mapping_manifold.cc

Go to the documentation of this file.

// ------------------------------------------------------------------------
//
// SPDX-License-Identifier: LGPL-2.1-or-later
// Copyright (C) 2016 - 2024 by the deal.II authors
//
// This file is part of the deal.II library.
//
// Part of the source code is dual licensed under Apache-2.0 WITH
// LLVM-exception OR LGPL-2.1-or-later. Detailed license information
// governing the source code and code contributions can be found in
// LICENSE.md and CONTRIBUTING.md at the top level directory of deal.II.
//
// ------------------------------------------------------------------------
 
 
#include <deal.II/base/array_view.h>
#include <deal.II/base/derivative_form.h>
#include <deal.II/base/memory_consumption.h>
#include <deal.II/base/qprojector.h>
#include <deal.II/base/quadrature.h>
#include <deal.II/base/quadrature_lib.h>
#include <deal.II/base/tensor_product_polynomials.h>
 
#include <deal.II/dofs/dof_accessor.h>
 
#include <deal.II/fe/fe.h>
#include <deal.II/fe/fe_tools.h>
#include <deal.II/fe/fe_values.h>
#include <deal.II/fe/mapping_manifold.h>
 
#include <deal.II/grid/manifold.h>
#include <deal.II/grid/tria.h>
#include <deal.II/grid/tria_iterator.h>
 
#include <deal.II/lac/full_matrix.h>
 
#include <algorithm>
#include <array>
#include <cmath>
#include <memory>
#include <numeric>
 
 
DEAL_II_NAMESPACE_OPEN
 
template <int dim, int spacedim>
std::size_t
MappingManifold<dim, spacedim>::InternalData::memory_consumption() const
{
  return (
    Mapping<dim, spacedim>::InternalDataBase::memory_consumption() +
    MemoryConsumption::memory_consumption(vertices) +
    MemoryConsumption::memory_consumption(cell) +
    MemoryConsumption::memory_consumption(quad) +
    MemoryConsumption::memory_consumption(cell_manifold_quadrature_weights) +
    MemoryConsumption::memory_consumption(vertex_weights) +
    MemoryConsumption::memory_consumption(unit_tangentials) +
    MemoryConsumption::memory_consumption(covariant) +
    MemoryConsumption::memory_consumption(contravariant) +
    MemoryConsumption::memory_consumption(aux) +
    MemoryConsumption::memory_consumption(volume_elements) +
    MemoryConsumption::memory_consumption(manifold));
}
 
 
 
template <int dim, int spacedim>
void
MappingManifold<dim, spacedim>::InternalData::reinit(
  const UpdateFlags      update_flags,
  const Quadrature<dim> &q)
{
  // store the flags in the internal data object so we can access them
  // in fill_fe_*_values()
  this->update_each = update_flags;
 
  const unsigned int n_q_points = q.size();
 
  // Store the quadrature
  this->quad.initialize(q.get_points(), q.get_weights());
 
  // Resize the weights
  this->vertex_weights.resize(GeometryInfo<dim>::vertices_per_cell);
 
  // see if we need the (transformation) shape function values
  // and/or gradients and resize the necessary arrays
  if (this->update_each &
      (update_quadrature_points | update_contravariant_transformation))
    compute_manifold_quadrature_weights(q);
 
  if (this->update_each & update_covariant_transformation)
    covariant.resize(n_q_points);
 
  if (this->update_each & update_contravariant_transformation)
    contravariant.resize(n_q_points);
 
  if (this->update_each & update_volume_elements)
    volume_elements.resize(n_q_points);
}
 
 
 
template <int dim, int spacedim>
void
MappingManifold<dim, spacedim>::InternalData::initialize_face(
  const UpdateFlags      update_flags,
  const Quadrature<dim> &q,
  const unsigned int     n_original_q_points)
{
  reinit(update_flags, q);
 
  // Set to the size of a single quadrature object for faces, as the size set
  // in in reinit() is for all points
  if (this->update_each & update_covariant_transformation)
    covariant.resize(n_original_q_points);
 
  if (this->update_each & update_contravariant_transformation)
    contravariant.resize(n_original_q_points);
 
  if (this->update_each & update_volume_elements)
    volume_elements.resize(n_original_q_points);
 
  if (dim > 1)
    {
      if (this->update_each & update_boundary_forms)
        {
          aux.resize(dim - 1,
                     std::vector<Tensor<1, spacedim>>(n_original_q_points));
 
          // Compute tangentials to the unit cell.
          for (const unsigned int i : GeometryInfo<dim>::face_indices())
            {
              unit_tangentials[i].resize(n_original_q_points);
              std::fill(unit_tangentials[i].begin(),
                        unit_tangentials[i].end(),
                        GeometryInfo<dim>::unit_tangential_vectors[i][0]);
              if (dim > 2)
                {
                  unit_tangentials[GeometryInfo<dim>::faces_per_cell + i]
                    .resize(n_original_q_points);
                  std::fill(
                    unit_tangentials[GeometryInfo<dim>::faces_per_cell + i]
                      .begin(),
                    unit_tangentials[GeometryInfo<dim>::faces_per_cell + i]
                      .end(),
                    GeometryInfo<dim>::unit_tangential_vectors[i][1]);
                }
            }
        }
    }
}
 
 
 
template <int dim, int spacedim>
void
MappingManifold<dim, spacedim>::InternalData::store_vertices(
  const typename Triangulation<dim, spacedim>::cell_iterator &cell) const
{
  vertices.resize(GeometryInfo<dim>::vertices_per_cell);
  for (const unsigned int i : GeometryInfo<dim>::vertex_indices())
    vertices[i] = cell->vertex(i);
  this->cell = cell;
}
 
