grid_generator.cc

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// ------------------------------------------------------------------------
//
// SPDX-License-Identifier: LGPL-2.1-or-later
// Copyright (C) 1999 - 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/ndarray.h>
#include <deal.II/base/parameter_handler.h>
 
#include <deal.II/distributed/fully_distributed_tria.h>
#include <deal.II/distributed/shared_tria.h>
#include <deal.II/distributed/tria.h>
 
#include <deal.II/grid/filtered_iterator.h>
#include <deal.II/grid/grid_generator.h>
#include <deal.II/grid/grid_tools.h>
#include <deal.II/grid/intergrid_map.h>
#include <deal.II/grid/manifold_lib.h>
#include <deal.II/grid/tria.h>
#include <deal.II/grid/tria_accessor.h>
#include <deal.II/grid/tria_iterator.h>
 
#include <deal.II/physics/transformations.h>
 
#include <array>
#include <cmath>
#include <limits>
 
 
DEAL_II_NAMESPACE_OPEN
 
// work around the problem that doxygen for some reason lists all template
// specializations in this file
#ifndef DOXYGEN
 
namespace GridGenerator
{
  namespace Airfoil
  {
    AdditionalData::AdditionalData()
      // airfoil configuration
      : airfoil_type("NACA")
      , naca_id("2412")
      , joukowski_center(-0.1, 0.14)
      , airfoil_length(1.0)
      // far field
      , height(30.0)
      , length_b2(15.0)
      // mesh
      , incline_factor(0.35)
      , bias_factor(2.5)
      , refinements(2)
      , n_subdivision_x_0(3)
      , n_subdivision_x_1(2)
      , n_subdivision_x_2(5)
      , n_subdivision_y(3)
      , airfoil_sampling_factor(2)
    {
      Assert(
        airfoil_length <= height,
        ExcMessage(
          "Mesh is to small to enclose airfoil! Choose larger field or smaller"
          " chord length!"));
      Assert(incline_factor < 1.0 && incline_factor >= 0.0,
             ExcMessage("incline_factor has to be in [0,1)!"));
    }
 
 
 
    void
    AdditionalData::add_parameters(ParameterHandler &prm)
    {
      prm.enter_subsection("FarField");
      {
        prm.add_parameter(
          "Height",
          height,
          "Mesh height measured from airfoil nose to horizontal boundaries");
        prm.add_parameter(
          "LengthB2",
          length_b2,
          "Length measured from airfoil leading edge to vertical outlet boundary");
        prm.add_parameter(
          "InclineFactor",
          incline_factor,
          "Define obliqueness of the vertical mesh around the airfoil");
      }
      prm.leave_subsection();
 
      prm.enter_subsection("AirfoilType");
      {
        prm.add_parameter(
          "Type",
          airfoil_type,
          "Type of airfoil geometry, either NACA or Joukowski airfoil",
          Patterns::Selection("NACA|Joukowski"));
      }
      prm.leave_subsection();
 
      prm.enter_subsection("NACA");
      {
        prm.add_parameter("NacaId", naca_id, "Naca serial number");
      }
      prm.leave_subsection();
 
      prm.enter_subsection("Joukowski");
      {
        prm.add_parameter("Center",
                          joukowski_center,
                          "Joukowski circle center coordinates");
        prm.add_parameter("AirfoilLength",
                          airfoil_length,
                          "Joukowski airfoil length leading to trailing edge");
      }
      prm.leave_subsection();
 
