeta;
            //   }
            // // Edge 7
            // else if ((i0[i] == 0) && (i1[i] == 1))
            //   {
            //     if (elem->node_id(3) != std::min(elem->node_id(3), elem->node_id(7)))
            //       zeta_mapped = -zeta;
            //   }
            // // Edge 8
            // else if ((i1[i] == 0) && (i2[i] == 1))
            //   {
            //     if (elem->node_id(4) != std::min(elem->node_id(4), elem->node_id(5)))
            //       xi_mapped = -xi;
            //   }
            // // Edge 9
            // else if ((i0[i] == 1) && (i2[i] == 1))
            //   {
            //     if (elem->node_id(5) != std::min(elem->node_id(5), elem->node_id(6)))
            //       eta_mapped = -eta;
            //   }
            // // Edge 10
            // else if ((i1[i] == 1) && (i2[i] == 1))
            //   {
            //     if (elem->node_id(7) != std::min(elem->node_id(7), elem->node_id(6)))
            //       xi_mapped = -xi;
            //   }
            // // Edge 11
            // else if ((i0[i] == 0) && (i2[i] == 1))
            //   {
            //     if (elem->node_id(4) != std::min(elem->node_id(4), elem->node_id(7)))
            //       eta_mapped = -eta;
            //   }
            //       }


            //       // handle the face orientation
            //       {
            // // Face 0
            // if ((i2[i] == 0) && (i0[i] >= 2) && (i1[i] >= 2))
            //   {
            //     const unsigned int min_node = std::min(elem->node_id(1),
            //    std::min(elem->node_id(2),
            //     std::min(elem->node_id(0),
            //      elem->node_id(3))));
            //     if (elem->node_id(0) == min_node)
            //       if (elem->node_id(1) == std::min(elem->node_id(1), elem->node_id(3)))
            // {
            //   // Case 1
            //   xi_mapped  = xi;
            //   eta_mapped = eta;
            // }
            //       else
            // {
            //   // Case 2
            //   xi_mapped  = eta;
            //   eta_mapped = xi;
            // }

            //     else if (elem->node_id(3) == min_node)
            //       if (elem->node_id(0) == std::min(elem->node_id(0), elem->node_id(2)))
            // {
            //   // Case 3
            //   xi_mapped  = -eta;
            //   eta_mapped = xi;
            // }
            //       else
            // {
            //   // Case 4
            //   xi_mapped  = xi;
            //   eta_mapped = -eta;
            // }

            //     else if (elem->node_id(2) == min_node)
            //       if (elem->node_id(3) == std::min(elem->node_id(3), elem->node_id(1)))
            // {
            //   // Case 5
            //   xi_mapped  = -xi;
            //   eta_mapped = -eta;
            // }
            //       else
            // {
            //   // Case 6
            //   xi_mapped  = -eta;
            //   eta_mapped = -xi;
            // }

            //     else if (elem->node_id(1) == min_node)
            //       if (elem->node_id(2) == std::min(elem->node_id(2), elem->node_id(0)))
            // {
            //   // Case 7
            //   xi_mapped  = eta;
            //   eta_mapped = -xi;
            // }
            //       else
            // {
            //   // Case 8
            //   xi_mapped  = -xi;
            //   eta_mapped = eta;
            // }
            //   }


            // // Face 1
            // else if ((i1[i] == 0) && (i0[i] >= 2) && (i2[i] >= 2))
            //   {
            //     const unsigned int min_node = std::min(elem->node_id(0),
            //    std::min(elem->node_id(1),
            //     std::min(elem->node_id(5),
            //      elem->node_id(4))));
            //     if (elem->node_id(0) == min_node)
            //       if (elem->node_id(1) == std::min(elem->node_id(1), elem->node_id(4)))
            // {
            //   // Case 1
            //   xi_mapped   = xi;
            //   zeta_mapped = zeta;
            // }
            //       else
            // {
            //   // Case 2
            //   xi_mapped   = zeta;
            //   zeta_mapped = xi;
            // }

            //     else if (elem->node_id(1) == min_node)
            //       if (elem->node_id(5) == std::min(elem->node_id(5), elem->node_id(0)))
            // {
            //   // Case 3
            //   xi_mapped   = zeta;
            //   zeta_mapped = -xi;
            // }
            //       else
            // {
            //   // Case 4
            //   xi_mapped   = -xi;
            //   zeta_mapped = zeta;
            // }

            //     else if (elem->node_id(5) == min_node)
            //       if (elem->node_id(4) == std::min(elem->node_id(4), elem->node_id(1)))
            // {
            //   // Case 5
            //   xi_mapped   = -xi;
            //   zeta_mapped = -zeta;
            // }
            //       else
            // {
            //   // Case 6
            //   xi_mapped   = -zeta;
            //   zeta_mapped = -xi;
            // }

            //     else if (elem->node_id(4) == min_node)
            //       if (elem->node_id(0) == std::min(elem->node_id(0), elem->node_id(5)))
            // {
            //   // Case 7
            //   xi_mapped   = -xi;
            //   zeta_mapped = zeta;
            // }
            //       else
            // {
            //   // Case 8
            //   xi_mapped   = xi;
            //   zeta_mapped = -zeta;
            // }
            //   }


            // // Face 2
            // else if ((i0[i] == 1) && (i1[i] >= 2) && (i2[i] >= 2))
            //   {
            //     const unsigned int min_node = std::min(elem->node_id(1),
            //    std::min(elem->node_id(2),
            //     std::min(elem->node_id(6),
            //      elem->node_id(5))));
            //     if (elem->node_id(1) == min_node)
            //       if (elem->node_id(2) == std::min(elem->node_id(2), elem->node_id(5)))
            // {
            //   // Case 1
            //   eta_mapped  = eta;
            //   zeta_mapped = zeta;
            // }
            //       else
            // {
            //   // Case 2
            //   eta_mapped  = zeta;
            //   zeta_mapped = eta;
            // }

            //     else if (elem->node_id(2) == min_node)
            //       if (elem->node_id(6) == std::min(elem->node_id(6), elem->node_id(1)))
            // {
            //   // Case 3
            //   eta_mapped  = zeta;
            //   zeta_mapped = -eta;
            // }
            //       else
            // {
            //   // Case 4
            //   eta_mapped  = -eta;
            //   zeta_mapped = zeta;
            // }

            //     else if (elem->node_id(6) == min_node)
            //       if (elem->node_id(5) == std::min(elem->node_id(5), elem->node_id(2)))
            // {
            //   // Case 5
            //   eta_mapped  = -eta;
            //   zeta_mapped = -zeta;
            // }
            //       else
            // {
            //   // Case 6
            //   eta_mapped  = -zeta;
            //   zeta_mapped = -eta;
            // }

            //     else if (elem->node_id(5) == min_node)
            //       if (elem->node_id(1) == std::min(elem->node_id(1), elem->node_id(6)))
            // {
            //   // Case 7
            //   eta_mapped  = -zeta;
            //   zeta_mapped = eta;
            // }
            //       else
            // {
            //   // Case 8
            //   eta_mapped   = eta;
            //   zeta_mapped = -zeta;
            // }
            //   }


            // // Face 3
            // else if ((i1[i] == 1) && (i0[i] >= 2) && (i2[i] >= 2))
            //   {
            //     const unsigned int min_node = std::min(elem->node_id(2),
            //    std::min(elem->node_id(3),
            //     std::min(elem->node_id(7),
            //      elem->node_id(6))));
            //     if (elem->node_id(3) == min_node)
            //       if (elem->node_id(2) == std::min(elem->node_id(2), elem->node_id(7)))
            // {
            //   // Case 1
            //   xi_mapped   = xi;
            //   zeta_mapped = zeta;
            // }
            //       else
            // {
            //   // Case 2
            //   xi_mapped   = zeta;
            //   zeta_mapped = xi;
            // }

            //     else if (elem->node_id(7) == min_node)
            //       if (elem->node_id(3) == std::min(elem->node_id(3), elem->node_id(6)))
            // {
            //   // Case 3
            //   xi_mapped   = -zeta;
            //   zeta_mapped = xi;
            // }
            //       else
            // {
            //   // Case 4
            //   xi_mapped   = xi;
            //   zeta_mapped = -zeta;
            // }

            //     else if (elem->node_id(6) == min_node)
            //       if (elem->node_id(7) == std::min(elem->node_id(7), elem->node_id(2)))
            // {
            //   // Case 5
            //   xi_mapped   = -xi;
            //   zeta_mapped = -zeta;
            // }
            //       else
            // {
            //   // Case 6
            //   xi_mapped   = -zeta;
            //   zeta_mapped = -xi;
            // }

            //     else if (elem->node_id(2) == min_node)
            //       if (elem->node_id(6) == std::min(elem->node_id(3), elem->node_id(6)))
            // {
            //   // Case 7
            //   xi_mapped   = zeta;
            //   zeta_mapped = -xi;
            // }
            //       else
            // {
            //   // Case 8
            //   xi_mapped   = -xi;
            //   zeta_mapped = zeta;
            // }
            //   }


            // // Face 4
            // else if ((i0[i] == 0) && (i1[i] >= 2) && (i2[i] >= 2))
            //   {
            //     const unsigned int min_node = std::min(elem->node_id(3),
            //    std::min(elem->node_id(0),
            //     std::min(elem->node_id(4),
            //      elem->node_id(7))));
            //     if (elem->node_id(0) == min_node)
            //       if (elem->node_id(3) == std::min(elem->node_id(3), elem->node_id(4)))
            // {
            //   // Case 1
            //   eta_mapped  = eta;
            //   zeta_mapped = zeta;
            // }
            //       else
            // {
            //   // Case 2
            //   eta_mapped  = zeta;
            //   zeta_mapped = eta;
            // }

            //     else if (elem->node_id(4) == min_node)
            //       if (elem->node_id(0) == std::min(elem->node_id(0), elem->node_id(7)))
            // {
            //   // Case 3
            //   eta_mapped  = -zeta;
            //   zeta_mapped = eta;
            // }
            //       else
            // {
            //   // Case 4
            //   eta_mapped  = eta;
            //   zeta_mapped = -zeta;
            // }

            //     else if (elem->node_id(7) == min_node)
            //       if (elem->node_id(4) == std::min(elem->node_id(4), elem->node_id(3)))
            // {
            //   // Case 5
            //   eta_mapped  = -eta;
            //   zeta_mapped = -zeta;
            // }
            //       else
            // {
            //   // Case 6
            //   eta_mapped  = -zeta;
            //   zeta_mapped = -eta;
            // }

            //     else if (elem->node_id(3) == min_node)
            //       if (elem->node_id(7) == std::min(elem->node_id(7), elem->node_id(0)))
            // {
            //   // Case 7
            //   eta_mapped   = zeta;
            //   zeta_mapped = -eta;
            // }
            //       else
            // {
            //   // Case 8
            //   eta_mapped  = -eta;
            //   zeta_mapped = zeta;
            // }
            //   }


            // // Face 5
            // else if ((i2[i] == 1) && (i0[i] >= 2) && (i1[i] >= 2))
            //   {
            //     const unsigned int min_node = std::min(elem->node_id(4),
            //    std::min(elem->node_id(5),
            //     std::min(elem->node_id(6),
            //      elem->node_id(7))));
            //     if (elem->node_id(4) == min_node)
            //       if (elem->node_id(5) == std::min(elem->node_id(5), elem->node_id(7)))
            // {
            //   // Case 1
            //   xi_mapped  = xi;
            //   eta_mapped = eta;
            // }
            //       else
            // {
            //   // Case 2
            //   xi_mapped  = eta;
            //   eta_mapped = xi;
            // }

            //     else if (elem->node_id(5) == min_node)
            //       if (elem->node_id(6) == std::min(elem->node_id(6), elem->node_id(4)))
            // {
            //   // Case 3
            //   xi_mapped  = eta;
            //   eta_mapped = -xi;
            // }
            //       else
            // {
            //   // Case 4
            //   xi_mapped  = -xi;
            //   eta_mapped = eta;
            // }

            //     else if (elem->node_id(6) == min_node)
            //       if (elem->node_id(7) == std::min(elem->node_id(7), elem->node_id(5)))
            // {
            //   // Case 5
            //   xi_mapped  = -xi;
            //   eta_mapped = -eta;
            // }
            //       else
            // {
            //   // Case 6
            //   xi_mapped  = -eta;
            //   eta_mapped = -xi;
            // }

            //     else if (elem->node_id(7) == min_node)
            //       if (elem->node_id(4) == std::min(elem->node_id(4), elem->node_id(6)))
            // {
            //   // Case 7
            //   xi_mapped  = -eta;
            //   eta_mapped = xi;
            // }
            //       else
            // {
            //   // Case 8
            //   xi_mapped  = xi;
            //   eta_mapped = eta;
            // }
            //   }


            //       }



            //       libmesh_assert_less (j, 3);

            //       switch (j)
            // {
            //   // d()/dxi
            // case 0:
            //   return (FE<1,BERNSTEIN>::shape_deriv(EDGE3, totalorder, i0[i], 0, xi_mapped)*
            //   FE<1,BERNSTEIN>::shape      (EDGE3, totalorder, i1[i],    eta_mapped)*
            //   FE<1,BERNSTEIN>::shape      (EDGE3, totalorder, i2[i],    zeta_mapped));

            //   // d()/deta
            // case 1:
            //   return (FE<1,BERNSTEIN>::shape      (EDGE3, totalorder, i0[i],    xi_mapped)*
            //   FE<1,BERNSTEIN>::shape_deriv(EDGE3, totalorder, i1[i], 0, eta_mapped)*
            //   FE<1,BERNSTEIN>::shape      (EDGE3, totalorder, i2[i],    zeta_mapped));

            //   // d()/dzeta
            // case 2:
            //   return (FE<1,BERNSTEIN>::shape      (EDGE3, totalorder, i0[i],    xi_mapped)*
            //   FE<1,BERNSTEIN>::shape      (EDGE3, totalorder, i1[i],    eta_mapped)*
            //   FE<1,BERNSTEIN>::shape_deriv(EDGE3, totalorder, i2[i], 0, zeta_mapped));

            // default:
            // libmesh_error_msg("Invalid derivative index j = " << j);
            // }
            //     }


          default:
            libmesh_error_msg("Invalid element type = " << type);
          }
      }

      // 4th-order Bernstein.
    case FOURTH:
      {
        switch (type)
          {

            // Bernstein shape functions derivatives on the hexahedral.
          case HEX27:
            {
              const Real eps = 1.e-6;

              libmesh_assert_less (i, 125);
              libmesh_assert_less (j, 3);

              switch (j)
                {
                  //  d()/dxi
                case 0:
                  {
                    const Point pp(p(0)+eps, p(1), p(2));
                    const Point pm(p(0)-eps, p(1), p(2));

                    return (FE<3,BERNSTEIN>::shape(elem, order, i, pp) -
                            FE<3,BERNSTEIN>::shape(elem, order, i, pm))/2./eps;
                  }

                  // d()/deta
                case 1:
                  {
                    const Point pp(p(0), p(1)+eps, p(2));
                    const Point pm(p(0), p(1)-eps, p(2));

                    return (FE<3,BERNSTEIN>::shape(elem, order, i, pp) -
                            FE<3,BERNSTEIN>::shape(elem, order, i, pm))/2./eps;
                  }
                  // d()/dzeta
                case 2:
                  {
                    const Point pp(p(0), p(1), p(2)+eps);
                    const Point pm(p(0), p(1), p(2)-eps);

                    return (FE<3,BERNSTEIN>::shape(elem, order, i, pp) -
                            FE<3,BERNSTEIN>::shape(elem, order, i, pm))/2./eps;
                  }
                default:
                  libmesh_error_msg("Invalid derivative index j = " << j);
                }
            }

            //       // Compute hex shape functions as a tensor-product
            //       const Real xi    = p(0);
            //       const Real eta   = p(1);
            //       const Real zeta  = p(2);
            //       Real xi_mapped   = p(0);
            //       Real eta_mapped  = p(1);
            //       Real zeta_mapped = p(2);

            //       // The only way to make any sense of this
            //       // is to look at the mgflo/mg2/mgf documentation
            //       // and make the cut-out cube!
            //       //  Nodes                         0  1  2  3  4  5  6  7  8  8  9  9 10 10 11 11 12 12 13 13 14 14 15 15 16 16 17 17 18 18 19 19 20 20 20 20 21 21 21 21 22 22 22 22 23 23 23 23 24 24 24 24 25 25 25 25 26 26 26 26 26 26 26 26
            //       //  DOFS                          0  1  2  3  4  5  6  7  8  9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 18 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 60 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 00 01 02 03 04 05 06 07 08 09 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24
            //       static const unsigned int i0[] = {0, 1, 1, 0, 0, 1, 1, 0, 2, 3, 4, 1, 1, 1, 2, 3, 4, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 0, 0, 0, 2, 3, 4, 1, 1, 1, 2, 3, 4, 0, 0, 0, 2, 3, 4, 2, 3, 4, 2, 3, 4, 2, 3, 4, 2, 3, 4, 2, 3, 4, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 3, 4, 2, 3, 4, 2, 3, 4, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 3, 4, 2, 3, 4, 2, 3, 4, 2, 3, 4, 2, 3, 4, 2, 3, 4, 2, 3, 4, 2, 3, 4, 2, 3, 4, 2, 3, 4, 2, 3, 4, 2, 3, 4};
            //       static const unsigned int i1[] = {0, 0, 1, 1, 0, 0, 1, 1, 0, 0, 0, 2, 3, 4, 1, 1, 1, 2, 3, 4, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 0, 0, 0, 2, 3, 4, 1, 1, 1, 2, 3, 4, 2, 2, 2, 3, 3, 3, 4, 4, 4, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 3, 4, 2, 3, 4, 2, 3, 4, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 3, 4, 2, 3, 4, 2, 3, 4, 2, 2, 2, 3, 3, 3, 4, 4, 4, 2, 2, 2, 3, 3, 3, 4, 4, 4, 2, 2, 2, 3, 3, 3, 4, 4, 4, 2, 2, 2, 3, 3, 3, 4, 4, 4};
            //       static const unsigned int i2[] = {0, 0, 0, 0, 1, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 3, 4, 2, 3, 4, 2, 3, 4, 2, 3, 4, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 2, 2, 3, 3, 3, 4, 4, 4, 2, 2, 2, 3, 3, 3, 4, 4, 4, 2, 2, 2, 3, 3, 3, 4, 4, 4, 2, 2, 2, 3, 3, 3, 4, 4, 4, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 3, 3, 3, 3, 3, 3, 3, 3, 3, 4, 4, 4, 4, 4, 4, 4, 4, 4};



            //       // handle the edge orientation
            //       {
            // // Edge 0
            // if ((i1[i] == 0) && (i2[i] == 0))
            //   {
            //     if (elem->node_id(0) != std::min(elem->node_id(0), elem->node_id(1)))
            //       xi_mapped = -xi;
            //   }
            // // Edge 1
            // else if ((i0[i] == 1) && (i2[i] == 0))
            //   {
            //     if (elem->node_id(1) != std::min(elem->node_id(1), elem->node_id(2)))
            //       eta_mapped = -eta;
            //   }
            // // Edge 2
            // else if ((i1[i] == 1) && (i2[i] == 0))
            //   {
            //     if (elem->node_id(3) != std::min(elem->node_id(3), elem->node_id(2)))
            //       xi_mapped = -xi;
            //   }
            // // Edge 3
            // else if ((i0[i] == 0) && (i2[i] == 0))
            //   {
            //     if (elem->node_id(0) != std::min(elem->node_id(0), elem->node_id(3)))
            //       eta_mapped = -eta;
            //   }
            // // Edge 4
            // else if ((i0[i] == 0) && (i1[i] == 0))
            //   {
            //     if (elem->node_id(0) != std::min(elem->node_id(0), elem->node_id(4)))
            //       zeta_mapped = -zeta;
            //   }
            // // Edge 5
            // else if ((i0[i] == 1) && (i1[i] == 0))
            //   {
            //     if (elem->node_id(1) != std::min(elem->node_id(1), elem->node_id(5)))
            //       zeta_mapped = -zeta;
            //   }
            // // Edge 6
            // else if ((i0[i] == 1) && (i1[i] == 1))
            //   {
            //     if (elem->node_id(2) != std::min(elem->node_id(2), elem->node_id(6)))
            //       zeta_mapped = -zeta;
            //   }
            // // Edge 7
            // else if ((i0[i] == 0) && (i1[i] == 1))
            //   {
            //     if (elem->node_id(3) != std::min(elem->node_id(3), elem->node_id(7)))
            //       zeta_mapped = -zeta;
            //   }
            // // Edge 8
            // else if ((i1[i] == 0) && (i2[i] == 1))
            //   {
            //     if (elem->node_id(4) != std::min(elem->node_id(4), elem->node_id(5)))
            //       xi_mapped = -xi;
            //   }
            // // Edge 9
            // else if ((i0[i] == 1) && (i2[i] == 1))
            //   {
            //     if (elem->node_id(5) != std::min(elem->node_id(5), elem->node_id(6)))
            //       eta_mapped = -eta;
            //   }
            // // Edge 10
            // else if ((i1[i] == 1) && (i2[i] == 1))
            //   {
            //     if (elem->node_id(7) != std::min(elem->node_id(7), elem->node_id(6)))
            //       xi_mapped = -xi;
            //   }
            // // Edge 11
            // else if ((i0[i] == 0) && (i2[i] == 1))
            //   {
            //     if (elem->node_id(4) != std::min(elem->node_id(4), elem->node_id(7)))
            //       eta_mapped = -eta;
            //   }
            //       }


            //       // handle the face orientation
            //       {
            // // Face 0
            // if ((i2[i] == 0) && (i0[i] >= 2) && (i1[i] >= 2))
            //   {
            //     const unsigned int min_node = std::min(elem->node_id(1),
            //    std::min(elem->node_id(2),
            //     std::min(elem->node_id(0),
            //      elem->node_id(3))));
            //     if (elem->node_id(0) == min_node)
            //       if (elem->node_id(1) == std::min(elem->node_id(1), elem->node_id(3)))
            // {
            //   // Case 1
            //   xi_mapped  = xi;
            //   eta_mapped = eta;
            // }
            //       else
            // {
            //   // Case 2
            //   xi_mapped  = eta;
            //   eta_mapped = xi;
            // }

            //     else if (elem->node_id(3) == min_node)
            //       if (elem->node_id(0) == std::min(elem->node_id(0), elem->node_id(2)))
            // {
            //   // Case 3
            //   xi_mapped  = -eta;
            //   eta_mapped = xi;
            // }
            //       else
            // {
            //   // Case 4
            //   xi_mapped  = xi;
            //   eta_mapped = -eta;
            // }

            //     else if (elem->node_id(2) == min_node)
            //       if (elem->node_id(3) == std::min(elem->node_id(3), elem->node_id(1)))
            // {
            //   // Case 5
            //   xi_mapped  = -xi;
            //   eta_mapped = -eta;
            // }
            //       else
            // {
            //   // Case 6
            //   xi_mapped  = -eta;
            //   eta_mapped = -xi;
            // }

            //     else if (elem->node_id(1) == min_node)
            //       if (elem->node_id(2) == std::min(elem->node_id(2), elem->node_id(0)))
            // {
            //   // Case 7
            //   xi_mapped  = eta;
            //   eta_mapped = -xi;
            // }
            //       else
            // {
            //   // Case 8
            //   xi_mapped  = -xi;
            //   eta_mapped = eta;
            // }
            //   }


            // // Face 1
            // else if ((i1[i] == 0) && (i0[i] >= 2) && (i2[i] >= 2))
            //   {
            //     const unsigned int min_node = std::min(elem->node_id(0),
            //    std::min(elem->node_id(1),
            //     std::min(elem->node_id(5),
            //      elem->node_id(4))));
            //     if (elem->node_id(0) == min_node)
            //       if (elem->node_id(1) == std::min(elem->node_id(1), elem->node_id(4)))
            // {
            //   // Case 1
            //   xi_mapped   = xi;
            //   zeta_mapped = zeta;
            // }
            //       else
            // {
            //   // Case 2
            //   xi_mapped   = zeta;
            //   zeta_mapped = xi;
            // }

            //     else if (elem->node_id(1) == min_node)
            //       if (elem->node_id(5) == std::min(elem->node_id(5), elem->node_id(0)))
            // {
            //   // Case 3
            //   xi_mapped   = zeta;
            //   zeta_mapped = -xi;
            // }
            //       else
            // {
            //   // Case 4
            //   xi_mapped   = -xi;
            //   zeta_mapped = zeta;
            // }

            //     else if (elem->node_id(5) == min_node)
            //       if (elem->node_id(4) == std::min(elem->node_id(4), elem->node_id(1)))
            // {
            //   // Case 5
            //   xi_mapped   = -xi;
            //   zeta_mapped = -zeta;
            // }
            //       else
            // {
            //   // Case 6
            //   xi_mapped   = -zeta;
            //   zeta_mapped = -xi;
            // }

            //     else if (elem->node_id(4) == min_node)
            //       if (elem->node_id(0) == std::min(elem->node_id(0), elem->node_id(5)))
            // {
            //   // Case 7
            //   xi_mapped   = -xi;
            //   zeta_mapped = zeta;
            // }
            //       else
            // {
            //   // Case 8
            //   xi_mapped   = xi;
            //   zeta_mapped = -zeta;
            // }
            //   }


            // // Face 2
            // else if ((i0[i] == 1) && (i1[i] >= 2) && (i2[i] >= 2))
            //   {
            //     const unsigned int min_node = std::min(elem->node_id(1),
            //    std::min(elem->node_id(2),
            //     std::min(elem->node_id(6),
            //      elem->node_id(5))));
            //     if (elem->node_id(1) == min_node)
            //       if (elem->node_id(2) == std::min(elem->node_id(2), elem->node_id(5)))
            // {
            //   // Case 1
            //   eta_mapped  = eta;
            //   zeta_mapped = zeta;
            // }
            //       else
            // {
            //   // Case 2
            //   eta_mapped  = zeta;
            //   zeta_mapped = eta;
            // }

            //     else if (elem->node_id(2) == min_node)
            //       if (elem->node_id(6) == std::min(elem->node_id(6), elem->node_id(1)))
            // {
            //   // Case 3
            //   eta_mapped  = zeta;
            //   zeta_mapped = -eta;
            // }
            //       else
            // {
            //   // Case 4
            //   eta_mapped  = -eta;
            //   zeta_mapped = zeta;
            // }

            //     else if (elem->node_id(6) == min_node)
            //       if (elem->node_id(5) == std::min(elem->node_id(5), elem->node_id(2)))
            // {
            //   // Case 5
            //   eta_mapped  = -eta;
            //   zeta_mapped = -zeta;
            // }
            //       else
            // {
            //   // Case 6
            //   eta_mapped  = -zeta;
            //   zeta_mapped = -eta;
            // }

            //     else if (elem->node_id(5) == min_node)
            //       if (elem->node_id(1) == std::min(elem->node_id(1), elem->node_id(6)))
            // {
            //   // Case 7
            //   eta_mapped  = -zeta;
            //   zeta_mapped = eta;
            // }
            //       else
            // {
            //   // Case 8
            //   eta_mapped   = eta;
            //   zeta_mapped = -zeta;
            // }
            //   }


            // // Face 3
            // else if ((i1[i] == 1) && (i0[i] >= 2) && (i2[i] >= 2))
            //   {
            //     const unsigned int min_node = std::min(elem->node_id(2),
            //    std::min(elem->node_id(3),
            //     std::min(elem->node_id(7),
            //      elem->node_id(6))));
            //     if (elem->node_id(3) == min_node)
            //       if (elem->node_id(2) == std::min(elem->node_id(2), elem->node_id(7)))
            // {
            //   // Case 1
            //   xi_mapped   = xi;
            //   zeta_mapped = zeta;
            // }
            //       else
            // {
            //   // Case 2
            //   xi_mapped   = zeta;
            //   zeta_mapped = xi;
            // }

            //     else if (elem->node_id(7) == min_node)
            //       if (elem->node_id(3) == std::min(elem->node_id(3), elem->node_id(6)))
            // {
            //   // Case 3
            //   xi_mapped   = -zeta;
            //   zeta_mapped = xi;
            // }
            //       else
            // {
            //   // Case 4
            //   xi_mapped   = xi;
            //   zeta_mapped = -zeta;
            // }

            //     else if (elem->node_id(6) == min_node)
            //       if (elem->node_id(7) == std::min(elem->node_id(7), elem->node_id(2)))
            // {
            //   // Case 5
            //   xi_mapped   = -xi;
            //   zeta_mapped = -zeta;
            // }
            //       else
            // {
            //   // Case 6
            //   xi_mapped   = -zeta;
            //   zeta_mapped = -xi;
            // }

            //     else if (elem->node_id(2) == min_node)
            //       if (elem->node_id(6) == std::min(elem->node_id(3), elem->node_id(6)))
            // {
            //   // Case 7
            //   xi_mapped   = zeta;
            //   zeta_mapped = -xi;
            // }
            //       else
            // {
            //   // Case 8
            //   xi_mapped   = -xi;
            //   zeta_mapped = zeta;
            // }
            //   }


            // // Face 4
            // else if ((i0[i] == 0) && (i1[i] >= 2) && (i2[i] >= 2))
            //   {
            //     const unsigned int min_node = std::min(elem->node_id(3),
            //    std::min(elem->node_id(0),
            //     std::min(elem->node_id(4),
            //      elem->node_id(7))));
            //     if (elem->node_id(0) == min_node)
            //       if (elem->node_id(3) == std::min(elem->node_id(3), elem->node_id(4)))
            // {
            //   // Case 1
            //   eta_mapped  = eta;
            //   zeta_mapped = zeta;
            // }
            //       else
            // {
            //   // Case 2
            //   eta_mapped  = zeta;
            //   zeta_mapped = eta;
            // }

            //     else if (elem->node_id(4) == min_node)
            //       if (elem->node_id(0) == std::min(elem->node_id(0), elem->node_id(7)))
            // {
            //   // Case 3
            //   eta_mapped  = -zeta;
            //   zeta_mapped = eta;
            // }
            //       else
            // {
            //   // Case 4
            //   eta_mapped  = eta;
            //   zeta_mapped = -zeta;
            // }

            //     else if (elem->node_id(7) == min_node)
            //       if (elem->node_id(4) == std::min(elem->node_id(4), elem->node_id(3)))
            // {
            //   // Case 5
            //   eta_mapped  = -eta;
            //   zeta_mapped = -zeta;
            // }
            //       else
            // {
            //   // Case 6
            //   eta_mapped  = -zeta;
            //   zeta_mapped = -eta;
            // }

            //     else if (elem->node_id(3) == min_node)
            //       if (elem->node_id(7) == std::min(elem->node_id(7), elem->node_id(0)))
            // {
            //   // Case 7
            //   eta_mapped   = zeta;
            //   zeta_mapped = -eta;
            // }
            //       else
            // {
            //   // Case 8
            //   eta_mapped  = -eta;
            //   zeta_mapped = zeta;
            // }
            //   }


            // // Face 5
            // else if ((i2[i] == 1) && (i0[i] >= 2) && (i1[i] >= 2))
            //   {
            //     const unsigned int min_node = std::min(elem->node_id(4),
            //    std::min(elem->node_id(5),
            //     std::min(elem->node_id(6),
            //      elem->node_id(7))));
            //     if (elem->node_id(4) == min_node)
            //       if (elem->node_id(5) == std::min(elem->node_id(5), elem->node_id(7)))
            // {
            //   // Case 1
            //   xi_mapped  = xi;
            //   eta_mapped = eta;
            // }
            //       else
            // {
            //   // Case 2
            //   xi_mapped  = eta;
            //   eta_mapped = xi;
            // }

            //     else if (elem->node_id(5) == min_node)
            //       if (elem->node_id(6) == std::min(elem->node_id(6), elem->node_id(4)))
            // {
            //   // Case 3
            //   xi_mapped  = eta;
            //   eta_mapped = -xi;
            // }
            //       else
            // {
            //   // Case 4
            //   xi_mapped  = -xi;
            //   eta_mapped = eta;
            // }

            //     else if (elem->node_id(6) == min_node)
            //       if (elem->node_id(7) == std::min(elem->node_id(7), elem->node_id(5)))
            // {
            //   // Case 5
            //   xi_mapped  = -xi;
            //   eta_mapped = -eta;
            // }
            //       else
            // {
            //   // Case 6
            //   xi_mapped  = -eta;
            //   eta_mapped = -xi;
            // }

            //     else if (elem->node_id(7) == min_node)
            //       if (elem->node_id(4) == std::min(elem->node_id(4), elem->node_id(6)))
            // {
            //   // Case 7
            //   xi_mapped  = -eta;
            //   eta_mapped = xi;
            // }
            //       else
            // {
            //   // Case 8
            //   xi_mapped  = xi;
            //   eta_mapped = eta;
            // }
            //   }


            //       }



            //       libmesh_assert_less (j, 3);

            //       switch (j)
            // {
            //   // d()/dxi
            // case 0:
            //   return (FE<1,BERNSTEIN>::shape_deriv(EDGE3, totalorder, i0[i], 0, xi_mapped)*
            //   FE<1,BERNSTEIN>::shape      (EDGE3, totalorder, i1[i],    eta_mapped)*
            //   FE<1,BERNSTEIN>::shape      (EDGE3, totalorder, i2[i],    zeta_mapped));

            //   // d()/deta
            // case 1:
            //   return (FE<1,BERNSTEIN>::shape      (EDGE3, totalorder, i0[i],    xi_mapped)*
            //   FE<1,BERNSTEIN>::shape_deriv(EDGE3, totalorder, i1[i], 0, eta_mapped)*
            //   FE<1,BERNSTEIN>::shape      (EDGE3, totalorder, i2[i],    zeta_mapped));

            //   // d()/dzeta
            // case 2:
            //   return (FE<1,BERNSTEIN>::shape      (EDGE3, totalorder, i0[i],    xi_mapped)*
            //   FE<1,BERNSTEIN>::shape      (EDGE3, totalorder, i1[i],    eta_mapped)*
            //   FE<1,BERNSTEIN>::shape_deriv(EDGE3, totalorder, i2[i], 0, zeta_mapped));

            // default:
            //   libmesh_error_msg("Invalid derivative index j = " << j);
            // }
            //    }


          default:
            libmesh_error_msg("Invalid element type = " << type);
          }
      }


    default:
      libmesh_error_msg("Invalid totalorder = " << totalorder);
    }

#endif
}

shape_deriv() [107/109]

Real libMesh::FE< 3, LAGRANGE >::shape_deriv ( const Elem *  elem, const Order  order, const unsigned int  i, const unsigned int  j, const Point &  p  )

inherited

Definition at line 1914 of file fe_lagrange_shape_3D.C.

