manifold.cc

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//
// Copyright (C) 1998 - 2022 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.md at
// the top level directory of deal.II.
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#include <deal.II/base/table.h>
#include <deal.II/base/tensor.h>
 
#include <deal.II/fe/fe_q.h>
 
#include <deal.II/grid/manifold.h>
#include <deal.II/grid/reference_cell.h>
#include <deal.II/grid/tria.h>
#include <deal.II/grid/tria_accessor.h>
#include <deal.II/grid/tria_iterator.h>
 
DEAL_II_DISABLE_EXTRA_DIAGNOSTICS
#include <boost/container/small_vector.hpp>
DEAL_II_DISABLE_EXTRA_DIAGNOSTICS
 
#include <cmath>
#include <memory>
 
DEAL_II_NAMESPACE_OPEN
 
/* -------------------------- Manifold --------------------- */
template <int dim, int spacedim>
Point<spacedim>
Manifold<dim, spacedim>::project_to_manifold(
  const ArrayView<const Point<spacedim>> &,
  const Point<spacedim> &) const
{
  Assert(false, ExcPureFunctionCalled());
  return {};
}
 
 
 
template <int dim, int spacedim>
Point<spacedim>
Manifold<dim, spacedim>::get_intermediate_point(const Point<spacedim> &p1,
                                                const Point<spacedim> &p2,
                                                const double           w) const
{
  const std::array<Point<spacedim>, 2> vertices{{p1, p2}};
  return project_to_manifold(make_array_view(vertices.begin(), vertices.end()),
                             w * p2 + (1 - w) * p1);
}
 
 
 
template <int dim, int spacedim>
Point<spacedim>
Manifold<dim, spacedim>::get_new_point(
  const ArrayView<const Point<spacedim>> &surrounding_points,
  const ArrayView<const double> &         weights) const
{
  const double       tol      = 1e-10;
  const unsigned int n_points = surrounding_points.size();
 
  Assert(n_points > 0, ExcMessage("There should be at least one point."));
 
  Assert(n_points == weights.size(),
         ExcMessage(
           "There should be as many surrounding points as weights given."));
 
  Assert(std::abs(std::accumulate(weights.begin(), weights.end(), 0.0) - 1.0) <
           tol,
         ExcMessage("The weights for the individual points should sum to 1!"));
 
  // First sort points in the order of their weights. This is done to
  // produce unique points even if get_intermediate_points is not
  // associative (as for the SphericalManifold).
  boost::container::small_vector<unsigned int, 100> permutation(n_points);
  std::iota(permutation.begin(), permutation.end(), 0u);
  std::sort(permutation.begin(),
            permutation.end(),
            [&weights](const std::size_t a, const std::size_t b) {
              return weights[a] < weights[b];
            });
 
  // Now loop over points in the order of their associated weight
  Point<spacedim> p = surrounding_points[permutation[0]];
  double          w = weights[permutation[0]];
 
  for (unsigned int i = 1; i < n_points; ++i)
    {
      double weight = 0.0;
      if (std::abs(weights[permutation[i]] + w) < tol)
        weight = 0.0;
      else
        weight = w / (weights[permutation[i]] + w);
 
      if (std::abs(weight) > 1e-14)
        {
          p = get_intermediate_point(p,
                                     surrounding_points[permutation[i]],
                                     1.0 - weight);
        }
      else
        {
          p = surrounding_points[permutation[i]];
        }
      w += weights[permutation[i]];
    }
 
  return p;
}
 
 
 
template <int dim, int spacedim>
void
Manifold<dim, spacedim>::get_new_points(
  const ArrayView<const Point<spacedim>> &surrounding_points,
  const Table<2, double> &                weights,
  ArrayView<Point<spacedim>>              new_points) const
{
  AssertDimension(surrounding_points.size(), weights.size(1));
 
  for (unsigned int row = 0; row < weights.size(0); ++row)
    {
      new_points[row] =
        get_new_point(make_array_view(surrounding_points.begin(),
                                      surrounding_points.end()),
                      make_array_view(weights, row));
    }
}
 
 
 
template <>
Tensor<1, 2>
Manifold<2, 2>::normal_vector(const Triangulation<2, 2>::face_iterator &face,
                              const Point<2> &                          p) const
{
  const int spacedim = 2;
 
  // get the tangent vector from the point 'p' in the direction of the further
  // one of the two vertices that make up the face of this 2d cell
  const Tensor<1, spacedim> tangent =
    ((p - face->vertex(0)).norm_square() > (p - face->vertex(1)).norm_square() ?
       -get_tangent_vector(p, face->vertex(0)) :
       get_tangent_vector(p, face->vertex(1)));
 