 
 
template <int dim, int spacedim>
void
MappingManifold<dim, spacedim>::InternalData::
  compute_manifold_quadrature_weights(const Quadrature<dim> &quad)
{
  cell_manifold_quadrature_weights.resize(
    quad.size(), std::vector<double>(GeometryInfo<dim>::vertices_per_cell));
  for (unsigned int q = 0; q < quad.size(); ++q)
    {
      for (const unsigned int i : GeometryInfo<dim>::vertex_indices())
        {
          cell_manifold_quadrature_weights[q][i] =
            GeometryInfo<dim>::d_linear_shape_function(quad.point(q), i);
        }
    }
}
 
 
 
template <int dim, int spacedim>
MappingManifold<dim, spacedim>::MappingManifold(
  const MappingManifold<dim, spacedim> &)
{}
 
 
 
template <int dim, int spacedim>
std::unique_ptr<Mapping<dim, spacedim>>
MappingManifold<dim, spacedim>::clone() const
{
  return std::make_unique<MappingManifold<dim, spacedim>>(*this);
}
 
 
 
template <int dim, int spacedim>
Point<dim>
MappingManifold<dim, spacedim>::transform_real_to_unit_cell(
  const typename Triangulation<dim, spacedim>::cell_iterator &,
  const Point<spacedim> &) const
{
  DEAL_II_NOT_IMPLEMENTED();
  return {};
}
 
 
 
template <int dim, int spacedim>
Point<spacedim>
MappingManifold<dim, spacedim>::transform_unit_to_real_cell(
  const typename Triangulation<dim, spacedim>::cell_iterator &cell,
  const Point<dim>                                           &p) const
{
  std::array<Point<spacedim>, GeometryInfo<dim>::vertices_per_cell> vertices;
  std::array<double, GeometryInfo<dim>::vertices_per_cell>          weights;
 
  for (const unsigned int v : GeometryInfo<dim>::vertex_indices())
    {
      vertices[v] = cell->vertex(v);
      weights[v]  = GeometryInfo<dim>::d_linear_shape_function(p, v);
    }
  return cell->get_manifold().get_new_point(
    make_array_view(vertices.begin(), vertices.end()),
    make_array_view(weights.begin(), weights.end()));
}
 
 
 
// In the code below, GCC tries to instantiate MappingManifold<3,4> when
// seeing which of the overloaded versions of
// do_transform_real_to_unit_cell_internal() to call. This leads to bad
// error messages and, generally, nothing very good. Avoid this by ensuring
// that this class exists, but does not have an inner InternalData
// type, thereby ruling out the codim-1 version of the function
// below when doing overload resolution.
template <>
class MappingManifold<3, 4>
{};
 
 
 
template <int dim, int spacedim>
UpdateFlags
MappingManifold<dim, spacedim>::requires_update_flags(
  const UpdateFlags in) const
{
  // add flags if the respective quantities are necessary to compute
  // what we need. note that some flags appear in both the conditions
  // and in subsequent set operations. this leads to some circular
  // logic. the only way to treat this is to iterate. since there are
  // 5 if-clauses in the loop, it will take at most 5 iterations to
  // converge. do them:
  UpdateFlags out = in;
  for (unsigned int i = 0; i < 5; ++i)
    {
      // The following is a little incorrect:
      // If not applied on a face,
      // update_boundary_forms does not
      // make sense. On the other hand,
      // it is necessary on a
      // face. Currently,
      // update_boundary_forms is simply
      // ignored for the interior of a
      // cell.
      if (out & (update_JxW_values | update_normal_vectors))
        out |= update_boundary_forms;
 
      if (out & (update_covariant_transformation | update_JxW_values |
                 update_jacobians | update_jacobian_grads |
                 update_boundary_forms | update_normal_vectors))
        out |= update_contravariant_transformation;
 
      if (out &
          (update_inverse_jacobians | update_jacobian_pushed_forward_grads |
           update_jacobian_pushed_forward_2nd_derivatives |
           update_jacobian_pushed_forward_3rd_derivatives))
        out |= update_covariant_transformation;
 
      // The contravariant transformation used in the Piola
      // transformation, which requires the determinant of the Jacobi
      // matrix of the transformation.  Because we have no way of
      // knowing here whether the finite elements wants to use the
      // contravariant of the Piola transforms, we add the JxW values
      // to the list of flags to be updated for each cell.
      if (out & update_contravariant_transformation)
        out |= update_JxW_values;
 
      if (out & update_normal_vectors)
        out |= update_JxW_values;
    }
 
  // Now throw an exception if we stumble upon something that was not
  // implemented yet
  Assert(!(out & (update_jacobian_grads | update_jacobian_pushed_forward_grads |
                  update_jacobian_pushed_forward_2nd_derivatives |
                  update_jacobian_pushed_forward_3rd_derivatives)),
         ExcNotImplemented());
 
  return out;
}
 
 
 
template <int dim, int spacedim>
std::unique_ptr<typename Mapping<dim, spacedim>::InternalDataBase>
MappingManifold<dim, spacedim>::get_data(const UpdateFlags      update_flags,
                                         const Quadrature<dim> &q) const
{
  std::unique_ptr<typename Mapping<dim, spacedim>::InternalDataBase> data_ptr =
    std::make_unique<InternalData>();
  data_ptr->reinit(this->requires_update_flags(update_flags), q);
 
  return data_ptr;
}
 
 
 
