mapping_cartesian.cc

// ---------------------------------------------------------------------
//
// Copyright (C) 2001 - 2015 by the deal.II authors
//
// This file is part of the deal.II library.
//
// The deal.II library is free software; you can use it, 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.
// The full text of the license can be found in the file LICENSE at
// the top level of the deal.II distribution.
//
// ---------------------------------------------------------------------


#include <deal.II/base/tensor.h>
#include <deal.II/base/quadrature.h>
#include <deal.II/base/qprojector.h>
#include <deal.II/base/memory_consumption.h>
#include <deal.II/lac/full_matrix.h>
#include <deal.II/grid/tria.h>
#include <deal.II/grid/tria_iterator.h>
#include <deal.II/dofs/dof_accessor.h>
#include <deal.II/fe/mapping_cartesian.h>
#include <deal.II/fe/fe_values.h>

#include <cmath>
#include <algorithm>


DEAL_II_NAMESPACE_OPEN


template<int dim, int spacedim>
const unsigned int MappingCartesian<dim,spacedim>::invalid_face_number;



template<int dim, int spacedim>
MappingCartesian<dim, spacedim>::InternalData::InternalData (const Quadrature<dim> &q)
  :
  quadrature_points (q.get_points ())
{}



template<int dim, int spacedim>
std::size_t
MappingCartesian<dim, spacedim>::InternalData::memory_consumption () const
{
  return (Mapping<dim, spacedim>::InternalDataBase::memory_consumption() +
          MemoryConsumption::memory_consumption (cell_extents) +
          MemoryConsumption::memory_consumption (volume_element) +
          MemoryConsumption::memory_consumption (quadrature_points));
}



template <int dim, int spacedim>
bool
MappingCartesian<dim,spacedim>::preserves_vertex_locations () const
{
  return true;
}



template<int dim, int spacedim>
UpdateFlags
MappingCartesian<dim, spacedim>::requires_update_flags (const UpdateFlags in) const
{
  // this mapping is pretty simple in that it can basically compute
  // every piece of information wanted by FEValues without requiring
  // computing any other quantities. boundary forms are one exception
  // since they can be computed from the normal vectors without much
  // further ado
  UpdateFlags out = in;
  if (out & update_boundary_forms)
    out |= update_normal_vectors;

  return out;
}



template<int dim, int spacedim>
typename Mapping<dim, spacedim>::InternalDataBase *
MappingCartesian<dim, spacedim>::get_data (const UpdateFlags      update_flags,
                                           const Quadrature<dim> &q) const
{
  InternalData *data = new InternalData (q);

  // store the flags in the internal data object so we can access them
  // in fill_fe_*_values(). use the transitive hull of the required
  // flags
  data->update_each = requires_update_flags(update_flags);

  return data;
}



template<int dim, int spacedim>
typename Mapping<dim, spacedim>::InternalDataBase *
MappingCartesian<dim, spacedim>::get_face_data (const UpdateFlags update_flags,
                                                const Quadrature<dim-1>& quadrature) const
{
  InternalData *data
    = new InternalData (QProjector<dim>::project_to_all_faces(quadrature));

  // verify that we have computed the transitive hull of the required
  // flags and that FEValues has faithfully passed them on to us
  Assert (update_flags == requires_update_flags (update_flags),
          ExcInternalError());

  // store the flags in the internal data object so we can access them
  // in fill_fe_*_values()
  data->update_each = update_flags;

  return data;
}



template<int dim, int spacedim>
typename Mapping<dim, spacedim>::InternalDataBase *
MappingCartesian<dim, spacedim>::get_subface_data (const UpdateFlags update_flags,
                                                   const Quadrature<dim-1> &quadrature) const
{
  InternalData *data
    = new InternalData (QProjector<dim>::project_to_all_subfaces(quadrature));

  // verify that we have computed the transitive hull of the required
  // flags and that FEValues has faithfully passed them on to us
  Assert (update_flags == requires_update_flags (update_flags),
          ExcInternalError());

  // store the flags in the internal data object so we can access them
  // in fill_fe_*_values()
  data->update_each = update_flags;

  return data;
}




template<int dim, int spacedim>
void
MappingCartesian<dim, spacedim>::compute_fill (const typename Triangulation<dim,spacedim>::cell_iterator &cell,
                                               const unsigned int        face_no,
                                               const unsigned int        sub_no,
                                               const CellSimilarity::Similarity cell_similarity,
                                               const InternalData             &data,
                                               std::vector<Point<dim> > &quadrature_points,
                                               std::vector<Tensor<1,dim> > &normal_vectors) const
{
  const UpdateFlags update_flags = data.update_each;

  // some more sanity checks
  if (face_no != invalid_face_number)
    {
      // Add 1 on both sides of
      // assertion to avoid compiler
      // warning about testing
      // unsigned int < 0 in 1d.
      Assert (face_no+1 < GeometryInfo<dim>::faces_per_cell+1,
              ExcIndexRange (face_no, 0, GeometryInfo<dim>::faces_per_cell));