{
  libmesh_assert(elem);

  // call the orientation-independent shape function derivatives
  return FE<3,LAGRANGE>::shape_deriv(elem->type(), static_cast<Order>(order + elem->p_level()), i, j, p);
}

shape_deriv() [108/109]

Real libMesh::FE< 2, CLOUGH >::shape_deriv ( const ElemType  , const Order  , const unsigned  int, const unsigned  int, const Point &    )

inherited

Definition at line 1994 of file fe_clough_shape_2D.C.

{
  libmesh_error_msg("Clough-Tocher elements require the real element \nto construct gradient-based degrees of freedom.");
  return 0.;
}

shape_deriv() [109/109]

Real libMesh::FE< 2, CLOUGH >::shape_deriv ( const Elem *  elem, const Order  order, const unsigned int  i, const unsigned int  j, const Point &  p  )

inherited

Definition at line 2007 of file fe_clough_shape_2D.C.

{
  libmesh_assert(elem);

  clough_compute_coefs(elem);

  const ElemType type = elem->type();

  const Order totalorder = static_cast<Order>(order + elem->p_level());

  switch (totalorder)
    {
      // 2nd-order restricted Clough-Tocher element
    case SECOND:
      {
        // There may be a bug in the 2nd order case; the 3rd order
        // Clough-Tocher elements are pretty uniformly better anyways
        // so use those instead.
        libmesh_experimental();
        switch (type)
          {
            // C1 functions on the Clough-Tocher triangle.
          case TRI6:
            {
              libmesh_assert_less (i, 9);
              // FIXME: it would be nice to calculate (and cache)
              // clough_raw_shape(j,p) only once per triangle, not 1-7
              // times
              switch (i)
                {
                  // Note: these DoF numbers are "scrambled" because my
                  // initial numbering conventions didn't match libMesh
                case 0:
                  return clough_raw_shape_deriv(0, j, p)
                    + d1d2n * clough_raw_shape_deriv(10, j, p)
                    + d1d3n * clough_raw_shape_deriv(11, j, p);
                case 3:
                  return clough_raw_shape_deriv(1, j, p)
                    + d2d3n * clough_raw_shape_deriv(11, j, p)
                    + d2d1n * clough_raw_shape_deriv(9, j, p);
                case 6:
                  return clough_raw_shape_deriv(2, j, p)
                    + d3d1n * clough_raw_shape_deriv(9, j, p)
                    + d3d2n * clough_raw_shape_deriv(10, j, p);
                case 1:
                  return d1xd1x * clough_raw_shape_deriv(3, j, p)
                    + d1xd1y * clough_raw_shape_deriv(4, j, p)
                    + d1xd2n * clough_raw_shape_deriv(10, j, p)
                    + d1xd3n * clough_raw_shape_deriv(11, j, p)
                    + 0.5 * N01x * d3nd3n * clough_raw_shape_deriv(11, j, p)
                    + 0.5 * N02x * d2nd2n * clough_raw_shape_deriv(10, j, p);
                case 2:
                  return d1yd1y * clough_raw_shape_deriv(4, j, p)
                    + d1yd1x * clough_raw_shape_deriv(3, j, p)
                    + d1yd2n * clough_raw_shape_deriv(10, j, p)
                    + d1yd3n * clough_raw_shape_deriv(11, j, p)
                    + 0.5 * N01y * d3nd3n * clough_raw_shape_deriv(11, j, p)
                    + 0.5 * N02y * d2nd2n * clough_raw_shape_deriv(10, j, p);
                case 4:
                  return d2xd2x * clough_raw_shape_deriv(5, j, p)
                    + d2xd2y * clough_raw_shape_deriv(6, j, p)
                    + d2xd3n * clough_raw_shape_deriv(11, j, p)
                    + d2xd1n * clough_raw_shape_deriv(9, j, p)
                    + 0.5 * N10x * d3nd3n * clough_raw_shape_deriv(11, j, p)
                    + 0.5 * N12x * d1nd1n * clough_raw_shape_deriv(9, j, p);
                case 5:
                  return d2yd2y * clough_raw_shape_deriv(6, j, p)
                    + d2yd2x * clough_raw_shape_deriv(5, j, p)
                    + d2yd3n * clough_raw_shape_deriv(11, j, p)
                    + d2yd1n * clough_raw_shape_deriv(9, j, p)
                    + 0.5 * N10y * d3nd3n * clough_raw_shape_deriv(11, j, p)
                    + 0.5 * N12y * d1nd1n * clough_raw_shape_deriv(9, j, p);
                case 7:
                  return d3xd3x * clough_raw_shape_deriv(7, j, p)
                    + d3xd3y * clough_raw_shape_deriv(8, j, p)
                    + d3xd1n * clough_raw_shape_deriv(9, j, p)
                    + d3xd2n * clough_raw_shape_deriv(10, j, p)
                    + 0.5 * N20x * d2nd2n * clough_raw_shape_deriv(10, j, p)
                    + 0.5 * N21x * d1nd1n * clough_raw_shape_deriv(9, j, p);
                case 8:
                  return d3yd3y * clough_raw_shape_deriv(8, j, p)
                    + d3yd3x * clough_raw_shape_deriv(7, j, p)
                    + d3yd1n * clough_raw_shape_deriv(9, j, p)
                    + d3yd2n * clough_raw_shape_deriv(10, j, p)
                    + 0.5 * N20y * d2nd2n * clough_raw_shape_deriv(10, j, p)
                    + 0.5 * N21y * d1nd1n * clough_raw_shape_deriv(9, j, p);
                default:
                  libmesh_error_msg("Invalid shape function index i = " << i);
                }
            }
          default:
            libmesh_error_msg("ERROR: Unsupported element type = " << type);
          }
      }
      // 3rd-order Clough-Tocher element
    case THIRD:
      {
        switch (type)
          {
            // C1 functions on the Clough-Tocher triangle.
          case TRI6:
            {
              libmesh_assert_less (i, 12);

              // FIXME: it would be nice to calculate (and cache)
              // clough_raw_shape(j,p) only once per triangle, not 1-7
              // times
              switch (i)
                {
                  // Note: these DoF numbers are "scrambled" because my
                  // initial numbering conventions didn't match libMesh
                case 0:
                  return clough_raw_shape_deriv(0, j, p)
                    + d1d2n * clough_raw_shape_deriv(10, j, p)
                    + d1d3n * clough_raw_shape_deriv(11, j, p);
                case 3:
                  return clough_raw_shape_deriv(1, j, p)
                    + d2d3n * clough_raw_shape_deriv(11, j, p)
                    + d2d1n * clough_raw_shape_deriv(9, j, p);
                case 6:
                  return clough_raw_shape_deriv(2, j, p)
                    + d3d1n * clough_raw_shape_deriv(9, j, p)
                    + d3d2n * clough_raw_shape_deriv(10, j, p);
                case 1:
                  return d1xd1x * clough_raw_shape_deriv(3, j, p)
                    + d1xd1y * clough_raw_shape_deriv(4, j, p)
                    + d1xd2n * clough_raw_shape_deriv(10, j, p)
                    + d1xd3n * clough_raw_shape_deriv(11, j, p);
                case 2:
                  return d1yd1y * clough_raw_shape_deriv(4, j, p)
                    + d1yd1x * clough_raw_shape_deriv(3, j, p)
                    + d1yd2n * clough_raw_shape_deriv(10, j, p)
                    + d1yd3n * clough_raw_shape_deriv(11, j, p);
                case 4:
                  return d2xd2x * clough_raw_shape_deriv(5, j, p)
                    + d2xd2y * clough_raw_shape_deriv(6, j, p)
                    + d2xd3n * clough_raw_shape_deriv(11, j, p)
                    + d2xd1n * clough_raw_shape_deriv(9, j, p);
                case 5:
                  return d2yd2y * clough_raw_shape_deriv(6, j, p)
                    + d2yd2x * clough_raw_shape_deriv(5, j, p)
                    + d2yd3n * clough_raw_shape_deriv(11, j, p)
                    + d2yd1n * clough_raw_shape_deriv(9, j, p);
                case 7:
                  return d3xd3x * clough_raw_shape_deriv(7, j, p)
                    + d3xd3y * clough_raw_shape_deriv(8, j, p)
                    + d3xd1n * clough_raw_shape_deriv(9, j, p)
                    + d3xd2n * clough_raw_shape_deriv(10, j, p);
                case 8:
                  return d3yd3y * clough_raw_shape_deriv(8, j, p)
                    + d3yd3x * clough_raw_shape_deriv(7, j, p)
                    + d3yd1n * clough_raw_shape_deriv(9, j, p)
                    + d3yd2n * clough_raw_shape_deriv(10, j, p);
                case 10:
                  return d1nd1n * clough_raw_shape_deriv(9, j, p);
                case 11:
                  return d2nd2n * clough_raw_shape_deriv(10, j, p);
                case 9:
                  return d3nd3n * clough_raw_shape_deriv(11, j, p);

                default:
                  libmesh_error_msg("Invalid shape function index i = " << i);
                }
            }
          default:
            libmesh_error_msg("ERROR: Unsupported element type = " << type);
          }
      }
      // by default throw an error
    default:
      libmesh_error_msg("ERROR: Unsupported polynomial order = " << order);
    }
}

shape_second_deriv() [1/103]

Real libMesh::FE< 3, SCALAR >::shape_second_deriv ( const ElemType  , const Order  , const unsigned  int, const unsigned  int, const Point &    )

inherited

Definition at line 67 of file fe_scalar_shape_3D.C.

{
  return 0.;
}

shape_second_deriv() [2/103]

Real libMesh::FE< 1, SCALAR >::shape_second_deriv ( const ElemType  , const Order  , const unsigned  int, const unsigned  int, const Point &    )

inherited

Definition at line 67 of file fe_scalar_shape_1D.C.

{
  return 0.;
}

shape_second_deriv() [3/103]

Real libMesh::FE< 0, SCALAR >::shape_second_deriv ( const ElemType  , const Order  , const unsigned  int, const unsigned  int, const Point &    )

inherited

Definition at line 67 of file fe_scalar_shape_0D.C.

{
  return 0.;
}

shape_second_deriv() [4/103]

Real libMesh::FE< 2, SCALAR >::shape_second_deriv ( const ElemType  , const Order  , const unsigned  int, const unsigned  int, const Point &    )

inherited

Definition at line 69 of file fe_scalar_shape_2D.C.

{
  return 0.;
}

shape_second_deriv() [5/103]

Real libMesh::FE< 1, SCALAR >::shape_second_deriv ( const Elem *  , const Order  , const unsigned  int, const unsigned  int, const Point &    )

inherited

Definition at line 77 of file fe_scalar_shape_1D.C.

{
  return 0.;
}

shape_second_deriv() [6/103]

Real libMesh::FE< 0, SCALAR >::shape_second_deriv ( const Elem *  , const Order  , const unsigned  int, const unsigned  int, const Point &    )

inherited

Definition at line 77 of file fe_scalar_shape_0D.C.

{
  return 0.;
}

shape_second_deriv() [7/103]

Real libMesh::FE< 3, SCALAR >::shape_second_deriv ( const Elem *  , const Order  , const unsigned  int, const unsigned  int, const Point &    )

inherited

Definition at line 77 of file fe_scalar_shape_3D.C.

{
  return 0.;
}

shape_second_deriv() [8/103]

Real libMesh::FE< 2, SCALAR >::shape_second_deriv ( const Elem *  , const Order  , const unsigned  int, const unsigned  int, const Point &    )

inherited

Definition at line 79 of file fe_scalar_shape_2D.C.

{
  return 0.;
}

shape_second_deriv() [9/103]

Real libMesh::FE< 3, SZABAB >::shape_second_deriv ( const ElemType  , const Order  , const unsigned  int, const unsigned  int, const Point &    )

inherited

Definition at line 81 of file fe_szabab_shape_3D.C.

{
  libmesh_error_msg("Szabo-Babuska polynomials are not defined in 3D");
  return 0.;
}

shape_second_deriv() [10/103]

Real libMesh::FE< 0, BERNSTEIN >::shape_second_deriv ( const ElemType  , const Order  , const unsigned  int, const unsigned  int, const Point &    )

inherited

Definition at line 83 of file fe_bernstein_shape_0D.C.

{
  libmesh_error_msg("No spatial derivatives in 0D!");
  return 0.;
}

shape_second_deriv() [11/103]

Real libMesh::FE< 0, LAGRANGE >::shape_second_deriv ( const ElemType  , const Order  , const unsigned  int, const unsigned  int, const Point &    )

inherited

Definition at line 83 of file fe_lagrange_shape_0D.C.

{
  libmesh_error_msg("No spatial derivatives in 0D!");
  return 0.;
}

shape_second_deriv() [12/103]

Real libMesh::FE< 0, SZABAB >::shape_second_deriv ( const ElemType  , const Order  , const unsigned  int, const unsigned  int, const Point &    )

inherited

Definition at line 83 of file fe_szabab_shape_0D.C.

{
  libmesh_error_msg("No spatial derivatives in 0D!");
  return 0.;
}

shape_second_deriv() [13/103]

Real libMesh::FE< 0, XYZ >::shape_second_deriv ( const ElemType  , const Order  , const unsigned  int, const unsigned  int, const Point &    )

inherited

Definition at line 83 of file fe_xyz_shape_0D.C.

{
  libmesh_error_msg("No spatial derivatives in 0D!");
  return 0.;
}

shape_second_deriv() [14/103]

Real libMesh::FE< 0, MONOMIAL >::shape_second_deriv ( const ElemType  , const Order  , const unsigned  int, const unsigned  int, const Point &    )

inherited

Definition at line 83 of file fe_monomial_shape_0D.C.

{
  libmesh_error_msg("No spatial derivatives in 0D!");
  return 0.;
}

shape_second_deriv() [15/103]

Real libMesh::FE< 0, L2_HIERARCHIC >::shape_second_deriv ( const ElemType  , const Order  , const unsigned  int, const unsigned  int, const Point &    )

inherited

Definition at line 83 of file fe_l2_hierarchic_shape_0D.C.

{
  libmesh_error_msg("No spatial derivatives in 0D!");
  return 0.;
}

shape_second_deriv() [16/103]

Real libMesh::FE< 0, CLOUGH >::shape_second_deriv ( const ElemType  , const Order  , const unsigned  int, const unsigned  int, const Point &    )

inherited

Definition at line 83 of file fe_clough_shape_0D.C.

{
  libmesh_error_msg("No spatial derivatives in 0D!");
  return 0.;
}

shape_second_deriv() [17/103]

Real libMesh::FE< 0, L2_LAGRANGE >::shape_second_deriv ( const ElemType  , const Order  , const unsigned  int, const unsigned  int, const Point &    )

inherited

Definition at line 83 of file fe_l2_lagrange_shape_0D.C.

{
  libmesh_error_msg("No spatial derivatives in 0D!");
  return 0.;
}

shape_second_deriv() [18/103]

Real libMesh::FE< 0, HERMITE >::shape_second_deriv ( const ElemType  , const Order  , const unsigned  int, const unsigned  int, const Point &    )

inherited

Definition at line 83 of file fe_hermite_shape_0D.C.

{
  libmesh_error_msg("No spatial derivatives in 0D!");
  return 0.;
}

shape_second_deriv() [19/103]

Real libMesh::FE< 0, HIERARCHIC >::shape_second_deriv ( const ElemType  , const Order  , const unsigned  int, const unsigned  int, const Point &    )

inherited

Definition at line 83 of file fe_hierarchic_shape_0D.C.

{
  libmesh_error_msg("No spatial derivatives in 0D!");
  return 0.;
}

shape_second_deriv() [20/103]

Real libMesh::FE< 3, CLOUGH >::shape_second_deriv ( const Elem *  libmesh_dbg_varelem, const Order  , const unsigned  int, const unsigned  int, const Point &    )

inherited

Definition at line 86 of file fe_clough_shape_3D.C.

{
  libmesh_assert(elem);
  libmesh_not_implemented();
  return 0.;
}

shape_second_deriv() [21/103]

Real libMesh::FE< 3, SZABAB >::shape_second_deriv ( const Elem *  , const Order  , const unsigned  int, const unsigned  int, const Point &    )

inherited

Definition at line 94 of file fe_szabab_shape_3D.C.

{
  libmesh_error_msg("Szabo-Babuska polynomials are not defined in 3D");
  return 0.;
}

shape_second_deriv() [22/103]

Real libMesh::FE< 0, L2_LAGRANGE >::shape_second_deriv ( const Elem *  , const Order  , const unsigned  int, const unsigned  int, const Point &    )

inherited

Definition at line 96 of file fe_l2_lagrange_shape_0D.C.

{
  libmesh_error_msg("No spatial derivatives in 0D!");
  return 0.;
}

shape_second_deriv() [23/103]

Real libMesh::FE< 0, HIERARCHIC >::shape_second_deriv ( const Elem *  , const Order  , const unsigned  int, const unsigned  int, const Point &    )

inherited

Definition at line 96 of file fe_hierarchic_shape_0D.C.

{
  libmesh_error_msg("No spatial derivatives in 0D!");
  return 0.;
}

shape_second_deriv() [24/103]

Real libMesh::FE< 0, MONOMIAL >::shape_second_deriv ( const Elem *  , const Order  , const unsigned  int, const unsigned  int, const Point &    )

inherited

Definition at line 96 of file fe_monomial_shape_0D.C.

{
  libmesh_error_msg("No spatial derivatives in 0D!");
  return 0.;
}

shape_second_deriv() [25/103]

Real libMesh::FE< 0, XYZ >::shape_second_deriv ( const Elem *  , const Order  , const unsigned  int, const unsigned  int, const Point &    )

inherited

Definition at line 96 of file fe_xyz_shape_0D.C.

{
  libmesh_error_msg("No spatial derivatives in 0D!");
  return 0.;
}

shape_second_deriv() [26/103]

Real libMesh::FE< 0, SZABAB >::shape_second_deriv ( const Elem *  , const Order  , const unsigned  int, const unsigned  int, const Point &    )

inherited

Definition at line 96 of file fe_szabab_shape_0D.C.

{
  libmesh_error_msg("No spatial derivatives in 0D!");
  return 0.;
}

shape_second_deriv() [27/103]

Real libMesh::FE< 0, LAGRANGE >::shape_second_deriv ( const Elem *  , const Order  , const unsigned  int, const unsigned  int, const Point &    )

inherited

Definition at line 96 of file fe_lagrange_shape_0D.C.

{
  libmesh_error_msg("No spatial derivatives in 0D!");
  return 0.;
}

shape_second_deriv() [28/103]

Real libMesh::FE< 0, BERNSTEIN >::shape_second_deriv ( const Elem *  , const Order  , const unsigned  int, const unsigned  int, const Point &    )

inherited

Definition at line 96 of file fe_bernstein_shape_0D.C.

{
  libmesh_error_msg("No spatial derivatives in 0D!");
  return 0.;
}

shape_second_deriv() [29/103]

Real libMesh::FE< 0, L2_HIERARCHIC >::shape_second_deriv ( const Elem *  , const Order  , const unsigned  int, const unsigned  int, const Point &    )

inherited

Definition at line 96 of file fe_l2_hierarchic_shape_0D.C.

{
  libmesh_error_msg("No spatial derivatives in 0D!");
  return 0.;
}

shape_second_deriv() [30/103]

Real libMesh::FE< 0, CLOUGH >::shape_second_deriv ( const Elem *  , const Order  , const unsigned  int, const unsigned  int, const Point &    )

inherited

Definition at line 96 of file fe_clough_shape_0D.C.

{
  libmesh_error_msg("No spatial derivatives in 0D!");
  return 0.;
}

shape_second_deriv() [31/103]

Real libMesh::FE< 0, HERMITE >::shape_second_deriv ( const Elem *  , const Order  , const unsigned  int, const unsigned  int, const Point &    )

inherited

Definition at line 96 of file fe_hermite_shape_0D.C.

{
  libmesh_error_msg("No spatial derivatives in 0D!");
  return 0.;
}

shape_second_deriv() [32/103]

Real libMesh::FE< 1, MONOMIAL >::shape_second_deriv ( const ElemType  , const Order  libmesh_dbg_varorder, const unsigned int  i, const unsigned int  libmesh_dbg_varj, const Point &  p  )

inherited

Definition at line 141 of file fe_monomial_shape_1D.C.

{
  // only d()/dxi in 1D!

  libmesh_assert_equal_to (j, 0);

  const Real xi = p(0);

  libmesh_assert_less_equal (i, static_cast<unsigned int>(order));

  switch (i)
    {
    case 0:
    case 1:
      return 0.;

    case 2:
      return 2.;

    case 3:
      return 6.*xi;

    case 4:
      return 12.*xi*xi;

    default:
      Real val = 2.;
      for (unsigned int index = 2; index != i; ++index)
        val *= (index+1) * xi;
      return val;
    }
}

shape_second_deriv() [33/103]

Real libMesh::FE< 1, SZABAB >::shape_second_deriv ( const ElemType  , const Order  , const unsigned  int, const unsigned  int, const Point &    )

inherited

Definition at line 153 of file fe_szabab_shape_1D.C.

{
  static bool warning_given = false;

  if (!warning_given)
    libMesh::err << "Second derivatives for Szabab elements "
                 << " are not yet implemented!"
                 << std::endl;

  warning_given = true;
  return 0.;
}

shape_second_deriv() [34/103]

Real libMesh::FE< 1, XYZ >::shape_second_deriv ( const ElemType  , const Order  , const unsigned  int, const unsigned  int, const Point &    )

inherited

Definition at line 160 of file fe_xyz_shape_1D.C.

{
  libmesh_error_msg("XYZ polynomials require the element \nbecause the centroid is needed.");
  return 0.;
}

shape_second_deriv() [35/103]

Real libMesh::FE< 1, XYZ >::shape_second_deriv ( const Elem *  elem, const Order  libmesh_dbg_varorder, const unsigned int  i, const unsigned int  libmesh_dbg_varj, const Point &  point_in  )

inherited

Definition at line 173 of file fe_xyz_shape_1D.C.

{
  libmesh_assert(elem);
  libmesh_assert_less_equal (i, order + elem->p_level());

  // only d2()/dxi2 in 1D!

  libmesh_assert_equal_to (j, 0);

  Point centroid = elem->centroid();
  Real max_distance = 0.;
  for (unsigned int p = 0; p < elem->n_nodes(); p++)
    {
      const Real distance = std::abs(centroid(0) - elem->point(p)(0));
      max_distance = std::max(distance, max_distance);
    }

  const Real x  = point_in(0);
  const Real xc = centroid(0);
  const Real dx = (x - xc)/max_distance;
  const Real dist2 = pow(max_distance,2.);

  // monomials. since they are hierarchic we only need one case block.
  switch (i)
    {
    case 0:
    case 1:
      return 0.;

    case 2:
      return 2./dist2;

    case 3:
      return 6.*dx/dist2;

    case 4:
      return 12.*dx*dx/dist2;

    default:
      Real val = 2.;
      for (unsigned int index = 2; index != i; ++index)
        val *= (index+1) * dx;
      return val/dist2;
    }
}

shape_second_deriv() [36/103]

Real libMesh::FE< 1, SZABAB >::shape_second_deriv ( const Elem *  , const Order  , const unsigned  int, const unsigned  int, const Point &    )

inherited

Definition at line 173 of file fe_szabab_shape_1D.C.

{
  static bool warning_given = false;

  if (!warning_given)
    libMesh::err << "Second derivatives for Szabab elements "
                 << " are not yet implemented!"
                 << std::endl;

  warning_given = true;
  return 0.;
}

shape_second_deriv() [37/103]

static OutputShape libMesh::FE< Dim, T >::shape_second_deriv ( const ElemType  t, const Order  o, const unsigned int  i, const unsigned int  j, const Point &  p  )

staticinherited

Returns

The second derivative of the shape function at the point p.

Note

Cross-derivatives are indexed according to: j = 0 ==> d^2 phi / dxi^2 j = 1 ==> d^2 phi / dxi deta j = 2 ==> d^2 phi / deta^2 j = 3 ==> d^2 phi / dxi dzeta j = 4 ==> d^2 phi / deta dzeta j = 5 ==> d^2 phi / dzeta^2

Computing second derivatives is not currently supported for all element types: (Clough, Hermite and Subdivision), Lagrange, Hierarchic, L2_Hierarchic, and Monomial are supported. All other element types return an error when asked for second derivatives.

On a p-refined element, o should be the total order of the element.

shape_second_deriv() [38/103]

Real libMesh::FE< 1, MONOMIAL >::shape_second_deriv ( const Elem *  elem, const Order  order, const unsigned int  i, const unsigned int  j, const Point &  p  )

inherited

Definition at line 181 of file fe_monomial_shape_1D.C.

{
  libmesh_assert(elem);

  return FE<1,MONOMIAL>::shape_second_deriv(elem->type(),
                                            static_cast<Order>(order + elem->p_level()), i, j, p);
}

shape_second_deriv() [39/103]

static OutputShape libMesh::FE< Dim, T >::shape_second_deriv ( const Elem *  elem, const Order  o, const unsigned int  i, const unsigned int  j, const Point &  p  )

staticinherited

Returns

The second derivative of the shape function at the point p.

Note

Cross-derivatives are indexed according to: j = 0 ==> d^2 phi / dxi^2 j = 1 ==> d^2 phi / dxi deta j = 2 ==> d^2 phi / deta^2 j = 3 ==> d^2 phi / dxi dzeta j = 4 ==> d^2 phi / deta dzeta j = 5 ==> d^2 phi / dzeta^2

Computing second derivatives is not currently supported for all element types: (Clough, Hermite and Subdivision), Lagrange, Hierarchic, L2_Hierarchic, and Monomial are supported. All other element types return an error when asked for second derivatives.

On a p-refined element, o should be the base order of the element.

shape_second_deriv() [40/103]

Real libMesh::FE< 1, L2_HIERARCHIC >::shape_second_deriv ( const ElemType  , const Order  libmesh_dbg_varorder, const unsigned int  i, const unsigned int  libmesh_dbg_varj, const Point &  p  )

inherited

Definition at line 200 of file fe_l2_hierarchic_shape_1D.C.

{
  // only d2()/d2xi in 1D!

  libmesh_assert_equal_to (j, 0);
  libmesh_assert_less (i, order+1u);

  // Declare that we are using our own special power function
  // from the Utility namespace.  This saves typing later.
  using Utility::pow;

  const Real xi = p(0);

  Real returnval = 1.;

  switch (i)
    {
    case 0:
    case 1:
      returnval = 0;
      break;
      // All terms have the same form.
      // xi^(p-2)/(p-2)!
    case 2:
      returnval = 1;
      break;
    case 3:
      returnval = xi;
      break;
    case 4:
      returnval = pow<2>(xi)/2.;
      break;
    case 5:
      returnval = pow<3>(xi)/6.;
      break;
    case 6:
      returnval = pow<4>(xi)/24.;
      break;
    case 7:
      returnval = pow<5>(xi)/120.;
      break;

    default:
      Real denominator = 1.;
      for (unsigned int n=1; n != i; ++n)
        {
          returnval *= xi;
          denominator *= n;
        }
      // Odd:
      if (i % 2)
        returnval = (i * returnval - 1.)/denominator/i;
      // Even:
      else
        returnval = returnval/denominator;
      break;
    }

  return returnval;
}

shape_second_deriv() [41/103]

Real libMesh::FE< 1, HIERARCHIC >::shape_second_deriv ( const ElemType  , const Order  libmesh_dbg_varorder, const unsigned int  i, const unsigned int  libmesh_dbg_varj, const Point &  p  )

inherited

Definition at line 200 of file fe_hierarchic_shape_1D.C.

{
  // only d2()/d2xi in 1D!

  libmesh_assert_equal_to (j, 0);
  libmesh_assert_less (i, order+1u);

  // Declare that we are using our own special power function
  // from the Utility namespace.  This saves typing later.
  using Utility::pow;

  const Real xi = p(0);

  Real returnval = 1.;

  switch (i)
    {
    case 0:
    case 1:
      returnval = 0;
      break;
      // All terms have the same form.
      // xi^(p-2)/(p-2)!
    case 2:
      returnval = 1;
      break;
    case 3:
      returnval = xi;
      break;
    case 4:
      returnval = pow<2>(xi)/2.;
      break;
    case 5:
      returnval = pow<3>(xi)/6.;
      break;
    case 6:
      returnval = pow<4>(xi)/24.;
      break;
    case 7:
      returnval = pow<5>(xi)/120.;
      break;

    default:
      Real denominator = 1.;
      for (unsigned int n=1; n != i; ++n)
        {
          returnval *= xi;
          denominator *= n;
        }
      // Odd:
      if (i % 2)
        returnval = (i * returnval - 1.)/denominator/i;
      // Even:
      else
        returnval = returnval/denominator;
      break;
    }

  return returnval;
}

shape_second_deriv() [42/103]

Real libMesh::FE< 1, L2_LAGRANGE >::shape_second_deriv ( const ElemType  , const Order  order, const unsigned int  i, const unsigned int  libmesh_dbg_varj, const Point &  p  )

inherited

Definition at line 235 of file fe_l2_lagrange_shape_1D.C.

{
  // Don't need to switch on j.  1D shape functions
  // depend on xi only!

  const Real xi = p(0);
  libmesh_assert_equal_to (j, 0);

  switch (order)
    {
      // linear Lagrange shape functions
    case FIRST:
      {
        // All second derivatives of linears are zero....
        return 0.;
      }

      // quadratic Lagrange shape functions
    case SECOND:
      {
        switch (i)
          {
          case 0:
            return 1.;

          case 1:
            return 1.;

          case 2:
            return -2.;

          default:
            libmesh_error_msg("Invalid shape function index i = " << i);
          }
      } // end case SECOND

    case THIRD:
      {
        switch (i)
          {
          case 0:
            return -9./16.*(6.*xi-2);

          case 1:
            return -9./16.*(-6*xi-2.);

          case 2:
            return 27./16.*(6*xi-2./3.);

          case 3:
            return 27./16.*(-6*xi-2./3.);

          default:
            libmesh_error_msg("Invalid shape function index i = " << i);
          }
      } // end case THIRD


    default:
      libmesh_error_msg("ERROR: Unsupported polynomial order = " << order);
    } // end switch (order)
}

shape_second_deriv() [43/103]

Real libMesh::FE< 1, LAGRANGE >::shape_second_deriv ( const ElemType  , const Order  order, const unsigned int  i, const unsigned int  libmesh_dbg_varj, const Point &  p  )

inherited

Definition at line 235 of file fe_lagrange_shape_1D.C.

{
  // Don't need to switch on j.  1D shape functions
  // depend on xi only!

  const Real xi = p(0);
  libmesh_assert_equal_to (j, 0);

  switch (order)
    {
      // linear Lagrange shape functions
    case FIRST:
      {
        // All second derivatives of linears are zero....
        return 0.;
      }

      // quadratic Lagrange shape functions
    case SECOND:
      {
        switch (i)
          {
          case 0:
            return 1.;

          case 1:
            return 1.;

          case 2:
            return -2.;

          default:
            libmesh_error_msg("Invalid shape function index i = " << i);
          }
      } // end case SECOND

    case THIRD:
      {
        switch (i)
          {
          case 0:
            return -9./16.*(6.*xi-2);

          case 1:
            return -9./16.*(-6*xi-2.);

          case 2:
            return 27./16.*(6*xi-2./3.);

          case 3:
            return 27./16.*(-6*xi-2./3.);

          default:
            libmesh_error_msg("Invalid shape function index i = " << i);
          }
      } // end case THIRD


    default:
      libmesh_error_msg("ERROR: Unsupported polynomial order = " << order);
    } // end switch (order)
}

shape_second_deriv() [44/103]

Real libMesh::FE< 1, HIERARCHIC >::shape_second_deriv ( const Elem *  elem, const Order  order, const unsigned int  i, const unsigned int  j, const Point &  p  )

inherited

Definition at line 268 of file fe_hierarchic_shape_1D.C.

{
  libmesh_assert(elem);

  return FE<1,HIERARCHIC>::shape_second_deriv(elem->type(),
                                              static_cast<Order>(order + elem->p_level()), i, j, p);
}

shape_second_deriv() [45/103]

Real libMesh::FE< 1, L2_HIERARCHIC >::shape_second_deriv ( const Elem *  elem, const Order  order, const unsigned int  i, const unsigned int  j, const Point &  p  )

inherited

Definition at line 268 of file fe_l2_hierarchic_shape_1D.C.

{
  libmesh_assert(elem);

  return FE<1,L2_HIERARCHIC>::shape_second_deriv(elem->type(),
                                                 static_cast<Order>(order + elem->p_level()), i, j, p);
}

shape_second_deriv() [46/103]

Real libMesh::FE< 1, HERMITE >::shape_second_deriv ( const Elem *  elem, const Order  libmesh_dbg_varorder, const unsigned int  i, const unsigned  int, const Point &  p  )

inherited

Definition at line 301 of file fe_hermite_shape_1D.C.