  // then rotate it by 90 degrees
  const Tensor<1, spacedim> normal = cross_product_2d(tangent);
  return normal / normal.norm();
}
 
 
 
template <>
Tensor<1, 3>
Manifold<3, 3>::normal_vector(const Triangulation<3, 3>::face_iterator &face,
                              const Point<3> &                          p) const
{
  const int spacedim = 3;
 
  const std::array<Point<spacedim>, 4> vertices{
    {face->vertex(0), face->vertex(1), face->vertex(2), face->vertex(3)}};
  const std::array<double, 4> distances{{vertices[0].distance(p),
                                         vertices[1].distance(p),
                                         vertices[2].distance(p),
                                         vertices[3].distance(p)}};
  const double max_distance = std::max(std::max(distances[0], distances[1]),
                                       std::max(distances[2], distances[3]));
 
  // We need to find two tangential vectors to the given point p, but we do
  // not know how the point is oriented against the face. We guess the two
  // directions by assuming a flat topology and take the two directions that
  // indicate the angle closest to a perpendicular one (i.e., cos(theta) close
  // to zero). We start with an invalid value but the loops should always find
  // a value.
  double       abs_cos_angle = std::numeric_limits<double>::max();
  unsigned int first_index   = numbers::invalid_unsigned_int,
               second_index  = numbers::invalid_unsigned_int;
  for (unsigned int i = 0; i < 3; ++i)
    if (distances[i] > 1e-8 * max_distance)
      for (unsigned int j = i + 1; j < 4; ++j)
        if (distances[j] > 1e-8 * max_distance)
          {
            const double new_angle = (p - vertices[i]) * (p - vertices[j]) /
                                     (distances[i] * distances[j]);
            // multiply by factor 0.999 to bias the search in a way that
            // avoids trouble with roundoff
            if (std::abs(new_angle) < 0.999 * abs_cos_angle)
              {
                abs_cos_angle = std::abs(new_angle);
                first_index   = i;
                second_index  = j;
              }
          }
  Assert(first_index != numbers::invalid_unsigned_int,
         ExcMessage("The search for possible directions did not succeed."));
 
  // Compute tangents and normal for selected vertices
  Tensor<1, spacedim> t1     = get_tangent_vector(p, vertices[first_index]);
  Tensor<1, spacedim> t2     = get_tangent_vector(p, vertices[second_index]);
  Tensor<1, spacedim> normal = cross_product_3d(t1, t2);
 
  Assert(
    normal.norm_square() > 1e4 * std::numeric_limits<double>::epsilon() *
                             t1.norm_square() * t2.norm_square(),
    ExcMessage(
      "Manifold::normal_vector was unable to find a suitable combination "
      "of vertices to compute a normal on this face. We chose the secants "
      "that are as orthogonal as possible, but tangents appear to be "
      "linearly dependent. Check for distorted faces in your triangulation."));
 
  // Now figure out if we need to flip the direction, we do this by comparing
  // to a reference normal that would be the correct result if all vertices
  // would lie in a plane
  const Tensor<1, spacedim> rt1              = vertices[3] - vertices[0];
  const Tensor<1, spacedim> rt2              = vertices[2] - vertices[1];
  const Tensor<1, spacedim> reference_normal = cross_product_3d(rt1, rt2);
 
  if (reference_normal * normal < 0.0)
    normal *= -1.0;
 
  return normal / normal.norm();
}
 
 
 
template <int dim, int spacedim>
Tensor<1, spacedim>
Manifold<dim, spacedim>::normal_vector(
  const typename Triangulation<dim, spacedim>::face_iterator & /*face*/,
  const Point<spacedim> & /*p*/) const
{
  Assert(false, ExcPureFunctionCalled());
  return Tensor<1, spacedim>();
}
 
 
 
template <>
void
Manifold<2, 2>::get_normals_at_vertices(
  const Triangulation<2, 2>::face_iterator &face,
  FaceVertexNormals &                       n) const
{
  n[0] = cross_product_2d(get_tangent_vector(face->vertex(0), face->vertex(1)));
  n[1] =
    -cross_product_2d(get_tangent_vector(face->vertex(1), face->vertex(0)));
 
  for (unsigned int i = 0; i < 2; ++i)
    {
      Assert(n[i].norm() != 0,
             ExcInternalError("Something went wrong. The "
                              "computed normals have "
                              "zero length."));
      n[i] /= n[i].norm();
    }
}
 