      // We would like to check for
      // sub_no < cell->face(face_no)->n_children(),
      // but unfortunately the current
      // function is also called for
      // faces without children (see
      // tests/fe/mapping.cc). Therefore,
      // we must use following workaround
      // of two separate assertions
      Assert ((sub_no == invalid_face_number) ||
              cell->face(face_no)->has_children() ||
              (sub_no+1 < GeometryInfo<dim>::max_children_per_face+1),
              ExcIndexRange (sub_no, 0,
                             GeometryInfo<dim>::max_children_per_face));
      Assert ((sub_no == invalid_face_number) ||
              !cell->face(face_no)->has_children() ||
              (sub_no < cell->face(face_no)->n_children()),
              ExcIndexRange (sub_no, 0, cell->face(face_no)->n_children()));
    }
  else
    // invalid face number, so
    // subface should be invalid as
    // well
    Assert (sub_no == invalid_face_number, ExcInternalError());

  // let @p{start} be the origin of a
  // local coordinate system. it is
  // chosen as the (lower) left
  // vertex
  const Point<dim> start = cell->vertex(0);

  // Compute start point and sizes
  // along axes.  Strange vertex
  // numbering makes this complicated
  // again.
  if (cell_similarity != CellSimilarity::translation)
    {
      switch (dim)
        {
        case 1:
          data.cell_extents[0] = cell->vertex(1)(0) - start(0);
          break;
        case 2:
          data.cell_extents[0] = cell->vertex(1)(0) - start(0);
          data.cell_extents[1] = cell->vertex(2)(1) - start(1);
          break;
        case 3:
          data.cell_extents[0] = cell->vertex(1)(0) - start(0);
          data.cell_extents[1] = cell->vertex(2)(1) - start(1);
          data.cell_extents[2] = cell->vertex(4)(2) - start(2);
          break;
        default:
          Assert(false, ExcNotImplemented());
        }
    }


  // transform quadrature point. this
  // is obtained simply by scaling
  // unit coordinates with lengths in
  // each direction
  if (update_flags & update_quadrature_points)
    {
      const typename QProjector<dim>::DataSetDescriptor offset
        = (face_no == invalid_face_number
           ?
           QProjector<dim>::DataSetDescriptor::cell()
           :
           (sub_no == invalid_face_number
            ?
            // called from FEFaceValues
            QProjector<dim>::DataSetDescriptor::face (face_no,
                                                      cell->face_orientation(face_no),
                                                      cell->face_flip(face_no),
                                                      cell->face_rotation(face_no),
                                                      quadrature_points.size())
            :
            // called from FESubfaceValues
            QProjector<dim>::DataSetDescriptor::subface (face_no, sub_no,
                                                         cell->face_orientation(face_no),
                                                         cell->face_flip(face_no),
                                                         cell->face_rotation(face_no),
                                                         quadrature_points.size(),
                                                         cell->subface_case(face_no))
           ));

      for (unsigned int i=0; i<quadrature_points.size(); ++i)
        {
          quadrature_points[i] = start;
          for (unsigned int d=0; d<dim; ++d)
            quadrature_points[i](d) += data.cell_extents[d] *
                                       data.quadrature_points[i+offset](d);
        }
    }


  // compute normal vectors. since
  // cells are aligned to coordinate
  // axes, they are simply vectors
  // with exactly one entry equal to
  // 1 or -1. Furthermore, all
  // normals on a face have the same
  // value
  if (update_flags & update_normal_vectors)
    {
      Assert (face_no < GeometryInfo<dim>::faces_per_cell,
              ExcInternalError());

      switch (dim)
        {
        case 1:
        {
          static const Point<dim>
          normals[GeometryInfo<1>::faces_per_cell]
            = { Point<dim>(-1.),
                Point<dim>( 1.)
              };
          std::fill (normal_vectors.begin(),
                     normal_vectors.end(),
                     normals[face_no]);
          break;
        }

        case 2:
        {
          static const Point<dim>
          normals[GeometryInfo<2>::faces_per_cell]
            = { Point<dim>(-1, 0),
                Point<dim>( 1, 0),
                Point<dim>( 0,-1),
                Point<dim>( 0, 1)
              };
          std::fill (normal_vectors.begin(),
                     normal_vectors.end(),
                     normals[face_no]);
          break;
        }

        case 3:
        {
          static const Point<dim>
          normals[GeometryInfo<3>::faces_per_cell]
            = { Point<dim>(-1, 0, 0),
                Point<dim>( 1, 0, 0),
                Point<dim>( 0,-1, 0),
                Point<dim>( 0, 1, 0),
                Point<dim>( 0, 0,-1),
                Point<dim>( 0, 0, 1)
              };
          std::fill (normal_vectors.begin(),
                     normal_vectors.end(),
                     normals[face_no]);
          break;
        }

        default:
          Assert (false, ExcNotImplemented());
        }
    }
}



template<int dim, int spacedim>
CellSimilarity::Similarity
MappingCartesian<dim, spacedim>::
fill_fe_values (const typename Triangulation<dim,spacedim>::cell_iterator &cell,
                const CellSimilarity::Similarity                           cell_similarity,
                const Quadrature<dim>                                     &quadrature,
                const typename Mapping<dim,spacedim>::InternalDataBase    &internal_data,
                internal::FEValues::MappingRelatedData<dim,spacedim>      &output_data) const
{
  // convert data object to internal data for this class. fails with
  // an exception if that is not possible
  Assert (dynamic_cast<const InternalData *> (&internal_data) != 0, ExcInternalError());
  const InternalData &data = static_cast<const InternalData &> (internal_data);

  std::vector<Tensor<1,dim> > dummy;

  compute_fill (cell, invalid_face_number, invalid_face_number, cell_similarity,
                data,
                output_data.quadrature_points,
                dummy);