{
  libmesh_assert(elem);

  // Coefficient naming: d(1)d(2n) is the coefficient of the
  // global shape function corresponding to value 1 in terms of the
  // local shape function corresponding to normal derivative 2
  Real d1xd1x, d2xd2x;

  hermite_compute_coefs(elem, d1xd1x, d2xd2x);

  const ElemType type = elem->type();

#ifndef NDEBUG
  const unsigned int totalorder = order + elem->p_level();
#endif

  switch (type)
    {
      // C1 functions on the C1 cubic edge
    case EDGE2:
    case EDGE3:
      {
        libmesh_assert_less (i, totalorder+1);

        switch (i)
          {
          case 0:
            return FEHermite<1>::hermite_raw_shape_second_deriv(0, p(0));
          case 1:
            return d1xd1x * FEHermite<1>::hermite_raw_shape_second_deriv(2, p(0));
          case 2:
            return FEHermite<1>::hermite_raw_shape_second_deriv(1, p(0));
          case 3:
            return d2xd2x * FEHermite<1>::hermite_raw_shape_second_deriv(3, p(0));
          default:
            return FEHermite<1>::hermite_raw_shape_second_deriv(i, p(0));
          }
      }
    default:
      libmesh_error_msg("ERROR: Unsupported element type = " << type);
    }
}

shape_second_deriv() [47/103]

Real libMesh::FE< 1, LAGRANGE >::shape_second_deriv ( const Elem *  elem, const Order  order, const unsigned int  i, const unsigned int  j, const Point &  p  )

inherited

Definition at line 305 of file fe_lagrange_shape_1D.C.

{
  libmesh_assert(elem);

  return FE<1,LAGRANGE>::shape_second_deriv(elem->type(),
                                            static_cast<Order>(order + elem->p_level()), i, j, p);
}

shape_second_deriv() [48/103]

Real libMesh::FE< 1, L2_LAGRANGE >::shape_second_deriv ( const Elem *  elem, const Order  order, const unsigned int  i, const unsigned int  j, const Point &  p  )

inherited

Definition at line 305 of file fe_l2_lagrange_shape_1D.C.

{
  libmesh_assert(elem);

  return FE<1,L2_LAGRANGE>::shape_second_deriv(elem->type(),
                                               static_cast<Order>(order + elem->p_level()), i, j, p);
}

shape_second_deriv() [49/103]

Real libMesh::FE< 2, MONOMIAL >::shape_second_deriv ( const ElemType  , const Order  libmesh_dbg_varorder, const unsigned int  i, const unsigned int  j, const Point &  p  )

inherited

Definition at line 319 of file fe_monomial_shape_2D.C.

{
#if LIBMESH_DIM > 1


  libmesh_assert_less_equal (j, 2);

  libmesh_assert_less (i, (static_cast<unsigned int>(order)+1)*
                       (static_cast<unsigned int>(order)+2)/2);

  const Real xi  = p(0);
  const Real eta = p(1);

  // monomials. since they are hierarchic we only need one case block.

  switch (j)
    {
      // d^2()/dxi^2
    case 0:
      {
        switch (i)
          {
            // constants
          case 0:
            // linears
          case 1:
          case 2:
            return 0.;

            // quadratics
          case 3:
            return 2.;

          case 4:
          case 5:
            return 0.;

            // cubics
          case 6:
            return 6.*xi;

          case 7:
            return 2.*eta;

          case 8:
          case 9:
            return 0.;

            // quartics
          case 10:
            return 12.*xi*xi;

          case 11:
            return 6.*xi*eta;

          case 12:
            return 2.*eta*eta;

          case 13:
          case 14:
            return 0.;

          default:
            unsigned int o = 0;
            for (; i >= (o+1)*(o+2)/2; o++) { }
            unsigned int ny = i - (o*(o+1)/2);
            unsigned int nx = o - ny;
            Real val = nx * (nx - 1);
            for (unsigned int index=2; index < nx; index++)
              val *= xi;
            for (unsigned int index=0; index != ny; index++)
              val *= eta;
            return val;
          }
      }

      // d^2()/dxideta
    case 1:
      {
        switch (i)
          {
            // constants
          case 0:

            // linears
          case 1:
          case 2:
            return 0.;

            // quadratics
          case 3:
            return 0.;

          case 4:
            return 1.;

          case 5:
            return 0.;

            // cubics
          case 6:
            return 0.;
          case 7:
            return 2.*xi;

          case 8:
            return 2.*eta;

          case 9:
            return 0.;

            // quartics
          case 10:
            return 0.;

          case 11:
            return 3.*xi*xi;

          case 12:
            return 4.*xi*eta;

          case 13:
            return 3.*eta*eta;

          case 14:
            return 0.;

          default:
            unsigned int o = 0;
            for (; i >= (o+1)*(o+2)/2; o++) { }
            unsigned int ny = i - (o*(o+1)/2);
            unsigned int nx = o - ny;
            Real val = nx * ny;
            for (unsigned int index=1; index < nx; index++)
              val *= xi;
            for (unsigned int index=1; index < ny; index++)
              val *= eta;
            return val;
          }
      }

      // d^2()/deta^2
    case 2:
      {
        switch (i)
          {
            // constants
          case 0:

            // linears
          case 1:
          case 2:
            return 0.;

            // quadratics
          case 3:
          case 4:
            return 0.;

          case 5:
            return 2.;

            // cubics
          case 6:
            return 0.;

          case 7:
            return 0.;

          case 8:
            return 2.*xi;

          case 9:
            return 6.*eta;

            // quartics
          case 10:
          case 11:
            return 0.;

          case 12:
            return 2.*xi*xi;

          case 13:
            return 6.*xi*eta;

          case 14:
            return 12.*eta*eta;

          default:
            unsigned int o = 0;
            for (; i >= (o+1)*(o+2)/2; o++) { }
            unsigned int ny = i - (o*(o+1)/2);
            unsigned int nx = o - ny;
            Real val = ny * (ny - 1);
            for (unsigned int index=0; index != nx; index++)
              val *= xi;
            for (unsigned int index=2; index < ny; index++)
              val *= eta;
            return val;
          }
      }

    default:
      libmesh_error_msg("Invalid shape function derivative j = " << j);
    }

#else
  return 0.;
#endif
}

shape_second_deriv() [50/103]

Real libMesh::FE< 2, HERMITE >::shape_second_deriv ( const Elem *  elem, const Order  order, const unsigned int  i, const unsigned int  j, const Point &  p  )

inherited

Definition at line 327 of file fe_hermite_shape_2D.C.

{
  libmesh_assert(elem);
  libmesh_assert (j == 0 || j == 1 || j == 2);

  std::vector<std::vector<Real>> dxdxi(2, std::vector<Real>(2, 0));

#ifdef DEBUG
  std::vector<Real> dxdeta(2), dydxi(2);
#endif

  hermite_compute_coefs(elem,dxdxi
#ifdef DEBUG
                        ,dxdeta,dydxi
#endif
                        );

  const ElemType type = elem->type();

  const Order totalorder = static_cast<Order>(order + elem->p_level());

  switch (type)
    {
    case QUAD4:
    case QUADSHELL4:
      libmesh_assert_less (totalorder, 4);
      libmesh_fallthrough();
    case QUAD8:
    case QUADSHELL8:
    case QUAD9:
      {
        libmesh_assert_less (i, (totalorder+1u)*(totalorder+1u));

        std::vector<unsigned int> bases1D;

        Real coef = hermite_bases_2D(bases1D, dxdxi, totalorder, i);

        switch (j)
          {
          case 0:
            return coef *
              FEHermite<1>::hermite_raw_shape_second_deriv(bases1D[0],p(0)) *
              FEHermite<1>::hermite_raw_shape(bases1D[1],p(1));
          case 1:
            return coef *
              FEHermite<1>::hermite_raw_shape_deriv(bases1D[0],p(0)) *
              FEHermite<1>::hermite_raw_shape_deriv(bases1D[1],p(1));
          case 2:
            return coef *
              FEHermite<1>::hermite_raw_shape(bases1D[0],p(0)) *
              FEHermite<1>::hermite_raw_shape_second_deriv(bases1D[1],p(1));
          default:
            libmesh_error_msg("Invalid derivative index j = " << j);
          }
      }
    default:
      libmesh_error_msg("ERROR: Unsupported element type = " << type);
    }
}

shape_second_deriv() [51/103]

RealGradient libMesh::FE< 2, NEDELEC_ONE >::shape_second_deriv ( const ElemType  , const Order  , const unsigned  int, const unsigned  int, const Point &    )

inherited

Definition at line 330 of file fe_nedelec_one_shape_2D.C.

{
  libmesh_error_msg("Nedelec elements require the element type \nbecause edge orientation is needed.");
  return RealGradient();
}

shape_second_deriv() [52/103]

Real libMesh::FE< 1, CLOUGH >::shape_second_deriv ( const Elem *  elem, const Order  order, const unsigned int  i, const unsigned int  j, const Point &  p  )

inherited

Definition at line 341 of file fe_clough_shape_1D.C.

{
  libmesh_assert(elem);

  clough_compute_coefs(elem);

  const ElemType type = elem->type();

  const Order totalorder = static_cast<Order>(order + elem->p_level());

  switch (totalorder)
    {
      // 3rd-order C1 cubic element
    case THIRD:
      {
        switch (type)
          {
            // C1 functions on the C1 cubic edge
          case EDGE2:
          case EDGE3:
            {
              switch (i)
                {
                case 0:
                  return clough_raw_shape_second_deriv(0, j, p);
                case 1:
                  return clough_raw_shape_second_deriv(1, j, p);
                case 2:
                  return d1xd1x * clough_raw_shape_second_deriv(2, j, p);
                case 3:
                  return d2xd2x * clough_raw_shape_second_deriv(3, j, p);
                default:
                  libmesh_error_msg("Invalid shape function index i = " << i);
                }
            }
          default:
            libmesh_error_msg("ERROR: Unsupported element type = " << type);
          }
      }
      // by default throw an error
    default:
      libmesh_error_msg("ERROR: Unsupported polynomial order = " << totalorder);
    }
}

shape_second_deriv() [53/103]

RealGradient libMesh::FE< 2, NEDELEC_ONE >::shape_second_deriv ( const Elem *  elem, const Order  order, const unsigned int  libmesh_dbg_vari, const unsigned int  libmesh_dbg_varj, const Point &    )

inherited

Definition at line 343 of file fe_nedelec_one_shape_2D.C.

{
#if LIBMESH_DIM > 1
  libmesh_assert(elem);

  // j = 0 ==> d^2 phi / dxi^2
  // j = 1 ==> d^2 phi / dxi deta
  // j = 2 ==> d^2 phi / deta^2
  libmesh_assert_less (j, 3);

  const Order total_order = static_cast<Order>(order + elem->p_level());

  switch (total_order)
    {
      // linear Lagrange shape functions
    case FIRST:
      {
        switch (elem->type())
          {
          case QUAD8:
          case QUAD9:
            {
              libmesh_assert_less (i, 4);
              // All second derivatives for linear quads are zero.
              return RealGradient();
            }

          case TRI6:
            {
              libmesh_assert_less (i, 3);
              // All second derivatives for linear triangles are zero.
              return RealGradient();
            }

          default:
            libmesh_error_msg("ERROR: Unsupported 2D element type!: " << elem->type());

          } // end switch (type)
      } // end case FIRST

      // unsupported order
    default:
      libmesh_error_msg("ERROR: Unsupported 2D FE order!: " << total_order);

    } // end switch (order)

#else // LIBMESH_DIM > 1
  return RealGradient();
#endif
}

shape_second_deriv() [54/103]

Real libMesh::FE< 2, XYZ >::shape_second_deriv ( const ElemType  , const Order  , const unsigned  int, const unsigned  int, const Point &    )

inherited

Definition at line 356 of file fe_xyz_shape_2D.C.

{
  libmesh_error_msg("XYZ polynomials require the element \nbecause the centroid is needed.");
  return 0.;
}

shape_second_deriv() [55/103]

Real libMesh::FE< 2, XYZ >::shape_second_deriv ( const Elem *  elem, const Order  libmesh_dbg_varorder, const unsigned int  i, const unsigned int  j, const Point &  point_in  )

inherited

Definition at line 369 of file fe_xyz_shape_2D.C.

{
#if LIBMESH_DIM > 1

  libmesh_assert_less_equal (j, 2);
  libmesh_assert(elem);

  Point centroid = elem->centroid();
  Point max_distance = Point(0.,0.,0.);
  for (unsigned int p = 0; p < elem->n_nodes(); p++)
    for (unsigned int d = 0; d < 2; d++)
      {
        const Real distance = std::abs(centroid(d) - elem->point(p)(d));
        max_distance(d) = std::max(distance, max_distance(d));
      }

  const Real x  = point_in(0);
  const Real y  = point_in(1);
  const Real xc = centroid(0);
  const Real yc = centroid(1);
  const Real distx = max_distance(0);
  const Real disty = max_distance(1);
  const Real dx = (x - xc)/distx;
  const Real dy = (y - yc)/disty;
  const Real dist2x = pow(distx,2.);
  const Real dist2y = pow(disty,2.);
  const Real distxy = distx * disty;

#ifndef NDEBUG
  // totalorder is only used in the assertion below, so
  // we avoid declaring it when asserts are not active.
  const unsigned int totalorder = order + elem->p_level();
#endif
  libmesh_assert_less (i, (totalorder+1)*(totalorder+2)/2);

  // monomials. since they are hierarchic we only need one case block.

  switch (j)
    {
      // d^2()/dx^2
    case 0:
      {
        switch (i)
          {
            // constants
          case 0:
            // linears
          case 1:
          case 2:
            return 0.;

            // quadratics
          case 3:
            return 2./dist2x;

          case 4:
          case 5:
            return 0.;

            // cubics
          case 6:
            return 6.*dx/dist2x;

          case 7:
            return 2.*dy/dist2x;

          case 8:
          case 9:
            return 0.;

            // quartics
          case 10:
            return 12.*dx*dx/dist2x;

          case 11:
            return 6.*dx*dy/dist2x;

          case 12:
            return 2.*dy*dy/dist2x;

          case 13:
          case 14:
            return 0.;

          default:
            unsigned int o = 0;
            for (; i >= (o+1)*(o+2)/2; o++) { }
            unsigned int i2 = i - (o*(o+1)/2);
            Real val = (o - i2) * (o - i2 - 1);
            for (unsigned int index=i2+2; index < o; index++)
              val *= dx;
            for (unsigned int index=0; index != i2; index++)
              val *= dy;
            return val/dist2x;
          }
      }

      // d^2()/dxdy
    case 1:
      {
        switch (i)
          {
            // constants
          case 0:

            // linears
          case 1:
          case 2:
            return 0.;

            // quadratics
          case 3:
            return 0.;

          case 4:
            return 1./distxy;

          case 5:
            return 0.;

            // cubics
          case 6:
            return 0.;
          case 7:
            return 2.*dx/distxy;

          case 8:
            return 2.*dy/distxy;

          case 9:
            return 0.;

            // quartics
          case 10:
            return 0.;

          case 11:
            return 3.*dx*dx/distxy;

          case 12:
            return 4.*dx*dy/distxy;

          case 13:
            return 3.*dy*dy/distxy;

          case 14:
            return 0.;

          default:
            unsigned int o = 0;
            for (; i >= (o+1)*(o+2)/2; o++) { }
            unsigned int i2 = i - (o*(o+1)/2);
            Real val = (o - i2) * i2;
            for (unsigned int index=i2+1; index < o; index++)
              val *= dx;
            for (unsigned int index=1; index < i2; index++)
              val *= dy;
            return val/distxy;
          }
      }

      // d^2()/dy^2
    case 2:
      {
        switch (i)
          {
            // constants
          case 0:

            // linears
          case 1:
          case 2:
            return 0.;

            // quadratics
          case 3:
          case 4:
            return 0.;

          case 5:
            return 2./dist2y;

            // cubics
          case 6:
            return 0.;

          case 7:
            return 0.;

          case 8:
            return 2.*dx/dist2y;

          case 9:
            return 6.*dy/dist2y;

            // quartics
          case 10:
          case 11:
            return 0.;

          case 12:
            return 2.*dx*dx/dist2y;

          case 13:
            return 6.*dx*dy/dist2y;

          case 14:
            return 12.*dy*dy/dist2y;

          default:
            unsigned int o = 0;
            for (; i >= (o+1)*(o+2)/2; o++) { }
            unsigned int i2 = i - (o*(o+1)/2);
            Real val = i2 * (i2 - 1);
            for (unsigned int index=i2; index != o; index++)
              val *= dx;
            for (unsigned int index=2; index < i2; index++)
              val *= dy;
            return val/dist2y;
          }
      }

    default:
      libmesh_error_msg("Invalid shape function derivative j = " << j);
    }

#else
  return 0.;
#endif
}

shape_second_deriv() [56/103]

Real libMesh::FE< 1, BERNSTEIN >::shape_second_deriv ( const ElemType  , const Order  , const unsigned  int, const unsigned  int, const Point &    )

inherited

Definition at line 376 of file fe_bernstein_shape_1D.C.

{
  static bool warning_given = false;

  if (!warning_given)
    libMesh::err << "Second derivatives for Bernstein elements "
                 << "are not yet implemented!"
                 << std::endl;

  warning_given = true;
  return 0.;
}

shape_second_deriv() [57/103]

Real libMesh::FE< 2, HIERARCHIC >::shape_second_deriv ( const ElemType  , const Order  , const unsigned  int, const unsigned  int, const Point &    )

inherited

Definition at line 383 of file fe_hierarchic_shape_2D.C.

{
  libmesh_error_msg("Hierarchic polynomials require the element type \nbecause edge orientation is needed.");
  return 0.;
}

shape_second_deriv() [58/103]

Real libMesh::FE< 2, L2_HIERARCHIC >::shape_second_deriv ( const ElemType  , const Order  , const unsigned  int, const unsigned  int, const Point &    )

inherited

Definition at line 384 of file fe_l2_hierarchic_shape_2D.C.

{
  libmesh_error_msg("Hierarchic polynomials require the element type \nbecause edge orientation is needed.");
  return 0.;
}

shape_second_deriv() [59/103]

Real libMesh::FE< 2, HIERARCHIC >::shape_second_deriv ( const Elem *  elem, const Order  order, const unsigned int  i, const unsigned int  j, const Point &  p  )

inherited

Definition at line 396 of file fe_hierarchic_shape_2D.C.

{
  libmesh_assert(elem);

  // I have been lazy here and am using finite differences
  // to compute the derivatives!
  const Real eps = 1.e-6;
  Point pp, pm;
  unsigned int prevj = libMesh::invalid_uint;

  switch (j)
    {
      //  d^2()/dxi^2
    case 0:
      {
        pp = Point(p(0)+eps, p(1));
        pm = Point(p(0)-eps, p(1));
        prevj = 0;
        break;
      }

      // d^2()/dxideta
    case 1:
      {
        pp = Point(p(0), p(1)+eps);
        pm = Point(p(0), p(1)-eps);
        prevj = 0;
        break;
      }

      // d^2()/deta^2
    case 2:
      {
        pp = Point(p(0), p(1)+eps);
        pm = Point(p(0), p(1)-eps);
        prevj = 1;
        break;
      }
    default:
      libmesh_error_msg("Invalid derivative index j = " << j);
    }

  return (FE<2,HIERARCHIC>::shape_deriv(elem, order, i, prevj, pp) -
          FE<2,HIERARCHIC>::shape_deriv(elem, order, i, prevj, pm)
          )/2./eps;
}

shape_second_deriv() [60/103]

Real libMesh::FE< 1, BERNSTEIN >::shape_second_deriv ( const Elem *  , const Order  , const unsigned  int, const unsigned  int, const Point &    )

inherited

Definition at line 397 of file fe_bernstein_shape_1D.C.

{
  static bool warning_given = false;

  if (!warning_given)
    libMesh::err << "Second derivatives for Bernstein elements "
                 << "are not yet implemented!"
                 << std::endl;

  warning_given = true;
  return 0.;
}

shape_second_deriv() [61/103]

Real libMesh::FE< 2, L2_HIERARCHIC >::shape_second_deriv ( const Elem *  elem, const Order  order, const unsigned int  i, const unsigned int  j, const Point &  p  )

inherited

Definition at line 397 of file fe_l2_hierarchic_shape_2D.C.

{
  libmesh_assert(elem);

  // I have been lazy here and am using finite differences
  // to compute the derivatives!
  const Real eps = 1.e-6;
  Point pp, pm;
  unsigned int prevj = libMesh::invalid_uint;

  switch (j)
    {
      //  d^2()/dxi^2
    case 0:
      {
        pp = Point(p(0)+eps, p(1));
        pm = Point(p(0)-eps, p(1));
        prevj = 0;
        break;
      }

      // d^2()/dxideta
    case 1:
      {
        pp = Point(p(0), p(1)+eps);
        pm = Point(p(0), p(1)-eps);
        prevj = 0;
        break;
      }

      // d^2()/deta^2
    case 2:
      {
        pp = Point(p(0), p(1)+eps);
        pm = Point(p(0), p(1)-eps);
        prevj = 1;
        break;
      }
    default:
      libmesh_error_msg("Invalid derivative index j = " << j);
    }
  return (FE<2,L2_HIERARCHIC>::shape_deriv(elem, order, i, prevj, pp) -
          FE<2,L2_HIERARCHIC>::shape_deriv(elem, order, i, prevj, pm)
          )/2./eps;
}

shape_second_deriv() [62/103]

RealGradient libMesh::FE< 3, NEDELEC_ONE >::shape_second_deriv ( const ElemType  , const Order  , const unsigned  int, const unsigned  int, const Point &    )

inherited

Definition at line 491 of file fe_nedelec_one_shape_3D.C.

{
  libmesh_error_msg("Nedelec elements require the element type \nbecause edge orientation is needed.");
  return RealGradient();
}

shape_second_deriv() [63/103]

RealGradient libMesh::FE< 3, NEDELEC_ONE >::shape_second_deriv ( const Elem *  elem, const Order  order, const unsigned int  i, const unsigned int  j, const Point &  libmesh_dbg_varp  )

inherited

Definition at line 504 of file fe_nedelec_one_shape_3D.C.

{
#if LIBMESH_DIM == 3

  libmesh_assert(elem);

  // j = 0 ==> d^2 phi / dxi^2
  // j = 1 ==> d^2 phi / dxi deta
  // j = 2 ==> d^2 phi / deta^2
  // j = 3 ==> d^2 phi / dxi dzeta
  // j = 4 ==> d^2 phi / deta dzeta
  // j = 5 ==> d^2 phi / dzeta^2
  libmesh_assert_less (j, 6);

  const Order totalorder = static_cast<Order>(order + elem->p_level());

  switch (totalorder)
    {
      // linear Lagrange shape functions
    case FIRST:
      {
        switch (elem->type())
          {
          case HEX20:
          case HEX27:
            {
              libmesh_assert_less (i, 12);

#ifndef NDEBUG
              const Real xi   = p(0);
              const Real eta  = p(1);
              const Real zeta = p(2);
#endif

              libmesh_assert_less_equal ( std::fabs(xi),   1.0+TOLERANCE );
              libmesh_assert_less_equal ( std::fabs(eta),  1.0+TOLERANCE );
              libmesh_assert_less_equal ( std::fabs(zeta), 1.0+TOLERANCE );

              switch (j)
                {
                  // d^2()/dxi^2
                case 0:
                  {
                    // All d^2()/dxi^2 derivatives for linear hexes are zero.
                    return RealGradient();
                  } // j=0

                  // d^2()/dxideta
                case 1:
                  {
                    switch(i)
                      {
                      case 0:
                      case 1:
                      case 2:
                      case 3:
                      case 8:
                      case 9:
                      case 10:
                      case 11:
                        return RealGradient();
                      case 4:
                        {
                          if (elem->point(0) > elem->point(4))
                            return RealGradient( 0.0, 0.0, -0.125 );
                          else
                            return RealGradient( 0.0, 0.0,  0.125 );
                        }
                      case 5:
                        {
                          if (elem->point(1) > elem->point(5))
                            return RealGradient( 0.0, 0.0,  0.125 );
                          else
                            return RealGradient( 0.0, 0.0, -0.125 );
                        }
                      case 6:
                        {
                          if (elem->point(2) > elem->point(6))
                            return RealGradient( 0.0, 0.0, -0.125 );
                          else
                            return RealGradient( 0.0, 0.0,  0.125 );
                        }
                      case 7:
                        {
                          if (elem->point(3) > elem->point(7))
                            return RealGradient( 0.0, 0.0,  0.125 );
                          else
                            return RealGradient( 0.0, 0.0, -0.125 );
                        }
                      default:
                        libmesh_error_msg("Invalid i = " << i);
                      } // switch(i)

                  } // j=1

                  // d^2()/deta^2
                case 2:
                  {
                    // All d^2()/deta^2 derivatives for linear hexes are zero.
                    return RealGradient();
                  } // j = 2

                  // d^2()/dxidzeta
                case 3:
                  {
                    switch(i)
                      {
                      case 0:
                      case 2:
                      case 4:
                      case 5:
                      case 6:
                      case 7:
                      case 8:
                      case 10:
                        return RealGradient();

                      case 1:
                        {
                          if (elem->point(1) > elem->point(2))
                            return RealGradient( 0.0,  0.125 );
                          else
                            return RealGradient( 0.0, -0.125 );
                        }
                      case 3:
                        {
                          if (elem->point(3) > elem->point(0))
                            return RealGradient( 0.0, -0.125 );
                          else
                            return RealGradient( 0.0,  0.125 );
                        }
                      case 9:
                        {
                          if (elem->point(5) > elem->point(6))
                            return RealGradient( 0.0, -0.125, 0.0 );
                          else
                            return RealGradient( 0.0,  0.125, 0.0 );
                        }
                      case 11:
                        {
                          if (elem->point(4) > elem->point(7))
                            return RealGradient( 0.0,  0.125, 0.0 );
                          else
                            return RealGradient( 0.0, -0.125, 0.0 );
                        }
                      default:
                        libmesh_error_msg("Invalid i = " << i);
                      } // switch(i)

                  } // j = 3

                  // d^2()/detadzeta
                case 4:
                  {
                    switch(i)
                      {
                      case 1:
                      case 3:
                      case 4:
                      case 5:
                      case 6:
                      case 7:
                      case 9:
                      case 11:
                        return RealGradient();

                      case 0:
                        {
                          if (elem->point(0) > elem->point(1))
                            return RealGradient( -0.125, 0.0, 0.0 );
                          else
                            return RealGradient(  0.125, 0.0, 0.0 );
                        }
                      case 2:
                        {
                          if (elem->point(2) > elem->point(3))
                            return RealGradient(  0.125, 0.0, 0.0 );
                          else
                            return RealGradient( -0.125, 0.0, 0.0 );
                        }
                      case 8:
                        {
                          if (elem->point(4) > elem->point(5))
                            return RealGradient(  0.125, 0.0, 0.0 );
                          else
                            return RealGradient( -0.125, 0.0, 0.0 );
                        }
                      case 10:
                        {
                          if (elem->point(7) > elem->point(6))
                            return RealGradient( -0.125, 0.0, 0.0 );
                          else
                            return RealGradient(  0.125, 0.0, 0.0 );
                        }
                      default:
                        libmesh_error_msg("Invalid i = " << i);
                      } // switch(i)

                  } // j = 4

                  // d^2()/dzeta^2
                case 5:
                  {
                    // All d^2()/dzeta^2 derivatives for linear hexes are zero.
                    return RealGradient();
                  } // j = 5

                default:
                  libmesh_error_msg("Invalid j = " << j);
                }

              return RealGradient();
            }

          case TET10:
            {
              libmesh_assert_less (i, 6);

              libmesh_not_implemented();
              return RealGradient();
            }

          default:
            libmesh_error_msg("ERROR: Unsupported 3D element type!: " << elem->type());

          } //switch(type)

      } // case FIRST:
      // unsupported order
    default:
      libmesh_error_msg("ERROR: Unsupported 3D FE order!: " << totalorder);
    }

#else
  return RealGradient();
#endif
}

shape_second_deriv() [64/103]

Real libMesh::FE< 2, MONOMIAL >::shape_second_deriv ( const Elem *  elem, const Order  order, const unsigned int  i, const unsigned int  j, const Point &  p  )

inherited

Definition at line 538 of file fe_monomial_shape_2D.C.

{
  libmesh_assert(elem);

  // by default call the orientation-independent shape functions
  return FE<2,MONOMIAL>::shape_second_deriv(elem->type(), static_cast<Order>(order + elem->p_level()), i, j, p);
}

shape_second_deriv() [65/103]

Real libMesh::FE< 3, HERMITE >::shape_second_deriv ( const Elem *  elem, const Order  order, const unsigned int  i, const unsigned int  j, const Point &  p  )

inherited

Definition at line 541 of file fe_hermite_shape_3D.C.

{
  libmesh_assert(elem);

  std::vector<std::vector<Real>> dxdxi(3, std::vector<Real>(2, 0));

#ifdef DEBUG
  std::vector<Real> dydxi(2), dzdeta(2), dxdzeta(2);
  std::vector<Real> dzdxi(2), dxdeta(2), dydzeta(2);
#endif //DEBUG

  hermite_compute_coefs(elem, dxdxi
#ifdef DEBUG
                        , dydxi, dzdeta, dxdzeta, dzdxi, dxdeta, dydzeta
#endif
                        );

  const ElemType type = elem->type();

  const Order totalorder = static_cast<Order>(order + elem->p_level());

  switch (totalorder)
    {
      // 3rd-order tricubic Hermite functions
    case THIRD:
      {
        switch (type)
          {
          case HEX8:
          case HEX20:
          case HEX27:
            {
              libmesh_assert_less (i, 64);

              std::vector<unsigned int> bases1D;

              Real coef = hermite_bases_3D(bases1D, dxdxi, totalorder, i);

              switch (j) // Derivative type
                {
                case 0:
                  return coef *
                    FEHermite<1>::hermite_raw_shape_second_deriv(bases1D[0],p(0)) *
                    FEHermite<1>::hermite_raw_shape(bases1D[1],p(1)) *
                    FEHermite<1>::hermite_raw_shape(bases1D[2],p(2));
                  break;
                case 1:
                  return coef *
                    FEHermite<1>::hermite_raw_shape_deriv(bases1D[0],p(0)) *
                    FEHermite<1>::hermite_raw_shape_deriv(bases1D[1],p(1)) *
                    FEHermite<1>::hermite_raw_shape(bases1D[2],p(2));
                  break;
                case 2:
                  return coef *
                    FEHermite<1>::hermite_raw_shape(bases1D[0],p(0)) *
                    FEHermite<1>::hermite_raw_shape_second_deriv(bases1D[1],p(1)) *
                    FEHermite<1>::hermite_raw_shape(bases1D[2],p(2));
                  break;
                case 3:
                  return coef *
                    FEHermite<1>::hermite_raw_shape_deriv(bases1D[0],p(0)) *
                    FEHermite<1>::hermite_raw_shape(bases1D[1],p(1)) *
                    FEHermite<1>::hermite_raw_shape_deriv(bases1D[2],p(2));
                  break;
                case 4:
                  return coef *
                    FEHermite<1>::hermite_raw_shape(bases1D[0],p(0)) *
                    FEHermite<1>::hermite_raw_shape_deriv(bases1D[1],p(1)) *
                    FEHermite<1>::hermite_raw_shape_deriv(bases1D[2],p(2));
                  break;
                case 5:
                  return coef *
                    FEHermite<1>::hermite_raw_shape(bases1D[0],p(0)) *
                    FEHermite<1>::hermite_raw_shape(bases1D[1],p(1)) *
                    FEHermite<1>::hermite_raw_shape_second_deriv(bases1D[2],p(2));
                  break;
                default:
                  libmesh_error_msg("Invalid shape function derivative j = " << j);
                }

            }
          default:
            libmesh_error_msg("ERROR: Unsupported element type " << type);
          }
      }
      // by default throw an error
    default:
      libmesh_error_msg("ERROR: Unsupported polynomial order " << totalorder);
    }
}

shape_second_deriv() [66/103]

RealGradient libMesh::FE< 0, LAGRANGE_VEC >::shape_second_deriv ( const ElemType  type, const Order  order, const unsigned int  i, const unsigned int  j, const Point &  p  )

inherited

Definition at line 551 of file fe_lagrange_vec.C.

{
  Real value = FE<0,LAGRANGE>::shape_second_deriv( type, order, i, j, p );
  return libMesh::RealGradient( value );
}

shape_second_deriv() [67/103]

Real libMesh::FE< 2, BERNSTEIN >::shape_second_deriv ( const ElemType  , const Order  , const unsigned  int, const unsigned  int, const Point &    )

inherited

Definition at line 563 of file fe_bernstein_shape_2D.C.

{
  static bool warning_given = false;

  if (!warning_given)
    libMesh::err << "Second derivatives for Bernstein elements "
                 << "are not yet implemented!"
                 << std::endl;

  warning_given = true;
  return 0.;
}

shape_second_deriv() [68/103]

Real libMesh::FE< 2, SUBDIVISION >::shape_second_deriv ( const Elem *  elem, const Order  order, const unsigned int  i, const unsigned int  j, const Point &  p  )

inherited

shape_second_deriv() [69/103]

Real libMesh::FE< 2, L2_LAGRANGE >::shape_second_deriv ( const ElemType  type, const Order  order, const unsigned int  i, const unsigned int  j, const Point &  p  )

inherited

Definition at line 567 of file fe_l2_lagrange_shape_2D.C.