 
 
template <>
void
Manifold<3, 3>::get_normals_at_vertices(
  const Triangulation<3, 3>::face_iterator &face,
  FaceVertexNormals &                       n) const
{
  n[0] = cross_product_3d(get_tangent_vector(face->vertex(0), face->vertex(1)),
                          get_tangent_vector(face->vertex(0), face->vertex(2)));
 
  n[1] = cross_product_3d(get_tangent_vector(face->vertex(1), face->vertex(3)),
                          get_tangent_vector(face->vertex(1), face->vertex(0)));
 
  n[2] = cross_product_3d(get_tangent_vector(face->vertex(2), face->vertex(0)),
                          get_tangent_vector(face->vertex(2), face->vertex(3)));
 
  n[3] = cross_product_3d(get_tangent_vector(face->vertex(3), face->vertex(2)),
                          get_tangent_vector(face->vertex(3), face->vertex(1)));
 
  for (unsigned int i = 0; i < 4; ++i)
    {
      Assert(n[i].norm() != 0,
             ExcInternalError("Something went wrong. The "
                              "computed normals have "
                              "zero length."));
      n[i] /= n[i].norm();
    }
}
 
 
 
template <int dim, int spacedim>
void
Manifold<dim, spacedim>::get_normals_at_vertices(
  const typename Triangulation<dim, spacedim>::face_iterator &face,
  FaceVertexNormals &                                         n) const
{
  for (unsigned int v = 0; v < face->reference_cell().n_vertices(); ++v)
    {
      n[v] = normal_vector(face, face->vertex(v));
      n[v] /= n[v].norm();
    }
}
 
 
 
template <int dim, int spacedim>
Point<spacedim>
Manifold<dim, spacedim>::get_new_point_on_line(
  const typename Triangulation<dim, spacedim>::line_iterator &line) const
{
  const auto points_weights = Manifolds::get_default_points_and_weights(line);
  return get_new_point(make_array_view(points_weights.first.begin(),
                                       points_weights.first.end()),
                       make_array_view(points_weights.second.begin(),
                                       points_weights.second.end()));
}
 
 
 
template <int dim, int spacedim>
Point<spacedim>
Manifold<dim, spacedim>::get_new_point_on_quad(
  const typename Triangulation<dim, spacedim>::quad_iterator &quad) const
{
  const auto points_weights = Manifolds::get_default_points_and_weights(quad);
  return get_new_point(make_array_view(points_weights.first.begin(),
                                       points_weights.first.end()),
                       make_array_view(points_weights.second.begin(),
                                       points_weights.second.end()));
}
 
 
 
template <int dim, int spacedim>
Point<spacedim>
Manifold<dim, spacedim>::get_new_point_on_face(
  const typename Triangulation<dim, spacedim>::face_iterator &face) const
{
  Assert(dim > 1, ExcImpossibleInDim(dim));
 
  switch (dim)
    {
      case 2:
        return get_new_point_on_line(face);
      case 3:
        return get_new_point_on_quad(face);
    }
 
  return {};
}
 
 
 
template <int dim, int spacedim>
Point<spacedim>
Manifold<dim, spacedim>::get_new_point_on_cell(
  const typename Triangulation<dim, spacedim>::cell_iterator &cell) const
{
  switch (dim)
    {
      case 1:
        return get_new_point_on_line(cell);
      case 2:
        return get_new_point_on_quad(cell);
      case 3:
        return get_new_point_on_hex(cell);
    }
 
  return {};
}
 
 
 
template <>
Point<1>
Manifold<1, 1>::get_new_point_on_face(
  const Triangulation<1, 1>::face_iterator &) const
{
  Assert(false, ExcImpossibleInDim(1));
  return {};
}
 
 
 
template <>
Point<2>
Manifold<1, 2>::get_new_point_on_face(
  const Triangulation<1, 2>::face_iterator &) const
{
  Assert(false, ExcImpossibleInDim(1));
  return {};
}
 
 
 
template <>
Point<3>
Manifold<1, 3>::get_new_point_on_face(
  const Triangulation<1, 3>::face_iterator &) const
{
  Assert(false, ExcImpossibleInDim(1));
  return {};
}
 
 
 
template <>
Point<1>
Manifold<1, 1>::get_new_point_on_quad(
  const Triangulation<1, 1>::quad_iterator &) const
{
  Assert(false, ExcImpossibleInDim(1));
  return {};
}
 
 
 
template <>
Point<2>
Manifold<1, 2>::get_new_point_on_quad(
  const Triangulation<1, 2>::quad_iterator &) const
{
  Assert(false, ExcImpossibleInDim(1));
  return {};
}
 