  // compute Jacobian determinant. all values are equal and are the
  // product of the local lengths in each coordinate direction
  if (data.update_each & (update_JxW_values | update_volume_elements))
    if (cell_similarity != CellSimilarity::translation)
      {
        double J = data.cell_extents[0];
        for (unsigned int d=1; d<dim; ++d)
          J *= data.cell_extents[d];
        data.volume_element = J;
        if (data.update_each & update_JxW_values)
          for (unsigned int i=0; i<output_data.JxW_values.size(); ++i)
            output_data.JxW_values[i] = J * quadrature.weight(i);
      }
  // "compute" Jacobian at the quadrature points, which are all the
  // same
  if (data.update_each & update_jacobians)
    if (cell_similarity != CellSimilarity::translation)
      for (unsigned int i=0; i<output_data.jacobians.size(); ++i)
        {
          output_data.jacobians[i] =  DerivativeForm<1,dim,spacedim>();
          for (unsigned int j=0; j<dim; ++j)
            output_data.jacobians[i][j][j] = data.cell_extents[j];
        }
  // "compute" the derivative of the Jacobian at the quadrature
  // points, which are all zero of course
  if (data.update_each & update_jacobian_grads)
    if (cell_similarity != CellSimilarity::translation)
      for (unsigned int i=0; i<output_data.jacobian_grads.size(); ++i)
        output_data.jacobian_grads[i] = DerivativeForm<2,dim,spacedim>();

  if (data.update_each & update_jacobian_pushed_forward_grads)
    if (cell_similarity != CellSimilarity::translation)
      for (unsigned int i=0; i<output_data.jacobian_pushed_forward_grads.size(); ++i)
        output_data.jacobian_pushed_forward_grads[i] = Tensor<3,spacedim>();

  // "compute" the hessian of the Jacobian at the quadrature points,
  // which are all also zero of course
  if (data.update_each & update_jacobian_2nd_derivatives)
    if (cell_similarity != CellSimilarity::translation)
      for (unsigned int i=0; i<output_data.jacobian_2nd_derivatives.size(); ++i)
        output_data.jacobian_2nd_derivatives[i] = DerivativeForm<3,dim,spacedim>();

  if (data.update_each & update_jacobian_pushed_forward_2nd_derivatives)
    if (cell_similarity != CellSimilarity::translation)
      for (unsigned int i=0; i<output_data.jacobian_pushed_forward_2nd_derivatives.size(); ++i)
        output_data.jacobian_pushed_forward_2nd_derivatives[i] = Tensor<4,spacedim>();

  if (data.update_each & update_jacobian_3rd_derivatives)
    if (cell_similarity != CellSimilarity::translation)
      for (unsigned int i=0; i<output_data.jacobian_3rd_derivatives.size(); ++i)
        output_data.jacobian_3rd_derivatives[i] = DerivativeForm<4,dim,spacedim>();

  if (data.update_each & update_jacobian_pushed_forward_3rd_derivatives)
    if (cell_similarity != CellSimilarity::translation)
      for (unsigned int i=0; i<output_data.jacobian_pushed_forward_3rd_derivatives.size(); ++i)
        output_data.jacobian_pushed_forward_3rd_derivatives[i] = Tensor<5,spacedim>();

  // "compute" inverse Jacobian at the quadrature points, which are
  // all the same
  if (data.update_each & update_inverse_jacobians)
    if (cell_similarity != CellSimilarity::translation)
      for (unsigned int i=0; i<output_data.inverse_jacobians.size(); ++i)
        {
          output_data.inverse_jacobians[i] = Tensor<2,dim>();
          for (unsigned int j=0; j<dim; ++j)
            output_data.inverse_jacobians[j][j]=1./data.cell_extents[j];
        }

  return cell_similarity;
}



template<int dim, int spacedim>
void
MappingCartesian<dim, spacedim>::
fill_fe_face_values (const typename Triangulation<dim,spacedim>::cell_iterator &cell,
                     const unsigned int                                         face_no,
                     const Quadrature<dim-1>                                   &quadrature,
                     const typename Mapping<dim,spacedim>::InternalDataBase    &internal_data,
                     internal::FEValues::MappingRelatedData<dim,spacedim>      &output_data) const
{
  // convert data object to internal
  // data for this class. fails with
  // an exception if that is not
  // possible
  Assert (dynamic_cast<const InternalData *> (&internal_data) != 0,
          ExcInternalError());
  const InternalData &data = static_cast<const InternalData &> (internal_data);

  compute_fill (cell, face_no, invalid_face_number,
                CellSimilarity::none,
                data,
                output_data.quadrature_points,
                output_data.normal_vectors);