{
#if LIBMESH_DIM > 1

  // j = 0 ==> d^2 phi / dxi^2
  // j = 1 ==> d^2 phi / dxi deta
  // j = 2 ==> d^2 phi / deta^2
  libmesh_assert_less (j, 3);

  switch (order)
    {
      // linear Lagrange shape functions
    case FIRST:
      {
        switch (type)
          {
          case QUAD4:
          case QUADSHELL4:
          case QUAD8:
          case QUADSHELL8:
          case QUAD9:
            {
              // Compute quad shape functions as a tensor-product
              const Real xi  = p(0);
              const Real eta = p(1);

              libmesh_assert_less (i, 4);

              //                                0  1  2  3
              static const unsigned int i0[] = {0, 1, 1, 0};
              static const unsigned int i1[] = {0, 0, 1, 1};

              switch (j)
                {
                  // d^2() / dxi^2
                case 0:
                  return 0.;

                  // d^2() / dxi deta
                case 1:
                  return (FE<1,L2_LAGRANGE>::shape_deriv(EDGE2, FIRST, i0[i], 0, xi)*
                          FE<1,L2_LAGRANGE>::shape_deriv(EDGE2, FIRST, i1[i], 0, eta));

                  // d^2() / deta^2
                case 2:
                  return 0.;

                default:
                  libmesh_error_msg("ERROR: Invalid derivative index j = " << j);
                }
            }

          case TRI3:
          case TRISHELL3:
          case TRI6:
            {
              // All second derivatives for linear triangles are zero.
              return 0.;
            }

          default:
            libmesh_error_msg("ERROR: Unsupported 2D element type: " << type);

          } // end switch (type)
      } // end case FIRST


      // quadratic Lagrange shape functions
    case SECOND:
      {
        switch (type)
          {
          case QUAD8:
            {
              const Real xi  = p(0);
              const Real eta = p(1);

              libmesh_assert_less (j, 3);

              switch (j)
                {
                  // d^2() / dxi^2
                case 0:
                  {
                    switch (i)
                      {
                      case 0:
                      case 1:
                        return 0.5*(1.-eta);

                      case 2:
                      case 3:
                        return 0.5*(1.+eta);

                      case 4:
                        return eta - 1.;

                      case 5:
                      case 7:
                        return 0.0;

                      case 6:
                        return -1. - eta;

                      default:
                        libmesh_error_msg("Invalid shape function index i = " << i);
                      }
                  }

                  // d^2() / dxi deta
                case 1:
                  {
                    switch (i)
                      {
                      case 0:
                        return 0.25*( 1. - 2.*xi - 2.*eta);

                      case 1:
                        return 0.25*(-1. - 2.*xi + 2.*eta);

                      case 2:
                        return 0.25*( 1. + 2.*xi + 2.*eta);

                      case 3:
                        return 0.25*(-1. + 2.*xi - 2.*eta);

                      case 4:
                        return xi;

                      case 5:
                        return -eta;

                      case 6:
                        return -xi;

                      case 7:
                        return eta;

                      default:
                        libmesh_error_msg("Invalid shape function index i = " << i);
                      }
                  }

                  // d^2() / deta^2
                case 2:
                  {
                    switch (i)
                      {
                      case 0:
                      case 3:
                        return 0.5*(1.-xi);

                      case 1:
                      case 2:
                        return 0.5*(1.+xi);

                      case 4:
                      case 6:
                        return 0.0;

                      case 5:
                        return -1.0 - xi;

                      case 7:
                        return xi - 1.0;

                      default:
                        libmesh_error_msg("Invalid shape function index i = " << i);
                      }
                  }


                default:
                  libmesh_error_msg("ERROR: Invalid derivative index j = " << j);
                } // end switch (j)
            } // end case QUAD8

          case QUAD9:
            {
              // Compute QUAD9 second derivatives as tensor product
              const Real xi  = p(0);
              const Real eta = p(1);

              libmesh_assert_less (i, 9);

              //                                0  1  2  3  4  5  6  7  8
              static const unsigned int i0[] = {0, 1, 1, 0, 2, 1, 2, 0, 2};
              static const unsigned int i1[] = {0, 0, 1, 1, 0, 2, 1, 2, 2};

              switch (j)
                {
                  // d^2() / dxi^2
                case 0:
                  return (FE<1,L2_LAGRANGE>::shape_second_deriv(EDGE3, SECOND, i0[i], 0, xi)*
                          FE<1,L2_LAGRANGE>::shape             (EDGE3, SECOND, i1[i], eta));

                  // d^2() / dxi deta
                case 1:
                  return (FE<1,L2_LAGRANGE>::shape_deriv(EDGE3, SECOND, i0[i], 0, xi)*
                          FE<1,L2_LAGRANGE>::shape_deriv(EDGE3, SECOND, i1[i], 0, eta));

                  // d^2() / deta^2
                case 2:
                  return (FE<1,L2_LAGRANGE>::shape             (EDGE3, SECOND, i0[i], xi)*
                          FE<1,L2_LAGRANGE>::shape_second_deriv(EDGE3, SECOND, i1[i], 0, eta));

                default:
                  libmesh_error_msg("ERROR: Invalid derivative index j = " << j);
                }  // end switch (j)
            } // end case QUAD9

          case TRI6:
            {
              const Real dzeta0dxi  = -1.;
              const Real dzeta1dxi  = 1.;
              const Real dzeta2dxi  = 0.;

              const Real dzeta0deta = -1.;
              const Real dzeta1deta = 0.;
              const Real dzeta2deta = 1.;

              libmesh_assert_less (j, 3);

              switch (j)
                {
                  // d^2() / dxi^2
                case 0:
                  {
                    switch (i)
                      {
                      case 0:
                        return 4.*dzeta0dxi*dzeta0dxi;

                      case 1:
                        return 4.*dzeta1dxi*dzeta1dxi;

                      case 2:
                        return 4.*dzeta2dxi*dzeta2dxi;

                      case 3:
                        return 8.*dzeta0dxi*dzeta1dxi;

                      case 4:
                        return 8.*dzeta1dxi*dzeta2dxi;

                      case 5:
                        return 8.*dzeta0dxi*dzeta2dxi;

                      default:
                        libmesh_error_msg("Invalid shape function index i = " << i);
                      }
                  }

                  // d^2() / dxi deta
                case 1:
                  {
                    switch (i)
                      {
                      case 0:
                        return 4.*dzeta0dxi*dzeta0deta;

                      case 1:
                        return 4.*dzeta1dxi*dzeta1deta;

                      case 2:
                        return 4.*dzeta2dxi*dzeta2deta;

                      case 3:
                        return 4.*dzeta1deta*dzeta0dxi + 4.*dzeta0deta*dzeta1dxi;

                      case 4:
                        return 4.*dzeta2deta*dzeta1dxi + 4.*dzeta1deta*dzeta2dxi;

                      case 5:
                        return 4.*dzeta2deta*dzeta0dxi + 4.*dzeta0deta*dzeta2dxi;

                      default:
                        libmesh_error_msg("Invalid shape function index i = " << i);
                      }
                  }

                  // d^2() / deta^2
                case 2:
                  {
                    switch (i)
                      {
                      case 0:
                        return 4.*dzeta0deta*dzeta0deta;

                      case 1:
                        return 4.*dzeta1deta*dzeta1deta;

                      case 2:
                        return 4.*dzeta2deta*dzeta2deta;

                      case 3:
                        return 8.*dzeta0deta*dzeta1deta;

                      case 4:
                        return 8.*dzeta1deta*dzeta2deta;

                      case 5:
                        return 8.*dzeta0deta*dzeta2deta;

                      default:
                        libmesh_error_msg("Invalid shape function index i = " << i);
                      }
                  }

                default:
                  libmesh_error_msg("ERROR: Invalid derivative index j = " << j);
                } // end switch (j)
            }  // end case TRI6

          default:
            libmesh_error_msg("ERROR: Unsupported 2D element type: " << type);
          }
      } // end case SECOND



      // unsupported order
    default:
      libmesh_error_msg("ERROR: Unsupported 2D FE order: " << order);

    } // end switch (order)

#endif // LIBMESH_DIM > 1
}

shape_second_deriv() [70/103]

Real libMesh::FE< 2, LAGRANGE >::shape_second_deriv ( const ElemType  type, const Order  order, const unsigned int  i, const unsigned int  j, const Point &  p  )

inherited

Definition at line 569 of file fe_lagrange_shape_2D.C.

{
#if LIBMESH_DIM > 1

  // j = 0 ==> d^2 phi / dxi^2
  // j = 1 ==> d^2 phi / dxi deta
  // j = 2 ==> d^2 phi / deta^2
  libmesh_assert_less (j, 3);

  switch (order)
    {
      // linear Lagrange shape functions
    case FIRST:
      {
        switch (type)
          {
          case QUAD4:
          case QUADSHELL4:
          case QUAD8:
          case QUADSHELL8:
          case QUAD9:
            {
              // Compute quad shape functions as a tensor-product
              const Real xi  = p(0);
              const Real eta = p(1);

              libmesh_assert_less (i, 4);

              //                                0  1  2  3
              static const unsigned int i0[] = {0, 1, 1, 0};
              static const unsigned int i1[] = {0, 0, 1, 1};

              switch (j)
                {
                  // d^2() / dxi^2
                case 0:
                  return 0.;

                  // d^2() / dxi deta
                case 1:
                  return (FE<1,LAGRANGE>::shape_deriv(EDGE2, FIRST, i0[i], 0, xi)*
                          FE<1,LAGRANGE>::shape_deriv(EDGE2, FIRST, i1[i], 0, eta));

                  // d^2() / deta^2
                case 2:
                  return 0.;

                default:
                  libmesh_error_msg("ERROR: Invalid derivative index j = " << j);
                }
            }

          case TRI3:
          case TRISHELL3:
          case TRI6:
            {
              // All second derivatives for linear triangles are zero.
              return 0.;
            }

          default:
            libmesh_error_msg("ERROR: Unsupported 2D element type: " << type);

          } // end switch (type)
      } // end case FIRST


      // quadratic Lagrange shape functions
    case SECOND:
      {
        switch (type)
          {
          case QUAD8:
          case QUADSHELL8:
            {
              const Real xi  = p(0);
              const Real eta = p(1);

              libmesh_assert_less (j, 3);

              switch (j)
                {
                  // d^2() / dxi^2
                case 0:
                  {
                    switch (i)
                      {
                      case 0:
                      case 1:
                        return 0.5*(1.-eta);

                      case 2:
                      case 3:
                        return 0.5*(1.+eta);

                      case 4:
                        return eta - 1.;

                      case 5:
                      case 7:
                        return 0.0;

                      case 6:
                        return -1. - eta;

                      default:
                        libmesh_error_msg("Invalid shape function index i = " << i);
                      }
                  }

                  // d^2() / dxi deta
                case 1:
                  {
                    switch (i)
                      {
                      case 0:
                        return 0.25*( 1. - 2.*xi - 2.*eta);

                      case 1:
                        return 0.25*(-1. - 2.*xi + 2.*eta);

                      case 2:
                        return 0.25*( 1. + 2.*xi + 2.*eta);

                      case 3:
                        return 0.25*(-1. + 2.*xi - 2.*eta);

                      case 4:
                        return xi;

                      case 5:
                        return -eta;

                      case 6:
                        return -xi;

                      case 7:
                        return eta;

                      default:
                        libmesh_error_msg("Invalid shape function index i = " << i);
                      }
                  }

                  // d^2() / deta^2
                case 2:
                  {
                    switch (i)
                      {
                      case 0:
                      case 3:
                        return 0.5*(1.-xi);

                      case 1:
                      case 2:
                        return 0.5*(1.+xi);

                      case 4:
                      case 6:
                        return 0.0;

                      case 5:
                        return -1.0 - xi;

                      case 7:
                        return xi - 1.0;

                      default:
                        libmesh_error_msg("Invalid shape function index i = " << i);
                      }
                  }

                default:
                  libmesh_error_msg("ERROR: Invalid derivative index j = " << j);
                } // end switch (j)
            } // end case QUAD8

          case QUAD9:
            {
              // Compute QUAD9 second derivatives as tensor product
              const Real xi  = p(0);
              const Real eta = p(1);

              libmesh_assert_less (i, 9);

              //                                0  1  2  3  4  5  6  7  8
              static const unsigned int i0[] = {0, 1, 1, 0, 2, 1, 2, 0, 2};
              static const unsigned int i1[] = {0, 0, 1, 1, 0, 2, 1, 2, 2};

              switch (j)
                {
                  // d^2() / dxi^2
                case 0:
                  return (FE<1,LAGRANGE>::shape_second_deriv(EDGE3, SECOND, i0[i], 0, xi)*
                          FE<1,LAGRANGE>::shape             (EDGE3, SECOND, i1[i], eta));

                  // d^2() / dxi deta
                case 1:
                  return (FE<1,LAGRANGE>::shape_deriv(EDGE3, SECOND, i0[i], 0, xi)*
                          FE<1,LAGRANGE>::shape_deriv(EDGE3, SECOND, i1[i], 0, eta));

                  // d^2() / deta^2
                case 2:
                  return (FE<1,LAGRANGE>::shape             (EDGE3, SECOND, i0[i], xi)*
                          FE<1,LAGRANGE>::shape_second_deriv(EDGE3, SECOND, i1[i], 0, eta));

                default:
                  libmesh_error_msg("ERROR: Invalid derivative index j = " << j);
                }  // end switch (j)
            } // end case QUAD9

          case TRI6:
            {
              const Real dzeta0dxi  = -1.;
              const Real dzeta1dxi  = 1.;
              const Real dzeta2dxi  = 0.;

              const Real dzeta0deta = -1.;
              const Real dzeta1deta = 0.;
              const Real dzeta2deta = 1.;

              libmesh_assert_less (j, 3);

              switch (j)
                {
                  // d^2() / dxi^2
                case 0:
                  {
                    switch (i)
                      {
                      case 0:
                        return 4.*dzeta0dxi*dzeta0dxi;

                      case 1:
                        return 4.*dzeta1dxi*dzeta1dxi;

                      case 2:
                        return 4.*dzeta2dxi*dzeta2dxi;

                      case 3:
                        return 8.*dzeta0dxi*dzeta1dxi;

                      case 4:
                        return 8.*dzeta1dxi*dzeta2dxi;

                      case 5:
                        return 8.*dzeta0dxi*dzeta2dxi;

                      default:
                        libmesh_error_msg("Invalid shape function index i = " << i);
                      }
                  }

                  // d^2() / dxi deta
                case 1:
                  {
                    switch (i)
                      {
                      case 0:
                        return 4.*dzeta0dxi*dzeta0deta;

                      case 1:
                        return 4.*dzeta1dxi*dzeta1deta;

                      case 2:
                        return 4.*dzeta2dxi*dzeta2deta;

                      case 3:
                        return 4.*dzeta1deta*dzeta0dxi + 4.*dzeta0deta*dzeta1dxi;

                      case 4:
                        return 4.*dzeta2deta*dzeta1dxi + 4.*dzeta1deta*dzeta2dxi;

                      case 5:
                        return 4.*dzeta2deta*dzeta0dxi + 4.*dzeta0deta*dzeta2dxi;

                      default:
                        libmesh_error_msg("Invalid shape function index i = " << i);
                      }
                  }

                  // d^2() / deta^2
                case 2:
                  {
                    switch (i)
                      {
                      case 0:
                        return 4.*dzeta0deta*dzeta0deta;

                      case 1:
                        return 4.*dzeta1deta*dzeta1deta;

                      case 2:
                        return 4.*dzeta2deta*dzeta2deta;

                      case 3:
                        return 8.*dzeta0deta*dzeta1deta;

                      case 4:
                        return 8.*dzeta1deta*dzeta2deta;

                      case 5:
                        return 8.*dzeta0deta*dzeta2deta;

                      default:
                        libmesh_error_msg("Invalid shape function index i = " << i);
                      }
                  }

                default:
                  libmesh_error_msg("ERROR: Invalid derivative index j = " << j);
                } // end switch (j)
            }  // end case TRI6

          default:
            libmesh_error_msg("ERROR: Unsupported 2D element type: " << type);
          }
      } // end case SECOND



      // unsupported order
    default:
      libmesh_error_msg("ERROR: Unsupported 2D FE order: " << order);

    } // end switch (order)

#else // LIBMESH_DIM > 1
  return 0.;
#endif
}

shape_second_deriv() [71/103]

RealGradient libMesh::FE< 1, LAGRANGE_VEC >::shape_second_deriv ( const ElemType  type, const Order  order, const unsigned int  i, const unsigned int  j, const Point &  p  )

inherited

Definition at line 573 of file fe_lagrange_vec.C.

{
  Real value = FE<1,LAGRANGE>::shape_second_deriv( type, order, i, j, p );
  return libMesh::RealGradient( value );
}

shape_second_deriv() [72/103]

Real libMesh::FE< 2, BERNSTEIN >::shape_second_deriv ( const Elem *  , const Order  , const unsigned  int, const unsigned  int, const Point &    )

inherited

Definition at line 583 of file fe_bernstein_shape_2D.C.

{
  static bool warning_given = false;

  if (!warning_given)
    libMesh::err << "Second derivatives for Bernstein elements "
                 << "are not yet implemented!"
                 << std::endl;

  warning_given = true;
  return 0.;
}

shape_second_deriv() [73/103]

RealGradient libMesh::FE< 0, NEDELEC_ONE >::shape_second_deriv ( const ElemType  , const Order  , const unsigned  int, const unsigned  int, const Point &    )

inherited

Definition at line 607 of file fe_nedelec_one.C.

{ NEDELEC_LOW_D_ERROR_MESSAGE }

shape_second_deriv() [74/103]

RealGradient libMesh::FE< 0, NEDELEC_ONE >::shape_second_deriv ( const Elem *  , const Order  , const unsigned  int, const unsigned  int, const Point &    )

inherited

Definition at line 611 of file fe_nedelec_one.C.

{ NEDELEC_LOW_D_ERROR_MESSAGE }

shape_second_deriv() [75/103]

RealGradient libMesh::FE< 2, LAGRANGE_VEC >::shape_second_deriv ( const ElemType  type, const Order  order, const unsigned int  i, const unsigned int  j, const Point &  p  )

inherited

Definition at line 623 of file fe_lagrange_vec.C.

{
  Real value = FE<2,LAGRANGE>::shape_second_deriv( type, order, i/2, j, p );

  switch( i%2 )
    {
    case 0:
      return libMesh::RealGradient( value );

    case 1:
      return libMesh::RealGradient( Real(0), value );

    default:
      libmesh_error_msg("i%2 must be either 0 or 1!");
    }

  //dummy
  return libMesh::RealGradient();
}

shape_second_deriv() [76/103]

RealGradient libMesh::FE< 1, NEDELEC_ONE >::shape_second_deriv ( const ElemType  , const Order  , const unsigned  int, const unsigned  int, const Point &    )

inherited

Definition at line 630 of file fe_nedelec_one.C.

{ NEDELEC_LOW_D_ERROR_MESSAGE }

shape_second_deriv() [77/103]

RealGradient libMesh::FE< 1, NEDELEC_ONE >::shape_second_deriv ( const Elem *  , const Order  , const unsigned  int, const unsigned  int, const Point &    )

inherited

Definition at line 634 of file fe_nedelec_one.C.

{ NEDELEC_LOW_D_ERROR_MESSAGE }

shape_second_deriv() [78/103]

Real libMesh::FE< 3, MONOMIAL >::shape_second_deriv ( const ElemType  , const Order  libmesh_dbg_varorder, const unsigned int  i, const unsigned int  j, const Point &  p  )

inherited

Definition at line 653 of file fe_monomial_shape_3D.C.

{
#if LIBMESH_DIM == 3

  libmesh_assert_less (j, 6);

  libmesh_assert_less (i, (static_cast<unsigned int>(order)+1)*
                       (static_cast<unsigned int>(order)+2)*
                       (static_cast<unsigned int>(order)+3)/6);

  const Real xi   = p(0);
  const Real eta  = p(1);
  const Real zeta = p(2);

  // monomials. since they are hierarchic we only need one case block.
  switch (j)
    {
      // d^2()/dxi^2
    case 0:
      {
        switch (i)
          {
            // constant
          case 0:

            // linear
          case 1:
          case 2:
          case 3:
            return 0.;

            // quadratic
          case 4:
            return 2.;

          case 5:
          case 6:
          case 7:
          case 8:
          case 9:
            return 0.;

            // cubic
          case 10:
            return 6.*xi;

          case 11:
            return 2.*eta;

          case 12:
          case 13:
            return 0.;

          case 14:
            return 2.*zeta;

          case 15:
          case 16:
          case 17:
          case 18:
          case 19:
            return 0.;

            // quartics
          case 20:
            return 12.*xi*xi;

          case 21:
            return 6.*xi*eta;

          case 22:
            return 2.*eta*eta;

          case 23:
          case 24:
            return 0.;

          case 25:
            return 6.*xi*zeta;

          case 26:
            return 2.*eta*zeta;

          case 27:
          case 28:
            return 0.;

          case 29:
            return 2.*zeta*zeta;

          case 30:
          case 31:
          case 32:
          case 33:
          case 34:
            return 0.;

          default:
            unsigned int o = 0;
            for (; i >= (o+1)*(o+2)*(o+3)/6; o++) { }
            unsigned int i2 = i - (o*(o+1)*(o+2)/6);
            unsigned int block=o, nz = 0;
            for (; block < i2; block += (o-nz+1)) { nz++; }
            const unsigned int nx = block - i2;
            const unsigned int ny = o - nx - nz;
            Real val = nx * (nx - 1);
            for (unsigned int index=2; index < nx; index++)
              val *= xi;
            for (unsigned int index=0; index != ny; index++)
              val *= eta;
            for (unsigned int index=0; index != nz; index++)
              val *= zeta;
            return val;
          }
      }


      // d^2()/dxideta
    case 1:
      {
        switch (i)
          {
            // constant
          case 0:

            // linear
          case 1:
          case 2:
          case 3:
            return 0.;

            // quadratic
          case 4:
            return 0.;

          case 5:
            return 1.;

          case 6:
          case 7:
          case 8:
          case 9:
            return 0.;

            // cubic
          case 10:
            return 0.;

          case 11:
            return 2.*xi;

          case 12:
            return 2.*eta;

          case 13:
          case 14:
            return 0.;

          case 15:
            return zeta;

          case 16:
          case 17:
          case 18:
          case 19:
            return 0.;

            // quartics
          case 20:
            return 0.;

          case 21:
            return 3.*xi*xi;

          case 22:
            return 4.*xi*eta;

          case 23:
            return 3.*eta*eta;

          case 24:
          case 25:
            return 0.;

          case 26:
            return 2.*xi*zeta;

          case 27:
            return 2.*eta*zeta;

          case 28:
          case 29:
            return 0.;

          case 30:
            return zeta*zeta;

          case 31:
          case 32:
          case 33:
          case 34:
            return 0.;

          default:
            unsigned int o = 0;
            for (; i >= (o+1)*(o+2)*(o+3)/6; o++) { }
            unsigned int i2 = i - (o*(o+1)*(o+2)/6);
            unsigned int block=o, nz = 0;
            for (; block < i2; block += (o-nz+1)) { nz++; }
            const unsigned int nx = block - i2;
            const unsigned int ny = o - nx - nz;
            Real val = nx * ny;
            for (unsigned int index=1; index < nx; index++)
              val *= xi;
            for (unsigned int index=1; index < ny; index++)
              val *= eta;
            for (unsigned int index=0; index != nz; index++)
              val *= zeta;
            return val;
          }
      }


      // d^2()/deta^2
    case 2:
      {
        switch (i)
          {
            // constant
          case 0:

            // linear
          case 1:
          case 2:
          case 3:
            return 0.;

            // quadratic
          case 4:
          case 5:
            return 0.;

          case 6:
            return 2.;

          case 7:
          case 8:
          case 9:
            return 0.;

            // cubic
          case 10:
          case 11:
            return 0.;

          case 12:
            return 2.*xi;
          case 13:
            return 6.*eta;

          case 14:
          case 15:
            return 0.;

          case 16:
            return 2.*zeta;

          case 17:
          case 18:
          case 19:
            return 0.;

            // quartics
          case 20:
          case 21:
            return 0.;

          case 22:
            return 2.*xi*xi;

          case 23:
            return 6.*xi*eta;

          case 24:
            return 12.*eta*eta;

          case 25:
          case 26:
            return 0.;

          case 27:
            return 2.*xi*zeta;

          case 28:
            return 6.*eta*zeta;

          case 29:
          case 30:
            return 0.;

          case 31:
            return 2.*zeta*zeta;

          case 32:
          case 33:
          case 34:
            return 0.;

          default:
            unsigned int o = 0;
            for (; i >= (o+1)*(o+2)*(o+3)/6; o++) { }
            unsigned int i2 = i - (o*(o+1)*(o+2)/6);
            unsigned int block=o, nz = 0;
            for (; block < i2; block += (o-nz+1)) { nz++; }
            const unsigned int nx = block - i2;
            const unsigned int ny = o - nx - nz;
            Real val = ny * (ny - 1);
            for (unsigned int index=0; index != nx; index++)
              val *= xi;
            for (unsigned int index=2; index < ny; index++)
              val *= eta;
            for (unsigned int index=0; index != nz; index++)
              val *= zeta;
            return val;
          }
      }


      // d^2()/dxidzeta
    case 3:
      {
        switch (i)
          {
            // constant
          case 0:

            // linear
          case 1:
          case 2:
          case 3:
            return 0.;

            // quadratic
          case 4:
          case 5:
          case 6:
            return 0.;

          case 7:
            return 1.;

          case 8:
          case 9:
            return 0.;

            // cubic
          case 10:
          case 11:
          case 12:
          case 13:
            return 0.;

          case 14:
            return 2.*xi;

          case 15:
            return eta;

          case 16:
            return 0.;

          case 17:
            return 2.*zeta;

          case 18:
          case 19:
            return 0.;

            // quartics
          case 20:
          case 21:
          case 22:
          case 23:
          case 24:
            return 0.;

          case 25:
            return 3.*xi*xi;

          case 26:
            return 2.*xi*eta;

          case 27:
            return eta*eta;

          case 28:
            return 0.;

          case 29:
            return 4.*xi*zeta;

          case 30:
            return 2.*eta*zeta;

          case 31:
            return 0.;

          case 32:
            return 3.*zeta*zeta;

          case 33:
          case 34:
            return 0.;

          default:
            unsigned int o = 0;
            for (; i >= (o+1)*(o+2)*(o+3)/6; o++) { }
            unsigned int i2 = i - (o*(o+1)*(o+2)/6);
            unsigned int block=o, nz = 0;
            for (; block < i2; block += (o-nz+1)) { nz++; }
            const unsigned int nx = block - i2;
            const unsigned int ny = o - nx - nz;
            Real val = nx * nz;
            for (unsigned int index=1; index < nx; index++)
              val *= xi;
            for (unsigned int index=0; index != ny; index++)
              val *= eta;
            for (unsigned int index=1; index < nz; index++)
              val *= zeta;
            return val;
          }
      }

      // d^2()/detadzeta
    case 4:
      {
        switch (i)
          {
            // constant
          case 0:

            // linear
          case 1:
          case 2:
          case 3:
            return 0.;

            // quadratic
          case 4:
          case 5:
          case 6:
          case 7:
            return 0.;

          case 8:
            return 1.;

          case 9:
            return 0.;

            // cubic
          case 10:
          case 11:
          case 12:
          case 13:
          case 14:
            return 0.;

          case 15:
            return xi;

          case 16:
            return 2.*eta;

          case 17:
            return 0.;

          case 18:
            return 2.*zeta;

          case 19:
            return 0.;

            // quartics
          case 20:
          case 21:
          case 22:
          case 23:
          case 24:
          case 25:
            return 0.;

          case 26:
            return xi*xi;

          case 27:
            return 2.*xi*eta;

          case 28:
            return 3.*eta*eta;

          case 29:
            return 0.;

          case 30:
            return 2.*xi*zeta;

          case 31:
            return 4.*eta*zeta;

          case 32:
            return 0.;

          case 33:
            return 3.*zeta*zeta;

          case 34:
            return 0.;

          default:
            unsigned int o = 0;
            for (; i >= (o+1)*(o+2)*(o+3)/6; o++) { }
            unsigned int i2 = i - (o*(o+1)*(o+2)/6);
            unsigned int block=o, nz = 0;
            for (; block < i2; block += (o-nz+1)) { nz++; }
            const unsigned int nx = block - i2;
            const unsigned int ny = o - nx - nz;
            Real val = ny * nz;
            for (unsigned int index=0; index != nx; index++)
              val *= xi;
            for (unsigned int index=1; index < ny; index++)
              val *= eta;
            for (unsigned int index=1; index < nz; index++)
              val *= zeta;
            return val;
          }
      }


      // d^2()/dzeta^2
    case 5:
      {
        switch (i)
          {
            // constant
          case 0:

            // linear
          case 1:
          case 2:
          case 3:
            return 0.;

            // quadratic
          case 4:
          case 5:
          case 6:
          case 7:
          case 8:
            return 0.;

          case 9:
            return 2.;

            // cubic
          case 10:
          case 11:
          case 12:
          case 13:
          case 14:
          case 15:
          case 16:
            return 0.;

          case 17:
            return 2.*xi;

          case 18:
            return 2.*eta;

          case 19:
            return 6.*zeta;

            // quartics
          case 20:
          case 21:
          case 22:
          case 23:
          case 24:
          case 25:
          case 26:
          case 27:
          case 28:
            return 0.;

          case 29:
            return 2.*xi*xi;

          case 30:
            return 2.*xi*eta;

          case 31:
            return 2.*eta*eta;

          case 32:
            return 6.*xi*zeta;

          case 33:
            return 6.*eta*zeta;

          case 34:
            return 12.*zeta*zeta;

          default:
            unsigned int o = 0;
            for (; i >= (o+1)*(o+2)*(o+3)/6; o++) { }
            unsigned int i2 = i - (o*(o+1)*(o+2)/6);
            unsigned int block=o, nz = 0;
            for (; block < i2; block += (o-nz+1)) { nz++; }
            const unsigned int nx = block - i2;
            const unsigned int ny = o - nx - nz;
            Real val = nz * (nz - 1);
            for (unsigned int index=0; index != nx; index++)
              val *= xi;
            for (unsigned int index=0; index != ny; index++)
              val *= eta;
            for (unsigned int index=2; index < nz; index++)
              val *= zeta;
            return val;
          }
      }

    default:
      libmesh_error_msg("Invalid j = " << j);
    }

#else
  return 0.;
#endif
}

shape_second_deriv() [79/103]

RealGradient libMesh::FE< 3, LAGRANGE_VEC >::shape_second_deriv ( const ElemType  type, const Order  order, const unsigned int  i, const unsigned int  j, const Point &  p  )

inherited

Definition at line 694 of file fe_lagrange_vec.C.

{
  Real value = FE<3,LAGRANGE>::shape_second_deriv( type, order, i/3, j, p );

  switch( i%3 )
    {
    case 0:
      return libMesh::RealGradient( value );

    case 1:
      return libMesh::RealGradient( Real(0), value );

    case 2:
      return libMesh::RealGradient( Real(0), Real(0), value );

    default:
      libmesh_error_msg("i%3 must be 0, 1, or 2!");
    }

  //dummy
  return libMesh::RealGradient();
}

shape_second_deriv() [80/103]

Real libMesh::FE< 3, XYZ >::shape_second_deriv ( const ElemType  , const Order  , const unsigned  int, const unsigned  int, const Point &    )

inherited

Definition at line 696 of file fe_xyz_shape_3D.C.

{
  libmesh_error_msg("XYZ polynomials require the element \nbecause the centroid is needed.");
  return 0.;
}

shape_second_deriv() [81/103]

Real libMesh::FE< 3, XYZ >::shape_second_deriv ( const Elem *  elem, const Order  libmesh_dbg_varorder, const unsigned int  i, const unsigned int  j, const Point &  point_in  )

inherited

Definition at line 709 of file fe_xyz_shape_3D.C.