 
 
template <>
Point<3>
Manifold<1, 3>::get_new_point_on_quad(
  const Triangulation<1, 3>::quad_iterator &) const
{
  Assert(false, ExcImpossibleInDim(1));
  return {};
}
 
 
 
template <int dim, int spacedim>
Point<spacedim>
Manifold<dim, spacedim>::get_new_point_on_hex(
  const typename Triangulation<dim, spacedim>::hex_iterator & /*hex*/) const
{
  Assert(false, ExcImpossibleInDim(dim));
  return {};
}
 
 
 
template <>
Point<3>
Manifold<3, 3>::get_new_point_on_hex(
  const Triangulation<3, 3>::hex_iterator &hex) const
{
  const auto points_weights =
    Manifolds::get_default_points_and_weights(hex, true);
  return get_new_point(make_array_view(points_weights.first.begin(),
                                       points_weights.first.end()),
                       make_array_view(points_weights.second.begin(),
                                       points_weights.second.end()));
}
 
 
 
template <int dim, int spacedim>
Tensor<1, spacedim>
Manifold<dim, spacedim>::get_tangent_vector(const Point<spacedim> &x1,
                                            const Point<spacedim> &x2) const
{
  const double epsilon = 1e-8;
 
  const std::array<Point<spacedim>, 2> points{{x1, x2}};
  const std::array<double, 2>          weights{{epsilon, 1.0 - epsilon}};
  const Point<spacedim>                neighbor_point =
    get_new_point(make_array_view(points.begin(), points.end()),
                  make_array_view(weights.begin(), weights.end()));
  return (neighbor_point - x1) / epsilon;
}
 
/* -------------------------- FlatManifold --------------------- */
 
namespace internal
{
  namespace
  {
    Tensor<1, 3>
    normalized_alternating_product(const Tensor<1, 3> (&)[1])
    {
      // we get here from FlatManifold<2,3>::normal_vector, but
      // the implementation below is bogus for this case anyway
      // (see the assert at the beginning of that function).
      Assert(false, ExcNotImplemented());
      return {};
    }
 
 
 
    Tensor<1, 3>
    normalized_alternating_product(const Tensor<1, 3> (&basis_vectors)[2])
    {
      Tensor<1, 3> tmp = cross_product_3d(basis_vectors[0], basis_vectors[1]);
      return tmp / tmp.norm();
    }
 
  } // namespace
} // namespace internal
 
template <int dim, int spacedim>
FlatManifold<dim, spacedim>::FlatManifold(
  const Tensor<1, spacedim> &periodicity,
  const double               tolerance)
  : periodicity(periodicity)
  , tolerance(tolerance)
{}
 
 
 
template <int dim, int spacedim>
std::unique_ptr<Manifold<dim, spacedim>>
FlatManifold<dim, spacedim>::clone() const
{
  return std::make_unique<FlatManifold<dim, spacedim>>(periodicity, tolerance);
}
 
 
 
template <int dim, int spacedim>
Point<spacedim>
FlatManifold<dim, spacedim>::get_new_point(
  const ArrayView<const Point<spacedim>> &surrounding_points,
  const ArrayView<const double> &         weights) const
{
  Assert(std::abs(std::accumulate(weights.begin(), weights.end(), 0.0) - 1.0) <
           1e-10,
         ExcMessage("The weights for the individual points should sum to 1!"));
 
  Point<spacedim> p;
 
  // if there is no periodicity, use a shortcut
  if (periodicity == Tensor<1, spacedim>())
    {
      for (unsigned int i = 0; i < surrounding_points.size(); ++i)
        p += surrounding_points[i] * weights[i];
    }
  else
    {
      Tensor<1, spacedim> minP = periodicity;
 
      for (unsigned int d = 0; d < spacedim; ++d)
        if (periodicity[d] > 0)
          for (unsigned int i = 0; i < surrounding_points.size(); ++i)
            {
              minP[d] = std::min(minP[d], surrounding_points[i][d]);
              Assert((surrounding_points[i][d] <
                      periodicity[d] + tolerance * periodicity[d]) ||
                       (surrounding_points[i][d] >=
                        -tolerance * periodicity[d]),
                     ExcPeriodicBox(d, surrounding_points[i], periodicity[d]));
            }
 