template <int dim, int spacedim>
std::unique_ptr<typename Mapping<dim, spacedim>::InternalDataBase>
MappingManifold<dim, spacedim>::get_face_data(
  const UpdateFlags               update_flags,
  const hp::QCollection<dim - 1> &quadrature) const
{
  AssertDimension(quadrature.size(), 1);
 
  std::unique_ptr<typename Mapping<dim, spacedim>::InternalDataBase> data_ptr =
    std::make_unique<InternalData>();
  auto &data = dynamic_cast<InternalData &>(*data_ptr);
  data.initialize_face(this->requires_update_flags(update_flags),
                       QProjector<dim>::project_to_all_faces(
                         ReferenceCells::get_hypercube<dim>(), quadrature[0]),
                       quadrature[0].size());
 
  return data_ptr;
}
 
 
 
template <int dim, int spacedim>
std::unique_ptr<typename Mapping<dim, spacedim>::InternalDataBase>
MappingManifold<dim, spacedim>::get_subface_data(
  const UpdateFlags          update_flags,
  const Quadrature<dim - 1> &quadrature) const
{
  std::unique_ptr<typename Mapping<dim, spacedim>::InternalDataBase> data_ptr =
    std::make_unique<InternalData>();
  auto &data = dynamic_cast<InternalData &>(*data_ptr);
  data.initialize_face(this->requires_update_flags(update_flags),
                       QProjector<dim>::project_to_all_subfaces(
                         ReferenceCells::get_hypercube<dim>(), quadrature),
                       quadrature.size());
 
  return data_ptr;
}
 
 
 
namespace internal
{
  namespace MappingManifoldImplementation
  {
    namespace
    {
      template <int dim, int spacedim>
      void
      maybe_compute_q_points(
        const typename QProjector<dim>::DataSetDescriptor data_set,
        const typename ::MappingManifold<dim, spacedim>::InternalData
                                     &data,
        std::vector<Point<spacedim>> &quadrature_points)
      {
        const UpdateFlags update_flags = data.update_each;
 
        AssertDimension(data.vertices.size(),
                        GeometryInfo<dim>::vertices_per_cell);
 
        if (update_flags & update_quadrature_points)
          {
            for (unsigned int point = 0; point < quadrature_points.size();
                 ++point)
              {
                quadrature_points[point] = data.manifold->get_new_point(
                  make_array_view(data.vertices),
                  make_array_view(
                    data.cell_manifold_quadrature_weights[point + data_set]));
              }
          }
      }
 
 
 
      template <int dim, int spacedim>
      void
      maybe_update_Jacobians(
        const typename ::QProjector<dim>::DataSetDescriptor data_set,
        const typename ::MappingManifold<dim, spacedim>::InternalData
          &data)
      {
        const UpdateFlags update_flags = data.update_each;
 
        if (update_flags & update_contravariant_transformation)
          {
            const unsigned int n_q_points = data.contravariant.size();
 
            std::fill(data.contravariant.begin(),
                      data.contravariant.end(),
                      DerivativeForm<1, dim, spacedim>());
 
            AssertDimension(GeometryInfo<dim>::vertices_per_cell,
                            data.vertices.size());
            for (unsigned int point = 0; point < n_q_points; ++point)
              {
                // Start by figuring out how to compute the direction in
                // the reference space:
                const Point<dim> &p = data.quad.point(point + data_set);
 
                // And get its image on the manifold:
                const Point<spacedim> P = data.manifold->get_new_point(
                  make_array_view(data.vertices),
                  make_array_view(
                    data.cell_manifold_quadrature_weights[point + data_set]));
 
                // To compute the Jacobian, we choose dim points aligned
                // with the dim reference axes, which are still in the
                // given cell, and ask for the tangent vector in these
                // directions. Choosing the points is somewhat arbitrary,
                // so we try to be smart and we pick points which are
                // on the opposite quadrant w.r.t. the evaluation
                // point.
                for (unsigned int i = 0; i < dim; ++i)
                  {
                    const Point<dim> ei = Point<dim>::unit_vector(i);
                    const double     pi = p[i];
                    Assert(pi >= 0 && pi <= 1.0,
                           ExcInternalError(
                             "Was expecting a quadrature point "
                             "inside the unit reference element."));
 
                    // In the length L, we store also the direction sign,
                    // which is positive, if the coordinate is < .5,
                    const double L = pi > .5 ? -pi : 1 - pi;
 
                    const Point<dim> np(p + L * ei);
 
                    // Get the weights to compute the np point in real space
                    for (const unsigned int j :
                         GeometryInfo<dim>::vertex_indices())
                      data.vertex_weights[j] =
                        GeometryInfo<dim>::d_linear_shape_function(np, j);
 
                    const Point<spacedim> NP = data.manifold->get_new_point(
                      make_array_view(data.vertices),
                      make_array_view(data.vertex_weights));
 
                    const Tensor<1, spacedim> T =
                      data.manifold->get_tangent_vector(P, NP);
 
                    for (unsigned int d = 0; d < spacedim; ++d)
                      data.contravariant[point][d][i] = T[d] / L;
                  }
              }
 
            if (update_flags & update_covariant_transformation)
              {
                const unsigned int n_q_points = data.contravariant.size();
                for (unsigned int point = 0; point < n_q_points; ++point)
                  {
                    data.covariant[point] =
                      (data.contravariant[point]).covariant_form();
                  }
              }
 
            if (update_flags & update_volume_elements)
              {
                const unsigned int n_q_points = data.contravariant.size();
                for (unsigned int point = 0; point < n_q_points; ++point)
                  data.volume_elements[point] =
                    data.contravariant[point].determinant();
              }
          }
      }
    } // namespace
  }   // namespace MappingManifoldImplementation
} // namespace internal
 