{
#if LIBMESH_DIM == 3

  libmesh_assert(elem);
  libmesh_assert_less (j, 6);

  Point centroid = elem->centroid();
  Point max_distance = Point(0.,0.,0.);
  for (unsigned int p = 0; p < elem->n_nodes(); p++)
    for (unsigned int d = 0; d < 3; d++)
      {
        const Real distance = std::abs(centroid(d) - elem->point(p)(d));
        max_distance(d) = std::max(distance, max_distance(d));
      }

  const Real x  = point_in(0);
  const Real y  = point_in(1);
  const Real z  = point_in(2);
  const Real xc = centroid(0);
  const Real yc = centroid(1);
  const Real zc = centroid(2);
  const Real distx = max_distance(0);
  const Real disty = max_distance(1);
  const Real distz = max_distance(2);
  const Real dx = (x - xc)/distx;
  const Real dy = (y - yc)/disty;
  const Real dz = (z - zc)/distz;
  const Real dist2x = pow(distx,2.);
  const Real dist2y = pow(disty,2.);
  const Real dist2z = pow(distz,2.);
  const Real distxy = distx * disty;
  const Real distxz = distx * distz;
  const Real distyz = disty * distz;

#ifndef NDEBUG
  // totalorder is only used in the assertion below, so
  // we avoid declaring it when asserts are not active.
  const unsigned int totalorder = static_cast<Order>(order + elem->p_level());
#endif
  libmesh_assert_less (i, (static_cast<unsigned int>(totalorder)+1)*
                       (static_cast<unsigned int>(totalorder)+2)*
                       (static_cast<unsigned int>(totalorder)+3)/6);

  // monomials. since they are hierarchic we only need one case block.
  switch (j)
    {
      // d^2()/dx^2
    case 0:
      {
        switch (i)
          {
            // constant
          case 0:

            // linear
          case 1:
          case 2:
          case 3:
            return 0.;

            // quadratic
          case 4:
            return 2./dist2x;

          case 5:
          case 6:
          case 7:
          case 8:
          case 9:
            return 0.;

            // cubic
          case 10:
            return 6.*dx/dist2x;

          case 11:
            return 2.*dy/dist2x;

          case 12:
          case 13:
            return 0.;

          case 14:
            return 2.*dz/dist2x;

          case 15:
          case 16:
          case 17:
          case 18:
          case 19:
            return 0.;

            // quartics
          case 20:
            return 12.*dx*dx/dist2x;

          case 21:
            return 6.*dx*dy/dist2x;

          case 22:
            return 2.*dy*dy/dist2x;

          case 23:
          case 24:
            return 0.;

          case 25:
            return 6.*dx*dz/dist2x;

          case 26:
            return 2.*dy*dz/dist2x;

          case 27:
          case 28:
            return 0.;

          case 29:
            return 2.*dz*dz/dist2x;

          case 30:
          case 31:
          case 32:
          case 33:
          case 34:
            return 0.;

          default:
            unsigned int o = 0;
            for (; i >= (o+1)*(o+2)*(o+3)/6; o++) { }
            unsigned int i2 = i - (o*(o+1)*(o+2)/6);
            unsigned int block=o, nz = 0;
            for (; block < i2; block += (o-nz+1)) { nz++; }
            const unsigned int nx = block - i2;
            const unsigned int ny = o - nx - nz;
            Real val = nx * (nx - 1);
            for (unsigned int index=2; index < nx; index++)
              val *= dx;
            for (unsigned int index=0; index != ny; index++)
              val *= dy;
            for (unsigned int index=0; index != nz; index++)
              val *= dz;
            return val/dist2x;
          }
      }


      // d^2()/dxdy
    case 1:
      {
        switch (i)
          {
            // constant
          case 0:

            // linear
          case 1:
          case 2:
          case 3:
            return 0.;

            // quadratic
          case 4:
            return 0.;

          case 5:
            return 1./distxy;

          case 6:
          case 7:
          case 8:
          case 9:
            return 0.;

            // cubic
          case 10:
            return 0.;

          case 11:
            return 2.*dx/distxy;

          case 12:
            return 2.*dy/distxy;

          case 13:
          case 14:
            return 0.;

          case 15:
            return dz/distxy;

          case 16:
          case 17:
          case 18:
          case 19:
            return 0.;

            // quartics
          case 20:
            return 0.;

          case 21:
            return 3.*dx*dx/distxy;

          case 22:
            return 4.*dx*dy/distxy;

          case 23:
            return 3.*dy*dy/distxy;

          case 24:
          case 25:
            return 0.;

          case 26:
            return 2.*dx*dz/distxy;

          case 27:
            return 2.*dy*dz/distxy;

          case 28:
          case 29:
            return 0.;

          case 30:
            return dz*dz/distxy;

          case 31:
          case 32:
          case 33:
          case 34:
            return 0.;

          default:
            unsigned int o = 0;
            for (; i >= (o+1)*(o+2)*(o+3)/6; o++) { }
            unsigned int i2 = i - (o*(o+1)*(o+2)/6);
            unsigned int block=o, nz = 0;
            for (; block < i2; block += (o-nz+1)) { nz++; }
            const unsigned int nx = block - i2;
            const unsigned int ny = o - nx - nz;
            Real val = nx * ny;
            for (unsigned int index=1; index < nx; index++)
              val *= dx;
            for (unsigned int index=1; index < ny; index++)
              val *= dy;
            for (unsigned int index=0; index != nz; index++)
              val *= dz;
            return val/distxy;
          }
      }


      // d^2()/dy^2
    case 2:
      {
        switch (i)
          {
            // constant
          case 0:

            // linear
          case 1:
          case 2:
          case 3:
            return 0.;

            // quadratic
          case 4:
          case 5:
            return 0.;

          case 6:
            return 2./dist2y;

          case 7:
          case 8:
          case 9:
            return 0.;

            // cubic
          case 10:
          case 11:
            return 0.;

          case 12:
            return 2.*dx/dist2y;
          case 13:
            return 6.*dy/dist2y;

          case 14:
          case 15:
            return 0.;

          case 16:
            return 2.*dz/dist2y;

          case 17:
          case 18:
          case 19:
            return 0.;

            // quartics
          case 20:
          case 21:
            return 0.;

          case 22:
            return 2.*dx*dx/dist2y;

          case 23:
            return 6.*dx*dy/dist2y;

          case 24:
            return 12.*dy*dy/dist2y;

          case 25:
          case 26:
            return 0.;

          case 27:
            return 2.*dx*dz/dist2y;

          case 28:
            return 6.*dy*dz/dist2y;

          case 29:
          case 30:
            return 0.;

          case 31:
            return 2.*dz*dz/dist2y;

          case 32:
          case 33:
          case 34:
            return 0.;

          default:
            unsigned int o = 0;
            for (; i >= (o+1)*(o+2)*(o+3)/6; o++) { }
            unsigned int i2 = i - (o*(o+1)*(o+2)/6);
            unsigned int block=o, nz = 0;
            for (; block < i2; block += (o-nz+1)) { nz++; }
            const unsigned int nx = block - i2;
            const unsigned int ny = o - nx - nz;
            Real val = ny * (ny - 1);
            for (unsigned int index=0; index != nx; index++)
              val *= dx;
            for (unsigned int index=2; index < ny; index++)
              val *= dy;
            for (unsigned int index=0; index != nz; index++)
              val *= dz;
            return val/dist2y;
          }
      }


      // d^2()/dxdz
    case 3:
      {
        switch (i)
          {
            // constant
          case 0:

            // linear
          case 1:
          case 2:
          case 3:
            return 0.;

            // quadratic
          case 4:
          case 5:
          case 6:
            return 0.;

          case 7:
            return 1./distxz;

          case 8:
          case 9:
            return 0.;

            // cubic
          case 10:
          case 11:
          case 12:
          case 13:
            return 0.;

          case 14:
            return 2.*dx/distxz;

          case 15:
            return dy/distxz;

          case 16:
            return 0.;

          case 17:
            return 2.*dz/distxz;

          case 18:
          case 19:
            return 0.;

            // quartics
          case 20:
          case 21:
          case 22:
          case 23:
          case 24:
            return 0.;

          case 25:
            return 3.*dx*dx/distxz;

          case 26:
            return 2.*dx*dy/distxz;

          case 27:
            return dy*dy/distxz;

          case 28:
            return 0.;

          case 29:
            return 4.*dx*dz/distxz;

          case 30:
            return 2.*dy*dz/distxz;

          case 31:
            return 0.;

          case 32:
            return 3.*dz*dz/distxz;

          case 33:
          case 34:
            return 0.;

          default:
            unsigned int o = 0;
            for (; i >= (o+1)*(o+2)*(o+3)/6; o++) { }
            unsigned int i2 = i - (o*(o+1)*(o+2)/6);
            unsigned int block=o, nz = 0;
            for (; block < i2; block += (o-nz+1)) { nz++; }
            const unsigned int nx = block - i2;
            const unsigned int ny = o - nx - nz;
            Real val = nx * nz;
            for (unsigned int index=1; index < nx; index++)
              val *= dx;
            for (unsigned int index=0; index != ny; index++)
              val *= dy;
            for (unsigned int index=1; index < nz; index++)
              val *= dz;
            return val/distxz;
          }
      }

      // d^2()/dydz
    case 4:
      {
        switch (i)
          {
            // constant
          case 0:

            // linear
          case 1:
          case 2:
          case 3:
            return 0.;

            // quadratic
          case 4:
          case 5:
          case 6:
          case 7:
            return 0.;

          case 8:
            return 1./distyz;

          case 9:
            return 0.;

            // cubic
          case 10:
          case 11:
          case 12:
          case 13:
          case 14:
            return 0.;

          case 15:
            return dx/distyz;

          case 16:
            return 2.*dy/distyz;

          case 17:
            return 0.;

          case 18:
            return 2.*dz/distyz;

          case 19:
            return 0.;

            // quartics
          case 20:
          case 21:
          case 22:
          case 23:
          case 24:
          case 25:
            return 0.;

          case 26:
            return dx*dx/distyz;

          case 27:
            return 2.*dx*dy/distyz;

          case 28:
            return 3.*dy*dy/distyz;

          case 29:
            return 0.;

          case 30:
            return 2.*dx*dz/distyz;

          case 31:
            return 4.*dy*dz/distyz;

          case 32:
            return 0.;

          case 33:
            return 3.*dz*dz/distyz;

          case 34:
            return 0.;

          default:
            unsigned int o = 0;
            for (; i >= (o+1)*(o+2)*(o+3)/6; o++) { }
            unsigned int i2 = i - (o*(o+1)*(o+2)/6);
            unsigned int block=o, nz = 0;
            for (; block < i2; block += (o-nz+1)) { nz++; }
            const unsigned int nx = block - i2;
            const unsigned int ny = o - nx - nz;
            Real val = ny * nz;
            for (unsigned int index=0; index != nx; index++)
              val *= dx;
            for (unsigned int index=1; index < ny; index++)
              val *= dy;
            for (unsigned int index=1; index < nz; index++)
              val *= dz;
            return val/distyz;
          }
      }


      // d^2()/dz^2
    case 5:
      {
        switch (i)
          {
            // constant
          case 0:

            // linear
          case 1:
          case 2:
          case 3:
            return 0.;

            // quadratic
          case 4:
          case 5:
          case 6:
          case 7:
          case 8:
            return 0.;

          case 9:
            return 2./dist2z;

            // cubic
          case 10:
          case 11:
          case 12:
          case 13:
          case 14:
          case 15:
          case 16:
            return 0.;

          case 17:
            return 2.*dx/dist2z;

          case 18:
            return 2.*dy/dist2z;

          case 19:
            return 6.*dz/dist2z;

            // quartics
          case 20:
          case 21:
          case 22:
          case 23:
          case 24:
          case 25:
          case 26:
          case 27:
          case 28:
            return 0.;

          case 29:
            return 2.*dx*dx/dist2z;

          case 30:
            return 2.*dx*dy/dist2z;

          case 31:
            return 2.*dy*dy/dist2z;

          case 32:
            return 6.*dx*dz/dist2z;

          case 33:
            return 6.*dy*dz/dist2z;

          case 34:
            return 12.*dz*dz/dist2z;

          default:
            unsigned int o = 0;
            for (; i >= (o+1)*(o+2)*(o+3)/6; o++) { }
            unsigned int i2 = i - (o*(o+1)*(o+2)/6);
            unsigned int block=o, nz = 0;
            for (; block < i2; block += (o-nz+1)) { nz++; }
            const unsigned int nx = block - i2;
            const unsigned int ny = o - nx - nz;
            Real val = nz * (nz - 1);
            for (unsigned int index=0; index != nx; index++)
              val *= dx;
            for (unsigned int index=0; index != ny; index++)
              val *= dy;
            for (unsigned int index=2; index < nz; index++)
              val *= dz;
            return val/dist2z;
          }
      }


    default:
      libmesh_error_msg("Invalid j = " << j);
    }

#else
  return 0.;
#endif
}

shape_second_deriv() [82/103]

RealGradient libMesh::FE< 0, LAGRANGE_VEC >::shape_second_deriv ( const Elem *  elem, const Order  order, const unsigned int  i, const unsigned int  j, const Point &  p  )

inherited

Definition at line 735 of file fe_lagrange_vec.C.

{
  Real value = FE<0,LAGRANGE>::shape_second_deriv( elem->type(), static_cast<Order>(order + elem->p_level()), i, j, p);
  return libMesh::RealGradient( value );
}

shape_second_deriv() [83/103]

RealGradient libMesh::FE< 1, LAGRANGE_VEC >::shape_second_deriv ( const Elem *  elem, const Order  order, const unsigned int  i, const unsigned int  j, const Point &  p  )

inherited

Definition at line 757 of file fe_lagrange_vec.C.

{
  Real value = FE<1,LAGRANGE>::shape_second_deriv( elem->type(), static_cast<Order>(order + elem->p_level()), i, j, p);
  return libMesh::RealGradient( value );
}

shape_second_deriv() [84/103]

Real libMesh::FE< 3, L2_HIERARCHIC >::shape_second_deriv ( const ElemType  , const Order  , const unsigned  int, const unsigned  int, const Point &    )

inherited

Definition at line 770 of file fe_l2_hierarchic_shape_3D.C.

{
  libmesh_error_msg("Hierarchic polynomials require the element type \nbecause edge and face orientation is needed.");
  return 0.;
}

shape_second_deriv() [85/103]

Real libMesh::FE< 3, HIERARCHIC >::shape_second_deriv ( const ElemType  , const Order  , const unsigned  int, const unsigned  int, const Point &    )

inherited

Definition at line 770 of file fe_hierarchic_shape_3D.C.

{
  libmesh_error_msg("Hierarchic polynomials require the element type \nbecause edge and face orientation is needed.");
  return 0.;
}

shape_second_deriv() [86/103]

Real libMesh::FE< 2, SUBDIVISION >::shape_second_deriv ( const ElemType  type, const Order  order, const unsigned int  i, const unsigned int  j, const Point &  p  )

inherited

Definition at line 778 of file fe_subdivision_2D.C.

{
  switch (order)
    {
    case FOURTH:
      {
        switch (type)
          {
          case TRI3SUBDIVISION:
            libmesh_assert_less(i, 12);
            return FESubdivision::regular_shape_second_deriv(i,j,p(0),p(1));
          default:
            libmesh_error_msg("ERROR: Unsupported element type!");
          }
      }
    default:
      libmesh_error_msg("ERROR: Unsupported polynomial order!");
    }
}

shape_second_deriv() [87/103]

Real libMesh::FE< 3, HIERARCHIC >::shape_second_deriv ( const Elem *  elem, const Order  order, const unsigned int  i, const unsigned int  j, const Point &  p  )

inherited

Definition at line 783 of file fe_hierarchic_shape_3D.C.

{
  libmesh_assert(elem);

  const Real eps = 1.e-6;
  Point pp, pm;
  unsigned int prevj = libMesh::invalid_uint;

  switch (j)
    {
      //  d^2()/dxi^2
    case 0:
      {
        pp = Point(p(0)+eps, p(1), p(2));
        pm = Point(p(0)-eps, p(1), p(2));
        prevj = 0;
        break;
      }

      //  d^2()/dxideta
    case 1:
      {
        pp = Point(p(0), p(1)+eps, p(2));
        pm = Point(p(0), p(1)-eps, p(2));
        prevj = 0;
        break;
      }

      //  d^2()/deta^2
    case 2:
      {
        pp = Point(p(0), p(1)+eps, p(2));
        pm = Point(p(0), p(1)-eps, p(2));
        prevj = 1;
        break;
      }

      //  d^2()/dxidzeta
    case 3:
      {
        pp = Point(p(0), p(1), p(2)+eps);
        pm = Point(p(0), p(1), p(2)-eps);
        prevj = 0;
        break;
      }

      //  d^2()/detadzeta
    case 4:
      {
        pp = Point(p(0), p(1), p(2)+eps);
        pm = Point(p(0), p(1), p(2)-eps);
        prevj = 1;
        break;
      }

      //  d^2()/dzeta^2
    case 5:
      {
        pp = Point(p(0), p(1), p(2)+eps);
        pm = Point(p(0), p(1), p(2)-eps);
        prevj = 2;
        break;
      }
    default:
      libmesh_error_msg("Invalid derivative index j = " << j);
    }

  return (FE<3,HIERARCHIC>::shape_deriv(elem, order, i, prevj, pp) -
          FE<3,HIERARCHIC>::shape_deriv(elem, order, i, prevj, pm))
    / 2. / eps;
}

shape_second_deriv() [88/103]

Real libMesh::FE< 3, L2_HIERARCHIC >::shape_second_deriv ( const Elem *  elem, const Order  order, const unsigned int  i, const unsigned int  j, const Point &  p  )

inherited

Definition at line 783 of file fe_l2_hierarchic_shape_3D.C.

{
  libmesh_assert(elem);

  const Real eps = 1.e-6;
  Point pp, pm;
  unsigned int prevj = libMesh::invalid_uint;

  switch (j)
    {
      //  d^2()/dxi^2
    case 0:
      {
        pp = Point(p(0)+eps, p(1), p(2));
        pm = Point(p(0)-eps, p(1), p(2));
        prevj = 0;
        break;
      }

      //  d^2()/dxideta
    case 1:
      {
        pp = Point(p(0), p(1)+eps, p(2));
        pm = Point(p(0), p(1)-eps, p(2));
        prevj = 0;
        break;
      }

      //  d^2()/deta^2
    case 2:
      {
        pp = Point(p(0), p(1)+eps, p(2));
        pm = Point(p(0), p(1)-eps, p(2));
        prevj = 1;
        break;
      }

      //  d^2()/dxidzeta
    case 3:
      {
        pp = Point(p(0), p(1), p(2)+eps);
        pm = Point(p(0), p(1), p(2)-eps);
        prevj = 0;
        break;
      }

      //  d^2()/detadzeta
    case 4:
      {
        pp = Point(p(0), p(1), p(2)+eps);
        pm = Point(p(0), p(1), p(2)-eps);
        prevj = 1;
        break;
      }

      //  d^2()/dzeta^2
    case 5:
      {
        pp = Point(p(0), p(1), p(2)+eps);
        pm = Point(p(0), p(1), p(2)-eps);
        prevj = 2;
        break;
      }
    default:
      libmesh_error_msg("Invalid derivative index j = " << j);
    }

  return (FE<3,L2_HIERARCHIC>::shape_deriv(elem, order, i, prevj, pp) -
          FE<3,L2_HIERARCHIC>::shape_deriv(elem, order, i, prevj, pm))
    / 2. / eps;
}

shape_second_deriv() [89/103]

Real libMesh::FE< 2, SUBDIVISION >::shape_second_deriv ( const Elem *  elem, const Order  order, const unsigned int  i, const unsigned int  j, const Point &  p  )

inherited

Definition at line 805 of file fe_subdivision_2D.C.

{
  libmesh_assert(elem);
  return FE<2,SUBDIVISION>::shape_second_deriv(elem->type(), order, i, j, p);
}

shape_second_deriv() [90/103]

RealGradient libMesh::FE< 2, LAGRANGE_VEC >::shape_second_deriv ( const Elem *  elem, const Order  order, const unsigned int  i, const unsigned int  j, const Point &  p  )

inherited

Definition at line 807 of file fe_lagrange_vec.C.

{
  Real value = FE<2,LAGRANGE>::shape_second_deriv( elem->type(), static_cast<Order>(order + elem->p_level()), i/2, j, p );

  switch( i%2 )
    {
    case 0:
      return libMesh::RealGradient( value );

    case 1:
      return libMesh::RealGradient( Real(0), value );

    default:
      libmesh_error_msg("i%2 must be either 0 or 1!");
    }

  //dummy
  return libMesh::RealGradient();
}

shape_second_deriv() [91/103]

RealGradient libMesh::FE< 3, LAGRANGE_VEC >::shape_second_deriv ( const Elem *  elem, const Order  order, const unsigned int  i, const unsigned int  j, const Point &  p  )

inherited

Definition at line 877 of file fe_lagrange_vec.C.

{
  Real value = FE<3,LAGRANGE>::shape_second_deriv( elem->type(), static_cast<Order>(order + elem->p_level()), i/3, j, p );

  switch( i%3 )
    {
    case 0:
      return libMesh::RealGradient( value );

    case 1:
      return libMesh::RealGradient( Real(0), value );

    case 2:
      return libMesh::RealGradient( Real(0), Real(0), value );

    default:
      libmesh_error_msg("i%3 must be 0, 1, or 2!");
    }

  //dummy
  return libMesh::RealGradient();
}

shape_second_deriv() [92/103]

Real libMesh::FE< 2, L2_LAGRANGE >::shape_second_deriv ( const Elem *  elem, const Order  order, const unsigned int  i, const unsigned int  j, const Point &  p  )

inherited

Definition at line 904 of file fe_l2_lagrange_shape_2D.C.

{
  libmesh_assert(elem);

  // call the orientation-independent shape functions
  return FE<2,L2_LAGRANGE>::shape_second_deriv(elem->type(), static_cast<Order>(order + elem->p_level()), i, j, p);
}

shape_second_deriv() [93/103]

Real libMesh::FE< 2, LAGRANGE >::shape_second_deriv ( const Elem *  elem, const Order  order, const unsigned int  i, const unsigned int  j, const Point &  p  )

inherited

Definition at line 908 of file fe_lagrange_shape_2D.C.

{
  libmesh_assert(elem);

  // call the orientation-independent shape functions
  return FE<2,LAGRANGE>::shape_second_deriv(elem->type(), static_cast<Order>(order + elem->p_level()), i, j, p);
}

shape_second_deriv() [94/103]

Real libMesh::FE< 3, L2_LAGRANGE >::shape_second_deriv ( const ElemType  type, const Order  order, const unsigned int  i, const unsigned int  j, const Point &  p  )

inherited

Definition at line 1210 of file fe_l2_lagrange_shape_3D.C.

{
#if LIBMESH_DIM == 3

  libmesh_assert_less (j, 6);

  switch (order)
    {
      // linear Lagrange shape functions
    case FIRST:
      {
        return 0.;
      }

      // quadratic Lagrange shape functions
    case SECOND:
      {
        switch (type)
          {

            // serendipity hexahedral quadratic shape functions
          case HEX20:
            {
              static bool warning_given_HEX20 = false;

              if (!warning_given_HEX20)
                libMesh::err << "Second derivatives for 3D Lagrangian HEX20"
                             << " elements are not yet implemented!"
                             << std::endl;
              warning_given_HEX20 = true;
            }
            libmesh_fallthrough();

          case HEX27:
            // triquadratic hexahedral shape functions
            {
              libmesh_assert_less (i, 27);

              // Compute hex shape functions as a tensor-product
              const Real xi   = p(0);
              const Real eta  = p(1);
              const Real zeta = p(2);

              // The only way to make any sense of this
              // is to look at the mgflo/mg2/mgf documentation
              // and make the cut-out cube!
              //                                0  1  2  3  4  5  6  7  8  9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26
              static const unsigned int i0[] = {0, 1, 1, 0, 0, 1, 1, 0, 2, 1, 2, 0, 0, 1, 1, 0, 2, 1, 2, 0, 2, 2, 1, 2, 0, 2, 2};
              static const unsigned int i1[] = {0, 0, 1, 1, 0, 0, 1, 1, 0, 2, 1, 2, 0, 0, 1, 1, 0, 2, 1, 2, 2, 0, 2, 1, 2, 2, 2};
              static const unsigned int i2[] = {0, 0, 0, 0, 1, 1, 1, 1, 0, 0, 0, 0, 2, 2, 2, 2, 1, 1, 1, 1, 0, 2, 2, 2, 2, 1, 2};

              switch(j)
                {
                  // d^2()/dxi^2
                case 0:
                  return (FE<1,L2_LAGRANGE>::shape_second_deriv(EDGE3, SECOND, i0[i], 0, xi)*
                          FE<1,L2_LAGRANGE>::shape      (EDGE3, SECOND, i1[i], eta)*
                          FE<1,L2_LAGRANGE>::shape      (EDGE3, SECOND, i2[i], zeta));

                  // d^2()/dxideta
                case 1:
                  return (FE<1,L2_LAGRANGE>::shape_deriv(EDGE3, SECOND, i0[i], 0, xi)*
                          FE<1,L2_LAGRANGE>::shape_deriv(EDGE3, SECOND, i1[i], 0, eta)*
                          FE<1,L2_LAGRANGE>::shape      (EDGE3, SECOND, i2[i], zeta));

                  // d^2()/deta^2
                case 2:
                  return (FE<1,L2_LAGRANGE>::shape      (EDGE3, SECOND, i0[i], xi)*
                          FE<1,L2_LAGRANGE>::shape_second_deriv(EDGE3, SECOND, i1[i], 0, eta)*
                          FE<1,L2_LAGRANGE>::shape      (EDGE3, SECOND, i2[i], zeta));

                  // d^2()/dxidzeta
                case 3:
                  return (FE<1,L2_LAGRANGE>::shape_deriv(EDGE3, SECOND, i0[i], 0, xi)*
                          FE<1,L2_LAGRANGE>::shape      (EDGE3, SECOND, i1[i], eta)*
                          FE<1,L2_LAGRANGE>::shape_deriv(EDGE3, SECOND, i2[i], 0, zeta));

                  // d^2()/detadzeta
                case 4:
                  return (FE<1,L2_LAGRANGE>::shape      (EDGE3, SECOND, i0[i], xi)*
                          FE<1,L2_LAGRANGE>::shape_deriv(EDGE3, SECOND, i1[i], 0, eta)*
                          FE<1,L2_LAGRANGE>::shape_deriv(EDGE3, SECOND, i2[i], 0, zeta));

                  // d^2()/dzeta^2
                case 5:
                  return (FE<1,L2_LAGRANGE>::shape      (EDGE3, SECOND, i0[i], xi)*
                          FE<1,L2_LAGRANGE>::shape      (EDGE3, SECOND, i1[i], eta)*
                          FE<1,L2_LAGRANGE>::shape_second_deriv(EDGE3, SECOND, i2[i], 0, zeta));

                default:
                  libmesh_error_msg("Invalid j = " << j);
                }
            }

            // quadratic tetrahedral shape functions
          case TET10:
            {
              // The area coordinates are the same as used for the
              // shape() and shape_deriv() functions.
              // const Real zeta0 = 1. - zeta1 - zeta2 - zeta3;
              // const Real zeta1 = p(0);
              // const Real zeta2 = p(1);
              // const Real zeta3 = p(2);
              static const Real dzetadxi[4][3] =
                {
                  {-1., -1., -1.},
                  {1.,   0.,  0.},
                  {0.,   1.,  0.},
                  {0.,   0.,  1.}
                };

              // Convert from j -> (j,k) indices for independent variable
              // (0=xi, 1=eta, 2=zeta)
              static const unsigned short int independent_var_indices[6][2] =
                {
                  {0, 0}, // d^2 phi / dxi^2
                  {0, 1}, // d^2 phi / dxi deta
                  {1, 1}, // d^2 phi / deta^2
                  {0, 2}, // d^2 phi / dxi dzeta
                  {1, 2}, // d^2 phi / deta dzeta
                  {2, 2}  // d^2 phi / dzeta^2
                };

              // Convert from i -> zeta indices.  Each quadratic shape
              // function for the Tet10 depends on up to two of the zeta
              // area coordinate functions (see the shape() function above).
              // This table just tells which two area coords it uses.
              static const unsigned short int zeta_indices[10][2] =
                {
                  {0, 0},
                  {1, 1},
                  {2, 2},
                  {3, 3},
                  {0, 1},
                  {1, 2},
                  {2, 0},
                  {0, 3},
                  {1, 3},
                  {2, 3},
                };

              // Look up the independent variable indices for this value of j.
              const unsigned int my_j = independent_var_indices[j][0];
              const unsigned int my_k = independent_var_indices[j][1];

              if (i<4)
                {
                  return 4.*dzetadxi[i][my_j]*dzetadxi[i][my_k];
                }

              else if (i<10)
                {
                  const unsigned short int my_m = zeta_indices[i][0];
                  const unsigned short int my_n = zeta_indices[i][1];

                  return 4.*(dzetadxi[my_n][my_j]*dzetadxi[my_m][my_k] +
                             dzetadxi[my_m][my_j]*dzetadxi[my_n][my_k] );
                }
              else
                libmesh_error_msg("Invalid shape function index " << i);
            }


            // quadratic prism shape functions
          case PRISM18:
            {
              libmesh_assert_less (i, 18);

              // Compute prism shape functions as a tensor-product
              // of a triangle and an edge

              Point p2d(p(0),p(1));
              Point p1d(p(2));

              //                                0  1  2  3  4  5  6  7  8  9 10 11 12 13 14 15 16 17
              static const unsigned int i0[] = {0, 0, 0, 1, 1, 1, 0, 0, 0, 2, 2, 2, 1, 1, 1, 2, 2, 2};
              static const unsigned int i1[] = {0, 1, 2, 0, 1, 2, 3, 4, 5, 0, 1, 2, 3, 4, 5, 3, 4, 5};

              switch (j)
                {
                  // d^2()/dxi^2
                case 0:
                  return (FE<2,L2_LAGRANGE>::shape_second_deriv(TRI6, SECOND, i1[i], 0, p2d)*
                          FE<1,L2_LAGRANGE>::shape(EDGE3, SECOND, i0[i], p1d));

                  // d^2()/dxideta
                case 1:
                  return (FE<2,L2_LAGRANGE>::shape_second_deriv(TRI6, SECOND, i1[i], 1, p2d)*
                          FE<1,L2_LAGRANGE>::shape(EDGE3, SECOND, i0[i], p1d));

                  // d^2()/deta^2
                case 2:
                  return (FE<2,L2_LAGRANGE>::shape_second_deriv(TRI6, SECOND, i1[i], 2, p2d)*
                          FE<1,L2_LAGRANGE>::shape(EDGE3, SECOND, i0[i], p1d));

                  // d^2()/dxidzeta
                case 3:
                  return (FE<2,L2_LAGRANGE>::shape_deriv(TRI6,  SECOND, i1[i], 0, p2d)*
                          FE<1,L2_LAGRANGE>::shape_deriv(EDGE3, SECOND, i0[i], 0, p1d));

                  // d^2()/detadzeta
                case 4:
                  return (FE<2,L2_LAGRANGE>::shape_deriv(TRI6,  SECOND, i1[i], 1, p2d)*
                          FE<1,L2_LAGRANGE>::shape_deriv(EDGE3, SECOND, i0[i], 0, p1d));

                  // d^2()/dzeta^2
                case 5:
                  return (FE<2,L2_LAGRANGE>::shape(TRI6,  SECOND, i1[i], p2d)*
                          FE<1,L2_LAGRANGE>::shape_second_deriv(EDGE3, SECOND, i0[i], 0, p1d));

                default:
                  libmesh_error_msg("Invalid shape function derivative j = " << j);
                }
            }



          default:
            libmesh_error_msg("ERROR: Unsupported 3D element type!: " << type);
          }
      }


      // unsupported order
    default:
      libmesh_error_msg("ERROR: Unsupported 3D FE order!: " << order);
    }

#endif
}

shape_second_deriv() [95/103]

Real libMesh::FE< 3, MONOMIAL >::shape_second_deriv ( const Elem *  elem, const Order  order, const unsigned int  i, const unsigned int  j, const Point &  p  )

inherited

Definition at line 1301 of file fe_monomial_shape_3D.C.

{
  libmesh_assert(elem);

  // call the orientation-independent shape function derivatives
  return FE<3,MONOMIAL>::shape_second_deriv(elem->type(), static_cast<Order>(order + elem->p_level()), i, j, p);
}

shape_second_deriv() [96/103]

Real libMesh::FE< 2, SZABAB >::shape_second_deriv ( const ElemType  , const Order  , const unsigned  int, const unsigned  int, const Point &    )

inherited

Definition at line 1400 of file fe_szabab_shape_2D.C.

{
  static bool warning_given = false;

  if (!warning_given)
    libMesh::err << "Second derivatives for Szabab elements "
                 << " are not yet implemented!"
                 << std::endl;

  warning_given = true;
  return 0.;
}

shape_second_deriv() [97/103]

Real libMesh::FE< 2, SZABAB >::shape_second_deriv ( const Elem *  , const Order  , const unsigned  int, const unsigned  int, const Point &    )

inherited

Definition at line 1420 of file fe_szabab_shape_2D.C.

{
  static bool warning_given = false;

  if (!warning_given)
    libMesh::err << "Second derivatives for Szabab elements "
                 << " are not yet implemented!"
                 << std::endl;

  warning_given = true;
  return 0.;
}

shape_second_deriv() [98/103]

Real libMesh::FE< 3, L2_LAGRANGE >::shape_second_deriv ( const Elem *  elem, const Order  order, const unsigned int  i, const unsigned int  j, const Point &  p  )

inherited

Definition at line 1448 of file fe_l2_lagrange_shape_3D.C.

{
  libmesh_assert(elem);

  // call the orientation-independent shape function derivatives
  return FE<3,L2_LAGRANGE>::shape_second_deriv(elem->type(), static_cast<Order>(order + elem->p_level()), i, j, p);
}

shape_second_deriv() [99/103]

Real libMesh::FE< 3, LAGRANGE >::shape_second_deriv ( const ElemType  type, const Order  order, const unsigned int  i, const unsigned int  j, const Point &  p  )

inherited

Definition at line 1929 of file fe_lagrange_shape_3D.C.

