fe.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/elem.h"
#include "libmesh/fe.h"
#include "libmesh/fe_interface.h"
#include "libmesh/fe_macro.h"
#include "libmesh/libmesh_logging.h"
#include "libmesh/quadrature.h"
#include "libmesh/tensor_value.h"
#include "libmesh/enum_elem_type.h"
 
namespace libMesh
{
 
 
// ------------------------------------------------------------
// FE class members
template <unsigned int Dim, FEFamily T>
FE<Dim,T>::FE (const FEType & fet) :
  FEGenericBase<typename FEOutputType<T>::type> (Dim,fet),
  last_side(INVALID_ELEM),
  last_edge(libMesh::invalid_uint)
{
  // Sanity check.  Make sure the
  // Family specified in the template instantiation
  // matches the one in the FEType object
  libmesh_assert_equal_to (T, this->get_family());
}
 
 
template <unsigned int Dim, FEFamily T>
unsigned int FE<Dim,T>::n_shape_functions () const
{
  return FE<Dim,T>::n_dofs (this->elem_type,
                            static_cast<Order>(this->fe_type.order + this->_p_level));
}
 
 
template <unsigned int Dim, FEFamily T>
void FE<Dim,T>::attach_quadrature_rule (QBase * q)
{
  libmesh_assert(q);
  this->qrule = q;
  // make sure we don't cache results from a previous quadrature rule
  this->elem_type = INVALID_ELEM;
  return;
}
 
 
template <unsigned int Dim, FEFamily T>
unsigned int FE<Dim,T>::n_quadrature_points () const
{
  libmesh_assert(this->qrule);
  return this->qrule->n_points();
}
 
 
template <unsigned int Dim, FEFamily T>
void FE<Dim,T>::dofs_on_side(const Elem * const elem,
                             const Order o,
                             unsigned int s,
                             std::vector<unsigned int> & di)
{
  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;
    }
}
 
 
 
template <unsigned int Dim, FEFamily T>
void FE<Dim,T>::dofs_on_edge(const Elem * const elem,
                             const Order o,
                             unsigned int e,
                             std::vector<unsigned int> & di)
{
  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;
    }
}
 
 
 
template <unsigned int Dim, FEFamily T>
void FE<Dim,T>::reinit(const Elem * elem,
                       const std::vector<Point> * const pts,
                       const std::vector<Real> * const weights)
{
  // We can be called with no element.  If we're evaluating SCALAR
  // dofs we'll still have work to do.
  // libmesh_assert(elem);
 
  // We're calculating now!  Time to determine what.
  this->determine_calculations();
 
  // Try to avoid calling init_shape_functions
  // even when shapes_need_reinit
  bool cached_nodes_still_fit = false;
 
  // Most of the hard work happens when we have an actual element
  if (elem)
    {
      // Initialize the shape functions at the user-specified
      // points
      if (pts != nullptr)
        {
          // Set the type and p level for this element
          this->elem_type = elem->type();
          this->_p_level = elem->p_level();
 
          // Initialize the shape functions
          this->_fe_map->template init_reference_to_physical_map<Dim>
            (*pts, elem);
          this->init_shape_functions (*pts, elem);
 
          // The shape functions do not correspond to the qrule
          this->shapes_on_quadrature = false;
        }
 
      // If there are no user specified points, we use the
      // quadrature rule
 
      // update the type in accordance to the current cell
      // and reinit if the cell type has changed or (as in
      // the case of the hierarchics) the shape functions need
      // reinit, since they depend on the particular element shape
      else
        {
          libmesh_assert(this->qrule);
          this->qrule->init(elem->type(), elem->p_level());
 
          if (this->qrule->shapes_need_reinit())
            this->shapes_on_quadrature = false;
 
          if (this->elem_type != elem->type() ||
              this->_p_level != elem->p_level() ||
              !this->shapes_on_quadrature)
            {
              // Set the type and p level for this element
              this->elem_type = elem->type();
              this->_p_level = elem->p_level();
              // Initialize the shape functions
              this->_fe_map->template init_reference_to_physical_map<Dim>
                (this->qrule->get_points(), elem);
              this->init_shape_functions (this->qrule->get_points(), elem);
 
              if (this->shapes_need_reinit())
                {
                  cached_nodes.resize(elem->n_nodes());
                  for (auto n : elem->node_index_range())
                    cached_nodes[n] = elem->point(n);
                }
            }
          else
            {
              // libmesh_assert_greater (elem->n_nodes(), 1);
 