 
 
template <int dim, int spacedim>
CellSimilarity::Similarity
MappingManifold<dim, spacedim>::fill_fe_values(
  const typename Triangulation<dim, spacedim>::cell_iterator &cell,
  const CellSimilarity::Similarity,
  const Quadrature<dim>                                   &quadrature,
  const typename Mapping<dim, spacedim>::InternalDataBase &internal_data,
  internal::FEValuesImplementation::MappingRelatedData<dim, spacedim>
    &output_data) const
{
  // ensure that the following static_cast is really correct:
  Assert(dynamic_cast<const InternalData *>(&internal_data) != nullptr,
         ExcInternalError());
  const InternalData &data = static_cast<const InternalData &>(internal_data);
 
  const unsigned int n_q_points = quadrature.size();
 
  data.store_vertices(cell);
  data.manifold = &(cell->get_manifold());
 
  internal::MappingManifoldImplementation::maybe_compute_q_points<dim,
                                                                  spacedim>(
    QProjector<dim>::DataSetDescriptor::cell(),
    data,
    output_data.quadrature_points);
 
  internal::MappingManifoldImplementation::maybe_update_Jacobians<dim,
                                                                  spacedim>(
    QProjector<dim>::DataSetDescriptor::cell(), data);
 
  const UpdateFlags          update_flags = data.update_each;
  const std::vector<double> &weights      = quadrature.get_weights();
 
  // Multiply quadrature weights by absolute value of Jacobian determinants or
  // the area element g=sqrt(DX^t DX) in case of codim > 0
 
  if (update_flags & (update_normal_vectors | update_JxW_values))
    {
      AssertDimension(output_data.JxW_values.size(), n_q_points);
 
      Assert(!(update_flags & update_normal_vectors) ||
               (output_data.normal_vectors.size() == n_q_points),
             ExcDimensionMismatch(output_data.normal_vectors.size(),
                                  n_q_points));
 
 
      for (unsigned int point = 0; point < n_q_points; ++point)
        {
          if (dim == spacedim)
            {
              const double det = data.contravariant[point].determinant();
 
              // check for distorted cells.
 
              // TODO: this allows for anisotropies of up to 1e6 in 3d and
              // 1e12 in 2d. might want to find a finer
              // (dimension-independent) criterion
              Assert(det > 1e-12 * Utilities::fixed_power<dim>(
                                     cell->diameter() / std::sqrt(double(dim))),
                     (typename Mapping<dim, spacedim>::ExcDistortedMappedCell(
                       cell->center(), det, point)));
 
              output_data.JxW_values[point] = weights[point] * det;
            }
          // if dim==spacedim, then there is no cell normal to
          // compute. since this is for FEValues (and not FEFaceValues),
          // there are also no face normals to compute
          else // codim>0 case
            {
              Tensor<1, spacedim> DX_t[dim];
              for (unsigned int i = 0; i < spacedim; ++i)
                for (unsigned int j = 0; j < dim; ++j)
                  DX_t[j][i] = data.contravariant[point][i][j];
 
              Tensor<2, dim> G; // First fundamental form
              for (unsigned int i = 0; i < dim; ++i)
                for (unsigned int j = 0; j < dim; ++j)
                  G[i][j] = DX_t[i] * DX_t[j];
 
              output_data.JxW_values[point] =
                std::sqrt(determinant(G)) * weights[point];
 
              if (update_flags & update_normal_vectors)
                {
                  Assert(spacedim == dim + 1,
                         ExcMessage(
                           "There is no (unique) cell normal for " +
                           Utilities::int_to_string(dim) +
                           "-dimensional cells in " +
                           Utilities::int_to_string(spacedim) +
                           "-dimensional space. This only works if the "
                           "space dimension is one greater than the "
                           "dimensionality of the mesh cells."));
 
                  if (dim == 1)
                    output_data.normal_vectors[point] =
                      cross_product_2d(-DX_t[0]);
                  else // dim == 2
                    output_data.normal_vectors[point] =
                      cross_product_3d(DX_t[0], DX_t[1]);
 
                  output_data.normal_vectors[point] /=
                    output_data.normal_vectors[point].norm();
 
                  if (cell->direction_flag() == false)
                    output_data.normal_vectors[point] *= -1.;
                }
            } // codim>0 case
        }
    }
 
 
 
  // copy values from InternalData to vector given by reference
  if (update_flags & update_jacobians)
    {
      AssertDimension(output_data.jacobians.size(), n_q_points);
      for (unsigned int point = 0; point < n_q_points; ++point)
        output_data.jacobians[point] = data.contravariant[point];
    }
 
  // copy values from InternalData to vector given by reference
  if (update_flags & update_inverse_jacobians)
    {
      AssertDimension(output_data.inverse_jacobians.size(), n_q_points);
      for (unsigned int point = 0; point < n_q_points; ++point)
        output_data.inverse_jacobians[point] =
          data.covariant[point].transpose();
    }
 
  return CellSimilarity::invalid_next_cell;
}
 
 
 
namespace internal
{
  namespace MappingManifoldImplementation
  {
    namespace
    {
      template <int dim, int spacedim>
      void
      maybe_compute_face_data(
        const ::MappingManifold<dim, spacedim> &mapping,
        const typename ::Triangulation<dim, spacedim>::cell_iterator
                                  &cell,
        const unsigned int         face_no,
        const unsigned int         subface_no,
        const unsigned int         n_q_points,
        const std::vector<double> &weights,
        const typename ::MappingManifold<dim, spacedim>::InternalData
          &data,
        internal::FEValuesImplementation::MappingRelatedData<dim, spacedim>
          &output_data)
      {
        const UpdateFlags update_flags = data.update_each;
 
        if (update_flags & update_boundary_forms)
          {
            AssertDimension(output_data.boundary_forms.size(), n_q_points);
            if (update_flags & update_normal_vectors)
              AssertDimension(output_data.normal_vectors.size(), n_q_points);
            if (update_flags & update_JxW_values)
              AssertDimension(output_data.JxW_values.size(), n_q_points);
 
