fe_rational_shape_2D.C

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// The libMesh Finite Element Library.
// Copyright (C) 2002-2019 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
 
 
// Local includes
#include "libmesh/libmesh_config.h"
#ifdef LIBMESH_ENABLE_HIGHER_ORDER_SHAPES
 
#include "libmesh/elem.h"
#include "libmesh/fe.h"
#include "libmesh/fe_interface.h"
 
 
namespace {
static const libMesh::FEFamily _underlying_fe_family = libMesh::BERNSTEIN;
}
 
 
namespace libMesh
{
 
template <>
Real FE<2,RATIONAL_BERNSTEIN>::shape(const ElemType,
                                     const Order,
                                     const unsigned int,
                                     const Point &)
{
  libmesh_error_msg("Rational bases require the real element \nto query nodal weighting.");
  return 0.;
}
 
 
 
template <>
Real FE<2,RATIONAL_BERNSTEIN>::shape(const Elem * elem,
                                     const Order order,
                                     const unsigned int i,
                                     const Point & p,
                                     const bool add_p_level)
{
  libmesh_assert(elem);
 
  const ElemType elem_type = elem->type();
 
  const Order totalorder = static_cast<Order>(order + add_p_level * elem->p_level());
 
  // FEType object to be passed to various FEInterface functions below.
  FEType fe_type(totalorder, _underlying_fe_family);
 
  const unsigned int n_sf =
    FEInterface::n_shape_functions(2, fe_type, elem_type);
 
  const unsigned int n_nodes = elem->n_nodes();
  libmesh_assert_equal_to (n_sf, n_nodes);
 
  std::vector<Real> node_weights(n_nodes);
 
  const unsigned char datum_index = elem->mapping_data();
  for (unsigned int n=0; n<n_nodes; n++)
    node_weights[n] =
      elem->node_ref(n).get_extra_datum<Real>(datum_index);
 
  Real weighted_shape_i = 0, weighted_sum = 0;
 
  for (unsigned int sf=0; sf<n_sf; sf++)
    {
      Real weighted_shape = node_weights[sf] *
        FEInterface::shape(2, fe_type, elem, sf, p);
      weighted_sum += weighted_shape;
      if (sf == i)
        weighted_shape_i = weighted_shape;
    }
 
  return weighted_shape_i / weighted_sum;
}
 
 
 
template <>
Real FE<2,RATIONAL_BERNSTEIN>::shape_deriv(const ElemType,
                                           const Order,
                                           const unsigned int,
                                           const unsigned int,
                                           const Point &)
{
  libmesh_error_msg("Rational bases require the real element \nto query nodal weighting.");
  return 0.;
}
 
 
 
template <>
Real FE<2,RATIONAL_BERNSTEIN>::shape_deriv(const Elem * elem,
                                           const Order order,
                                           const unsigned int i,
                                           const unsigned int j,
                                           const Point & p,
                                           const bool add_p_level)
{
  libmesh_assert(elem);
 
  const ElemType elem_type = elem->type();
 
  const Order totalorder = static_cast<Order>(order + add_p_level * elem->p_level());
 
  // FEType object to be passed to various FEInterface functions below.
  FEType fe_type(totalorder, _underlying_fe_family);
 
  const unsigned int n_sf =
    FEInterface::n_shape_functions(2, fe_type, elem_type);
 
  const unsigned int n_nodes = elem->n_nodes();
  libmesh_assert_equal_to (n_sf, n_nodes);
 
  std::vector<Real> node_weights(n_nodes);
 
  const unsigned char datum_index = elem->mapping_data();
  for (unsigned int n=0; n<n_nodes; n++)
    node_weights[n] =
      elem->node_ref(n).get_extra_datum<Real>(datum_index);
 
  Real weighted_shape_i = 0, weighted_sum = 0,
       weighted_grad_i = 0, weighted_grad_sum = 0;
 
  for (unsigned int sf=0; sf<n_sf; sf++)
    {
      Real weighted_shape = node_weights[sf] *
        FEInterface::shape(2, fe_type, elem, sf, p);
      Real weighted_grad = node_weights[sf] *
        FEInterface::shape_deriv(2, fe_type, elem, sf, j, p);
      weighted_sum += weighted_shape;
      weighted_grad_sum += weighted_grad;
      if (sf == i)
        {
          weighted_shape_i = weighted_shape;
          weighted_grad_i = weighted_grad;
        }
    }
 
  return (weighted_sum * weighted_grad_i - weighted_shape_i * weighted_grad_sum) /
         weighted_sum / weighted_sum;
}
 