              cached_nodes_still_fit = true;
              if (cached_nodes.size() != elem->n_nodes())
                cached_nodes_still_fit = false;
              else
                for (auto n : IntRange<unsigned int>(1, elem->n_nodes()))
                  {
                    if (!(elem->point(n) - elem->point(0)).relative_fuzzy_equals
                        ((cached_nodes[n] - cached_nodes[0]), 1e-13))
                      {
                        cached_nodes_still_fit = false;
                        break;
                      }
                  }
 
              if (this->shapes_need_reinit() && !cached_nodes_still_fit)
                {
                  this->_fe_map->template init_reference_to_physical_map<Dim>
                    (this->qrule->get_points(), elem);
                  this->init_shape_functions (this->qrule->get_points(), elem);
                  cached_nodes.resize(elem->n_nodes());
                  for (auto n : elem->node_index_range())
                    cached_nodes[n] = elem->point(n);
                }
            }
 
          // The shape functions correspond to the qrule
          this->shapes_on_quadrature = true;
        }
    }
  else // With no defined elem, so mapping or caching to
       // be done, and our "quadrature rule" is one point for nonlocal
       // (SCALAR) variables and zero points for local variables.
    {
      this->elem_type = INVALID_ELEM;
      this->_p_level = 0;
 
      if (!pts)
        {
          if (T == SCALAR)
            {
              this->qrule->get_points() =
                std::vector<Point>(1,Point(0));
 
              this->qrule->get_weights() =
                std::vector<Real>(1,1);
            }
          else
            {
              this->qrule->get_points().clear();
              this->qrule->get_weights().clear();
            }
 
          this->init_shape_functions (this->qrule->get_points(), elem);
        }
      else
        this->init_shape_functions (*pts, elem);
    }
 
  // Compute the map for this element.
  if (pts != nullptr)
    {
      if (weights != nullptr)
        {
          this->_fe_map->compute_map (this->dim, *weights, elem, this->calculate_d2phi);
        }
      else
        {
          std::vector<Real> dummy_weights (pts->size(), 1.);
          this->_fe_map->compute_map (this->dim, dummy_weights, elem, this->calculate_d2phi);
        }
    }
  else
    {
      this->_fe_map->compute_map (this->dim, this->qrule->get_weights(), elem, this->calculate_d2phi);
    }
 
  // Compute the shape functions and the derivatives at all of the
  // quadrature points.
  if (!cached_nodes_still_fit)
    {
      if (pts != nullptr)
        this->compute_shape_functions (elem,*pts);
      else
        this->compute_shape_functions(elem,this->qrule->get_points());
    }
}
 
 
 
template <unsigned int Dim, FEFamily T>
void FE<Dim,T>::init_shape_functions(const std::vector<Point> & qp,
                                     const Elem * elem)
{
  // Start logging the shape function initialization
  LOG_SCOPE("init_shape_functions()", "FE");
 
  // The number of quadrature points.
  const unsigned int n_qp = cast_int<unsigned int>(qp.size());
 
  // Number of shape functions in the finite element approximation
  // space.
  const unsigned int n_approx_shape_functions =
    this->n_shape_functions(this->get_type(),
                            this->get_order());
 
  // resize the vectors to hold current data
  // Phi are the shape functions used for the FE approximation
  // Phi_map are the shape functions used for the FE mapping
  if (this->calculate_phi)
    this->phi.resize     (n_approx_shape_functions);
 
  if (this->calculate_dphi)
    {
      this->dphi.resize    (n_approx_shape_functions);
      this->dphidx.resize  (n_approx_shape_functions);
      this->dphidy.resize  (n_approx_shape_functions);
      this->dphidz.resize  (n_approx_shape_functions);
    }
 
  if (this->calculate_dphiref)
    {
      if (Dim > 0)
        this->dphidxi.resize (n_approx_shape_functions);
 
      if (Dim > 1)
        this->dphideta.resize      (n_approx_shape_functions);
 
      if (Dim > 2)
        this->dphidzeta.resize     (n_approx_shape_functions);
    }
 
  if (this->calculate_curl_phi && (FEInterface::field_type(T) == TYPE_VECTOR))
    this->curl_phi.resize(n_approx_shape_functions);
 
  if (this->calculate_div_phi && (FEInterface::field_type(T) == TYPE_VECTOR))
    this->div_phi.resize(n_approx_shape_functions);
 