            // map the unit tangentials to the real cell. checking for d!=dim-1
            // eliminates compiler warnings regarding unsigned int expressions <
            // 0.
            for (unsigned int d = 0; d != dim - 1; ++d)
              {
                Assert(face_no + GeometryInfo<dim>::faces_per_cell * d <
                         data.unit_tangentials.size(),
                       ExcInternalError());
                Assert(
                  data.aux[d].size() <=
                    data
                      .unit_tangentials[face_no +
                                        GeometryInfo<dim>::faces_per_cell * d]
                      .size(),
                  ExcInternalError());
 
                mapping.transform(
                  make_array_view(
                    data
                      .unit_tangentials[face_no +
                                        GeometryInfo<dim>::faces_per_cell * d]),
                  mapping_contravariant,
                  data,
                  make_array_view(data.aux[d]));
              }
 
            // if dim==spacedim, we can use the unit tangentials to compute the
            // boundary form by simply taking the cross product
            if (dim == spacedim)
              {
                for (unsigned int i = 0; i < n_q_points; ++i)
                  switch (dim)
                    {
                      case 1:
                        // in 1d, we don't have access to any of the data.aux
                        // fields (because it has only dim-1 components), but we
                        // can still compute the boundary form by simply
                        // looking at the number of the face
                        output_data.boundary_forms[i][0] =
                          (face_no == 0 ? -1 : +1);
                        break;
                      case 2:
                        output_data.boundary_forms[i] =
                          cross_product_2d(data.aux[0][i]);
                        break;
                      case 3:
                        output_data.boundary_forms[i] =
                          cross_product_3d(data.aux[0][i], data.aux[1][i]);
                        break;
                      default:
                        DEAL_II_NOT_IMPLEMENTED();
                    }
              }
            else //(dim < spacedim)
              {
                // in the codim-one case, the boundary form results from the
                // cross product of all the face tangential vectors and the cell
                // normal vector
                //
                // to compute the cell normal, use the same method used in
                // fill_fe_values for cells above
                AssertDimension(data.contravariant.size(), n_q_points);
 
                for (unsigned int point = 0; point < n_q_points; ++point)
                  {
                    switch (dim)
                      {
                        case 1:
                          {
                            // J is a tangent vector
                            output_data.boundary_forms[point] =
                              data.contravariant[point].transpose()[0];
                            output_data.boundary_forms[point] /=
                              (face_no == 0 ? -1. : +1.) *
                              output_data.boundary_forms[point].norm();
 
                            break;
                          }
 
                        case 2:
                          {
                            const DerivativeForm<1, spacedim, dim> DX_t =
                              data.contravariant[point].transpose();
 
                            Tensor<1, spacedim> cell_normal =
                              cross_product_3d(DX_t[0], DX_t[1]);
                            cell_normal /= cell_normal.norm();
 
                            // then compute the face normal from the face
                            // tangent and the cell normal:
                            output_data.boundary_forms[point] =
                              cross_product_3d(data.aux[0][point], cell_normal);
 
                            break;
                          }
 
                        default:
                          DEAL_II_NOT_IMPLEMENTED();
                      }
                  }
              }
 
            if (update_flags & (update_normal_vectors | update_JxW_values))
              for (unsigned int i = 0; i < output_data.boundary_forms.size();
                   ++i)
                {
                  if (update_flags & update_JxW_values)
                    {
                      output_data.JxW_values[i] =
                        output_data.boundary_forms[i].norm() * weights[i];
 
                      if (subface_no != numbers::invalid_unsigned_int)
                        {
                          const double area_ratio =
                            GeometryInfo<dim>::subface_ratio(
                              cell->subface_case(face_no), subface_no);
                          output_data.JxW_values[i] *= area_ratio;
                        }
                    }
 
                  if (update_flags & update_normal_vectors)
                    output_data.normal_vectors[i] =
                      Point<spacedim>(output_data.boundary_forms[i] /
                                      output_data.boundary_forms[i].norm());
                }
 
            if (update_flags & update_jacobians)
              for (unsigned int point = 0; point < n_q_points; ++point)
                output_data.jacobians[point] = data.contravariant[point];
 
            if (update_flags & update_inverse_jacobians)
              for (unsigned int point = 0; point < n_q_points; ++point)
                output_data.inverse_jacobians[point] =
                  data.covariant[point].transpose();
          }
      }
 
 
      template <int dim, int spacedim>
      void
      do_fill_fe_face_values(
        const ::MappingManifold<dim, spacedim> &mapping,
        const typename ::Triangulation<dim, spacedim>::cell_iterator
                                                         &cell,
        const unsigned int                                face_no,
        const unsigned int                                subface_no,
        const typename QProjector<dim>::DataSetDescriptor data_set,
        const Quadrature<dim - 1>                        &quadrature,
        const typename ::MappingManifold<dim, spacedim>::InternalData
          &data,
        internal::FEValuesImplementation::MappingRelatedData<dim, spacedim>
          &output_data)
      {
        data.store_vertices(cell);
 