  // first compute Jacobian determinant, which is simply the product
  // of the local lengths since the jacobian is diagonal
  double J = 1.;
  for (unsigned int d=0; d<dim; ++d)
    if (d != GeometryInfo<dim>::unit_normal_direction[face_no])
      J *= data.cell_extents[d];

  if (data.update_each & update_JxW_values)
    for (unsigned int i=0; i<output_data.JxW_values.size(); ++i)
      output_data.JxW_values[i] = J * quadrature.weight(i);

  if (data.update_each & update_boundary_forms)
    for (unsigned int i=0; i<output_data.boundary_forms.size(); ++i)
      output_data.boundary_forms[i] = J * output_data.normal_vectors[i];

  if (data.update_each & update_volume_elements)
    {
      J = data.cell_extents[0];
      for (unsigned int d=1; d<dim; ++d)
        J *= data.cell_extents[d];
      data.volume_element = J;
    }

  if (data.update_each & update_jacobians)
    for (unsigned int i=0; i<output_data.jacobians.size(); ++i)
      {
        output_data.jacobians[i] = DerivativeForm<1,dim,spacedim>();
        for (unsigned int d=0; d<dim; ++d)
          output_data.jacobians[i][d][d] = data.cell_extents[d];
      }

  if (data.update_each & update_jacobian_grads)
    for (unsigned int i=0; i<output_data.jacobian_grads.size(); ++i)
      output_data.jacobian_grads[i] = DerivativeForm<2,dim,spacedim>();

  if (data.update_each & update_jacobian_pushed_forward_grads)
    for (unsigned int i=0; i<output_data.jacobian_pushed_forward_grads.size(); ++i)
      output_data.jacobian_pushed_forward_grads[i] = Tensor<3,spacedim>();

  if (data.update_each & update_jacobian_2nd_derivatives)
    for (unsigned int i=0; i<output_data.jacobian_2nd_derivatives.size(); ++i)
      output_data.jacobian_2nd_derivatives[i] = DerivativeForm<3,dim,spacedim>();

  if (data.update_each & update_jacobian_pushed_forward_2nd_derivatives)
    for (unsigned int i=0; i<output_data.jacobian_pushed_forward_2nd_derivatives.size(); ++i)
      output_data.jacobian_pushed_forward_2nd_derivatives[i] = Tensor<4,spacedim>();

  if (data.update_each & update_jacobian_3rd_derivatives)
    for (unsigned int i=0; i<output_data.jacobian_3rd_derivatives.size(); ++i)
      output_data.jacobian_3rd_derivatives[i] = DerivativeForm<4,dim,spacedim>();

  if (data.update_each & update_jacobian_pushed_forward_3rd_derivatives)
    for (unsigned int i=0; i<output_data.jacobian_pushed_forward_3rd_derivatives.size(); ++i)
      output_data.jacobian_pushed_forward_3rd_derivatives[i] = Tensor<5,spacedim>();

  if (data.update_each & update_inverse_jacobians)
    for (unsigned int i=0; i<output_data.inverse_jacobians.size(); ++i)
      {
        output_data.inverse_jacobians[i] = DerivativeForm<1,dim,spacedim>();
        for (unsigned int d=0; d<dim; ++d)
          output_data.inverse_jacobians[i][d][d] = 1./data.cell_extents[d];
      }
}



template<int dim, int spacedim>
void
MappingCartesian<dim, spacedim>::
fill_fe_subface_values (const typename Triangulation<dim,spacedim>::cell_iterator &cell,
                        const unsigned int                                         face_no,
                        const unsigned int                                         subface_no,
                        const Quadrature<dim-1>                                   &quadrature,
                        const typename Mapping<dim,spacedim>::InternalDataBase    &internal_data,
                        internal::FEValues::MappingRelatedData<dim,spacedim>      &output_data) const
{
  // convert data object to internal data for this class. fails with
  // an exception if that is not possible
  Assert (dynamic_cast<const InternalData *> (&internal_data) != 0, ExcInternalError());
  const InternalData &data = static_cast<const InternalData &> (internal_data);

  compute_fill (cell, face_no, subface_no, CellSimilarity::none,
                data,
                output_data.quadrature_points,
                output_data.normal_vectors);