{
#if LIBMESH_DIM == 3

  libmesh_assert_less (j, 6);

  switch (order)
    {
      // linear Lagrange shape functions
    case FIRST:
      {
        switch (type)
          {
            // Linear tets have all second derivatives = 0
          case TET4:
          case TET10:
            {
              return 0.;
            }

            // The following elements use either tensor product or
            // rational basis functions, and therefore probably have
            // second derivatives, but we have not implemented them
            // yet...
          case PRISM6:
          case PRISM15:
          case PRISM18:
            {
              libmesh_assert_less (i, 6);

              // Compute prism shape functions as a tensor-product
              // of a triangle and an edge

              Point p2d(p(0),p(1));
              Point p1d(p(2));

              //                                0  1  2  3  4  5
              static const unsigned int i0[] = {0, 0, 0, 1, 1, 1};
              static const unsigned int i1[] = {0, 1, 2, 0, 1, 2};

              switch (j)
                {
                  // All repeated second derivatives and the xi-eta derivative are zero on PRISMs
                case 0: // d^2()/dxi^2
                case 1: // d^2()/dxideta
                case 2: // d^2()/deta^2
                case 5: // d^2()/dzeta^2
                  {
                    return 0.;
                  }

                case 3: // d^2()/dxidzeta
                  return (FE<2,LAGRANGE>::shape_deriv(TRI3,  FIRST, i1[i], 0, p2d)*
                          FE<1,LAGRANGE>::shape_deriv(EDGE2, FIRST, i0[i], 0, p1d));

                case 4: // d^2()/detadzeta
                  return (FE<2,LAGRANGE>::shape_deriv(TRI3,  FIRST, i1[i], 1, p2d)*
                          FE<1,LAGRANGE>::shape_deriv(EDGE2, FIRST, i0[i], 0, p1d));

                default:
                  libmesh_error_msg("Invalid j = " << j);
                }
            }

          case PYRAMID5:
          case PYRAMID13:
          case PYRAMID14:
            {
              libmesh_assert_less (i, 5);

              const Real xi   = p(0);
              const Real eta  = p(1);
              const Real zeta = p(2);
              const Real eps  = 1.e-35;

              switch (j)
                {
                  // xi-xi and eta-eta derivatives are all zero for PYRAMID5.
                case 0: // d^2()/dxi^2
                case 2: // d^2()/deta^2
                  return 0.;

                case 1: // d^2()/dxideta
                  {
                    switch (i)
                      {
                      case 0:
                      case 2:
                        return 0.25/(1. - zeta + eps);
                      case 1:
                      case 3:
                        return -0.25/(1. - zeta + eps);
                      case 4:
                        return 0.;
                      default:
                        libmesh_error_msg("Invalid i = " << i);
                      }
                  }

                case 3: // d^2()/dxidzeta
                  {
                    Real den = (1. - zeta + eps)*(1. - zeta + eps);

                    switch (i)
                      {
                      case 0:
                      case 2:
                        return 0.25*eta/den;
                      case 1:
                      case 3:
                        return -0.25*eta/den;
                      case 4:
                        return 0.;
                      default:
                        libmesh_error_msg("Invalid i = " << i);
                      }
                  }

                case 4: // d^2()/detadzeta
                  {
                    Real den = (1. - zeta + eps)*(1. - zeta + eps);

                    switch (i)
                      {
                      case 0:
                      case 2:
                        return 0.25*xi/den;
                      case 1:
                      case 3:
                        return -0.25*xi/den;
                      case 4:
                        return 0.;
                      default:
                        libmesh_error_msg("Invalid i = " << i);
                      }
                  }

                case 5: // d^2()/dzeta^2
                  {
                    Real den = (1. - zeta + eps)*(1. - zeta + eps)*(1. - zeta + eps);

                    switch (i)
                      {
                      case 0:
                      case 2:
                        return 0.5*xi*eta/den;
                      case 1:
                      case 3:
                        return -0.5*xi*eta/den;
                      case 4:
                        return 0.;
                      default:
                        libmesh_error_msg("Invalid i = " << i);
                      }
                  }

                default:
                  libmesh_error_msg("Invalid j = " << j);
                }
            }

            // Trilinear shape functions on HEX8s have nonzero mixed second derivatives
          case HEX8:
          case HEX20:
          case HEX27:
            {
              libmesh_assert_less (i, 8);

              // Compute hex shape functions as a tensor-product
              const Real xi   = p(0);
              const Real eta  = p(1);
              const Real zeta = p(2);

              static const unsigned int i0[] = {0, 1, 1, 0, 0, 1, 1, 0};
              static const unsigned int i1[] = {0, 0, 1, 1, 0, 0, 1, 1};
              static const unsigned int i2[] = {0, 0, 0, 0, 1, 1, 1, 1};

              switch (j)
                {
                  // All repeated second derivatives are zero on HEX8
                case 0: // d^2()/dxi^2
                case 2: // d^2()/deta^2
                case 5: // d^2()/dzeta^2
                  {
                    return 0.;
                  }

                case 1: // d^2()/dxideta
                  return (FE<1,LAGRANGE>::shape_deriv(EDGE2, FIRST, i0[i], 0, xi)*
                          FE<1,LAGRANGE>::shape_deriv(EDGE2, FIRST, i1[i], 0, eta)*
                          FE<1,LAGRANGE>::shape      (EDGE2, FIRST, i2[i], zeta));

                case 3: // d^2()/dxidzeta
                  return (FE<1,LAGRANGE>::shape_deriv(EDGE2, FIRST, i0[i], 0, xi)*
                          FE<1,LAGRANGE>::shape      (EDGE2, FIRST, i1[i], eta)*
                          FE<1,LAGRANGE>::shape_deriv(EDGE2, FIRST, i2[i], 0, zeta));

                case 4: // d^2()/detadzeta
                  return (FE<1,LAGRANGE>::shape      (EDGE2, FIRST, i0[i], xi)*
                          FE<1,LAGRANGE>::shape_deriv(EDGE2, FIRST, i1[i], 0, eta)*
                          FE<1,LAGRANGE>::shape_deriv(EDGE2, FIRST, i2[i], 0, zeta));

                default:
                  libmesh_error_msg("Invalid j = " << j);
                }
            }

          default:
            libmesh_error_msg("ERROR: Unsupported 3D element type!: " << type);
          }

      }

      // quadratic Lagrange shape functions
    case SECOND:
      {
        switch (type)
          {

            // serendipity hexahedral quadratic shape functions
          case HEX20:
            {
              libmesh_assert_less (i, 20);

              const Real xi   = p(0);
              const Real eta  = p(1);
              const Real zeta = p(2);

              // these functions are defined for (x,y,z) in [0,1]^3
              // so transform the locations
              const Real x = .5*(xi   + 1.);
              const Real y = .5*(eta  + 1.);
              const Real z = .5*(zeta + 1.);

              switch(j)
                {
                case 0: // d^2()/dxi^2
                  {
                    switch(i)
                      {
                      case 0:
                      case 1:
                        return (1. - y) * (1. - z);
                      case 2:
                      case 3:
                        return y * (1. - z);
                      case 4:
                      case 5:
                        return (1. - y) * z;
                      case 6:
                      case 7:
                        return y * z;
                      case 8:
                        return -2. * (1. - y) * (1. - z);
                      case 10:
                        return -2. * y * (1. - z);
                      case 16:
                        return -2. * (1. - y) * z;
                      case 18:
                        return -2. * y * z;
                      case 9:
                      case 11:
                      case 12:
                      case 13:
                      case 14:
                      case 15:
                      case 17:
                      case 19:
                        return 0;
                      default:
                        libmesh_error_msg("Invalid i = " << i);
                      }
                  }
                case 1: // d^2()/dxideta
                  {
                    switch(i)
                      {
                      case 0:
                        return (1.25 - x - y - .5*z) * (1. - z);
                      case 1:
                        return (-x + y + .5*z - .25) * (1. - z);
                      case 2:
                        return (x + y - .5*z - .75) * (1. - z);
                      case 3:
                        return (-y + x + .5*z - .25) * (1. - z);
                      case 4:
                        return -.25*z * (4.*x + 4.*y - 2.*z - 3);
                      case 5:
                        return -.25*z * (-4.*y + 4.*x + 2.*z - 1.);
                      case 6:
                        return .25*z * (-5 + 4.*x + 4.*y + 2.*z);
                      case 7:
                        return .25*z * (4.*x - 4.*y - 2.*z + 1.);
                      case 8:
                        return (-1. + 2.*x) * (1. - z);
                      case 9:
                        return (1. - 2.*y) * (1. - z);
                      case 10:
                        return (1. - 2.*x) * (1. - z);
                      case 11:
                        return (-1. + 2.*y) * (1. - z);
                      case 12:
                        return z * (1. - z);
                      case 13:
                        return -z * (1. - z);
                      case 14:
                        return z * (1. - z);
                      case 15:
                        return -z * (1. - z);
                      case 16:
                        return (-1. + 2.*x) * z;
                      case 17:
                        return (1. - 2.*y) * z;
                      case 18:
                        return (1. - 2.*x) * z;
                      case 19:
                        return (-1. + 2.*y) * z;
                      default:
                        libmesh_error_msg("Invalid i = " << i);
                      }
                  }
                case 2: // d^2()/deta^2
                  switch(i)
                    {
                    case 0:
                    case 3:
                      return (1. - x) * (1. - z);
                    case 1:
                    case 2:
                      return x * (1. - z);
                    case 4:
                    case 7:
                      return (1. - x) * z;
                    case 5:
                    case 6:
                      return x * z;
                    case 9:
                      return -2. * x * (1. - z);
                    case 11:
                      return -2. * (1. - x) * (1. - z);
                    case 17:
                      return -2. * x * z;
                    case 19:
                      return -2. * (1. - x) * z;
                    case 8:
                    case 10:
                    case 12:
                    case 13:
                    case 14:
                    case 15:
                    case 16:
                    case 18:
                      return 0.;
                    default:
                      libmesh_error_msg("Invalid i = " << i);
                    }
                case 3: // d^2()/dxidzeta
                  switch(i)
                    {
                    case 0:
                      return (1.25 - x - .5*y - z) * (1. - y);
                    case 1:
                      return (-x + .5*y + z - .25) * (1. - y);
                    case 2:
                      return -.25*y * (2.*y + 4.*x - 4.*z - 1.);
                    case 3:
                      return -.25*y * (-2.*y + 4.*x + 4.*z - 3);
                    case 4:
                      return (-z + x + .5*y - .25) * (1. - y);
                    case 5:
                      return (x - .5*y + z - .75) * (1. - y);
                    case 6:
                      return .25*y * (2.*y + 4.*x + 4.*z - 5);
                    case 7:
                      return .25*y * (-2.*y + 4.*x - 4.*z + 1.);
                    case 8:
                      return (-1. + 2.*x) * (1. - y);
                    case 9:
                      return -y * (1. - y);
                    case 10:
                      return (-1. + 2.*x) * y;
                    case 11:
                      return y * (1. - y);
                    case 12:
                      return (-1. + 2.*z) * (1. - y);
                    case 13:
                      return (1. - 2.*z) * (1. - y);
                    case 14:
                      return (1. - 2.*z) * y;
                    case 15:
                      return (-1. + 2.*z) * y;
                    case 16:
                      return (1. - 2.*x) * (1. - y);
                    case 17:
                      return y * (1. - y);
                    case 18:
                      return (1. - 2.*x) * y;
                    case 19:
                      return -y * (1. - y);
                    default:
                      libmesh_error_msg("Invalid i = " << i);
                    }
                case 4: // d^2()/detadzeta
                  switch(i)
                    {
                    case 0:
                      return (1.25 - .5*x - y - z) * (1. - x);
                    case 1:
                      return .25*x * (2.*x - 4.*y - 4.*z + 3.);
                    case 2:
                      return -.25*x * (2.*x + 4.*y - 4.*z - 1.);
                    case 3:
                      return (-y + .5*x + z - .25) * (1. - x);
                    case 4:
                      return (-z + .5*x + y - .25) * (1. - x);
                    case 5:
                      return -.25*x * (2.*x - 4.*y + 4.*z - 1.);
                    case 6:
                      return .25*x * (2.*x + 4.*y + 4.*z - 5);
                    case 7:
                      return (y - .5*x + z - .75) * (1. - x);
                    case 8:
                      return x * (1. - x);
                    case 9:
                      return (-1. + 2.*y) * x;
                    case 10:
                      return -x * (1. - x);
                    case 11:
                      return (-1. + 2.*y) * (1. - x);
                    case 12:
                      return (-1. + 2.*z) * (1. - x);
                    case 13:
                      return (-1. + 2.*z) * x;
                    case 14:
                      return (1. - 2.*z) * x;
                    case 15:
                      return (1. - 2.*z) * (1. - x);
                    case 16:
                      return -x * (1. - x);
                    case 17:
                      return (1. - 2.*y) * x;
                    case 18:
                      return x * (1. - x);
                    case 19:
                      return (1. - 2.*y) * (1. - x);
                    default:
                      libmesh_error_msg("Invalid i = " << i);
                    }
                case 5: // d^2()/dzeta^2
                  switch(i)
                    {
                    case 0:
                    case 4:
                      return (1. - x) * (1. - y);
                    case 1:
                    case 5:
                      return x * (1. - y);
                    case 2:
                    case 6:
                      return x * y;
                    case 3:
                    case 7:
                      return (1. - x) * y;
                    case 12:
                      return -2. * (1. - x) * (1. - y);
                    case 13:
                      return -2. * x * (1. - y);
                    case 14:
                      return -2. * x * y;
                    case 15:
                      return -2. * (1. - x) * y;
                    case 8:
                    case 9:
                    case 10:
                    case 11:
                    case 16:
                    case 17:
                    case 18:
                    case 19:
                      return 0.;
                    default:
                      libmesh_error_msg("Invalid i = " << i);
                    }
                default:
                  libmesh_error_msg("Invalid j = " << j);
                }
            }

            // triquadratic hexahedral shape functions
          case HEX27:
            {
              libmesh_assert_less (i, 27);

              // Compute hex shape functions as a tensor-product
              const Real xi   = p(0);
              const Real eta  = p(1);
              const Real zeta = p(2);

              // The only way to make any sense of this
              // is to look at the mgflo/mg2/mgf documentation
              // and make the cut-out cube!
              //                                0  1  2  3  4  5  6  7  8  9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26
              static const unsigned int i0[] = {0, 1, 1, 0, 0, 1, 1, 0, 2, 1, 2, 0, 0, 1, 1, 0, 2, 1, 2, 0, 2, 2, 1, 2, 0, 2, 2};
              static const unsigned int i1[] = {0, 0, 1, 1, 0, 0, 1, 1, 0, 2, 1, 2, 0, 0, 1, 1, 0, 2, 1, 2, 2, 0, 2, 1, 2, 2, 2};
              static const unsigned int i2[] = {0, 0, 0, 0, 1, 1, 1, 1, 0, 0, 0, 0, 2, 2, 2, 2, 1, 1, 1, 1, 0, 2, 2, 2, 2, 1, 2};

              switch(j)
                {
                  // d^2()/dxi^2
                case 0:
                  return (FE<1,LAGRANGE>::shape_second_deriv(EDGE3, SECOND, i0[i], 0, xi)*
                          FE<1,LAGRANGE>::shape      (EDGE3, SECOND, i1[i], eta)*
                          FE<1,LAGRANGE>::shape      (EDGE3, SECOND, i2[i], zeta));

                  // d^2()/dxideta
                case 1:
                  return (FE<1,LAGRANGE>::shape_deriv(EDGE3, SECOND, i0[i], 0, xi)*
                          FE<1,LAGRANGE>::shape_deriv(EDGE3, SECOND, i1[i], 0, eta)*
                          FE<1,LAGRANGE>::shape      (EDGE3, SECOND, i2[i], zeta));

                  // d^2()/deta^2
                case 2:
                  return (FE<1,LAGRANGE>::shape      (EDGE3, SECOND, i0[i], xi)*
                          FE<1,LAGRANGE>::shape_second_deriv(EDGE3, SECOND, i1[i], 0, eta)*
                          FE<1,LAGRANGE>::shape      (EDGE3, SECOND, i2[i], zeta));

                  // d^2()/dxidzeta
                case 3:
                  return (FE<1,LAGRANGE>::shape_deriv(EDGE3, SECOND, i0[i], 0, xi)*
                          FE<1,LAGRANGE>::shape      (EDGE3, SECOND, i1[i], eta)*
                          FE<1,LAGRANGE>::shape_deriv(EDGE3, SECOND, i2[i], 0, zeta));

                  // d^2()/detadzeta
                case 4:
                  return (FE<1,LAGRANGE>::shape      (EDGE3, SECOND, i0[i], xi)*
                          FE<1,LAGRANGE>::shape_deriv(EDGE3, SECOND, i1[i], 0, eta)*
                          FE<1,LAGRANGE>::shape_deriv(EDGE3, SECOND, i2[i], 0, zeta));

                  // d^2()/dzeta^2
                case 5:
                  return (FE<1,LAGRANGE>::shape      (EDGE3, SECOND, i0[i], xi)*
                          FE<1,LAGRANGE>::shape      (EDGE3, SECOND, i1[i], eta)*
                          FE<1,LAGRANGE>::shape_second_deriv(EDGE3, SECOND, i2[i], 0, zeta));

                default:
                  libmesh_error_msg("Invalid j = " << j);
                }
            }

            // quadratic tetrahedral shape functions
          case TET10:
            {
              // The area coordinates are the same as used for the
              // shape() and shape_deriv() functions.
              // const Real zeta0 = 1. - zeta1 - zeta2 - zeta3;
              // const Real zeta1 = p(0);
              // const Real zeta2 = p(1);
              // const Real zeta3 = p(2);
              static const Real dzetadxi[4][3] =
                {
                  {-1., -1., -1.},
                  {1.,   0.,  0.},
                  {0.,   1.,  0.},
                  {0.,   0.,  1.}
                };

              // Convert from j -> (j,k) indices for independent variable
              // (0=xi, 1=eta, 2=zeta)
              static const unsigned short int independent_var_indices[6][2] =
                {
                  {0, 0}, // d^2 phi / dxi^2
                  {0, 1}, // d^2 phi / dxi deta
                  {1, 1}, // d^2 phi / deta^2
                  {0, 2}, // d^2 phi / dxi dzeta
                  {1, 2}, // d^2 phi / deta dzeta
                  {2, 2}  // d^2 phi / dzeta^2
                };

              // Convert from i -> zeta indices.  Each quadratic shape
              // function for the Tet10 depends on up to two of the zeta
              // area coordinate functions (see the shape() function above).
              // This table just tells which two area coords it uses.
              static const unsigned short int zeta_indices[10][2] =
                {
                  {0, 0},
                  {1, 1},
                  {2, 2},
                  {3, 3},
                  {0, 1},
                  {1, 2},
                  {2, 0},
                  {0, 3},
                  {1, 3},
                  {2, 3},
                };

              // Look up the independent variable indices for this value of j.
              const unsigned int my_j = independent_var_indices[j][0];
              const unsigned int my_k = independent_var_indices[j][1];

              if (i<4)
                {
                  return 4.*dzetadxi[i][my_j]*dzetadxi[i][my_k];
                }

              else if (i<10)
                {
                  const unsigned short int my_m = zeta_indices[i][0];
                  const unsigned short int my_n = zeta_indices[i][1];

                  return 4.*(dzetadxi[my_n][my_j]*dzetadxi[my_m][my_k] +
                             dzetadxi[my_m][my_j]*dzetadxi[my_n][my_k] );
                }
              else
                libmesh_error_msg("Invalid shape function index " << i);
            }



            // "serendipity" prism
          case PRISM15:
            {
              libmesh_assert_less (i, 15);

              const Real xi   = p(0);
              const Real eta  = p(1);
              const Real zeta = p(2);

              switch (j)
                {
                  // d^2()/dxi^2
                case 0:
                  {
                    switch(i)
                      {
                      case 0:
                      case 1:
                        return 2.*(1. - zeta);
                      case 2:
                      case 5:
                      case 7:
                      case 8:
                      case 9:
                      case 10:
                      case 11:
                      case 13:
                      case 14:
                        return 0.;
                      case 3:
                      case 4:
                        return 2.*(1. + zeta);
                      case 6:
                        return 4.*(zeta - 1);
                      case 12:
                        return -4.*(1. + zeta);
                      default:
                        libmesh_error_msg("Invalid i = " << i);
                      }
                  }

                  // d^2()/dxideta
                case 1:
                  {
                    switch(i)
                      {
                      case 0:
                      case 7:
                        return 2.*(1. - zeta);
                      case 1:
                      case 2:
                      case 4:
                      case 5:
                      case 9:
                      case 10:
                      case 11:
                        return 0.;
                      case 3:
                      case 13:
                        return 2.*(1. + zeta);
                      case 6:
                      case 8:
                        return 2.*(zeta - 1.);
                      case 12:
                      case 14:
                        return -2.*(1. + zeta);
                      default:
                        libmesh_error_msg("Invalid i = " << i);
                      }
                  }

                  // d^2()/deta^2
                case 2:
                  {
                    switch(i)
                      {
                      case 0:
                      case 2:
                        return 2.*(1. - zeta);
                      case 1:
                      case 4:
                      case 6:
                      case 7:
                      case 9:
                      case 10:
                      case 11:
                      case 12:
                      case 13:
                        return 0.;
                      case 3:
                      case 5:
                        return 2.*(1. + zeta);
                      case 8:
                        return 4.*(zeta - 1.);
                      case 14:
                        return -4.*(1. + zeta);
                      default:
                        libmesh_error_msg("Invalid i = " << i);
                      }
                  }

                  // d^2()/dxidzeta
                case 3:
                  {
                    switch(i)
                      {
                      case 0:
                        return 1.5 - zeta - 2.*xi - 2.*eta;
                      case 1:
                        return 0.5 + zeta - 2.*xi;
                      case 2:
                      case 5:
                      case 11:
                        return 0.;
                      case 3:
                        return -1.5 - zeta + 2.*xi + 2.*eta;
                      case 4:
                        return -0.5 + zeta + 2.*xi;
                      case 6:
                        return 4.*xi + 2.*eta - 2.;
                      case 7:
                        return -2.*eta;
                      case 8:
                        return 2.*eta;
                      case 9:
                        return 2.*zeta;
                      case 10:
                        return -2.*zeta;
                      case 12:
                        return -4.*xi - 2.*eta + 2.;
                      case 13:
                        return 2.*eta;
                      case 14:
                        return -2.*eta;
                      default:
                        libmesh_error_msg("Invalid i = " << i);
                      }
                  }

                  // d^2()/detadzeta
                case 4:
                  {
                    switch(i)
                      {
                      case 0:
                        return 1.5 - zeta - 2.*xi - 2.*eta;
                      case 1:
                      case 4:
                      case 10:
                        return 0.;
                      case 2:
                        return .5 + zeta - 2.*eta;
                      case 3:
                        return -1.5 - zeta + 2.*xi + 2.*eta;
                      case 5:
                        return -.5 + zeta + 2.*eta;
                      case 6:
                        return 2.*xi;
                      case 7:
                        return -2.*xi;
                      case 8:
                        return 2.*xi + 4.*eta - 2.;
                      case 9:
                        return 2.*zeta;
                      case 11:
                        return -2.*zeta;
                      case 12:
                        return -2.*xi;
                      case 13:
                        return 2.*xi;
                      case 14:
                        return -2.*xi - 4.*eta + 2.;
                      default:
                        libmesh_error_msg("Invalid i = " << i);
                      }
                  }

                  // d^2()/dzeta^2
                case 5:
                  {
                    switch(i)
                      {
                      case 0:
                      case 3:
                        return 1. - xi - eta;
                      case 1:
                      case 4:
                        return xi;
                      case 2:
                      case 5:
                        return eta;
                      case 6:
                      case 7:
                      case 8:
                      case 12:
                      case 13:
                      case 14:
                        return 0.;
                      case 9:
                        return 2.*xi + 2.*eta - 2.;
                      case 10:
                        return -2.*xi;
                      case 11:
                        return -2.*eta;
                      default:
                        libmesh_error_msg("Invalid i = " << i);
                      }
                  }

                default:
                  libmesh_error_msg("Invalid j = " << j);
                }
            }



            // quadratic prism shape functions
          case PRISM18:
            {
              libmesh_assert_less (i, 18);

              // Compute prism shape functions as a tensor-product
              // of a triangle and an edge

              Point p2d(p(0),p(1));
              Point p1d(p(2));

              //                                0  1  2  3  4  5  6  7  8  9 10 11 12 13 14 15 16 17
              static const unsigned int i0[] = {0, 0, 0, 1, 1, 1, 0, 0, 0, 2, 2, 2, 1, 1, 1, 2, 2, 2};
              static const unsigned int i1[] = {0, 1, 2, 0, 1, 2, 3, 4, 5, 0, 1, 2, 3, 4, 5, 3, 4, 5};

              switch (j)
                {
                  // d^2()/dxi^2
                case 0:
                  return (FE<2,LAGRANGE>::shape_second_deriv(TRI6, SECOND, i1[i], 0, p2d)*
                          FE<1,LAGRANGE>::shape(EDGE3, SECOND, i0[i], p1d));

                  // d^2()/dxideta
                case 1:
                  return (FE<2,LAGRANGE>::shape_second_deriv(TRI6, SECOND, i1[i], 1, p2d)*
                          FE<1,LAGRANGE>::shape(EDGE3, SECOND, i0[i], p1d));

                  // d^2()/deta^2
                case 2:
                  return (FE<2,LAGRANGE>::shape_second_deriv(TRI6, SECOND, i1[i], 2, p2d)*
                          FE<1,LAGRANGE>::shape(EDGE3, SECOND, i0[i], p1d));

                  // d^2()/dxidzeta
                case 3:
                  return (FE<2,LAGRANGE>::shape_deriv(TRI6,  SECOND, i1[i], 0, p2d)*
                          FE<1,LAGRANGE>::shape_deriv(EDGE3, SECOND, i0[i], 0, p1d));

                  // d^2()/detadzeta
                case 4:
                  return (FE<2,LAGRANGE>::shape_deriv(TRI6,  SECOND, i1[i], 1, p2d)*
                          FE<1,LAGRANGE>::shape_deriv(EDGE3, SECOND, i0[i], 0, p1d));

                  // d^2()/dzeta^2
                case 5:
                  return (FE<2,LAGRANGE>::shape(TRI6,  SECOND, i1[i], p2d)*
                          FE<1,LAGRANGE>::shape_second_deriv(EDGE3, SECOND, i0[i], 0, p1d));

                default:
                  libmesh_error_msg("Invalid shape function derivative j = " << j);
                }
            }


            // Quadratic shape functions, as defined in R. Graglia, "Higher order
            // bases on pyramidal elements", IEEE Trans Antennas and Propagation,
            // vol 47, no 5, May 1999.
          case PYRAMID14:
            {
              libmesh_assert_less (i, 14);

              const Real xi   = p(0);
              const Real eta  = p(1);
              const Real zeta = p(2);
              const Real eps  = 1.e-35;

              // The "normalized coordinates" defined by Graglia.  These are
              // the planes which define the faces of the pyramid.
              Real
                p1 = 0.5*(1. - eta - zeta), // back
                p2 = 0.5*(1. + xi  - zeta), // left
                p3 = 0.5*(1. + eta - zeta), // front
                p4 = 0.5*(1. - xi  - zeta); // right

              // Denominators are perturbed by epsilon to avoid
              // divide-by-zero issues.
              Real
                den = (-1. + zeta + eps),
                den2 = den*den,
                den3 = den2*den,
                den4 = den2*den2;

              // These terms are used in several of the derivatives
              Real
                numer_mp = xi*eta - zeta + zeta*zeta,
                numer_pm = xi*eta + zeta - zeta*zeta;

              switch (j)
                {
                case 0: // d^2()/dxi^2
                  {
                    switch(i)
                      {
                      case 0:
                      case 1:
                        return -p1*eta/den2;
                      case 2:
                      case 3:
                        return p3*eta/den2;
                      case 4:
                      case 9:
                      case 10:
                      case 11:
                      case 12:
                        return 0.;
                      case 5:
                        return 2.*p1*eta/den2;
                      case 6:
                      case 8:
                        return 4.*p1*p3/den2;
                      case 7:
                        return -2.*p3*eta/den2;
                      case 13:
                        return -8.*p1*p3/den2;
                      default:
                        libmesh_error_msg("Invalid i = " << i);
                      }
                  }

                case 1: // d^2()/dxideta
                  {
                    switch(i)
                      {
                      case 0:
                        return 0.25*numer_mp/den2
                          - 0.5*p1*xi/den2
                          - 0.5*p4*eta/den2
                          + p4*p1/den2;

                      case 1:
                        return 0.25*numer_pm/den2
                          - 0.5*p1*xi/den2
                          + 0.5*p2*eta/den2
                          - p1*p2/den2;

                      case 2:
                        return 0.25*numer_mp/den2
                          + 0.5*p3*xi/den2
                          + 0.5*p2*eta/den2
                          + p2*p3/den2;

                      case 3:
                        return 0.25*numer_pm/den2
                          + 0.5*p3*xi/den2
                          - 0.5*p4*eta/den2
                          - p3*p4/den2;

                      case 4:
                        return 0.;

                      case 5:
                        return p4*eta/den2
                          - 2.*p4*p1/den2
                          - p2*eta/den2
                          + 2.*p1*p2/den2;

                      case 6:
                        return -p3*xi/den2
                          + p1*xi/den2
                          - 2.*p2*p3/den2
                          + 2.*p1*p2/den2;

                      case 7:
                        return p4*eta/den2
                          + 2.*p3*p4/den2
                          - p2*eta/den2
                          - 2.*p2*p3/den2;

                      case 8:
                        return -p3*xi/den2
                          + p1*xi/den2
                          - 2.*p4*p1/den2
                          + 2.*p3*p4/den2;

                      case 9:
                      case 11:
                        return -zeta/den;

                      case 10:
                      case 12:
                        return zeta/den;

                      case 13:
                        return 4.*p4*p1/den2
                          - 4.*p3*p4/den2
                          + 4.*p2*p3/den2
                          - 4.*p1*p2/den2;

                      default:
                        libmesh_error_msg("Invalid i = " << i);
                      }
                  }


                case 2: // d^2()/deta^2
                  {
                    switch(i)
                      {
                      case 0:
                      case 3:
                        return -p4*xi/den2;
                      case 1:
                      case 2:
                        return p2*xi/den2;
                      case 4:
                      case 9:
                      case 10:
                      case 11:
                      case 12:
                        return 0.;
                      case 5:
                      case 7:
                        return 4.*p2*p4/den2;
                      case 6:
                        return -2.*p2*xi/den2;
                      case 8:
                        return 2.*p4*xi/den2;
                      case 13:
                        return -8.*p2*p4/den2;
                      default:
                        libmesh_error_msg("Invalid i = " << i);
                      }
                  }


                case 3: // d^2()/dxidzeta
                  {
                    switch(i)
                      {
                      case 0:
                        return 0.25*numer_mp/den2
                          - 0.5*p1*(2.*zeta - 1.)/den2
                          + p1*numer_mp/den3
                          - 0.5*p1*eta/den2
                          - 0.5*p4*eta/den2
                          - 2.*p4*p1*eta/den3;

                      case 1:
                        return 0.25*numer_pm/den2
                          - 0.5*p1*(1 - 2.*zeta)/den2
                          + p1*numer_pm/den3
                          + 0.5*p2*eta/den2
                          + 0.5*p1*eta/den2
                          + 2.*p1*p2*eta/den3;

                      case 2:
                        return -0.25*numer_mp/den2
                          + 0.5*p3*(2.*zeta - 1.)/den2
                          - p3*numer_mp/den3
                          - 0.5*p3*eta/den2
                          - 0.5*p2*eta/den2
                          - 2.*p2*p3*eta/den3;

                      case 3:
                        return -0.25*numer_pm/den2
                          + 0.5*p3*(1 - 2.*zeta)/den2
                          - p3*numer_pm/den3
                          + 0.5*p4*eta/den2
                          + 0.5*p3*eta/den2
                          + 2.*p3*p4*eta/den3;

                      case 4:
                        return 0.;

                      case 5:
                        return p4*eta/den2
                          + 4.*p4*p1*eta/den3
                          - p2*eta/den2
                          - 4.*p1*p2*eta/den3;

                      case 6:
                        return -p3*xi/den2
                          - p1*xi/den2
                          - 4.*p1*p3*xi/den3
                          - 2.*p2*p3/den2
                          - 2.*p1*p3/den2
                          - 2.*p1*p2/den2
                          - 8.*p1*p2*p3/den3;

                      case 7:
                        return -p4*eta/den2
                          - 4.*p3*p4*eta/den3
                          + p2*eta/den2
                          + 4.*p2*p3*eta/den3;

                      case 8:
                        return -p3*xi/den2
                          - p1*xi/den2
                          - 4.*p1*p3*xi/den3
                          + 2.*p4*p1/den2
                          + 2.*p1*p3/den2
                          + 2.*p3*p4/den2
                          + 8.*p3*p4*p1/den3;

                      case 9:
                        return -zeta/den
                          + 2.*p1/den
                          - 2.*p1*zeta/den2;

                      case 10:
                        return zeta/den
                          - 2.*p1/den
                          + 2.*p1*zeta/den2;

                      case 11:
                        return zeta/den
                          - 2.*p3/den
                          + 2.*p3*zeta/den2;

                      case 12:
                        return -zeta/den
                          + 2.*p3/den
                          - 2.*p3*zeta/den2;

                      case 13:
                        return -4.*p4*p1/den2
                          - 4.*p3*p4/den2
                          - 16.*p3*p4*p1/den3
                          + 4.*p2*p3/den2
                          + 4.*p1*p2/den2
                          + 16.*p1*p2*p3/den3;

                      default:
                        libmesh_error_msg("Invalid i = " << i);
                      }
                  }

                case 4: // d^2()/detadzeta
                  {
                    switch(i)
                      {
                      case 0:
                        return 0.25*numer_mp/den2
                          - 0.5*p4*(2.*zeta - 1.)/den2
                          + p4*numer_mp/den3
                          - 0.5*p1*xi/den2
                          - 0.5*p4*xi/den2
                          - 2.*p4*p1*xi/den3;

                      case 1:
                        return -0.25*numer_pm/den2
                          + 0.5*p2*(1. - 2.*zeta)/den2
                          - p2*numer_pm/den3
                          + 0.5*p2*xi/den2
                          + 0.5*p1*xi/den2
                          + 2.*p1*p2*xi/den3;