        data.manifold = &cell->face(face_no)->get_manifold();
 
        maybe_compute_q_points<dim, spacedim>(data_set,
                                              data,
                                              output_data.quadrature_points);
        maybe_update_Jacobians<dim, spacedim>(data_set, data);
 
        maybe_compute_face_data(mapping,
                                cell,
                                face_no,
                                subface_no,
                                quadrature.size(),
                                quadrature.get_weights(),
                                data,
                                output_data);
      }
 
      template <int dim, int spacedim, int rank>
      void
      transform_fields(
        const ArrayView<const Tensor<rank, dim>>                &input,
        const MappingKind                                        mapping_kind,
        const typename Mapping<dim, spacedim>::InternalDataBase &mapping_data,
        const ArrayView<Tensor<rank, spacedim>>                 &output)
      {
        AssertDimension(input.size(), output.size());
        Assert((dynamic_cast<const typename ::
                               MappingManifold<dim, spacedim>::InternalData *>(
                  &mapping_data) != nullptr),
               ExcInternalError());
        const typename ::MappingManifold<dim, spacedim>::InternalData
          &data =
            static_cast<const typename ::MappingManifold<dim, spacedim>::
                          InternalData &>(mapping_data);
 
        switch (mapping_kind)
          {
            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)
                  output[i] =
                    apply_transformation(data.contravariant[i], input[i]);
 
                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"));
                Assert(rank == 1, ExcMessage("Only for rank 1"));
                if (rank != 1)
                  return;
 
                for (unsigned int i = 0; i < output.size(); ++i)
                  {
                    output[i] =
                      apply_transformation(data.contravariant[i], input[i]);
                    output[i] /= data.volume_elements[i];
                  }
                return;
              }
            // We still allow this operation as in the
            // reference cell Derivatives are Tensor
            // rather than DerivativeForm
            case mapping_covariant:
              {
                Assert(
                  data.update_each & update_contravariant_transformation,
                  typename FEValuesBase<dim>::ExcAccessToUninitializedField(
                    "update_covariant_transformation"));
 
                for (unsigned int i = 0; i < output.size(); ++i)
                  output[i] = apply_transformation(data.covariant[i], input[i]);
 
                return;
              }
 
            default:
              DEAL_II_NOT_IMPLEMENTED();
          }
      }
 
 
      template <int dim, int spacedim, int rank>
      void
      transform_gradients(
        const ArrayView<const Tensor<rank, dim>>                &input,
        const MappingKind                                        mapping_kind,
        const typename Mapping<dim, spacedim>::InternalDataBase &mapping_data,
        const ArrayView<Tensor<rank, spacedim>>                 &output)
      {
        AssertDimension(input.size(), output.size());
        Assert((dynamic_cast<const typename ::
                               MappingManifold<dim, spacedim>::InternalData *>(
                  &mapping_data) != nullptr),
               ExcInternalError());
        const typename ::MappingManifold<dim, spacedim>::InternalData
          &data =
            static_cast<const typename ::MappingManifold<dim, spacedim>::
                          InternalData &>(mapping_data);
 
        switch (mapping_kind)
          {
            case mapping_contravariant_gradient:
              {
                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(rank == 2, ExcMessage("Only for rank 2"));
 
                for (unsigned int i = 0; i < output.size(); ++i)
                  {
                    DerivativeForm<1, spacedim, dim> A =
                      apply_transformation(data.contravariant[i],
                                           transpose(input[i]));
                    output[i] =
                      apply_transformation(data.covariant[i], A.transpose());
                  }
 
                return;
              }
 
            case mapping_covariant_gradient:
              {
                Assert(
                  data.update_each & update_covariant_transformation,
                  typename FEValuesBase<dim>::ExcAccessToUninitializedField(
                    "update_covariant_transformation"));
                Assert(rank == 2, ExcMessage("Only for rank 2"));
 
                for (unsigned int i = 0; i < output.size(); ++i)
                  {
                    DerivativeForm<1, spacedim, dim> A =
                      apply_transformation(data.covariant[i],
                                           transpose(input[i]));
                    output[i] =
                      apply_transformation(data.covariant[i], A.transpose());
                  }
 
                return;
              }
 
            case mapping_piola_gradient:
              {
                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"));
                Assert(rank == 2, ExcMessage("Only for rank 2"));
 
                for (unsigned int i = 0; i < output.size(); ++i)
                  {
                    DerivativeForm<1, spacedim, dim> A =
                      apply_transformation(data.covariant[i], input[i]);
                    Tensor<2, spacedim> T =
                      apply_transformation(data.contravariant[i],
                                           A.transpose());
 
                    output[i] = transpose(T);
                    output[i] /= data.volume_elements[i];
                  }
 
                return;
              }
 
            default:
              DEAL_II_NOT_IMPLEMENTED();
          }
      }
 
 
 
      template <int dim, int spacedim>
      void
      transform_hessians(
        const ArrayView<const Tensor<3, dim>>                   &input,
        const MappingKind                                        mapping_kind,
        const typename Mapping<dim, spacedim>::InternalDataBase &mapping_data,
        const ArrayView<Tensor<3, spacedim>>                    &output)
      {
        AssertDimension(input.size(), output.size());
        Assert((dynamic_cast<const typename ::
                               MappingManifold<dim, spacedim>::InternalData *>(
                  &mapping_data) != nullptr),
               ExcInternalError());
        const typename ::MappingManifold<dim, spacedim>::InternalData
          &data =
            static_cast<const typename ::MappingManifold<dim, spacedim>::
                          InternalData &>(mapping_data);
 