  // first compute Jacobian determinant, which is simply the product
  // of the local lengths since the jacobian is diagonal
  double J = 1.;
  for (unsigned int d=0; d<dim; ++d)
    if (d != GeometryInfo<dim>::unit_normal_direction[face_no])
      J *= data.cell_extents[d];

  if (data.update_each & update_JxW_values)
    {
      // Here, cell->face(face_no)->n_children() would be the right
      // choice, but unfortunately the current function is also called
      // for faces without children (see tests/fe/mapping.cc). Add
      // following switch to avoid diffs in tests/fe/mapping.OK
      const unsigned int n_subfaces=
        cell->face(face_no)->has_children() ?
        cell->face(face_no)->n_children() :
        GeometryInfo<dim>::max_children_per_face;
      for (unsigned int i=0; i<output_data.JxW_values.size(); ++i)
        output_data.JxW_values[i] = J * quadrature.weight(i) / n_subfaces;
    }

  if (data.update_each & update_boundary_forms)
    for (unsigned int i=0; i<output_data.boundary_forms.size(); ++i)
      output_data.boundary_forms[i] = J * output_data.normal_vectors[i];

  if (data.update_each & update_volume_elements)
    {
      J = data.cell_extents[0];
      for (unsigned int d=1; d<dim; ++d)
        J *= data.cell_extents[d];
      data.volume_element = J;
    }

  if (data.update_each & update_jacobians)
    for (unsigned int i=0; i<output_data.jacobians.size(); ++i)
      {
        output_data.jacobians[i] = DerivativeForm<1,dim,spacedim>();
        for (unsigned int d=0; d<dim; ++d)
          output_data.jacobians[i][d][d] = data.cell_extents[d];
      }

  if (data.update_each & update_jacobian_grads)
    for (unsigned int i=0; i<output_data.jacobian_grads.size(); ++i)
      output_data.jacobian_grads[i] = DerivativeForm<2,dim,spacedim>();

  if (data.update_each & update_jacobian_pushed_forward_grads)
    for (unsigned int i=0; i<output_data.jacobian_pushed_forward_grads.size(); ++i)
      output_data.jacobian_pushed_forward_grads[i] = Tensor<3,spacedim>();

  if (data.update_each & update_jacobian_2nd_derivatives)
    for (unsigned int i=0; i<output_data.jacobian_2nd_derivatives.size(); ++i)
      output_data.jacobian_2nd_derivatives[i] = DerivativeForm<3,dim,spacedim>();

  if (data.update_each & update_jacobian_pushed_forward_2nd_derivatives)
    for (unsigned int i=0; i<output_data.jacobian_pushed_forward_2nd_derivatives.size(); ++i)
      output_data.jacobian_pushed_forward_2nd_derivatives[i] = Tensor<4,spacedim>();

  if (data.update_each & update_jacobian_3rd_derivatives)
    for (unsigned int i=0; i<output_data.jacobian_3rd_derivatives.size(); ++i)
      output_data.jacobian_3rd_derivatives[i] = DerivativeForm<4,dim,spacedim>();

  if (data.update_each & update_jacobian_pushed_forward_3rd_derivatives)
    for (unsigned int i=0; i<output_data.jacobian_pushed_forward_3rd_derivatives.size(); ++i)
      output_data.jacobian_pushed_forward_3rd_derivatives[i] = Tensor<5,spacedim>();

  if (data.update_each & update_inverse_jacobians)
    for (unsigned int i=0; i<output_data.inverse_jacobians.size(); ++i)
      {
        output_data.inverse_jacobians[i] = DerivativeForm<1,spacedim,dim>();
        for (unsigned int d=0; d<dim; ++d)
          output_data.inverse_jacobians[i][d][d] = 1./data.cell_extents[d];
      }
}




template<int dim, int spacedim>
void
MappingCartesian<dim,spacedim>::
transform (const ArrayView<const Tensor<1,dim> >                  &input,
           const MappingType                                       mapping_type,
           const typename Mapping<dim,spacedim>::InternalDataBase &mapping_data,
           const ArrayView<Tensor<1,spacedim> >                   &output) const
{
  AssertDimension (input.size(), output.size());
  Assert (dynamic_cast<const InternalData *>(&mapping_data) != 0,
          ExcInternalError());
  const InternalData &data = static_cast<const InternalData &> (mapping_data);

  switch (mapping_type)
    {
    case mapping_covariant:
    {
      Assert (data.update_each & update_covariant_transformation,
              typename FEValuesBase<dim>::ExcAccessToUninitializedField("update_covariant_transformation"));

      for (unsigned int i=0; i<output.size(); ++i)
        for (unsigned int d=0; d<dim; ++d)
          output[i][d] = input[i][d]/data.cell_extents[d];
      return;
    }

    case mapping_contravariant:
    {
      Assert (data.update_each & update_contravariant_transformation,
              typename FEValuesBase<dim>::ExcAccessToUninitializedField("update_contravariant_transformation"));

      for (unsigned int i=0; i<output.size(); ++i)
        for (unsigned int d=0; d<dim; ++d)
          output[i][d] = input[i][d]*data.cell_extents[d];
      return;
    }
    case mapping_piola:
    {
      Assert (data.update_each & update_contravariant_transformation,
              typename FEValuesBase<dim>::ExcAccessToUninitializedField("update_contravariant_transformation"));
      Assert (data.update_each & update_volume_elements,
              typename FEValuesBase<dim>::ExcAccessToUninitializedField("update_volume_elements"));