      prm.enter_subsection("Mesh");
      {
        prm.add_parameter("Refinements",
                          refinements,
                          "Number of global refinements");
        prm.add_parameter(
          "NumberSubdivisionX0",
          n_subdivision_x_0,
          "Number of subdivisions along the airfoil in blocks with material ID 1 and 4");
        prm.add_parameter(
          "NumberSubdivisionX1",
          n_subdivision_x_1,
          "Number of subdivisions along the airfoil in blocks with material ID 2 and 5");
        prm.add_parameter(
          "NumberSubdivisionX2",
          n_subdivision_x_2,
          "Number of subdivisions in horizontal direction on the right of the trailing edge, i.e., blocks with material ID 3 and 6");
        prm.add_parameter("NumberSubdivisionY",
                          n_subdivision_y,
                          "Number of subdivisions normal to airfoil");
        prm.add_parameter(
          "BiasFactor",
          bias_factor,
          "Factor to obtain a finer mesh at the airfoil surface");
      }
      prm.leave_subsection();
    }
 
 
    namespace
    {
      class MeshGenerator
      {
      public:
        // IDs of the mesh blocks
        static const unsigned int id_block_1 = 1;
        static const unsigned int id_block_2 = 2;
        static const unsigned int id_block_3 = 3;
        static const unsigned int id_block_4 = 4;
        static const unsigned int id_block_5 = 5;
        static const unsigned int id_block_6 = 6;
 
        MeshGenerator(const AdditionalData &data)
          : refinements(data.refinements)
          , n_subdivision_x_0(data.n_subdivision_x_0)
          , n_subdivision_x_1(data.n_subdivision_x_1)
          , n_subdivision_x_2(data.n_subdivision_x_2)
          , n_subdivision_y(data.n_subdivision_y)
          , height(data.height)
          , length_b2(data.length_b2)
          , incline_factor(data.incline_factor)
          , bias_factor(data.bias_factor)
          , edge_length(1.0)
          , n_cells_x_0(Utilities::pow(2, refinements) * n_subdivision_x_0)
          , n_cells_x_1(Utilities::pow(2, refinements) * n_subdivision_x_1)
          , n_cells_x_2(Utilities::pow(2, refinements) * n_subdivision_x_2)
          , n_cells_y(Utilities::pow(2, refinements) * n_subdivision_y)
          , n_points_on_each_side(n_cells_x_0 + n_cells_x_1 + 1)
          // create points on the airfoil
          , airfoil_1D(set_airfoil_length(
              // call either the 'joukowski' or 'naca' static member function
              data.airfoil_type == "Joukowski" ?
                joukowski(data.joukowski_center,
                          n_points_on_each_side,
                          data.airfoil_sampling_factor) :
                (data.airfoil_type == "NACA" ?
                   naca(data.naca_id,
                        n_points_on_each_side,
                        data.airfoil_sampling_factor) :
                   std::array<std::vector<Point<2>>, 2>{
                     {std::vector<Point<2>>{Point<2>(0), Point<2>(1)},
                      std::vector<Point<2>>{
                        Point<2>(0),
                        Point<2>(
                          1)}}} /* dummy vector since we are asserting later*/),
              data.airfoil_length))
          , end_b0_x_u(airfoil_1D[0][n_cells_x_0][0])
          , end_b0_x_l(airfoil_1D[1][n_cells_x_0][0])
          , nose_x(airfoil_1D[0].front()[0])
          , tail_x(airfoil_1D[0].back()[0])
          , tail_y(airfoil_1D[0].back()[1])
          , center_mesh(0.5 * std::abs(end_b0_x_u + end_b0_x_l))
          , length_b1_x(tail_x - center_mesh)
          , gamma(std::atan(height /
                            (edge_length + std::abs(nose_x - center_mesh))))
          // points on coarse grid
          // coarse grid has to be symmetric in respect to x-axis to allow
          // periodic BC and make sure that interpolate() works
          , A(nose_x - edge_length, 0)
          , B(nose_x, 0)
          , C(center_mesh, +std::abs(nose_x - center_mesh) * std::tan(gamma))
          , D(center_mesh, height)
          , E(center_mesh, -std::abs(nose_x - center_mesh) * std::tan(gamma))
          , F(center_mesh, -height)
          , G(tail_x, height)
          , H(tail_x, 0)
          , I(tail_x, -height)
          , J(tail_x + length_b2, 0)
          , K(J[0], G[1])
          , L(J[0], I[1])
        {
          Assert(data.airfoil_type == "Joukowski" ||
                   data.airfoil_type == "NACA",
                 ExcMessage("Unknown airfoil type."));
        }
 