        switch (mapping_kind)
          {
            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)
                    {
                      double tmp1[dim][dim];
                      for (unsigned int J = 0; J < dim; ++J)
                        for (unsigned int K = 0; K < dim; ++K)
                          {
                            tmp1[J][K] =
                              data.contravariant[q][i][0] * input[q][0][J][K];
                            for (unsigned int I = 1; I < dim; ++I)
                              tmp1[J][K] +=
                                data.contravariant[q][i][I] * input[q][I][J][K];
                          }
                      for (unsigned int j = 0; j < spacedim; ++j)
                        {
                          double tmp2[dim];
                          for (unsigned int K = 0; K < dim; ++K)
                            {
                              tmp2[K] = data.covariant[q][j][0] * tmp1[0][K];
                              for (unsigned int J = 1; J < dim; ++J)
                                tmp2[K] += data.covariant[q][j][J] * tmp1[J][K];
                            }
                          for (unsigned int k = 0; k < spacedim; ++k)
                            {
                              output[q][i][j][k] =
                                data.covariant[q][k][0] * tmp2[0];
                              for (unsigned int K = 1; K < dim; ++K)
                                output[q][i][j][k] +=
                                  data.covariant[q][k][K] * tmp2[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)
                    {
                      double tmp1[dim][dim];
                      for (unsigned int J = 0; J < dim; ++J)
                        for (unsigned int K = 0; K < dim; ++K)
                          {
                            tmp1[J][K] =
                              data.covariant[q][i][0] * input[q][0][J][K];
                            for (unsigned int I = 1; I < dim; ++I)
                              tmp1[J][K] +=
                                data.covariant[q][i][I] * input[q][I][J][K];
                          }
                      for (unsigned int j = 0; j < spacedim; ++j)
                        {
                          double tmp2[dim];
                          for (unsigned int K = 0; K < dim; ++K)
                            {
                              tmp2[K] = data.covariant[q][j][0] * tmp1[0][K];
                              for (unsigned int J = 1; J < dim; ++J)
                                tmp2[K] += data.covariant[q][j][J] * tmp1[J][K];
                            }
                          for (unsigned int k = 0; k < spacedim; ++k)
                            {
                              output[q][i][j][k] =
                                data.covariant[q][k][0] * tmp2[0];
                              for (unsigned int K = 1; K < dim; ++K)
                                output[q][i][j][k] +=
                                  data.covariant[q][k][K] * tmp2[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)
                    {
                      double factor[dim];
                      for (unsigned int I = 0; I < dim; ++I)
                        factor[I] =
                          data.contravariant[q][i][I] / data.volume_elements[q];
                      double tmp1[dim][dim];
                      for (unsigned int J = 0; J < dim; ++J)
                        for (unsigned int K = 0; K < dim; ++K)
                          {
                            tmp1[J][K] = factor[0] * input[q][0][J][K];
                            for (unsigned int I = 1; I < dim; ++I)
                              tmp1[J][K] += factor[I] * input[q][I][J][K];
                          }
                      for (unsigned int j = 0; j < spacedim; ++j)
                        {
                          double tmp2[dim];
                          for (unsigned int K = 0; K < dim; ++K)
                            {
                              tmp2[K] = data.covariant[q][j][0] * tmp1[0][K];
                              for (unsigned int J = 1; J < dim; ++J)
                                tmp2[K] += data.covariant[q][j][J] * tmp1[J][K];
                            }
                          for (unsigned int k = 0; k < spacedim; ++k)
                            {
                              output[q][i][j][k] =
                                data.covariant[q][k][0] * tmp2[0];
                              for (unsigned int K = 1; K < dim; ++K)
                                output[q][i][j][k] +=
                                  data.covariant[q][k][K] * tmp2[K];
                            }
                        }
                    }
 
                return;
              }
 
            default:
              DEAL_II_NOT_IMPLEMENTED();
          }
      }
 
 
 
      template <int dim, int spacedim, int rank>
      void
      transform_differential_forms(
        const ArrayView<const DerivativeForm<rank, dim, spacedim>> &input,
        const MappingKind                                        mapping_kind,
        const typename Mapping<dim, spacedim>::InternalDataBase &mapping_data,
        const ArrayView<Tensor<rank + 1, spacedim>>             &output)
      {
        AssertDimension(input.size(), output.size());
        Assert((dynamic_cast<const typename ::
                               MappingManifold<dim, spacedim>::InternalData *>(
                  &mapping_data) != nullptr),
               ExcInternalError());
        const typename ::MappingManifold<dim, spacedim>::InternalData
          &data =
            static_cast<const typename ::MappingManifold<dim, spacedim>::
                          InternalData &>(mapping_data);
 
        switch (mapping_kind)
          {
            case mapping_covariant:
              {
                Assert(
                  data.update_each & update_contravariant_transformation,
                  typename FEValuesBase<dim>::ExcAccessToUninitializedField(
                    "update_covariant_transformation"));
 
                for (unsigned int i = 0; i < output.size(); ++i)
                  output[i] = apply_transformation(data.covariant[i], input[i]);
 
                return;
              }
            default:
              DEAL_II_NOT_IMPLEMENTED();
          }
      }
    } // namespace
  }   // namespace MappingManifoldImplementation
} // namespace internal
 
 
 
template <int dim, int spacedim>
void
MappingManifold<dim, spacedim>::fill_fe_face_values(
  const typename Triangulation<dim, spacedim>::cell_iterator &cell,
  const unsigned int                                          face_no,
  const hp::QCollection<dim - 1>                             &quadrature,
  const typename Mapping<dim, spacedim>::InternalDataBase    &internal_data,
  internal::FEValuesImplementation::MappingRelatedData<dim, spacedim>
    &output_data) const
{
  AssertDimension(quadrature.size(), 1);
 