                      case 2:
                        return -0.25*numer_mp/den2
                          + 0.5*p2*(2.*zeta - 1.)/den2
                          - p2*numer_mp/den3
                          - 0.5*p3*xi/den2
                          - 0.5*p2*xi/den2
                          - 2.*p2*p3*xi/den3;

                      case 3:
                        return 0.25*numer_pm/den2
                          - 0.5*p4*(1. - 2.*zeta)/den2
                          + p4*numer_pm/den3
                          + 0.5*p4*xi/den2
                          + 0.5*p3*xi/den2
                          + 2.*p3*p4*xi/den3;

                      case 4:
                        return 0.;

                      case 5:
                        return -p4*eta/den2
                          - p2*eta/den2
                          - 4.*p2*p4*eta/den3
                          + 2.*p4*p1/den2
                          + 2.*p2*p4/den2
                          + 2.*p1*p2/den2
                          + 8.*p2*p1*p4/den3;

                      case 6:
                        return p3*xi/den2
                          + 4.*p2*p3*xi/den3
                          - p1*xi/den2
                          - 4.*p1*p2*xi/den3;

                      case 7:
                        return -p4*eta/den2
                          - p2*eta/den2
                          - 4.*p2*p4*eta/den3
                          - 2.*p3*p4/den2
                          - 2.*p2*p4/den2
                          - 2.*p2*p3/den2
                          - 8.*p2*p3*p4/den3;

                      case 8:
                        return p1*xi/den2
                          + 4.*p4*p1*xi/den3
                          - p3*xi/den2
                          - 4.*p3*p4*xi/den3;

                      case 9:
                        return -zeta/den
                          + 2.*p4/den
                          - 2.*p4*zeta/den2;

                      case 10:
                        return -zeta/den
                          + 2.*p2/den
                          - 2.*p2*zeta/den2;

                      case 11:
                        return zeta/den
                          - 2.*p2/den
                          + 2.*p2*zeta/den2;

                      case 12:
                        return zeta/den
                          - 2.*p4/den
                          + 2.*p4*zeta/den2;

                      case 13:
                        return 4.*p3*p4/den2
                          + 4.*p2*p3/den2
                          + 16.*p2*p3*p4/den3
                          - 4.*p4*p1/den2
                          - 4.*p1*p2/den2
                          - 16.*p2*p1*p4/den3;

                      default:
                        libmesh_error_msg("Invalid i = " << i);
                      }
                  }

                case 5: // d^2()/dzeta^2
                  {
                    switch(i)
                      {
                      case 0:
                        return 0.5*numer_mp/den2
                          - p1*(2.*zeta - 1.)/den2
                          + 2.*p1*numer_mp/den3
                          - p4*(2.*zeta - 1.)/den2
                          + 2.*p4*numer_mp/den3
                          + 2.*p4*p1/den2
                          - 4.*p4*p1*(2.*zeta - 1.)/den3
                          + 6.*p4*p1*numer_mp/den4;

                      case 1:
                        return -0.5*numer_pm/den2
                          + p2*(1 - 2.*zeta)/den2
                          - 2.*p2*numer_pm/den3
                          + p1*(1 - 2.*zeta)/den2
                          - 2.*p1*numer_pm/den3
                          + 2.*p1*p2/den2
                          + 4.*p1*p2*(1 - 2.*zeta)/den3
                          - 6.*p1*p2*numer_pm/den4;

                      case 2:
                        return 0.5*numer_mp/den2
                          - p3*(2.*zeta - 1.)/den2
                          + 2.*p3*numer_mp/den3
                          - p2*(2.*zeta - 1.)/den2
                          + 2.*p2*numer_mp/den3
                          + 2.*p2*p3/den2
                          - 4.*p2*p3*(2.*zeta - 1.)/den3
                          + 6.*p2*p3*numer_mp/den4;

                      case 3:
                        return -0.5*numer_pm/den2
                          + p4*(1 - 2.*zeta)/den2
                          - 2.*p4*numer_pm/den3
                          + p3*(1 - 2.*zeta)/den2
                          - 2.*p3*numer_pm/den3
                          + 2.*p3*p4/den2
                          + 4.*p3*p4*(1 - 2.*zeta)/den3
                          - 6.*p3*p4*numer_pm/den4;

                      case 4:
                        return 4.;

                      case 5:
                        return -2.*p1*eta/den2
                          - 2.*p4*eta/den2
                          - 8.*p4*p1*eta/den3
                          - 2.*p2*eta/den2
                          - 8.*p2*p4*eta/den3
                          - 8.*p1*p2*eta/den3
                          - 24.*p2*p1*p4*eta/den4;

                      case 6:
                        return 2.*p3*xi/den2
                          + 2.*p2*xi/den2
                          + 8.*p2*p3*xi/den3
                          + 2.*p1*xi/den2
                          + 8.*p1*p3*xi/den3
                          + 8.*p1*p2*xi/den3
                          + 24.*p1*p2*p3*xi/den4;

                      case 7:
                        return 2.*p4*eta/den2
                          + 2.*p3*eta/den2
                          + 8.*p3*p4*eta/den3
                          + 2.*p2*eta/den2
                          + 8.*p2*p4*eta/den3
                          + 8.*p2*p3*eta/den3
                          + 24.*p2*p3*p4*eta/den4;

                      case 8:
                        return -2.*p1*xi/den2
                          - 2.*p4*xi/den2
                          - 8.*p4*p1*xi/den3
                          - 2.*p3*xi/den2
                          - 8.*p1*p3*xi/den3
                          - 8.*p3*p4*xi/den3
                          - 24.*p3*p4*p1*xi/den4;

                      case 9:
                        return -2.*zeta/den
                          + 4.*p4/den
                          - 4.*p4*zeta/den2
                          + 4.*p1/den
                          - 4.*p1*zeta/den2
                          + 8.*p4*p1/den2
                          - 8.*p1*p4*zeta/den3;

                      case 10:
                        return -2.*zeta/den
                          + 4.*p1/den
                          - 4.*p1*zeta/den2
                          + 4.*p2/den
                          - 4.*p2*zeta/den2
                          + 8.*p1*p2/den2
                          - 8.*p2*p1*zeta/den3;

                      case 11:
                        return -2.*zeta/den
                          + 4.*p2/den
                          - 4.*p2*zeta/den2
                          + 4.*p3/den
                          - 4.*p3*zeta/den2
                          + 8.*p2*p3/den2
                          - 8.*p3*p2*zeta/den3;

                      case 12:
                        return -2.*zeta/den
                          + 4.*p3/den
                          - 4.*p3*zeta/den2
                          + 4.*p4/den
                          - 4.*p4*zeta/den2
                          + 8.*p3*p4/den2
                          - 8.*p4*p3*zeta/den3;

                      case 13:
                        return 8.*p3*p4/den2
                          + 8.*p2*p4/den2
                          + 8.*p2*p3/den2
                          + 32.*p2*p3*p4/den3
                          + 8.*p4*p1/den2
                          + 8.*p1*p3/den2
                          + 32.*p3*p4*p1/den3
                          + 8.*p1*p2/den2
                          + 32.*p2*p1*p4/den3
                          + 32.*p1*p2*p3/den3
                          + 96.*p1*p2*p3*p4/den4;

                      default:
                        libmesh_error_msg("Invalid i = " << i);
                      }
                  }

                default:
                  libmesh_error_msg("Invalid j = " << j);
                }
            }

            // G. Bedrosian, "Shape functions and integration formulas for
            // three-dimensional finite element analysis", Int. J. Numerical
            // Methods Engineering, vol 35, p. 95-108, 1992.
          case PYRAMID13:
            {
              libmesh_assert_less (i, 13);

              const Real xi   = p(0);
              const Real eta  = p(1);
              const Real zeta = p(2);
              const Real eps  = 1.e-35;

              // Denominators are perturbed by epsilon to avoid
              // divide-by-zero issues.
              Real
                den = (-1. + zeta + eps),
                den2 = den*den,
                den3 = den2*den,
                xi2 = xi*xi,
                eta2 = eta*eta,
                zeta2 = zeta*zeta,
                zeta3 = zeta2*zeta;

              switch (j)
                {
                case 0: // d^2()/dxi^2
                  {
                    switch(i)
                      {
                      case 0:
                      case 1:
                        return 0.5*(-1. + zeta + eta)/den;

                      case 2:
                      case 3:
                        return 0.5*(-1. + zeta - eta)/den;

                      case 4:
                      case 6:
                      case 8:
                      case 9:
                      case 10:
                      case 11:
                      case 12:
                        return 0.;

                      case 5:
                        return (1. - eta - zeta)/den;

                      case 7:
                        return (1. + eta - zeta)/den;

                      default:
                        libmesh_error_msg("Invalid i = " << i);
                      }
                  }

                case 1: // d^2()/dxideta
                  {
                    switch(i)
                      {
                      case 0:
                        return  0.25*(-1. + 2.*zeta + 2.*xi + 2.*eta)/den;

                      case 1:
                        return -0.25*(-1. + 2.*zeta - 2.*xi + 2.*eta)/den;

                      case 2:
                        return -0.25*(1. - 2.*zeta + 2.*xi + 2.*eta)/den;

                      case 3:
                        return  0.25*(1. - 2.*zeta - 2.*xi + 2.*eta)/den;

                      case 4:
                        return 0.;

                      case 5:
                        return -xi/den;

                      case 6:
                        return eta/den;

                      case 7:
                        return xi/den;

                      case 8:
                        return -eta/den;

                      case 9:
                        return -zeta/den;

                      case 10:
                        return zeta/den;

                      case 11:
                        return -zeta/den;

                      case 12:
                        return zeta/den;

                      default:
                        libmesh_error_msg("Invalid i = " << i);
                      }
                  }


                case 2: // d^2()/deta^2
                  {
                    switch(i)
                      {
                      case 0:
                      case 3:
                        return 0.5*(-1. + zeta + xi)/den;

                      case 1:
                      case 2:
                        return 0.5*(-1. + zeta - xi)/den;

                      case 4:
                      case 5:
                      case 7:
                      case 9:
                      case 10:
                      case 11:
                      case 12:
                        return 0.;

                      case 6:
                        return (1. + xi - zeta)/den;

                      case 8:
                        return (1. - xi - zeta)/den;

                      default:
                        libmesh_error_msg("Invalid i = " << i);
                      }
                  }


                case 3: // d^2()/dxidzeta
                  {
                    switch(i)
                      {
                      case 0:
                        return -0.25*(-1. + 2.*zeta - zeta2 + eta + 2.*eta*xi + eta2)/den2;

                      case 1:
                        return 0.25*(-1. + 2.*zeta - zeta2 + eta - 2.*eta*xi + eta2)/den2;

                      case 2:
                        return 0.25*(-1. + 2.*zeta - zeta2 - eta + 2.*eta*xi + eta2)/den2;

                      case 3:
                        return -0.25*(-1. + 2.*zeta - zeta2 - eta - 2.*eta*xi + eta2)/den2;

                      case 4:
                        return 0.;

                      case 5:
                        return eta*xi/den2;

                      case 6:
                        return -0.5*(1. + zeta2 + eta2 - 2.*zeta)/den2;

                      case 7:
                        return -eta*xi/den2;

                      case 8:
                        return 0.5*(1. + zeta2 + eta2 - 2.*zeta)/den2;

                      case 9:
                        return (-1. - zeta2 + eta + 2.*zeta)/den2;

                      case 10:
                        return -(-1. - zeta2 + eta + 2.*zeta)/den2;

                      case 11:
                        return (1. + zeta2 + eta - 2.*zeta)/den2;

                      case 12:
                        return -(1. + zeta2 + eta - 2.*zeta)/den2;

                      default:
                        libmesh_error_msg("Invalid i = " << i);
                      }
                  }

                case 4: // d^2()/detadzeta
                  {
                    switch(i)
                      {
                      case 0:
                        return -0.25*(-1. + 2.*zeta - zeta2 + xi + 2.*eta*xi + xi2)/den2;

                      case 1:
                        return 0.25*(1. - 2.*zeta + zeta2 + xi + 2.*eta*xi - xi2)/den2;

                      case 2:
                        return 0.25*(-1. + 2.*zeta - zeta2 - xi + 2.*eta*xi + xi2)/den2;

                      case 3:
                        return -0.25*(1. - 2.*zeta + zeta2 - xi + 2.*eta*xi - xi2)/den2;

                      case 4:
                        return 0.;

                      case 5:
                        return 0.5*(1. + xi2 + zeta2 - 2.*zeta)/den2;

                      case 6:
                        return -eta*xi/den2;

                      case 7:
                        return -0.5*(1. + xi2 + zeta2 - 2.*zeta)/den2;

                      case 8:
                        return eta*xi/den2;

                      case 9:
                        return (-1. - zeta2 + xi + 2.*zeta)/den2;

                      case 10:
                        return -(1. + zeta2 + xi - 2.*zeta)/den2;

                      case 11:
                        return (1. + zeta2 + xi - 2.*zeta)/den2;

                      case 12:
                        return -(-1. - zeta2 + xi + 2.*zeta)/den2;

                      default:
                        libmesh_error_msg("Invalid i = " << i);
                      }
                  }

                case 5: // d^2()/dzeta^2
                  {
                    switch(i)
                      {
                      case 0:
                        return 0.5*(xi + eta + 1.)*eta*xi/den3;

                      case 1:
                        return -0.5*(eta - xi + 1.)*eta*xi/den3;

                      case 2:
                        return -0.5*(xi + eta - 1.)*eta*xi/den3;

                      case 3:
                        return 0.5*(eta - xi - 1.)*eta*xi/den3;

                      case 4:
                        return 4.;

                      case 5:
                        return -(1. - 3.*zeta + 3.*zeta2 - zeta3 + eta*xi2)/den3;

                      case 6:
                        return (-1. + 3.*zeta - 3.*zeta2 + zeta3 + eta2*xi)/den3;

                      case 7:
                        return (-1. + 3.*zeta - 3.*zeta2 + zeta3 + eta*xi2)/den3;

                      case 8:
                        return -(1. - 3.*zeta + 3.*zeta2 - zeta3 + eta2*xi)/den3;

                      case 9:
                        return -2.*(-1. + 3.*zeta - 3.*zeta2 + zeta3 + eta*xi)/den3;

                      case 10:
                        return 2.*(1. - 3.*zeta + 3.*zeta2 - zeta3 + eta*xi)/den3;

                      case 11:
                        return -2.*(-1. + 3.*zeta - 3.*zeta2 + zeta3 + eta*xi)/den3;

                      case 12:
                        return 2.*(1. - 3.*zeta + 3.*zeta2 - zeta3 + eta*xi)/den3;

                      default:
                        libmesh_error_msg("Invalid i = " << i);
                      }
                  }

                default:
                  libmesh_error_msg("Invalid j = " << j);
                }
            }

          default:
            libmesh_error_msg("ERROR: Unsupported 3D element type!: " << type);
          }
      }


      // unsupported order
    default:
      libmesh_error_msg("ERROR: Unsupported 3D FE order!: " << order);
    }

#else
  return 0.;
#endif
}

shape_second_deriv() [100/103]

Real libMesh::FE< 2, CLOUGH >::shape_second_deriv ( const Elem *  elem, const Order  order, const unsigned int  i, const unsigned int  j, const Point &  p  )

inherited

Definition at line 2188 of file fe_clough_shape_2D.C.

{
  libmesh_assert(elem);

  clough_compute_coefs(elem);

  const ElemType type = elem->type();

  const Order totalorder = static_cast<Order>(order + elem->p_level());

  switch (totalorder)
    {
      // 2nd-order restricted Clough-Tocher element
    case SECOND:
      {
        switch (type)
          {
            // C1 functions on the Clough-Tocher triangle.
          case TRI6:
            {
              libmesh_assert_less (i, 9);
              // FIXME: it would be nice to calculate (and cache)
              // clough_raw_shape(j,p) only once per triangle, not 1-7
              // times
              switch (i)
                {
                  // Note: these DoF numbers are "scrambled" because my
                  // initial numbering conventions didn't match libMesh
                case 0:
                  return clough_raw_shape_second_deriv(0, j, p)
                    + d1d2n * clough_raw_shape_second_deriv(10, j, p)
                    + d1d3n * clough_raw_shape_second_deriv(11, j, p);
                case 3:
                  return clough_raw_shape_second_deriv(1, j, p)
                    + d2d3n * clough_raw_shape_second_deriv(11, j, p)
                    + d2d1n * clough_raw_shape_second_deriv(9, j, p);
                case 6:
                  return clough_raw_shape_second_deriv(2, j, p)
                    + d3d1n * clough_raw_shape_second_deriv(9, j, p)
                    + d3d2n * clough_raw_shape_second_deriv(10, j, p);
                case 1:
                  return d1xd1x * clough_raw_shape_second_deriv(3, j, p)
                    + d1xd1y * clough_raw_shape_second_deriv(4, j, p)
                    + d1xd2n * clough_raw_shape_second_deriv(10, j, p)
                    + d1xd3n * clough_raw_shape_second_deriv(11, j, p)
                    + 0.5 * N01x * d3nd3n * clough_raw_shape_second_deriv(11, j, p)
                    + 0.5 * N02x * d2nd2n * clough_raw_shape_second_deriv(10, j, p);
                case 2:
                  return d1yd1y * clough_raw_shape_second_deriv(4, j, p)
                    + d1yd1x * clough_raw_shape_second_deriv(3, j, p)
                    + d1yd2n * clough_raw_shape_second_deriv(10, j, p)
                    + d1yd3n * clough_raw_shape_second_deriv(11, j, p)
                    + 0.5 * N01y * d3nd3n * clough_raw_shape_second_deriv(11, j, p)
                    + 0.5 * N02y * d2nd2n * clough_raw_shape_second_deriv(10, j, p);
                case 4:
                  return d2xd2x * clough_raw_shape_second_deriv(5, j, p)
                    + d2xd2y * clough_raw_shape_second_deriv(6, j, p)
                    + d2xd3n * clough_raw_shape_second_deriv(11, j, p)
                    + d2xd1n * clough_raw_shape_second_deriv(9, j, p)
                    + 0.5 * N10x * d3nd3n * clough_raw_shape_second_deriv(11, j, p)
                    + 0.5 * N12x * d1nd1n * clough_raw_shape_second_deriv(9, j, p);
                case 5:
                  return d2yd2y * clough_raw_shape_second_deriv(6, j, p)
                    + d2yd2x * clough_raw_shape_second_deriv(5, j, p)
                    + d2yd3n * clough_raw_shape_second_deriv(11, j, p)
                    + d2yd1n * clough_raw_shape_second_deriv(9, j, p)
                    + 0.5 * N10y * d3nd3n * clough_raw_shape_second_deriv(11, j, p)
                    + 0.5 * N12y * d1nd1n * clough_raw_shape_second_deriv(9, j, p);
                case 7:
                  return d3xd3x * clough_raw_shape_second_deriv(7, j, p)
                    + d3xd3y * clough_raw_shape_second_deriv(8, j, p)
                    + d3xd1n * clough_raw_shape_second_deriv(9, j, p)
                    + d3xd2n * clough_raw_shape_second_deriv(10, j, p)
                    + 0.5 * N20x * d2nd2n * clough_raw_shape_second_deriv(10, j, p)
                    + 0.5 * N21x * d1nd1n * clough_raw_shape_second_deriv(9, j, p);
                case 8:
                  return d3yd3y * clough_raw_shape_second_deriv(8, j, p)
                    + d3yd3x * clough_raw_shape_second_deriv(7, j, p)
                    + d3yd1n * clough_raw_shape_second_deriv(9, j, p)
                    + d3yd2n * clough_raw_shape_second_deriv(10, j, p)
                    + 0.5 * N20y * d2nd2n * clough_raw_shape_second_deriv(10, j, p)
                    + 0.5 * N21y * d1nd1n * clough_raw_shape_second_deriv(9, j, p);
                default:
                  libmesh_error_msg("Invalid shape function index i = " << i);
                }
            }
          default:
            libmesh_error_msg("ERROR: Unsupported element type = " << type);
          }
      }
      // 3rd-order Clough-Tocher element
    case THIRD:
      {
        switch (type)
          {
            // C1 functions on the Clough-Tocher triangle.
          case TRI6:
            {
              libmesh_assert_less (i, 12);

              // FIXME: it would be nice to calculate (and cache)
              // clough_raw_shape(j,p) only once per triangle, not 1-7
              // times
              switch (i)
                {
                  // Note: these DoF numbers are "scrambled" because my
                  // initial numbering conventions didn't match libMesh
                case 0:
                  return clough_raw_shape_second_deriv(0, j, p)
                    + d1d2n * clough_raw_shape_second_deriv(10, j, p)
                    + d1d3n * clough_raw_shape_second_deriv(11, j, p);
                case 3:
                  return clough_raw_shape_second_deriv(1, j, p)
                    + d2d3n * clough_raw_shape_second_deriv(11, j, p)
                    + d2d1n * clough_raw_shape_second_deriv(9, j, p);
                case 6:
                  return clough_raw_shape_second_deriv(2, j, p)
                    + d3d1n * clough_raw_shape_second_deriv(9, j, p)
                    + d3d2n * clough_raw_shape_second_deriv(10, j, p);
                case 1:
                  return d1xd1x * clough_raw_shape_second_deriv(3, j, p)
                    + d1xd1y * clough_raw_shape_second_deriv(4, j, p)
                    + d1xd2n * clough_raw_shape_second_deriv(10, j, p)
                    + d1xd3n * clough_raw_shape_second_deriv(11, j, p);
                case 2:
                  return d1yd1y * clough_raw_shape_second_deriv(4, j, p)
                    + d1yd1x * clough_raw_shape_second_deriv(3, j, p)
                    + d1yd2n * clough_raw_shape_second_deriv(10, j, p)
                    + d1yd3n * clough_raw_shape_second_deriv(11, j, p);
                case 4:
                  return d2xd2x * clough_raw_shape_second_deriv(5, j, p)
                    + d2xd2y * clough_raw_shape_second_deriv(6, j, p)
                    + d2xd3n * clough_raw_shape_second_deriv(11, j, p)
                    + d2xd1n * clough_raw_shape_second_deriv(9, j, p);
                case 5:
                  return d2yd2y * clough_raw_shape_second_deriv(6, j, p)
                    + d2yd2x * clough_raw_shape_second_deriv(5, j, p)
                    + d2yd3n * clough_raw_shape_second_deriv(11, j, p)
                    + d2yd1n * clough_raw_shape_second_deriv(9, j, p);
                case 7:
                  return d3xd3x * clough_raw_shape_second_deriv(7, j, p)
                    + d3xd3y * clough_raw_shape_second_deriv(8, j, p)
                    + d3xd1n * clough_raw_shape_second_deriv(9, j, p)
                    + d3xd2n * clough_raw_shape_second_deriv(10, j, p);
                case 8:
                  return d3yd3y * clough_raw_shape_second_deriv(8, j, p)
                    + d3yd3x * clough_raw_shape_second_deriv(7, j, p)
                    + d3yd1n * clough_raw_shape_second_deriv(9, j, p)
                    + d3yd2n * clough_raw_shape_second_deriv(10, j, p);
                case 10:
                  return d1nd1n * clough_raw_shape_second_deriv(9, j, p);
                case 11:
                  return d2nd2n * clough_raw_shape_second_deriv(10, j, p);
                case 9:
                  return d3nd3n * clough_raw_shape_second_deriv(11, j, p);

                default:
                  libmesh_error_msg("Invalid shape function index i = " << i);
                }
            }
          default:
            libmesh_error_msg("ERROR: Unsupported element type = " << type);
          }
      }
      // by default throw an error
    default:
      libmesh_error_msg("ERROR: Unsupported polynomial order = " << order);
    }
}

shape_second_deriv() [101/103]

Real libMesh::FE< 3, BERNSTEIN >::shape_second_deriv ( const ElemType  , const Order  , const unsigned  int, const unsigned  int, const Point &    )

inherited

Definition at line 2965 of file fe_bernstein_shape_3D.C.

{
  static bool warning_given = false;

  if (!warning_given)
    libMesh::err << "Second derivatives for Bernstein elements "
                 << "are not yet implemented!"
                 << std::endl;

  warning_given = true;
  return 0.;
}

shape_second_deriv() [102/103]

Real libMesh::FE< 3, BERNSTEIN >::shape_second_deriv ( const Elem *  , const Order  , const unsigned  int, const unsigned  int, const Point &    )

inherited

Definition at line 2985 of file fe_bernstein_shape_3D.C.

{
  static bool warning_given = false;

  if (!warning_given)
    libMesh::err << "Second derivatives for Bernstein elements "
                 << "are not yet implemented!"
                 << std::endl;

  warning_given = true;
  return 0.;
}

shape_second_deriv() [103/103]

Real libMesh::FE< 3, LAGRANGE >::shape_second_deriv ( const Elem *  elem, const Order  order, const unsigned int  i, const unsigned int  j, const Point &  p  )

inherited

Definition at line 3645 of file fe_lagrange_shape_3D.C.

{
  libmesh_assert(elem);

  // call the orientation-independent shape function derivatives
  return FE<3,LAGRANGE>::shape_second_deriv(elem->type(), static_cast<Order>(order + elem->p_level()), i, j, p);
}

shapes_need_reinit() [1/54]

bool libMesh::FE< 0, SCALAR >::shapes_need_reinit ( ) const

virtualinherited

Returns

true when the shape functions (for this FEFamily) depend on the particular element, and therefore needs to be re-initialized for each new element. false otherwise.

Implements libMesh::FEAbstract.

Definition at line 137 of file fe_scalar.C.

{ return false; }

shapes_need_reinit() [2/54]

bool libMesh::FE< 1, SCALAR >::shapes_need_reinit ( ) const

virtualinherited

Returns

true when the shape functions (for this FEFamily) depend on the particular element, and therefore needs to be re-initialized for each new element. false otherwise.

Implements libMesh::FEAbstract.

Definition at line 138 of file fe_scalar.C.

{ return false; }

shapes_need_reinit() [3/54]

bool libMesh::FE< 2, SCALAR >::shapes_need_reinit ( ) const

virtualinherited

Returns

true when the shape functions (for this FEFamily) depend on the particular element, and therefore needs to be re-initialized for each new element. false otherwise.

Implements libMesh::FEAbstract.

Definition at line 139 of file fe_scalar.C.

{ return false; }

shapes_need_reinit() [4/54]

bool libMesh::FE< 3, SCALAR >::shapes_need_reinit ( ) const

virtualinherited

Returns

true when the shape functions (for this FEFamily) depend on the particular element, and therefore needs to be re-initialized for each new element. false otherwise.

Implements libMesh::FEAbstract.

Definition at line 140 of file fe_scalar.C.

{ return false; }

shapes_need_reinit() [5/54]

bool libMesh::FE< 0, L2_HIERARCHIC >::shapes_need_reinit ( ) const

virtualinherited

Returns

true when the shape functions (for this FEFamily) depend on the particular element, and therefore needs to be re-initialized for each new element. false otherwise.

Implements libMesh::FEAbstract.

Definition at line 222 of file fe_l2_hierarchic.C.

{ return true; }

shapes_need_reinit() [6/54]

bool libMesh::FE< 1, L2_HIERARCHIC >::shapes_need_reinit ( ) const

virtualinherited

Returns

true when the shape functions (for this FEFamily) depend on the particular element, and therefore needs to be re-initialized for each new element. false otherwise.

Implements libMesh::FEAbstract.

Definition at line 223 of file fe_l2_hierarchic.C.

{ return true; }

shapes_need_reinit() [7/54]

bool libMesh::FE< 2, L2_HIERARCHIC >::shapes_need_reinit ( ) const

virtualinherited

Returns

true when the shape functions (for this FEFamily) depend on the particular element, and therefore needs to be re-initialized for each new element. false otherwise.

Implements libMesh::FEAbstract.

Definition at line 224 of file fe_l2_hierarchic.C.

{ return true; }

shapes_need_reinit() [8/54]

bool libMesh::FE< 3, L2_HIERARCHIC >::shapes_need_reinit ( ) const

virtualinherited

Returns

true when the shape functions (for this FEFamily) depend on the particular element, and therefore needs to be re-initialized for each new element. false otherwise.

Implements libMesh::FEAbstract.

Definition at line 225 of file fe_l2_hierarchic.C.

{ return true; }

shapes_need_reinit() [9/54]

bool libMesh::FE< 0, CLOUGH >::shapes_need_reinit ( ) const

virtualinherited

Returns

true when the shape functions (for this FEFamily) depend on the particular element, and therefore needs to be re-initialized for each new element. false otherwise.

Implements libMesh::FEAbstract.

Definition at line 319 of file fe_clough.C.

{ return true; }

shapes_need_reinit() [10/54]

bool libMesh::FE< 1, CLOUGH >::shapes_need_reinit ( ) const

virtualinherited

Returns

true when the shape functions (for this FEFamily) depend on the particular element, and therefore needs to be re-initialized for each new element. false otherwise.

Implements libMesh::FEAbstract.

Definition at line 320 of file fe_clough.C.

{ return true; }

shapes_need_reinit() [11/54]

bool libMesh::FE< 2, CLOUGH >::shapes_need_reinit ( ) const

virtualinherited

Returns

true when the shape functions (for this FEFamily) depend on the particular element, and therefore needs to be re-initialized for each new element. false otherwise.

Implements libMesh::FEAbstract.

Definition at line 321 of file fe_clough.C.

{ return true; }

shapes_need_reinit() [12/54]

bool libMesh::FE< 3, CLOUGH >::shapes_need_reinit ( ) const

virtualinherited

Returns

true when the shape functions (for this FEFamily) depend on the particular element, and therefore needs to be re-initialized for each new element. false otherwise.

Implements libMesh::FEAbstract.

Definition at line 322 of file fe_clough.C.

{ return true; }

shapes_need_reinit() [13/54]

bool libMesh::FE< 0, HERMITE >::shapes_need_reinit ( ) const

virtualinherited

Returns

true when the shape functions (for this FEFamily) depend on the particular element, and therefore needs to be re-initialized for each new element. false otherwise.

Implements libMesh::FEAbstract.

Definition at line 370 of file fe_hermite.C.

{ return true; }

shapes_need_reinit() [14/54]

bool libMesh::FE< 1, HERMITE >::shapes_need_reinit ( ) const

virtualinherited

Returns

true when the shape functions (for this FEFamily) depend on the particular element, and therefore needs to be re-initialized for each new element. false otherwise.

Implements libMesh::FEAbstract.

Definition at line 371 of file fe_hermite.C.

{ return true; }

shapes_need_reinit() [15/54]

bool libMesh::FE< 2, HERMITE >::shapes_need_reinit ( ) const

virtualinherited

Returns

true when the shape functions (for this FEFamily) depend on the particular element, and therefore needs to be re-initialized for each new element. false otherwise.

Implements libMesh::FEAbstract.

Definition at line 372 of file fe_hermite.C.

{ return true; }

shapes_need_reinit() [16/54]

bool libMesh::FE< 3, HERMITE >::shapes_need_reinit ( ) const

virtualinherited

Returns

true when the shape functions (for this FEFamily) depend on the particular element, and therefore needs to be re-initialized for each new element. false otherwise.

Implements libMesh::FEAbstract.

Definition at line 373 of file fe_hermite.C.

{ return true; }

shapes_need_reinit() [17/54]

bool libMesh::FE< 0, HIERARCHIC >::shapes_need_reinit ( ) const

virtualinherited

Returns

true when the shape functions (for this FEFamily) depend on the particular element, and therefore needs to be re-initialized for each new element. false otherwise.

Implements libMesh::FEAbstract.

Definition at line 398 of file fe_hierarchic.C.

{ return true; }

shapes_need_reinit() [18/54]

bool libMesh::FE< 1, HIERARCHIC >::shapes_need_reinit ( ) const

virtualinherited

Returns

true when the shape functions (for this FEFamily) depend on the particular element, and therefore needs to be re-initialized for each new element. false otherwise.

Implements libMesh::FEAbstract.

Definition at line 399 of file fe_hierarchic.C.

{ return true; }

shapes_need_reinit() [19/54]

bool libMesh::FE< 2, HIERARCHIC >::shapes_need_reinit ( ) const

virtualinherited

Returns

true when the shape functions (for this FEFamily) depend on the particular element, and therefore needs to be re-initialized for each new element. false otherwise.

Implements libMesh::FEAbstract.

Definition at line 400 of file fe_hierarchic.C.

{ return true; }

shapes_need_reinit() [20/54]

bool libMesh::FE< 3, HIERARCHIC >::shapes_need_reinit ( ) const

virtualinherited

Returns

true when the shape functions (for this FEFamily) depend on the particular element, and therefore needs to be re-initialized for each new element. false otherwise.

Implements libMesh::FEAbstract.

Definition at line 401 of file fe_hierarchic.C.

{ return true; }

shapes_need_reinit() [21/54]

virtual bool libMesh::FE< Dim, T >::shapes_need_reinit ( ) const

overridevirtualinherited

Returns

true when the shape functions (for this FEFamily) depend on the particular element, and therefore needs to be re-initialized for each new element. false otherwise.

Implements libMesh::FEAbstract.

shapes_need_reinit() [22/54]

bool libMesh::FE< 0, MONOMIAL >::shapes_need_reinit ( ) const

virtualinherited

Returns

true when the shape functions (for this FEFamily) depend on the particular element, and therefore needs to be re-initialized for each new element. false otherwise.

Implements libMesh::FEAbstract.

Definition at line 425 of file fe_monomial.C.

{ return false; }

shapes_need_reinit() [23/54]

bool libMesh::FE< 1, MONOMIAL >::shapes_need_reinit ( ) const

virtualinherited

Returns

true when the shape functions (for this FEFamily) depend on the particular element, and therefore needs to be re-initialized for each new element. false otherwise.

Implements libMesh::FEAbstract.