 
 
#ifdef LIBMESH_ENABLE_SECOND_DERIVATIVES
template <>
Real FE<2,RATIONAL_BERNSTEIN>::shape_second_deriv(const ElemType,
                                                  const Order,
                                                  const unsigned int,
                                                  const unsigned int,
                                                  const Point &)
{
  libmesh_error_msg("Rational bases require the real element \nto query nodal weighting.");
  return 0.;
}
 
 
 
template <>
Real FE<2,RATIONAL_BERNSTEIN>::shape_second_deriv(const Elem * elem,
                                                  const Order order,
                                                  const unsigned int i,
                                                  const unsigned int j,
                                                  const Point & p,
                                                  const bool add_p_level)
{
  unsigned int j1, j2;
  switch (j)
    {
    case 0:
      // j = 0 ==> d^2 phi / dxi^2
      j1 = j2 = 0;
      break;
    case 1:
      // j = 1 ==> d^2 phi / dxi deta
      j1 = 0;
      j2 = 1;
      break;
    case 2:
      // j = 2 ==> d^2 phi / deta^2
      j1 = j2 = 1;
      break;
    default:
      libmesh_error();
    }
 
  libmesh_assert(elem);
 
  const ElemType elem_type = elem->type();
 
  const Order totalorder = static_cast<Order>(order + add_p_level * elem->p_level());
 
  // FEType object to be passed to various FEInterface functions below.
  FEType fe_type(totalorder, _underlying_fe_family);
 
  const unsigned int n_sf =
    FEInterface::n_shape_functions(2, fe_type, elem_type);
 
  const unsigned int n_nodes = elem->n_nodes();
  libmesh_assert_equal_to (n_sf, n_nodes);
 
  std::vector<Real> node_weights(n_nodes);
 
  for (unsigned int n=0; n<n_nodes; n++)
    node_weights[n] = elem->node_ref(n).get_extra_datum<Real>(0);
 
  Real weighted_shape_i = 0, weighted_sum = 0,
       weighted_grada_i = 0, weighted_grada_sum = 0,
       weighted_gradb_i = 0, weighted_gradb_sum = 0,
       weighted_hess_i = 0, weighted_hess_sum = 0;
 
  for (unsigned int sf=0; sf<n_sf; sf++)
    {
      Real weighted_shape = node_weights[sf] *
        FEInterface::shape(2, fe_type, elem, sf, p);
      Real weighted_grada = node_weights[sf] *
        FEInterface::shape_deriv(2, fe_type, elem, sf, j1, p);
      Real weighted_hess = node_weights[sf] *
        FEInterface::shape_second_deriv(2, fe_type, elem, sf, j, p);
      weighted_sum += weighted_shape;
      weighted_grada_sum += weighted_grada;
      Real weighted_gradb = weighted_grada;
      if (j1 != j2)
        {
          weighted_gradb = (j1 == j2) ? weighted_grada :
            node_weights[sf] *
             FEInterface::shape_deriv(2, fe_type, elem, sf, j2, p);
          weighted_grada_sum += weighted_grada;
        }
      weighted_hess_sum += weighted_hess;
      if (sf == i)
        {
          weighted_shape_i = weighted_shape;
          weighted_grada_i = weighted_grada;
          weighted_gradb_i = weighted_gradb;
          weighted_hess_i = weighted_hess;
        }
    }
 
  if (j1 == j2)
    weighted_gradb_sum = weighted_grada_sum;
 
  return (weighted_sum * weighted_hess_i - weighted_grada_i * weighted_gradb_sum -
          weighted_shape_i * weighted_hess_sum - weighted_gradb_i * weighted_grada_sum +
          2 * weighted_grada_sum * weighted_shape_i * weighted_gradb_sum / weighted_sum) /
         weighted_sum / weighted_sum;
}
 
#endif
 
} // namespace libMesh
 
 
#endif// LIBMESH_ENABLE_HIGHER_ORDER_SHAPES