  // ensure that the following cast is really correct:
  Assert((dynamic_cast<const InternalData *>(&internal_data) != nullptr),
         ExcInternalError());
  const InternalData &data = static_cast<const InternalData &>(internal_data);
 
  internal::MappingManifoldImplementation::do_fill_fe_face_values(
    *this,
    cell,
    face_no,
    numbers::invalid_unsigned_int,
    QProjector<dim>::DataSetDescriptor::face(
      ReferenceCells::get_hypercube<dim>(),
      face_no,
      cell->combined_face_orientation(face_no),
      quadrature[0].size()),
    quadrature[0],
    data,
    output_data);
}
 
 
 
template <int dim, int spacedim>
void
MappingManifold<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::FEValuesImplementation::MappingRelatedData<dim, spacedim>
    &output_data) const
{
  // ensure that the following cast is really correct:
  Assert((dynamic_cast<const InternalData *>(&internal_data) != nullptr),
         ExcInternalError());
  const InternalData &data = static_cast<const InternalData &>(internal_data);
 
  internal::MappingManifoldImplementation::do_fill_fe_face_values(
    *this,
    cell,
    face_no,
    subface_no,
    QProjector<dim>::DataSetDescriptor::subface(
      ReferenceCells::get_hypercube<dim>(),
      face_no,
      subface_no,
      cell->combined_face_orientation(face_no),
      quadrature.size(),
      cell->subface_case(face_no)),
    quadrature,
    data,
    output_data);
}
 
 
 
template <int dim, int spacedim>
void
MappingManifold<dim, spacedim>::transform(
  const ArrayView<const Tensor<1, dim>>                   &input,
  const MappingKind                                        mapping_kind,
  const typename Mapping<dim, spacedim>::InternalDataBase &mapping_data,
  const ArrayView<Tensor<1, spacedim>>                    &output) const
{
  internal::MappingManifoldImplementation::transform_fields(input,
                                                            mapping_kind,
                                                            mapping_data,
                                                            output);
}
 
 
 
template <int dim, int spacedim>
void
MappingManifold<dim, spacedim>::transform(
  const ArrayView<const DerivativeForm<1, dim, spacedim>> &input,
  const MappingKind                                        mapping_kind,
  const typename Mapping<dim, spacedim>::InternalDataBase &mapping_data,
  const ArrayView<Tensor<2, spacedim>>                    &output) const
{
  internal::MappingManifoldImplementation::transform_differential_forms(
    input, mapping_kind, mapping_data, output);
}
 
 
 
template <int dim, int spacedim>
void
MappingManifold<dim, spacedim>::transform(
  const ArrayView<const Tensor<2, dim>>                   &input,
  const MappingKind                                        mapping_kind,
  const typename Mapping<dim, spacedim>::InternalDataBase &mapping_data,
  const ArrayView<Tensor<2, spacedim>>                    &output) const
{
  switch (mapping_kind)
    {
      case mapping_contravariant:
        internal::MappingManifoldImplementation::transform_fields(input,
                                                                  mapping_kind,
                                                                  mapping_data,
                                                                  output);
        return;
 
      case mapping_piola_gradient:
      case mapping_contravariant_gradient:
      case mapping_covariant_gradient:
        internal::MappingManifoldImplementation::transform_gradients(
          input, mapping_kind, mapping_data, output);
        return;
      default:
        DEAL_II_NOT_IMPLEMENTED();
    }
}
 
 
 
template <int dim, int spacedim>
void
MappingManifold<dim, spacedim>::transform(
  const ArrayView<const DerivativeForm<2, dim, spacedim>> &input,
  const MappingKind                                        mapping_kind,
  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) != nullptr,
         ExcInternalError());
  const InternalData &data = static_cast<const InternalData &>(mapping_data);
 
  switch (mapping_kind)
    {
      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)
                {
                  double tmp[dim];
                  for (unsigned int K = 0; K < dim; ++K)
                    {
                      tmp[K] = data.covariant[q][j][0] * input[q][i][0][K];
                      for (unsigned int J = 1; J < dim; ++J)
                        tmp[K] += data.covariant[q][j][J] * input[q][i][J][K];
                    }
                  for (unsigned int k = 0; k < spacedim; ++k)
                    {
                      output[q][i][j][k] = data.covariant[q][k][0] * tmp[0];
                      for (unsigned int K = 1; K < dim; ++K)
                        output[q][i][j][k] += data.covariant[q][k][K] * tmp[K];
                    }
                }
          return;
        }
 
      default:
        DEAL_II_NOT_IMPLEMENTED();
    }
}
 
 
 
template <int dim, int spacedim>
void
MappingManifold<dim, spacedim>::transform(
  const ArrayView<const Tensor<3, dim>>                   &input,
  const MappingKind                                        mapping_kind,
  const typename Mapping<dim, spacedim>::InternalDataBase &mapping_data,
  const ArrayView<Tensor<3, spacedim>>                    &output) const
{
  switch (mapping_kind)
    {
      case mapping_piola_hessian:
      case mapping_contravariant_hessian:
      case mapping_covariant_hessian:
        internal::MappingManifoldImplementation::transform_hessians(
          input, mapping_kind, mapping_data, output);
        return;
      default:
        DEAL_II_NOT_IMPLEMENTED();
    }
}
 
//--------------------------- Explicit instantiations -----------------------
#include "mapping_manifold.inst"
 
 
DEAL_II_NAMESPACE_CLOSE