#ifdef LIBMESH_ENABLE_SECOND_DERIVATIVES
  if (this->calculate_d2phi)
    {
      this->d2phi.resize     (n_approx_shape_functions);
      this->d2phidx2.resize  (n_approx_shape_functions);
      this->d2phidxdy.resize (n_approx_shape_functions);
      this->d2phidxdz.resize (n_approx_shape_functions);
      this->d2phidy2.resize  (n_approx_shape_functions);
      this->d2phidydz.resize (n_approx_shape_functions);
      this->d2phidz2.resize  (n_approx_shape_functions);
 
      if (Dim > 0)
        this->d2phidxi2.resize (n_approx_shape_functions);
 
      if (Dim > 1)
        {
          this->d2phidxideta.resize (n_approx_shape_functions);
          this->d2phideta2.resize   (n_approx_shape_functions);
        }
      if (Dim > 2)
        {
          this->d2phidxidzeta.resize  (n_approx_shape_functions);
          this->d2phidetadzeta.resize (n_approx_shape_functions);
          this->d2phidzeta2.resize    (n_approx_shape_functions);
        }
    }
#endif // ifdef LIBMESH_ENABLE_SECOND_DERIVATIVES
 
  for (unsigned int i=0; i<n_approx_shape_functions; i++)
    {
      if (this->calculate_phi)
        this->phi[i].resize         (n_qp);
      if (this->calculate_dphi)
        {
          this->dphi[i].resize        (n_qp);
          this->dphidx[i].resize      (n_qp);
          this->dphidy[i].resize      (n_qp);
          this->dphidz[i].resize      (n_qp);
        }
 
      if (this->calculate_dphiref)
        {
          if (Dim > 0)
            this->dphidxi[i].resize(n_qp);
 
          if (Dim > 1)
            this->dphideta[i].resize(n_qp);
 
          if (Dim > 2)
            this->dphidzeta[i].resize(n_qp);
        }
 
      if (this->calculate_curl_phi && (FEInterface::field_type(T) == TYPE_VECTOR))
        this->curl_phi[i].resize(n_qp);
 
      if (this->calculate_div_phi && (FEInterface::field_type(T) == TYPE_VECTOR))
        this->div_phi[i].resize(n_qp);
 
#ifdef LIBMESH_ENABLE_SECOND_DERIVATIVES
      if (this->calculate_d2phi)
        {
          this->d2phi[i].resize     (n_qp);
          this->d2phidx2[i].resize  (n_qp);
          this->d2phidxdy[i].resize (n_qp);
          this->d2phidxdz[i].resize (n_qp);
          this->d2phidy2[i].resize  (n_qp);
          this->d2phidydz[i].resize (n_qp);
          this->d2phidz2[i].resize  (n_qp);
          if (Dim > 0)
            this->d2phidxi2[i].resize (n_qp);
          if (Dim > 1)
            {
              this->d2phidxideta[i].resize (n_qp);
              this->d2phideta2[i].resize   (n_qp);
            }
          if (Dim > 2)
            {
              this->d2phidxidzeta[i].resize  (n_qp);
              this->d2phidetadzeta[i].resize (n_qp);
              this->d2phidzeta2[i].resize    (n_qp);
            }
        }
#endif // ifdef LIBMESH_ENABLE_SECOND_DERIVATIVES
    }
 
 
#ifdef LIBMESH_ENABLE_INFINITE_ELEMENTS
  //------------------------------------------------------------
  // Initialize the data fields, which should only be used for infinite
  // elements, to some sensible values, so that using a FE with the
  // variational formulation of an InfFE, correct element matrices are
  // returned
 
  {
    this->weight.resize  (n_qp);
    this->dweight.resize (n_qp);
    this->dphase.resize  (n_qp);
 
    for (unsigned int p=0; p<n_qp; p++)
      {
        this->weight[p] = 1.;
        this->dweight[p].zero();
        this->dphase[p].zero();
      }
 
  }
#endif // ifdef LIBMESH_ENABLE_INFINITE_ELEMENTS
 
  switch (Dim)
    {
 
      //------------------------------------------------------------
      // 0D
    case 0:
      {
        break;
      }
 
      //------------------------------------------------------------
      // 1D
    case 1:
      {
        // Compute the value of the approximation shape function i at quadrature point p
        if (this->calculate_dphiref)
          for (unsigned int i=0; i<n_approx_shape_functions; i++)
            for (unsigned int p=0; p<n_qp; p++)
              this->dphidxi[i][p]  = FE<Dim,T>::shape_deriv (elem, this->fe_type.order, i, 0, qp[p]);
#ifdef LIBMESH_ENABLE_SECOND_DERIVATIVES
        if (this->calculate_d2phi)
          for (unsigned int i=0; i<n_approx_shape_functions; i++)
            for (unsigned int p=0; p<n_qp; p++)
              this->d2phidxi2[i][p] = FE<Dim,T>::shape_second_deriv (elem, this->fe_type.order, i, 0, qp[p]);
#endif // ifdef LIBMESH_ENABLE_SECOND_DERIVATIVES
 
        break;
      }
 
 
 