      // compute the weighted average point, possibly taking into account
      // periodicity
      for (unsigned int i = 0; i < surrounding_points.size(); ++i)
        {
          Point<spacedim> dp;
          for (unsigned int d = 0; d < spacedim; ++d)
            if (periodicity[d] > 0)
              dp[d] =
                ((surrounding_points[i][d] - minP[d]) > periodicity[d] / 2.0 ?
                   -periodicity[d] :
                   0.0);
 
          p += (surrounding_points[i] + dp) * weights[i];
        }
 
      // if necessary, also adjust the weighted point by the periodicity
      for (unsigned int d = 0; d < spacedim; ++d)
        if (periodicity[d] > 0)
          if (p[d] < 0)
            p[d] += periodicity[d];
    }
 
  return project_to_manifold(surrounding_points, p);
}
 
 
 
template <int dim, int spacedim>
void
FlatManifold<dim, spacedim>::get_new_points(
  const ArrayView<const Point<spacedim>> &surrounding_points,
  const Table<2, double> &                weights,
  ArrayView<Point<spacedim>>              new_points) const
{
  AssertDimension(surrounding_points.size(), weights.size(1));
  if (weights.size(0) == 0)
    return;
 
  const std::size_t n_points = surrounding_points.size();
 
  Tensor<1, spacedim> minP = periodicity;
  for (unsigned int d = 0; d < spacedim; ++d)
    if (periodicity[d] > 0)
      for (unsigned int i = 0; i < n_points; ++i)
        {
          minP[d] = std::min(minP[d], surrounding_points[i][d]);
          Assert((surrounding_points[i][d] <
                  periodicity[d] + tolerance * periodicity[d]) ||
                   (surrounding_points[i][d] >= -tolerance * periodicity[d]),
                 ExcPeriodicBox(d, surrounding_points[i], periodicity[i]));
        }
 
  // check whether periodicity shifts some of the points. Only do this if
  // necessary to avoid memory allocation
  const Point<spacedim> *surrounding_points_start = surrounding_points.data();
 
  boost::container::small_vector<Point<spacedim>, 200> modified_points;
  bool adjust_periodicity = false;
  for (unsigned int d = 0; d < spacedim; ++d)
    if (periodicity[d] > 0)
      for (unsigned int i = 0; i < n_points; ++i)
        if ((surrounding_points[i][d] - minP[d]) > periodicity[d] / 2.0)
          {
            adjust_periodicity = true;
            break;
          }
  if (adjust_periodicity == true)
    {
      modified_points.resize(surrounding_points.size());
      std::copy(surrounding_points.begin(),
                surrounding_points.end(),
                modified_points.begin());
      for (unsigned int d = 0; d < spacedim; ++d)
        if (periodicity[d] > 0)
          for (unsigned int i = 0; i < n_points; ++i)
            if ((surrounding_points[i][d] - minP[d]) > periodicity[d] / 2.0)
              modified_points[i][d] -= periodicity[d];
      surrounding_points_start = modified_points.data();
    }
 
  // Now perform the interpolation
  for (unsigned int row = 0; row < weights.size(0); ++row)
    {
      Assert(
        std::abs(
          std::accumulate(&weights(row, 0), &weights(row, 0) + n_points, 0.0) -
          1.0) < 1e-10,
        ExcMessage("The weights for the individual points should sum to 1!"));
      Point<spacedim> new_point;
      for (unsigned int p = 0; p < n_points; ++p)
        new_point += surrounding_points_start[p] * weights(row, p);
 
      // if necessary, also adjust the weighted point by the periodicity
      for (unsigned int d = 0; d < spacedim; ++d)
        if (periodicity[d] > 0)
          if (new_point[d] < 0)
            new_point[d] += periodicity[d];
 
      // TODO should this use surrounding_points_start or surrounding_points?
      // The older version used surrounding_points
      new_points[row] =
        project_to_manifold(make_array_view(surrounding_points.begin(),
                                            surrounding_points.end()),
                            new_point);
    }
}
 
 
 
template <int dim, int spacedim>
Point<spacedim>
FlatManifold<dim, spacedim>::project_to_manifold(
  const ArrayView<const Point<spacedim>> & /*vertices*/,
  const Point<spacedim> &candidate) const
{
  return candidate;
}
 
 
 
template <int dim, int spacedim>
const Tensor<1, spacedim> &
FlatManifold<dim, spacedim>::get_periodicity() const
{
  return periodicity;
}
 
 
 
template <int dim, int spacedim>
Tensor<1, spacedim>
FlatManifold<dim, spacedim>::get_tangent_vector(const Point<spacedim> &x1,
                                                const Point<spacedim> &x2) const
{
  Tensor<1, spacedim> direction = x2 - x1;
 