      for (unsigned int i=0; i<output.size(); ++i)
        for (unsigned int d=0; d<dim; ++d)
          output[i][d] = input[i][d] * data.cell_extents[d] / data.volume_element;
      return;
    }
    default:
      Assert(false, ExcNotImplemented());
    }
}



template<int dim, int spacedim>
void
MappingCartesian<dim,spacedim>::
transform (const ArrayView<const DerivativeForm<1, dim,spacedim> > &input,
           const MappingType                                        mapping_type,
           const typename Mapping<dim,spacedim>::InternalDataBase  &mapping_data,
           const ArrayView<Tensor<2,spacedim> >                    &output) const
{
  AssertDimension (input.size(), output.size());
  Assert (dynamic_cast<const InternalData *>(&mapping_data) != 0,
          ExcInternalError());
  const InternalData &data = static_cast<const InternalData &>(mapping_data);

  switch (mapping_type)
    {
    case mapping_covariant:
    {
      Assert (data.update_each & update_covariant_transformation,
              typename FEValuesBase<dim>::ExcAccessToUninitializedField("update_covariant_transformation"));

      for (unsigned int i=0; i<output.size(); ++i)
        for (unsigned int d1=0; d1<dim; ++d1)
          for (unsigned int d2=0; d2<dim; ++d2)
            output[i][d1][d2] = input[i][d1][d2] / data.cell_extents[d2];
      return;
    }

    case mapping_contravariant:
    {
      Assert (data.update_each & update_contravariant_transformation,
              typename FEValuesBase<dim>::ExcAccessToUninitializedField("update_contravariant_transformation"));

      for (unsigned int i=0; i<output.size(); ++i)
        for (unsigned int d1=0; d1<dim; ++d1)
          for (unsigned int d2=0; d2<dim; ++d2)
            output[i][d1][d2] = input[i][d1][d2] * data.cell_extents[d2];
      return;
    }

    case mapping_covariant_gradient:
    {
      Assert (data.update_each & update_covariant_transformation,
              typename FEValuesBase<dim>::ExcAccessToUninitializedField("update_covariant_transformation"));

      for (unsigned int i=0; i<output.size(); ++i)
        for (unsigned int d1=0; d1<dim; ++d1)
          for (unsigned int d2=0; d2<dim; ++d2)
            output[i][d1][d2] = input[i][d1][d2] / data.cell_extents[d2] / data.cell_extents[d1];
      return;
    }

    case mapping_contravariant_gradient:
    {
      Assert (data.update_each & update_contravariant_transformation,
              typename FEValuesBase<dim>::ExcAccessToUninitializedField("update_contravariant_transformation"));

      for (unsigned int i=0; i<output.size(); ++i)
        for (unsigned int d1=0; d1<dim; ++d1)
          for (unsigned int d2=0; d2<dim; ++d2)
            output[i][d1][d2] = input[i][d1][d2] * data.cell_extents[d2] / data.cell_extents[d1];
      return;
    }

    case mapping_piola:
    {
      Assert (data.update_each & update_contravariant_transformation,
              typename FEValuesBase<dim>::ExcAccessToUninitializedField("update_contravariant_transformation"));
      Assert (data.update_each & update_volume_elements,
              typename FEValuesBase<dim>::ExcAccessToUninitializedField("update_volume_elements"));

      for (unsigned int i=0; i<output.size(); ++i)
        for (unsigned int d1=0; d1<dim; ++d1)
          for (unsigned int d2=0; d2<dim; ++d2)
            output[i][d1][d2] = input[i][d1][d2] * data.cell_extents[d2]
                                / data.volume_element;
      return;
    }

    case mapping_piola_gradient:
    {
      Assert (data.update_each & update_contravariant_transformation,
              typename FEValuesBase<dim>::ExcAccessToUninitializedField("update_contravariant_transformation"));
      Assert (data.update_each & update_volume_elements,
              typename FEValuesBase<dim>::ExcAccessToUninitializedField("update_volume_elements"));

      for (unsigned int i=0; i<output.size(); ++i)
        for (unsigned int d1=0; d1<dim; ++d1)
          for (unsigned int d2=0; d2<dim; ++d2)
            output[i][d1][d2] = input[i][d1][d2] * data.cell_extents[d2]
                                / data.cell_extents[d1] / data.volume_element;
      return;
    }

    default:
      Assert(false, ExcNotImplemented());
    }
}




template<int dim, int spacedim>
void
MappingCartesian<dim,spacedim>::
transform (const ArrayView<const Tensor<2, dim> >                 &input,
           const MappingType                                       mapping_type,
           const typename Mapping<dim,spacedim>::InternalDataBase &mapping_data,
           const ArrayView<Tensor<2, spacedim> >                  &output) const
{

