fe_nedelec_one_shape_2D.C

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// The libMesh Finite Element Library.
// Copyright (C) 2002-2018 Benjamin S. Kirk, John W. Peterson, Roy H. Stogner

// This library is free software; you can redistribute it and/or
// modify it under the terms of the GNU Lesser General Public
// License as published by the Free Software Foundation; either
// version 2.1 of the License, or (at your option) any later version.

// This library is distributed in the hope that it will be useful,
// but WITHOUT ANY WARRANTY; without even the implied warranty of
// MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the GNU
// Lesser General Public License for more details.

// You should have received a copy of the GNU Lesser General Public
// License along with this library; if not, write to the Free Software
// Foundation, Inc., 59 Temple Place, Suite 330, Boston, MA  02111-1307  USA


// C++ includes

// Local includes
#include "libmesh/fe.h"
#include "libmesh/elem.h"

namespace libMesh
{

template <>
RealGradient FE<2,NEDELEC_ONE>::shape(const ElemType,
                                      const Order,
                                      const unsigned int,
                                      const Point &)
{
  libmesh_error_msg("Nedelec elements require the element type \nbecause edge orientation is needed.");
  return RealGradient();
}


// An excellent discussion of Nedelec shape functions is given in
// http://www.dealii.org/developer/reports/nedelec/nedelec.pdf
template <>
RealGradient FE<2,NEDELEC_ONE>::shape(const Elem * elem,
                                      const Order order,
                                      const unsigned int i,
                                      const Point & p)
{
#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
}



template <>
RealGradient FE<2,NEDELEC_ONE>::shape_deriv(const ElemType,
                                            const Order,
                                            const unsigned int,
                                            const unsigned int,
                                            const Point &)
{
  libmesh_error_msg("Nedelec elements require the element type \nbecause edge orientation is needed.");
  return RealGradient();
}



template <>
RealGradient FE<2,NEDELEC_ONE>::shape_deriv(const Elem * elem,
                                            const Order order,
                                            const unsigned int i,
                                            const unsigned int j,
                                            const Point &)
{
#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
}




template <>
RealGradient FE<2,NEDELEC_ONE>::shape_second_deriv(const ElemType,
                                                   const Order,
                                                   const unsigned int,
                                                   const unsigned int,
                                                   const Point &)
{
  libmesh_error_msg("Nedelec elements require the element type \nbecause edge orientation is needed.");
  return RealGradient();
}



template <>
RealGradient FE<2,NEDELEC_ONE>::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 &)
{
#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
}

} // namespace libMesh