  // see if we have to take into account periodicity. if so, we need
  // to make sure that if a distance in one coordinate direction
  // is larger than half of the box length, then go the other way
  // around (i.e., via the periodic box)
  for (unsigned int d = 0; d < spacedim; ++d)
    if (periodicity[d] > tolerance)
      {
        if (direction[d] < -periodicity[d] / 2)
          direction[d] += periodicity[d];
        else if (direction[d] > periodicity[d] / 2)
          direction[d] -= periodicity[d];
      }
 
  return direction;
}
 
 
 
template <>
void
FlatManifold<1>::get_normals_at_vertices(
  const Triangulation<1>::face_iterator &,
  Manifold<1, 1>::FaceVertexNormals &) const
{
  Assert(false, ExcImpossibleInDim(1));
}
 
 
 
template <>
void
FlatManifold<1, 2>::get_normals_at_vertices(
  const Triangulation<1, 2>::face_iterator &,
  Manifold<1, 2>::FaceVertexNormals &) const
{
  Assert(false, ExcNotImplemented());
}
 
 
 
template <>
void
FlatManifold<1, 3>::get_normals_at_vertices(
  const Triangulation<1, 3>::face_iterator &,
  Manifold<1, 3>::FaceVertexNormals &) const
{
  Assert(false, ExcNotImplemented());
}
 
 
 
template <>
void
FlatManifold<2>::get_normals_at_vertices(
  const Triangulation<2>::face_iterator &face,
  Manifold<2, 2>::FaceVertexNormals &    face_vertex_normals) const
{
  const Tensor<1, 2> tangent = face->vertex(1) - face->vertex(0);
  // We're in 2d. Faces are edges:
  for (const unsigned int vertex : ReferenceCells::Line.vertex_indices())
    // compute normals from tangent
    face_vertex_normals[vertex] = Tensor<1, 2>({tangent[1], -tangent[0]});
}
 
 
 
template <>
void
FlatManifold<2, 3>::get_normals_at_vertices(
  const Triangulation<2, 3>::face_iterator & /*face*/,
  Manifold<2, 3>::FaceVertexNormals & /*face_vertex_normals*/) const
{
  Assert(false, ExcNotImplemented());
}
 
 
 
template <>
void
FlatManifold<3>::get_normals_at_vertices(
  const Triangulation<3>::face_iterator &face,
  Manifold<3, 3>::FaceVertexNormals &    face_vertex_normals) const
{
  const unsigned int vertices_per_face = GeometryInfo<3>::vertices_per_face;
 
  static const unsigned int neighboring_vertices[4][2] = {{1, 2},
                                                          {3, 0},
                                                          {0, 3},
                                                          {2, 1}};
  for (unsigned int vertex = 0; vertex < vertices_per_face; ++vertex)
    {
      // first define the two tangent vectors at the vertex by using the
      // two lines radiating away from this vertex
      const Tensor<1, 3> tangents[2] = {
        face->vertex(neighboring_vertices[vertex][0]) - face->vertex(vertex),
        face->vertex(neighboring_vertices[vertex][1]) - face->vertex(vertex)};
 
      // then compute the normal by taking the cross product. since the
      // normal is not required to be normalized, no problem here
      face_vertex_normals[vertex] = cross_product_3d(tangents[0], tangents[1]);
    }
}
 
 
 
template <>
Tensor<1, 1>
FlatManifold<1, 1>::normal_vector(const Triangulation<1, 1>::face_iterator &,
                                  const Point<1> &) const
{
  Assert(false, ExcNotImplemented());
  return {};
}
 
 
 
template <>
Tensor<1, 2>
FlatManifold<1, 2>::normal_vector(const Triangulation<1, 2>::face_iterator &,
                                  const Point<2> &) const
{
  Assert(false, ExcNotImplemented());
  return {};
}
 
 
 
template <>
Tensor<1, 3>
FlatManifold<1, 3>::normal_vector(const Triangulation<1, 3>::face_iterator &,
                                  const Point<3> &) const
{
  Assert(false, ExcNotImplemented());
  return {};
}
 
 
 
template <>
Tensor<1, 2>
FlatManifold<2, 2>::normal_vector(
  const Triangulation<2, 2>::face_iterator &face,
  const Point<2> &                          p) const
{
  // In 2d, a face is just a straight line and
  // we can use the 'standard' implementation.
  return Manifold<2, 2>::normal_vector(face, p);
}
 