  AssertDimension (input.size(), output.size());
  Assert (dynamic_cast<const InternalData *>(&mapping_data) != 0,
          ExcInternalError());
  const InternalData &data = static_cast<const InternalData &>(mapping_data);

  switch (mapping_type)
    {
    case mapping_covariant:
    {
      Assert (data.update_each & update_covariant_transformation,
              typename FEValuesBase<dim>::ExcAccessToUninitializedField("update_covariant_transformation"));

      for (unsigned int i=0; i<output.size(); ++i)
        for (unsigned int d1=0; d1<dim; ++d1)
          for (unsigned int d2=0; d2<dim; ++d2)
            output[i][d1][d2] = input[i][d1][d2] / data.cell_extents[d2];
      return;
    }

    case mapping_contravariant:
    {
      Assert (data.update_each & update_contravariant_transformation,
              typename FEValuesBase<dim>::ExcAccessToUninitializedField("update_contravariant_transformation"));

      for (unsigned int i=0; i<output.size(); ++i)
        for (unsigned int d1=0; d1<dim; ++d1)
          for (unsigned int d2=0; d2<dim; ++d2)
            output[i][d1][d2] = input[i][d1][d2] * data.cell_extents[d2];
      return;
    }

    case mapping_covariant_gradient:
    {
      Assert (data.update_each & update_covariant_transformation,
              typename FEValuesBase<dim>::ExcAccessToUninitializedField("update_covariant_transformation"));

      for (unsigned int i=0; i<output.size(); ++i)
        for (unsigned int d1=0; d1<dim; ++d1)
          for (unsigned int d2=0; d2<dim; ++d2)
            output[i][d1][d2] = input[i][d1][d2] / data.cell_extents[d2] / data.cell_extents[d1];
      return;
    }

    case mapping_contravariant_gradient:
    {
      Assert (data.update_each & update_contravariant_transformation,
              typename FEValuesBase<dim>::ExcAccessToUninitializedField("update_contravariant_transformation"));

      for (unsigned int i=0; i<output.size(); ++i)
        for (unsigned int d1=0; d1<dim; ++d1)
          for (unsigned int d2=0; d2<dim; ++d2)
            output[i][d1][d2] = input[i][d1][d2] * data.cell_extents[d2] / data.cell_extents[d1];
      return;
    }

    case mapping_piola:
    {
      Assert (data.update_each & update_contravariant_transformation,
              typename FEValuesBase<dim>::ExcAccessToUninitializedField("update_contravariant_transformation"));
      Assert (data.update_each & update_volume_elements,
              typename FEValuesBase<dim>::ExcAccessToUninitializedField("update_volume_elements"));

      for (unsigned int i=0; i<output.size(); ++i)
        for (unsigned int d1=0; d1<dim; ++d1)
          for (unsigned int d2=0; d2<dim; ++d2)
            output[i][d1][d2] = input[i][d1][d2] * data.cell_extents[d2]
                                / data.volume_element;
      return;
    }

    case mapping_piola_gradient:
    {
      Assert (data.update_each & update_contravariant_transformation,
              typename FEValuesBase<dim>::ExcAccessToUninitializedField("update_contravariant_transformation"));
      Assert (data.update_each & update_volume_elements,
              typename FEValuesBase<dim>::ExcAccessToUninitializedField("update_volume_elements"));

      for (unsigned int i=0; i<output.size(); ++i)
        for (unsigned int d1=0; d1<dim; ++d1)
          for (unsigned int d2=0; d2<dim; ++d2)
            output[i][d1][d2] = input[i][d1][d2] * data.cell_extents[d2]
                                / data.cell_extents[d1] / data.volume_element;
      return;
    }

    default:
      Assert(false, ExcNotImplemented());
    }

}


template<int dim, int spacedim>
void
MappingCartesian<dim,spacedim>::
transform (const ArrayView<const  DerivativeForm<2, dim, spacedim> > &input,
           const MappingType                                          mapping_type,
           const typename Mapping<dim,spacedim>::InternalDataBase    &mapping_data,
           const ArrayView<Tensor<3,spacedim> >                      &output) const
{

  AssertDimension (input.size(), output.size());
  Assert (dynamic_cast<const InternalData *>(&mapping_data) != 0,
          ExcInternalError());
  const InternalData &data = static_cast<const InternalData &>(mapping_data);

  switch (mapping_type)
    {
    case mapping_covariant_gradient:
    {
      Assert (data.update_each & update_contravariant_transformation,
              typename FEValuesBase<dim>::ExcAccessToUninitializedField("update_covariant_transformation"));

      for (unsigned int q=0; q<output.size(); ++q)
        for (unsigned int i=0; i<spacedim; ++i)
          for (unsigned int j=0; j<spacedim; ++j)
            for (unsigned int k=0; k<spacedim; ++k)
              {

                output[q][i][j][k] = input[q][i][j][k] / data.cell_extents[j] / data.cell_extents[k];

              }
      return;
    }
    default:
      Assert(false, ExcNotImplemented());
    }

}

template<int dim, int spacedim>
void
MappingCartesian<dim,spacedim>::
transform (const ArrayView<const  Tensor<3,dim> >                 &input,
           const MappingType                                       mapping_type,
           const typename Mapping<dim,spacedim>::InternalDataBase &mapping_data,
           const ArrayView<Tensor<3,spacedim> >                   &output) const
{