        void
        create_triangulation(
          Triangulation<2>                             &tria_grid,
          std::vector<GridTools::PeriodicFacePair<
            typename Triangulation<2>::cell_iterator>> *periodic_faces) const
        {
          make_coarse_grid(tria_grid);
 
          set_boundary_ids(tria_grid);
 
          if (periodic_faces != nullptr)
            {
              GridTools::collect_periodic_faces(
                tria_grid, 5, 4, 1, *periodic_faces);
              tria_grid.add_periodicity(*periodic_faces);
            }
 
          tria_grid.refine_global(refinements);
          interpolate(tria_grid);
        }
 
        void
        create_triangulation(
          parallel::fullydistributed::Triangulation<2> &parallel_grid,
          std::vector<GridTools::PeriodicFacePair<
            typename Triangulation<2>::cell_iterator>> *periodic_faces) const
        {
          (void)parallel_grid;
          (void)periodic_faces;
 
          AssertThrow(false, ExcMessage("Not implemented, yet!")); // TODO [PM]
        }
 
      private:
        // number of global refinements
        const unsigned int refinements;
 
        // number of subdivisions of coarse grid in blocks 1 and 4
        const unsigned int n_subdivision_x_0;
 
        // number of subdivisions of coarse grid in blocks 2 and 5
        const unsigned int n_subdivision_x_1;
 
        // number of subdivisions of coarse grid in blocks 3 and 6
        const unsigned int n_subdivision_x_2;
 
        // number of subdivisions  of coarse grid in all blocks (normal to
        // airfoil or in y-direction, respectively)
        const unsigned int n_subdivision_y;
 
        // height of mesh, i.e. length JK or JL and radius of semicircle
        // (C-Mesh) that arises after interpolation in blocks 1 and 4
        const double height;
 
        // length block 3 and 6
        const double length_b2;
 
        // factor to move points G and I horizontal to the right, i.e. make
        // faces HG and HI inclined instead of vertical
        const double incline_factor;
 
        // bias factor (if factor goes to zero than equal y = x)
        const double bias_factor;
 
        // x-distance between coarse grid vertices A and B, i.e. used only once;
        const double edge_length;
 
        // number of cells (after refining) in block 1 and 4 along airfoil
        const unsigned int n_cells_x_0;
 
        // number of cells (after refining) in block 2 and 5 along airfoil
        const unsigned int n_cells_x_1;
 
        // number of cells (after refining) in block 3 and 6 in x-direction
        const unsigned int n_cells_x_2;
 
        // number of cells (after refining) in all blocks normal to airfoil or
        // in y-direction, respectively
        const unsigned int n_cells_y;
 
        // number of airfoil points on each side
        const unsigned int n_points_on_each_side;
 
        // vector containing upper/lower airfoil points. First and last point
        // are identical
        const std::array<std::vector<Point<2>>, 2> airfoil_1D;
 
        // x-coordinate of n-th airfoilpoint where n indicates number of cells
        // in block 1. end_b0_x_u = end_b0_x_l for symmetric airfoils
        const double end_b0_x_u;
 
        // x-coordinate of n-th airfoilpoint where n indicates number of cells
        // in block 4. end_b0_x_u = end_b0_x_l for symmetric airfoils
        const double end_b0_x_l;
 
        // x-coordinate of first airfoil point in airfoil_1d[0] and
        // airfoil_1d[1]
        const double nose_x;
 
        // x-coordinate of last airfoil point in airfoil_1d[0] and airfoil_1d[1]
        const double tail_x;
 
        // y-coordinate of last airfoil point in airfoil_1d[0] and airfoil_1d[1]
        const double tail_y;
 