      //------------------------------------------------------------
      // 2D
    case 2:
      {
        // Compute the value of the approximation shape function i at quadrature point p
        if (this->calculate_dphiref)
          for (unsigned int i=0; i<n_approx_shape_functions; i++)
            for (unsigned int p=0; p<n_qp; p++)
              {
                this->dphidxi[i][p]  = FE<Dim,T>::shape_deriv (elem, this->fe_type.order, i, 0, qp[p]);
                this->dphideta[i][p] = FE<Dim,T>::shape_deriv (elem, this->fe_type.order, i, 1, qp[p]);
              }
#ifdef LIBMESH_ENABLE_SECOND_DERIVATIVES
        if (this->calculate_d2phi)
          for (unsigned int i=0; i<n_approx_shape_functions; i++)
            for (unsigned int p=0; p<n_qp; p++)
              {
                this->d2phidxi2[i][p] = FE<Dim,T>::shape_second_deriv (elem, this->fe_type.order, i, 0, qp[p]);
                this->d2phidxideta[i][p] = FE<Dim,T>::shape_second_deriv (elem, this->fe_type.order, i, 1, qp[p]);
                this->d2phideta2[i][p] = FE<Dim,T>::shape_second_deriv (elem, this->fe_type.order, i, 2, qp[p]);
              }
#endif // ifdef LIBMESH_ENABLE_SECOND_DERIVATIVES
 
 
        break;
      }
 
 
 
      //------------------------------------------------------------
      // 3D
    case 3:
      {
        // Compute the value of the approximation shape function i at quadrature point p
        if (this->calculate_dphiref)
          for (unsigned int i=0; i<n_approx_shape_functions; i++)
            for (unsigned int p=0; p<n_qp; p++)
              {
                this->dphidxi[i][p]   = FE<Dim,T>::shape_deriv (elem, this->fe_type.order, i, 0, qp[p]);
                this->dphideta[i][p]  = FE<Dim,T>::shape_deriv (elem, this->fe_type.order, i, 1, qp[p]);
                this->dphidzeta[i][p] = FE<Dim,T>::shape_deriv (elem, this->fe_type.order, i, 2, qp[p]);
              }
#ifdef LIBMESH_ENABLE_SECOND_DERIVATIVES
        if (this->calculate_d2phi)
          for (unsigned int i=0; i<n_approx_shape_functions; i++)
            for (unsigned int p=0; p<n_qp; p++)
              {
                this->d2phidxi2[i][p] = FE<Dim,T>::shape_second_deriv (elem, this->fe_type.order, i, 0, qp[p]);
                this->d2phidxideta[i][p] = FE<Dim,T>::shape_second_deriv (elem, this->fe_type.order, i, 1, qp[p]);
                this->d2phideta2[i][p] = FE<Dim,T>::shape_second_deriv (elem, this->fe_type.order, i, 2, qp[p]);
                this->d2phidxidzeta[i][p] = FE<Dim,T>::shape_second_deriv (elem, this->fe_type.order, i, 3, qp[p]);
                this->d2phidetadzeta[i][p] = FE<Dim,T>::shape_second_deriv (elem, this->fe_type.order, i, 4, qp[p]);
                this->d2phidzeta2[i][p] = FE<Dim,T>::shape_second_deriv (elem, this->fe_type.order, i, 5, qp[p]);
              }
#endif // ifdef LIBMESH_ENABLE_SECOND_DERIVATIVES
 
        break;
      }
 
 
    default:
      libmesh_error_msg("Invalid dimension Dim = " << Dim);
    }
}
 
 
 
#ifdef LIBMESH_ENABLE_INFINITE_ELEMENTS
 
template <unsigned int Dim, FEFamily T>
void FE<Dim,T>::init_base_shape_functions(const std::vector<Point> & qp,
                                          const Elem * e)
{
  this->elem_type = e->type();
  this->_fe_map->template init_reference_to_physical_map<Dim>(qp, e);
  init_shape_functions(qp, e);
}
 
#endif // LIBMESH_ENABLE_INFINITE_ELEMENTS
 
 
 
//--------------------------------------------------------------
// Explicit instantiations using macro from fe_macro.h
 
INSTANTIATE_FE(0);
 
INSTANTIATE_FE(1);
 
INSTANTIATE_FE(2);
 
INSTANTIATE_FE(3);
 
INSTANTIATE_SUBDIVISION_FE;
 
} // namespace libMesh