Definition at line 426 of file fe_monomial.C.

{ return false; }

shapes_need_reinit() [24/54]

bool libMesh::FE< 2, MONOMIAL >::shapes_need_reinit ( ) const

virtualinherited

Returns

true when the shape functions (for this FEFamily) depend on the particular element, and therefore needs to be re-initialized for each new element. false otherwise.

Implements libMesh::FEAbstract.

Definition at line 427 of file fe_monomial.C.

{ return false; }

shapes_need_reinit() [25/54]

bool libMesh::FE< 3, MONOMIAL >::shapes_need_reinit ( ) const

virtualinherited

Returns

true when the shape functions (for this FEFamily) depend on the particular element, and therefore needs to be re-initialized for each new element. false otherwise.

Implements libMesh::FEAbstract.

Definition at line 428 of file fe_monomial.C.

{ return false; }

shapes_need_reinit() [26/54]

bool libMesh::FE< 0, BERNSTEIN >::shapes_need_reinit ( ) const

virtualinherited

Returns

true when the shape functions (for this FEFamily) depend on the particular element, and therefore needs to be re-initialized for each new element. false otherwise.

Implements libMesh::FEAbstract.

Definition at line 466 of file fe_bernstein.C.

{ return true; }

shapes_need_reinit() [27/54]

bool libMesh::FE< 1, BERNSTEIN >::shapes_need_reinit ( ) const

virtualinherited

Returns

true when the shape functions (for this FEFamily) depend on the particular element, and therefore needs to be re-initialized for each new element. false otherwise.

Implements libMesh::FEAbstract.

Definition at line 467 of file fe_bernstein.C.

{ return true; }

shapes_need_reinit() [28/54]

bool libMesh::FE< 2, BERNSTEIN >::shapes_need_reinit ( ) const

virtualinherited

Returns

true when the shape functions (for this FEFamily) depend on the particular element, and therefore needs to be re-initialized for each new element. false otherwise.

Implements libMesh::FEAbstract.

Definition at line 468 of file fe_bernstein.C.

{ return true; }

shapes_need_reinit() [29/54]

bool libMesh::FE< 3, BERNSTEIN >::shapes_need_reinit ( ) const

virtualinherited

Returns

true when the shape functions (for this FEFamily) depend on the particular element, and therefore needs to be re-initialized for each new element. false otherwise.

Implements libMesh::FEAbstract.

Definition at line 469 of file fe_bernstein.C.

{ return true; }

shapes_need_reinit() [30/54]

bool libMesh::FE< 0, L2_LAGRANGE >::shapes_need_reinit ( ) const

virtualinherited

Returns

true when the shape functions (for this FEFamily) depend on the particular element, and therefore needs to be re-initialized for each new element. false otherwise.

Implements libMesh::FEAbstract.

Definition at line 486 of file fe_l2_lagrange.C.

{ return false; }

shapes_need_reinit() [31/54]

bool libMesh::FE< 1, L2_LAGRANGE >::shapes_need_reinit ( ) const

virtualinherited

Returns

true when the shape functions (for this FEFamily) depend on the particular element, and therefore needs to be re-initialized for each new element. false otherwise.

Implements libMesh::FEAbstract.

Definition at line 487 of file fe_l2_lagrange.C.

{ return false; }

shapes_need_reinit() [32/54]

bool libMesh::FE< 2, L2_LAGRANGE >::shapes_need_reinit ( ) const

virtualinherited

Returns

true when the shape functions (for this FEFamily) depend on the particular element, and therefore needs to be re-initialized for each new element. false otherwise.

Implements libMesh::FEAbstract.

Definition at line 488 of file fe_l2_lagrange.C.

{ return false; }

shapes_need_reinit() [33/54]

bool libMesh::FE< 3, L2_LAGRANGE >::shapes_need_reinit ( ) const

virtualinherited

Returns

true when the shape functions (for this FEFamily) depend on the particular element, and therefore needs to be re-initialized for each new element. false otherwise.

Implements libMesh::FEAbstract.

Definition at line 489 of file fe_l2_lagrange.C.

{ return false; }

shapes_need_reinit() [34/54]

bool libMesh::FE< 0, NEDELEC_ONE >::shapes_need_reinit ( ) const

virtualinherited

Returns

true when the shape functions (for this FEFamily) depend on the particular element, and therefore needs to be re-initialized for each new element. false otherwise.

Implements libMesh::FEAbstract.

Definition at line 556 of file fe_nedelec_one.C.

{ NEDELEC_LOW_D_ERROR_MESSAGE }

shapes_need_reinit() [35/54]

bool libMesh::FE< 1, NEDELEC_ONE >::shapes_need_reinit ( ) const

virtualinherited

Returns

true when the shape functions (for this FEFamily) depend on the particular element, and therefore needs to be re-initialized for each new element. false otherwise.

Implements libMesh::FEAbstract.

Definition at line 557 of file fe_nedelec_one.C.

{ NEDELEC_LOW_D_ERROR_MESSAGE }

shapes_need_reinit() [36/54]

bool libMesh::FE< 2, NEDELEC_ONE >::shapes_need_reinit ( ) const

virtualinherited

Returns

true when the shape functions (for this FEFamily) depend on the particular element, and therefore needs to be re-initialized for each new element. false otherwise.

Implements libMesh::FEAbstract.

Definition at line 558 of file fe_nedelec_one.C.

{ return true; }

shapes_need_reinit() [37/54]

bool libMesh::FE< 3, NEDELEC_ONE >::shapes_need_reinit ( ) const

virtualinherited

Returns

true when the shape functions (for this FEFamily) depend on the particular element, and therefore needs to be re-initialized for each new element. false otherwise.

Implements libMesh::FEAbstract.

Definition at line 559 of file fe_nedelec_one.C.

{ return true; }

shapes_need_reinit() [38/54]

bool libMesh::FE< 0, LAGRANGE >::shapes_need_reinit ( ) const

virtualinherited

Returns

true when the shape functions (for this FEFamily) depend on the particular element, and therefore needs to be re-initialized for each new element. false otherwise.

Implements libMesh::FEAbstract.

Definition at line 889 of file fe_lagrange.C.

{ return false; }

shapes_need_reinit() [39/54]

bool libMesh::FE< 1, LAGRANGE >::shapes_need_reinit ( ) const

virtualinherited

Returns

true when the shape functions (for this FEFamily) depend on the particular element, and therefore needs to be re-initialized for each new element. false otherwise.

Implements libMesh::FEAbstract.

Definition at line 890 of file fe_lagrange.C.

{ return false; }

shapes_need_reinit() [40/54]

bool libMesh::FE< 2, LAGRANGE >::shapes_need_reinit ( ) const

virtualinherited

Returns

true when the shape functions (for this FEFamily) depend on the particular element, and therefore needs to be re-initialized for each new element. false otherwise.

Implements libMesh::FEAbstract.

Definition at line 891 of file fe_lagrange.C.

{ return false; }

shapes_need_reinit() [41/54]

bool libMesh::FE< 3, LAGRANGE >::shapes_need_reinit ( ) const

virtualinherited

Returns

true when the shape functions (for this FEFamily) depend on the particular element, and therefore needs to be re-initialized for each new element. false otherwise.

Implements libMesh::FEAbstract.

Definition at line 892 of file fe_lagrange.C.

{ return false; }

shapes_need_reinit() [42/54]

bool libMesh::FE< 2, SUBDIVISION >::shapes_need_reinit ( ) const

virtualinherited

Returns

true when the shape functions (for this FEFamily) depend on the particular element, and therefore needs to be re-initialized for each new element. false otherwise.

Implements libMesh::FEAbstract.

Definition at line 916 of file fe_subdivision_2D.C.

{ return true; }

shapes_need_reinit() [43/54]

bool libMesh::FE< 0, LAGRANGE_VEC >::shapes_need_reinit ( ) const

virtualinherited

Returns

true when the shape functions (for this FEFamily) depend on the particular element, and therefore needs to be re-initialized for each new element. false otherwise.

Implements libMesh::FEAbstract.

Definition at line 939 of file fe_lagrange_vec.C.

{ return false; }

shapes_need_reinit() [44/54]

bool libMesh::FE< 1, LAGRANGE_VEC >::shapes_need_reinit ( ) const

virtualinherited

Returns

true when the shape functions (for this FEFamily) depend on the particular element, and therefore needs to be re-initialized for each new element. false otherwise.

Implements libMesh::FEAbstract.

Definition at line 940 of file fe_lagrange_vec.C.

{ return false; }

shapes_need_reinit() [45/54]

bool libMesh::FE< 2, LAGRANGE_VEC >::shapes_need_reinit ( ) const

virtualinherited

Returns

true when the shape functions (for this FEFamily) depend on the particular element, and therefore needs to be re-initialized for each new element. false otherwise.

Implements libMesh::FEAbstract.

Definition at line 941 of file fe_lagrange_vec.C.

{ return false; }

shapes_need_reinit() [46/54]

bool libMesh::FE< 3, LAGRANGE_VEC >::shapes_need_reinit ( ) const

virtualinherited

Returns

true when the shape functions (for this FEFamily) depend on the particular element, and therefore needs to be re-initialized for each new element. false otherwise.

Implements libMesh::FEAbstract.

Definition at line 942 of file fe_lagrange_vec.C.

{ return false; }

shapes_need_reinit() [47/54]

bool libMesh::FE< 0, XYZ >::shapes_need_reinit ( ) const

virtualinherited

Returns

true when the shape functions (for this FEFamily) depend on the particular element, and therefore needs to be re-initialized for each new element. false otherwise.

Implements libMesh::FEAbstract.

Definition at line 948 of file fe_xyz.C.

{ return true; }

shapes_need_reinit() [48/54]

bool libMesh::FE< 1, XYZ >::shapes_need_reinit ( ) const

virtualinherited

Returns

true when the shape functions (for this FEFamily) depend on the particular element, and therefore needs to be re-initialized for each new element. false otherwise.

Implements libMesh::FEAbstract.

Definition at line 949 of file fe_xyz.C.

{ return true; }

shapes_need_reinit() [49/54]

bool libMesh::FE< 2, XYZ >::shapes_need_reinit ( ) const

virtualinherited

Returns

true when the shape functions (for this FEFamily) depend on the particular element, and therefore needs to be re-initialized for each new element. false otherwise.

Implements libMesh::FEAbstract.

Definition at line 950 of file fe_xyz.C.

{ return true; }

shapes_need_reinit() [50/54]

bool libMesh::FE< 3, XYZ >::shapes_need_reinit ( ) const

virtualinherited

Returns

true when the shape functions (for this FEFamily) depend on the particular element, and therefore needs to be re-initialized for each new element. false otherwise.

Implements libMesh::FEAbstract.

Definition at line 951 of file fe_xyz.C.

{ return true; }

shapes_need_reinit() [51/54]

bool libMesh::FE< 0, SZABAB >::shapes_need_reinit ( ) const

virtualinherited

Returns

true when the shape functions (for this FEFamily) depend on the particular element, and therefore needs to be re-initialized for each new element. false otherwise.

Implements libMesh::FEAbstract.

Definition at line 1303 of file fe_szabab.C.

{ return true; }

shapes_need_reinit() [52/54]

bool libMesh::FE< 1, SZABAB >::shapes_need_reinit ( ) const

virtualinherited

Returns

true when the shape functions (for this FEFamily) depend on the particular element, and therefore needs to be re-initialized for each new element. false otherwise.

Implements libMesh::FEAbstract.

Definition at line 1304 of file fe_szabab.C.

{ return true; }

shapes_need_reinit() [53/54]

bool libMesh::FE< 2, SZABAB >::shapes_need_reinit ( ) const

virtualinherited

Returns

true when the shape functions (for this FEFamily) depend on the particular element, and therefore needs to be re-initialized for each new element. false otherwise.

Implements libMesh::FEAbstract.

Definition at line 1305 of file fe_szabab.C.

{ return true; }

shapes_need_reinit() [54/54]

bool libMesh::FE< 3, SZABAB >::shapes_need_reinit ( ) const

virtualinherited

Returns

true when the shape functions (for this FEFamily) depend on the particular element, and therefore needs to be re-initialized for each new element. false otherwise.

Implements libMesh::FEAbstract.

Definition at line 1306 of file fe_szabab.C.

{ return true; }

side_map() [1/2]

void libMesh::FE< Dim, T >::side_map ( const Elem *  elem, const Elem *  side, const unsigned int  s, const std::vector< Point > &  reference_side_points, std::vector< Point > &  reference_points  )

overridevirtualinherited

Computes the reference space quadrature points on the side of an element based on the side quadrature points.

Implements libMesh::FEAbstract.

Definition at line 353 of file fe_boundary.C.

{
  // We're calculating mappings - we need at least first order info
  this->calculate_phi = true;
  this->determine_calculations();

  unsigned int side_p_level = elem->p_level();
  if (elem->neighbor_ptr(s) != nullptr)
    side_p_level = std::max(side_p_level, elem->neighbor_ptr(s)->p_level());

  if (side->type() != last_side ||
      side_p_level != this->_p_level ||
      !this->shapes_on_quadrature)
    {
      // Set the element type
      this->elem_type = elem->type();
      this->_p_level = side_p_level;

      // Set the last_side
      last_side = side->type();

      // Initialize the face shape functions
      this->_fe_map->template init_face_shape_functions<Dim>(reference_side_points, side);
    }

  const unsigned int n_points =
    cast_int<unsigned int>(reference_side_points.size());
  reference_points.resize(n_points);
  for (unsigned int i = 0; i < n_points; i++)
    reference_points[i].zero();

  std::vector<unsigned int> elem_nodes_map;
  elem_nodes_map.resize(side->n_nodes());
  for (unsigned int j = 0; j < side->n_nodes(); j++)
    for (unsigned int i = 0; i < elem->n_nodes(); i++)
      if (side->node_id(j) == elem->node_id(i))
        elem_nodes_map[j] = i;
  std::vector<Point> refspace_nodes;
  this->get_refspace_nodes(elem->type(), refspace_nodes);

  const std::vector<std::vector<Real>> & psi_map = this->_fe_map->get_psi();

  // sum over the nodes
  for (auto i : index_range(psi_map))
    {
      const Point & side_node = refspace_nodes[elem_nodes_map[i]];
      for (unsigned int p=0; p<n_points; p++)
        reference_points[p].add_scaled (side_node, psi_map[i][p]);
    }
}

side_map() [2/2]

void libMesh::FE< 2, SUBDIVISION >::side_map ( const Elem *  , const Elem *  , const unsigned int  , const std::vector< Point > &  , std::vector< Point > &    )

virtualinherited

Computes the reference space quadrature points on the side of an element based on the side quadrature points.

Implements libMesh::FEAbstract.

Definition at line 856 of file fe_subdivision_2D.C.

{
  libmesh_not_implemented();
}

Member Data Documentation

_counts

ReferenceCounter::Counts libMesh::ReferenceCounter::_counts

staticprotectedinherited

Actually holds the data.

Definition at line 122 of file reference_counter.h.

Referenced by libMesh::ReferenceCounter::get_info(), libMesh::ReferenceCounter::increment_constructor_count(), and libMesh::ReferenceCounter::increment_destructor_count().

_enable_print_counter

bool libMesh::ReferenceCounter::_enable_print_counter = true

staticprotectedinherited

Flag to control whether reference count information is printed when print_info is called.

Definition at line 141 of file reference_counter.h.

Referenced by libMesh::ReferenceCounter::disable_print_counter_info(), libMesh::ReferenceCounter::enable_print_counter_info(), and libMesh::ReferenceCounter::print_info().

_fe_map

std::unique_ptr<FEMap> libMesh::FEAbstract::_fe_map

protectedinherited

Definition at line 525 of file fe_abstract.h.

Referenced by libMesh::FEAbstract::get_curvatures(), libMesh::FEAbstract::get_d2xyzdeta2(), libMesh::FEAbstract::get_d2xyzdetadzeta(), libMesh::FEAbstract::get_d2xyzdxi2(), libMesh::FEAbstract::get_d2xyzdxideta(), libMesh::FEAbstract::get_d2xyzdxidzeta(), libMesh::FEAbstract::get_d2xyzdzeta2(), libMesh::FEAbstract::get_detadx(), libMesh::FEAbstract::get_detady(), libMesh::FEAbstract::get_detadz(), libMesh::FEAbstract::get_dxidx(), libMesh::FEAbstract::get_dxidy(), libMesh::FEAbstract::get_dxidz(), libMesh::FEAbstract::get_dxyzdeta(), libMesh::FEAbstract::get_dxyzdxi(), libMesh::FEAbstract::get_dxyzdzeta(), libMesh::FEAbstract::get_dzetadx(), libMesh::FEAbstract::get_dzetady(), libMesh::FEAbstract::get_dzetadz(), libMesh::FEAbstract::get_fe_map(), libMesh::FEAbstract::get_JxW(), libMesh::FEAbstract::get_normals(), libMesh::FEAbstract::get_tangents(), libMesh::FEAbstract::get_xyz(), init_shape_functions(), libMesh::FEAbstract::print_JxW(), and libMesh::FEAbstract::print_xyz().

_fe_trans

std::unique_ptr<FETransformationBase<FEOutputType< T >::type > > libMesh::FEGenericBase< FEOutputType< T >::type >::_fe_trans

protectedinherited

Object that handles computing shape function values, gradients, etc in the physical domain.

Definition at line 493 of file fe_base.h.

_mutex

Threads::spin_mutex libMesh::ReferenceCounter::_mutex

staticprotectedinherited

Mutual exclusion object to enable thread-safe reference counting.

Definition at line 135 of file reference_counter.h.

_n_objects

Threads::atomic< unsigned int > libMesh::ReferenceCounter::_n_objects

staticprotectedinherited

The number of objects. Print the reference count information when the number returns to 0.

Definition at line 130 of file reference_counter.h.

Referenced by libMesh::ReferenceCounter::n_objects(), libMesh::ReferenceCounter::ReferenceCounter(), and libMesh::ReferenceCounter::~ReferenceCounter().

_p_level

unsigned int libMesh::FEAbstract::_p_level

protectedinherited

The p refinement level the current data structures are set up for.

Definition at line 587 of file fe_abstract.h.

Referenced by libMesh::FEAbstract::get_order(), and libMesh::FEAbstract::get_p_level().

cached_nodes

std::vector<Point> libMesh::FE< Dim, T >::cached_nodes

protectedinherited

An array of the node locations on the last element we computed on

Definition at line 475 of file fe.h.

calculate_curl_phi

bool libMesh::FEAbstract::calculate_curl_phi

mutableprotectedinherited

Should we calculate shape function curls?

Definition at line 557 of file fe_abstract.h.

Referenced by libMesh::FEGenericBase< FEOutputType< T >::type >::get_curl_phi().

calculate_d2phi

bool libMesh::FEAbstract::calculate_d2phi

mutableprotectedinherited

Should we calculate shape function hessians?

Definition at line 552 of file fe_abstract.h.

Referenced by libMesh::FEGenericBase< FEOutputType< T >::type >::get_d2phi(), libMesh::FEGenericBase< FEOutputType< T >::type >::get_d2phideta2(), libMesh::FEGenericBase< FEOutputType< T >::type >::get_d2phidetadzeta(), libMesh::FEGenericBase< FEOutputType< T >::type >::get_d2phidx2(), libMesh::FEGenericBase< FEOutputType< T >::type >::get_d2phidxdy(), libMesh::FEGenericBase< FEOutputType< T >::type >::get_d2phidxdz(), libMesh::FEGenericBase< FEOutputType< T >::type >::get_d2phidxi2(), libMesh::FEGenericBase< FEOutputType< T >::type >::get_d2phidxideta(), libMesh::FEGenericBase< FEOutputType< T >::type >::get_d2phidxidzeta(), libMesh::FEGenericBase< FEOutputType< T >::type >::get_d2phidy2(), libMesh::FEGenericBase< FEOutputType< T >::type >::get_d2phidydz(), libMesh::FEGenericBase< FEOutputType< T >::type >::get_d2phidz2(), and libMesh::FEGenericBase< FEOutputType< T >::type >::get_d2phidzeta2().

calculate_div_phi

bool libMesh::FEAbstract::calculate_div_phi

mutableprotectedinherited

Should we calculate shape function divergences?

Definition at line 562 of file fe_abstract.h.

Referenced by libMesh::FEGenericBase< FEOutputType< T >::type >::get_div_phi().

calculate_dphi

bool libMesh::FEAbstract::calculate_dphi

mutableprotectedinherited

Should we calculate shape function gradients?

Definition at line 547 of file fe_abstract.h.

Referenced by libMesh::FEGenericBase< FEOutputType< T >::type >::get_dphi(), libMesh::FEGenericBase< FEOutputType< T >::type >::get_dphidx(), libMesh::FEGenericBase< FEOutputType< T >::type >::get_dphidy(), and libMesh::FEGenericBase< FEOutputType< T >::type >::get_dphidz().

calculate_dphiref

bool libMesh::FEAbstract::calculate_dphiref

mutableprotectedinherited

Should we calculate reference shape function gradients?

Definition at line 567 of file fe_abstract.h.

Referenced by libMesh::FEGenericBase< FEOutputType< T >::type >::get_curl_phi(), libMesh::FEGenericBase< FEOutputType< T >::type >::get_d2phi(), libMesh::FEGenericBase< FEOutputType< T >::type >::get_d2phideta2(), libMesh::FEGenericBase< FEOutputType< T >::type >::get_d2phidetadzeta(), libMesh::FEGenericBase< FEOutputType< T >::type >::get_d2phidx2(), libMesh::FEGenericBase< FEOutputType< T >::type >::get_d2phidxdy(), libMesh::FEGenericBase< FEOutputType< T >::type >::get_d2phidxdz(), libMesh::FEGenericBase< FEOutputType< T >::type >::get_d2phidxi2(), libMesh::FEGenericBase< FEOutputType< T >::type >::get_d2phidxideta(), libMesh::FEGenericBase< FEOutputType< T >::type >::get_d2phidxidzeta(), libMesh::FEGenericBase< FEOutputType< T >::type >::get_d2phidy2(), libMesh::FEGenericBase< FEOutputType< T >::type >::get_d2phidydz(), libMesh::FEGenericBase< FEOutputType< T >::type >::get_d2phidz2(), libMesh::FEGenericBase< FEOutputType< T >::type >::get_d2phidzeta2(), libMesh::FEGenericBase< FEOutputType< T >::type >::get_div_phi(), libMesh::FEGenericBase< FEOutputType< T >::type >::get_dphi(), libMesh::FEGenericBase< FEOutputType< T >::type >::get_dphideta(), libMesh::FEGenericBase< FEOutputType< T >::type >::get_dphidx(), libMesh::FEGenericBase< FEOutputType< T >::type >::get_dphidxi(), libMesh::FEGenericBase< FEOutputType< T >::type >::get_dphidy(), libMesh::FEGenericBase< FEOutputType< T >::type >::get_dphidz(), and libMesh::FEGenericBase< FEOutputType< T >::type >::get_dphidzeta().

calculate_phi

bool libMesh::FEAbstract::calculate_phi

mutableprotectedinherited

Should we calculate shape functions?

Definition at line 542 of file fe_abstract.h.

Referenced by libMesh::FEGenericBase< FEOutputType< T >::type >::get_phi().

calculations_started

bool libMesh::FEAbstract::calculations_started

mutableprotectedinherited

Have calculations with this object already been started? Then all get_* functions should already have been called.

Definition at line 537 of file fe_abstract.h.

Referenced by libMesh::FEGenericBase< FEOutputType< T >::type >::get_curl_phi(), libMesh::FEGenericBase< FEOutputType< T >::type >::get_d2phi(), libMesh::FEGenericBase< FEOutputType< T >::type >::get_d2phideta2(), libMesh::FEGenericBase< FEOutputType< T >::type >::get_d2phidetadzeta(), libMesh::FEGenericBase< FEOutputType< T >::type >::get_d2phidx2(), libMesh::FEGenericBase< FEOutputType< T >::type >::get_d2phidxdy(), libMesh::FEGenericBase< FEOutputType< T >::type >::get_d2phidxdz(), libMesh::FEGenericBase< FEOutputType< T >::type >::get_d2phidxi2(), libMesh::FEGenericBase< FEOutputType< T >::type >::get_d2phidxideta(), libMesh::FEGenericBase< FEOutputType< T >::type >::get_d2phidxidzeta(), libMesh::FEGenericBase< FEOutputType< T >::type >::get_d2phidy2(), libMesh::FEGenericBase< FEOutputType< T >::type >::get_d2phidydz(), libMesh::FEGenericBase< FEOutputType< T >::type >::get_d2phidz2(), libMesh::FEGenericBase< FEOutputType< T >::type >::get_d2phidzeta2(), libMesh::FEGenericBase< FEOutputType< T >::type >::get_div_phi(), libMesh::FEGenericBase< FEOutputType< T >::type >::get_dphi(), libMesh::FEGenericBase< FEOutputType< T >::type >::get_dphideta(), libMesh::FEGenericBase< FEOutputType< T >::type >::get_dphidx(), libMesh::FEGenericBase< FEOutputType< T >::type >::get_dphidxi(), libMesh::FEGenericBase< FEOutputType< T >::type >::get_dphidy(), libMesh::FEGenericBase< FEOutputType< T >::type >::get_dphidz(), libMesh::FEGenericBase< FEOutputType< T >::type >::get_dphidzeta(), libMesh::FEGenericBase< FEOutputType< T >::type >::get_phi(), and init_shape_functions().

curl_phi

std::vector<std::vector<OutputShape> > libMesh::FEGenericBase< FEOutputType< T >::type >::curl_phi

protectedinherited

Shape function curl values. Only defined for vector types.

Definition at line 508 of file fe_base.h.

d2phi

std::vector<std::vector<OutputTensor> > libMesh::FEGenericBase< FEOutputType< T >::type >::d2phi

protectedinherited

Shape function second derivative values.

Definition at line 551 of file fe_base.h.

Referenced by init_shape_functions().

d2phideta2

std::vector<std::vector<OutputShape> > libMesh::FEGenericBase< FEOutputType< T >::type >::d2phideta2

protectedinherited

Shape function second derivatives in the eta direction.

Definition at line 571 of file fe_base.h.

Referenced by init_shape_functions().

d2phidetadzeta

std::vector<std::vector<OutputShape> > libMesh::FEGenericBase< FEOutputType< T >::type >::d2phidetadzeta

protectedinherited

Shape function second derivatives in the eta-zeta direction.

Definition at line 576 of file fe_base.h.

d2phidx2

std::vector<std::vector<OutputShape> > libMesh::FEGenericBase< FEOutputType< T >::type >::d2phidx2

protectedinherited

Shape function second derivatives in the x direction.

Definition at line 586 of file fe_base.h.

d2phidxdy

std::vector<std::vector<OutputShape> > libMesh::FEGenericBase< FEOutputType< T >::type >::d2phidxdy

protectedinherited

Shape function second derivatives in the x-y direction.

Definition at line 591 of file fe_base.h.

d2phidxdz

std::vector<std::vector<OutputShape> > libMesh::FEGenericBase< FEOutputType< T >::type >::d2phidxdz

protectedinherited

Shape function second derivatives in the x-z direction.

Definition at line 596 of file fe_base.h.

d2phidxi2

std::vector<std::vector<OutputShape> > libMesh::FEGenericBase< FEOutputType< T >::type >::d2phidxi2

protectedinherited

Shape function second derivatives in the xi direction.

Definition at line 556 of file fe_base.h.

Referenced by init_shape_functions().

d2phidxideta

std::vector<std::vector<OutputShape> > libMesh::FEGenericBase< FEOutputType< T >::type >::d2phidxideta

protectedinherited

Shape function second derivatives in the xi-eta direction.

Definition at line 561 of file fe_base.h.

Referenced by init_shape_functions().

d2phidxidzeta

std::vector<std::vector<OutputShape> > libMesh::FEGenericBase< FEOutputType< T >::type >::d2phidxidzeta

protectedinherited

Shape function second derivatives in the xi-zeta direction.

Definition at line 566 of file fe_base.h.

d2phidy2

std::vector<std::vector<OutputShape> > libMesh::FEGenericBase< FEOutputType< T >::type >::d2phidy2

protectedinherited

Shape function second derivatives in the y direction.

Definition at line 601 of file fe_base.h.

d2phidydz

std::vector<std::vector<OutputShape> > libMesh::FEGenericBase< FEOutputType< T >::type >::d2phidydz

protectedinherited

Shape function second derivatives in the y-z direction.

Definition at line 606 of file fe_base.h.

d2phidz2

std::vector<std::vector<OutputShape> > libMesh::FEGenericBase< FEOutputType< T >::type >::d2phidz2

protectedinherited

Shape function second derivatives in the z direction.

Definition at line 611 of file fe_base.h.

d2phidzeta2

std::vector<std::vector<OutputShape> > libMesh::FEGenericBase< FEOutputType< T >::type >::d2phidzeta2

protectedinherited

Shape function second derivatives in the zeta direction.

Definition at line 581 of file fe_base.h.

dim

const unsigned int libMesh::FEAbstract::dim

protectedinherited

The dimensionality of the object

Definition at line 531 of file fe_abstract.h.

Referenced by libMesh::FEAbstract::build(), and libMesh::FEAbstract::get_dim().

div_phi

std::vector<std::vector<OutputDivergence> > libMesh::FEGenericBase< FEOutputType< T >::type >::div_phi

protectedinherited

Shape function divergence values. Only defined for vector types.

Definition at line 513 of file fe_base.h.

dphase

std::vector<OutputGradient> libMesh::FEGenericBase< FEOutputType< T >::type >::dphase

protectedinherited

Used for certain infinite element families: the first derivatives of the phase term in global coordinates, over all quadrature points.

Definition at line 629 of file fe_base.h.

dphi

std::vector<std::vector<OutputGradient> > libMesh::FEGenericBase< FEOutputType< T >::type >::dphi

protectedinherited

Shape function derivative values.

Definition at line 503 of file fe_base.h.

Referenced by init_shape_functions().

dphideta

std::vector<std::vector<OutputShape> > libMesh::FEGenericBase< FEOutputType< T >::type >::dphideta

protectedinherited

Shape function derivatives in the eta direction.

Definition at line 523 of file fe_base.h.

Referenced by init_shape_functions().

dphidx

std::vector<std::vector<OutputShape> > libMesh::FEGenericBase< FEOutputType< T >::type >::dphidx

protectedinherited

Shape function derivatives in the x direction.

Definition at line 533 of file fe_base.h.

dphidxi

std::vector<std::vector<OutputShape> > libMesh::FEGenericBase< FEOutputType< T >::type >::dphidxi

protectedinherited

Shape function derivatives in the xi direction.

Definition at line 518 of file fe_base.h.

Referenced by init_shape_functions().

dphidy

std::vector<std::vector<OutputShape> > libMesh::FEGenericBase< FEOutputType< T >::type >::dphidy

protectedinherited

Shape function derivatives in the y direction.

Definition at line 538 of file fe_base.h.

dphidz

std::vector<std::vector<OutputShape> > libMesh::FEGenericBase< FEOutputType< T >::type >::dphidz

protectedinherited

Shape function derivatives in the z direction.

Definition at line 543 of file fe_base.h.

dphidzeta

std::vector<std::vector<OutputShape> > libMesh::FEGenericBase< FEOutputType< T >::type >::dphidzeta

protectedinherited

Shape function derivatives in the zeta direction.

Definition at line 528 of file fe_base.h.

dweight

std::vector<RealGradient> libMesh::FEGenericBase< FEOutputType< T >::type >::dweight

protectedinherited

Used for certain infinite element families: the global derivative of the additional radial weight , over all quadrature points.

Definition at line 636 of file fe_base.h.

elem_type

ElemType libMesh::FEAbstract::elem_type

protectedinherited

The element type the current data structures are set up for.

Definition at line 581 of file fe_abstract.h.

Referenced by attach_quadrature_rule(), and libMesh::FEAbstract::get_type().

fe_type

FEType libMesh::FEAbstract::fe_type

protectedinherited

The finite element type for this object.

Note

This should be constant for the object.

Definition at line 575 of file fe_abstract.h.

Referenced by libMesh::FEAbstract::compute_node_constraints(), libMesh::FEAbstract::compute_periodic_node_constraints(), libMesh::FEAbstract::get_family(), libMesh::FEAbstract::get_fe_type(), libMesh::FEAbstract::get_order(), libMesh::InfFE< Dim, T_radial, T_map >::InfFE(), init_shape_functions(), and libMesh::FEAbstract::set_fe_order().

last_edge

unsigned int libMesh::FE< Dim, T >::last_edge

protectedinherited

Definition at line 482 of file fe.h.

last_side

ElemType libMesh::FE< Dim, T >::last_side

protectedinherited

The last side and last edge we did a reinit on

Definition at line 480 of file fe.h.

phi

std::vector<std::vector<OutputShape> > libMesh::FEGenericBase< FEOutputType< T >::type >::phi

protectedinherited

Shape function values.

Definition at line 498 of file fe_base.h.

Referenced by init_shape_functions().

qrule

QBase* libMesh::FEAbstract::qrule

protectedinherited

A pointer to the quadrature rule employed

Definition at line 592 of file fe_abstract.h.

Referenced by attach_quadrature_rule().

shapes_on_quadrature

bool libMesh::FEAbstract::shapes_on_quadrature

protectedinherited

A flag indicating if current data structures correspond to quadrature rule points

Definition at line 598 of file fe_abstract.h.

weight

std::vector<Real> libMesh::FEGenericBase< FEOutputType< T >::type >::weight

protectedinherited

Used for certain infinite element families: the additional radial weight in local coordinates, over all quadrature points.

Definition at line 643 of file fe_base.h.

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The documentation for this class was generated from the following files:

• fe.h
• fe_subdivision_2D.C