        // x-coordinate of C,D,E,F indicating ending of blocks 1 and 4 or
        // beginning of blocks 2 and 5, respectively
        const double center_mesh;
 
        // length of blocks 2 and 5
        const double length_b1_x;
 
        // angle enclosed between faces DAB and FAB
        const double gamma;
 
 
 
        const Point<2> A, B, C, D, E, F, G, H, I, J, K, L;
 
 
 
        static std::array<std::vector<Point<2>>, 2>
        joukowski(const Point<2>    &centerpoint,
                  const unsigned int number_points,
                  const unsigned int factor)
        {
          std::array<std::vector<Point<2>>, 2> airfoil_1D;
          const unsigned int total_points    = 2 * number_points - 2;
          const unsigned int n_airfoilpoints = factor * total_points;
          // joukowski points on the entire airfoil, i.e. upper and lower side
          const auto jouk_points =
            joukowski_transform(joukowski_circle(centerpoint, n_airfoilpoints));
 
          // vectors to collect airfoil points on either upper or lower side
          std::vector<Point<2>> upper_points;
          std::vector<Point<2>> lower_points;
 
          {
            // find point on nose and point on tail
            unsigned int nose_index        = 0;
            unsigned int tail_index        = 0;
            double       nose_x_coordinate = 0;
            double       tail_x_coordinate = 0;
 
 
            // find index in vector to nose point (min) and tail point (max)
            for (unsigned int i = 0; i < jouk_points.size(); ++i)
              {
                if (jouk_points[i][0] < nose_x_coordinate)
                  {
                    nose_x_coordinate = jouk_points[i][0];
                    nose_index        = i;
                  }
                if (jouk_points[i][0] > tail_x_coordinate)
                  {
                    tail_x_coordinate = jouk_points[i][0];
                    tail_index        = i;
                  }
              }
 
            // copy point on upper side of airfoil
            for (unsigned int i = tail_index; i < jouk_points.size(); ++i)
              upper_points.emplace_back(jouk_points[i]);
            for (unsigned int i = 0; i <= nose_index; ++i)
              upper_points.emplace_back(jouk_points[i]);
            std::reverse(upper_points.begin(), upper_points.end());
 
            // copy point on lower side of airfoil
            lower_points.insert(lower_points.end(),
                                jouk_points.begin() + nose_index,
                                jouk_points.begin() + tail_index + 1);
          }
 
          airfoil_1D[0] = make_points_equidistant(upper_points, number_points);
          airfoil_1D[1] = make_points_equidistant(lower_points, number_points);
 
          // move nose to origin
          auto move_nose_to_origin = [](std::vector<Point<2>> &vector) {
            const double nose_x_pos = vector.front()[0];
            for (auto &i : vector)
              i[0] -= nose_x_pos;
          };
 
          move_nose_to_origin(airfoil_1D[1]);
          move_nose_to_origin(airfoil_1D[0]);
 
          return airfoil_1D;
        }
 
        static std::vector<Point<2>>
        joukowski_circle(const Point<2>    &center,
                         const unsigned int number_points)
        {
          std::vector<Point<2>> circle_points;
 
          // Create Circle with number_points - points
          // unsigned int number_points = 2 * points_per_side - 2;
 
          // Calculate radius so that point (x=1|y=0) is enclosed - requirement
          //  for Joukowski transform
          const double radius      = std::sqrt(center[1] * center[1] +
                                          (1 - center[0]) * (1 - center[0]));
          const double radius_test = std::sqrt(
            center[1] * center[1] + (1 + center[0]) * (1 + center[0]));
          // Make sure point (x=-1|y=0) is enclosed by the circle
          (void)radius_test;
          AssertThrow(
            radius_test < radius,
            ExcMessage(
              "Error creating lower circle: Circle for Joukowski-transform does"
              " not enclose point zeta = -1! Choose different center "
              "coordinate."));
          // Create a full circle with radius 'radius' around Point 'center' of
          // (number_points) equidistant points.
          const double theta = 2 * numbers::PI / number_points;
          // first point is leading edge then counterclockwise
          for (unsigned int i = 0; i < number_points; ++i)
            circle_points.emplace_back(center[0] - radius * cos(i * theta),
                                       center[1] - radius * sin(i * theta));
 