  AssertDimension (input.size(), output.size());
  Assert (dynamic_cast<const InternalData *>(&mapping_data) != 0,
          ExcInternalError());
  const InternalData &data = static_cast<const InternalData &>(mapping_data);

  switch (mapping_type)
    {
    case mapping_contravariant_hessian:
    {
      Assert (data.update_each & update_covariant_transformation,
              typename FEValuesBase<dim>::ExcAccessToUninitializedField("update_covariant_transformation"));
      Assert (data.update_each & update_contravariant_transformation,
              typename FEValuesBase<dim>::ExcAccessToUninitializedField("update_contravariant_transformation"));

      for (unsigned int q=0; q<output.size(); ++q)
        for (unsigned int i=0; i<spacedim; ++i)
          for (unsigned int j=0; j<spacedim; ++j)
            for (unsigned int k=0; k<spacedim; ++k)
              {
                output[q][i][j][k] =    input[q][i][j][k]
                                        * data.cell_extents[i]
                                        / data.cell_extents[j]
                                        / data.cell_extents[k];
              }
      return;
    }

    case mapping_covariant_hessian:
    {
      Assert (data.update_each & update_covariant_transformation,
              typename FEValuesBase<dim>::ExcAccessToUninitializedField("update_covariant_transformation"));

      for (unsigned int q=0; q<output.size(); ++q)
        for (unsigned int i=0; i<spacedim; ++i)
          for (unsigned int j=0; j<spacedim; ++j)
            for (unsigned int k=0; k<spacedim; ++k)
              {
                output[q][i][j][k] =    input[q][i][j][k]
                                        / data.cell_extents[i]
                                        / data.cell_extents[j]
                                        / data.cell_extents[k];
              }

      return;
    }

    case mapping_piola_hessian:
    {
      Assert (data.update_each & update_covariant_transformation,
              typename FEValuesBase<dim>::ExcAccessToUninitializedField("update_covariant_transformation"));
      Assert (data.update_each & update_contravariant_transformation,
              typename FEValuesBase<dim>::ExcAccessToUninitializedField("update_contravariant_transformation"));
      Assert (data.update_each & update_volume_elements,
              typename FEValuesBase<dim>::ExcAccessToUninitializedField("update_volume_elements"));

      for (unsigned int q=0; q<output.size(); ++q)
        for (unsigned int i=0; i<spacedim; ++i)
          for (unsigned int j=0; j<spacedim; ++j)
            for (unsigned int k=0; k<spacedim; ++k)
              {
                output[q][i][j][k] =    input[q][i][j][k]
                                        * data.cell_extents[i]
                                        / data.volume_element
                                        / data.cell_extents[j]
                                        / data.cell_extents[k];
              }

      return;
    }

    default:
      Assert(false, ExcNotImplemented());
    }
}


template <int dim, int spacedim>
Point<spacedim>
MappingCartesian<dim, spacedim>::transform_unit_to_real_cell (
  const typename Triangulation<dim,spacedim>::cell_iterator &cell,
  const Point<dim>                                 &p) const
{
  Tensor<1,dim> length;
  const Point<dim> start = cell->vertex(0);
  switch (dim)
    {
    case 1:
      length[0] = cell->vertex(1)(0) - start(0);
      break;
    case 2:
      length[0] = cell->vertex(1)(0) - start(0);
      length[1] = cell->vertex(2)(1) - start(1);
      break;
    case 3:
      length[0] = cell->vertex(1)(0) - start(0);
      length[1] = cell->vertex(2)(1) - start(1);
      length[2] = cell->vertex(4)(2) - start(2);
      break;
    default:
      Assert(false, ExcNotImplemented());
    }

  Point<dim> p_real = cell->vertex(0);
  for (unsigned int d=0; d<dim; ++d)
    p_real(d) += length[d]*p(d);

  return p_real;
}



template<int dim, int spacedim>
Point<dim>
MappingCartesian<dim, spacedim>::transform_real_to_unit_cell (
  const typename Triangulation<dim,spacedim>::cell_iterator &cell,
  const Point<spacedim>                            &p) const
{

  if (dim != spacedim)
    Assert(false, ExcNotImplemented());
  const Point<dim> &start = cell->vertex(0);
  Point<dim> real = p;
  real -= start;

  switch (dim)
    {
    case 1:
      real(0) /= cell->vertex(1)(0) - start(0);
      break;
    case 2:
      real(0) /= cell->vertex(1)(0) - start(0);
      real(1) /= cell->vertex(2)(1) - start(1);
      break;
    case 3:
      real(0) /= cell->vertex(1)(0) - start(0);
      real(1) /= cell->vertex(2)(1) - start(1);
      real(2) /= cell->vertex(4)(2) - start(2);
      break;
    default:
      Assert(false, ExcNotImplemented());
    }
  return real;
}


template<int dim, int spacedim>
Mapping<dim, spacedim> *
MappingCartesian<dim, spacedim>::clone () const
{
  return new MappingCartesian<dim, spacedim>(*this);
}


//---------------------------------------------------------------------------
// explicit instantiations
#include "mapping_cartesian.inst"


DEAL_II_NAMESPACE_CLOSE