 
 
template <int dim, int spacedim>
Tensor<1, spacedim>
FlatManifold<dim, spacedim>::normal_vector(
  const typename Triangulation<dim, spacedim>::face_iterator &face,
  const Point<spacedim> &                                     p) const
{
  // I don't think the implementation below will work when dim!=spacedim;
  // in fact, I believe that we don't even have enough information here,
  // because we would need to know not only about the tangent vectors
  // of the face, but also of the cell, to compute the normal vector.
  // Someone will have to think about this some more.
  Assert(dim == spacedim, ExcNotImplemented());
 
  // in order to find out what the normal vector is, we first need to
  // find the reference coordinates of the point p on the given face,
  // or at least the reference coordinates of the closest point on the
  // face
  //
  // in other words, we need to find a point xi so that f(xi)=||F(xi)-p||^2->min
  // where F(xi) is the mapping. this algorithm is implemented in
  // MappingQ1<dim,spacedim>::transform_real_to_unit_cell but only for cells,
  // while we need it for faces here. it's also implemented in somewhat
  // more generality there using the machinery of the MappingQ1 class
  // while we really only need it for a specific case here
  //
  // in any case, the iteration we use here is a Gauss-Newton's iteration with
  //   xi^{n+1} = xi^n - H(xi^n)^{-1} J(xi^n)
  // where
  //   J(xi) = (grad F(xi))^T (F(xi)-p)
  // and
  //   H(xi) = [grad F(xi)]^T [grad F(xi)]
  // In all this,
  //   F(xi) = sum_v vertex[v] phi_v(xi)
  // We get the shape functions phi_v from an object of type FE_Q<dim-1>(1)
 
  // we start with the point xi=1/2, xi=(1/2,1/2), ...
  const unsigned int facedim = dim - 1;
 
  Point<facedim> xi;
 
  const auto face_reference_cell = face->reference_cell();
 
  if (face_reference_cell == ReferenceCells::get_hypercube<facedim>())
    {
      for (unsigned int i = 0; i < facedim; ++i)
        xi[i] = 1. / 2;
    }
  else
    {
      for (unsigned int i = 0; i < facedim; ++i)
        xi[i] = 1. / 3;
    }
 
  const double        eps = 1e-12;
  Tensor<1, spacedim> grad_F[facedim];
  unsigned int        iteration = 0;
  while (true)
    {
      Point<spacedim> F;
      for (const unsigned int v : face->vertex_indices())
        F +=
          face->vertex(v) * face_reference_cell.d_linear_shape_function(xi, v);
 
      for (unsigned int i = 0; i < facedim; ++i)
        {
          grad_F[i] = 0;
          for (const unsigned int v : face->vertex_indices())
            grad_F[i] +=
              face->vertex(v) *
              face_reference_cell.d_linear_shape_function_gradient(xi, v)[i];
        }
 
      Tensor<1, facedim> J;
      for (unsigned int i = 0; i < facedim; ++i)
        for (unsigned int j = 0; j < spacedim; ++j)
          J[i] += grad_F[i][j] * (F - p)[j];
 
      Tensor<2, facedim> H;
      for (unsigned int i = 0; i < facedim; ++i)
        for (unsigned int j = 0; j < facedim; ++j)
          for (unsigned int k = 0; k < spacedim; ++k)
            H[i][j] += grad_F[i][k] * grad_F[j][k];
 
      const Tensor<1, facedim> delta_xi = -invert(H) * J;
      xi += delta_xi;
      ++iteration;
 
      Assert(iteration < 10,
             ExcMessage("The Newton iteration to find the reference point "
                        "did not converge in 10 iterations. Do you have a "
                        "deformed cell? (See the glossary for a definition "
                        "of what a deformed cell is. You may want to output "
                        "the vertices of your cell."));
 
      // It turns out that the check in reference coordinates with an absolute
      // tolerance can cause a convergence failure of the Newton method as
      // seen in tests/manifold/flat_manifold_09.cc. To work around this, also
      // use a convergence check in world coordinates. This check has to be
      // relative to the size of the face of course. Here we decided to use
      // diameter because it works for non-planar faces and is cheap to
      // compute:
      const double normalized_delta_world = (F - p).norm() / face->diameter();
 
      if (delta_xi.norm() < eps || normalized_delta_world < eps)
        break;
    }
 
  // so now we have the reference coordinates xi of the point p.
  // we then have to compute the normal vector, which we can do
  // by taking the (normalize) alternating product of all the tangent
  // vectors given by grad_F
  return internal::normalized_alternating_product(grad_F);
}
 
 
/* -------------------------- ChartManifold --------------------- */
template <int dim, int spacedim, int chartdim>
ChartManifold<dim, spacedim, chartdim>::ChartManifold(
  const Tensor<1, chartdim> &periodicity)
  : sub_manifold(periodicity)
{}
 
 
 
template <int dim, int spacedim, int chartdim>
Point<spacedim>
ChartManifold<dim, spacedim, chartdim>::get_intermediate_point(
  const Point<spacedim> &p1,
  const Point<spacedim> &p2,
  const double           w) const
{
  const std::array<Point<spacedim>, 2> points{{p1, p2}};
  const std::array<double, 2>          weights{{1. - w, w}};
  return get_new_point(make_array_view(points.begin(), points.end()),
                       make_array_view(weights.begin(), weights.end()));
}
 
 
 
template <int dim, int spacedim, int chartdim>
Point<spacedim>
ChartManifold<dim, spacedim, chartdim>::get_new_point(
  const ArrayView<const Point<spacedim>> &surrounding_points,
  const ArrayView<const double> &         weights) const
{
  const std::size_t n_points = surrounding_points.size();
 
  boost::container::small_vector<Point<chartdim>, 200> chart_points(n_points);
 
  for (unsigned int i = 0; i < n_points; ++i)
    chart_points[i] = pull_back(surrounding_points[i]);
 
  const Point<chartdim> p_chart = sub_manifold.get_new_point(
    make_array_view(chart_points.begin(), chart_points.end()), weights);
 
  return push_forward(p_chart);
}
 
 
 
template <int dim, int spacedim, int chartdim>
void
ChartManifold<dim, spacedim, chartdim>::get_new_points(
  const ArrayView<const Point<spacedim>> &surrounding_points,
  const Table<2, double> &                weights,
  ArrayView<Point<spacedim>>              new_points) const
{
  Assert(weights.size(0) > 0, ExcEmptyObject());
  AssertDimension(surrounding_points.size(), weights.size(1));
 
  const std::size_t n_points = surrounding_points.size();
 
  boost::container::small_vector<Point<chartdim>, 200> chart_points(n_points);
  for (std::size_t i = 0; i < n_points; ++i)
    chart_points[i] = pull_back(surrounding_points[i]);
 
  boost::container::small_vector<Point<chartdim>, 200> new_points_on_chart(
    weights.size(0));
  sub_manifold.get_new_points(
    make_array_view(chart_points.begin(), chart_points.end()),
    weights,
    make_array_view(new_points_on_chart.begin(), new_points_on_chart.end()));
 
  for (std::size_t row = 0; row < weights.size(0); ++row)
    new_points[row] = push_forward(new_points_on_chart[row]);
}
 
 
 
template <int dim, int spacedim, int chartdim>
DerivativeForm<1, chartdim, spacedim>
ChartManifold<dim, spacedim, chartdim>::push_forward_gradient(
  const Point<chartdim> &) const
{
  // function must be implemented in a derived class to be usable,
  // as discussed in this function's documentation
  Assert(false, ExcPureFunctionCalled());
  return DerivativeForm<1, chartdim, spacedim>();
}
 
 
 
template <int dim, int spacedim, int chartdim>
Tensor<1, spacedim>
ChartManifold<dim, spacedim, chartdim>::get_tangent_vector(
  const Point<spacedim> &x1,
  const Point<spacedim> &x2) const
{
  const DerivativeForm<1, chartdim, spacedim> F_prime =
    push_forward_gradient(pull_back(x1));
 
  // ensure that the chart is not singular by asserting that its
  // derivative has a positive determinant. we need to make this
  // comparison relative to the size of the derivative. since the
  // determinant is the product of chartdim factors, take the
  // chartdim-th root of it in comparing against the size of the
  // derivative
  Assert(std::pow(std::abs(F_prime.determinant()), 1. / chartdim) >=
           1e-12 * F_prime.norm(),
         ExcMessage(
           "The derivative of a chart function must not be singular."));
 
  const Tensor<1, chartdim> delta =
    sub_manifold.get_tangent_vector(pull_back(x1), pull_back(x2));
 
  Tensor<1, spacedim> result;
  for (unsigned int i = 0; i < spacedim; ++i)
    result[i] += F_prime[i] * delta;
 
  return result;
}
 
 
 
template <int dim, int spacedim, int chartdim>
const Tensor<1, chartdim> &
ChartManifold<dim, spacedim, chartdim>::get_periodicity() const
{
  return sub_manifold.get_periodicity();
}
 
// explicit instantiations
#include "manifold.inst"
 
DEAL_II_NAMESPACE_CLOSE