          return circle_points;
        }
 
        static std::vector<Point<2>>
        joukowski_transform(const std::vector<Point<2>> &circle_points)
        {
          std::vector<Point<2>> joukowski_points(circle_points.size());
 
          // transform each point
          for (unsigned int i = 0; i < circle_points.size(); ++i)
            {
              const double               chi = circle_points[i][0];
              const double               eta = circle_points[i][1];
              const std::complex<double> zeta(chi, eta);
              const std::complex<double> z = zeta + 1. / zeta;
 
              joukowski_points[i] = {real(z), imag(z)};
            }
          return joukowski_points;
        }
 
        static std::array<std::vector<Point<2>>, 2>
        naca(const std::string &serialnumber,
             const unsigned int number_points,
             const unsigned int factor)
        {
          // number of non_equidistant airfoilpoints among which will be
          // interpolated
          const unsigned int n_airfoilpoints = factor * number_points;
 
          // create equidistant airfoil points for upper and lower side
          return {{make_points_equidistant(
                     naca_create_points(serialnumber, n_airfoilpoints, true),
                     number_points),
                   make_points_equidistant(
                     naca_create_points(serialnumber, n_airfoilpoints, false),
                     number_points)}};
        }
 
        static std::vector<Point<2>>
        naca_create_points(const std::string &serialnumber,
                           const unsigned int number_points,
                           const bool         is_upper)
        {
          Assert(serialnumber.size() == 4,
                 ExcMessage("This NACA-serial number is not implemented!"));
 
          return naca_create_points_4_digits(serialnumber,
                                             number_points,
                                             is_upper);
        }
 
        static std::vector<Point<2>>
        naca_create_points_4_digits(const std::string &serialnumber,
                                    const unsigned int number_points,
                                    const bool         is_upper)
        {
          // conversion string (char * ) to int
          const unsigned int digit_0 = (serialnumber[0] - '0');
          const unsigned int digit_1 = (serialnumber[1] - '0');
          const unsigned int digit_2 = (serialnumber[2] - '0');
          const unsigned int digit_3 = (serialnumber[3] - '0');
 
          const unsigned int digit_23 = 10 * digit_2 + digit_3;
 
          // maximum thickness in percentage of the cord
          const double t = static_cast<double>(digit_23) / 100.0;
 
          std::vector<Point<2>> naca_points;
 
          if (digit_0 == 0 && digit_1 == 0) // is symmetric
            for (unsigned int i = 0; i < number_points; ++i)
              {
                const double x = i * 1 / (1.0 * number_points - 1);
                const double y_t =
                  5 * t *
                  (0.2969 * std::sqrt(x) - 0.126 * x -
                   0.3516 * Utilities::fixed_power<2>(x) +
                   0.2843 * Utilities::fixed_power<3>(x) -
                   0.1036 * Utilities::fixed_power<4>(
                              x)); // half thickness at a position x
 
                if (is_upper)
                  naca_points.emplace_back(x, +y_t);
                else
                  naca_points.emplace_back(x, -y_t);
              }
          else // is asymmetric
            for (unsigned int i = 0; i < number_points; ++i)
              {
                const double m = 1.0 * digit_0 / 100; // max. chamber
                const double p = 1.0 * digit_1 / 10; // location of max. chamber
                const double x = i * 1 / (1.0 * number_points - 1);
 
                const double y_c =
                  (x <= p) ?
                    m / Utilities::fixed_power<2>(p) *
                      (2 * p * x - Utilities::fixed_power<2>(x)) :
                    m / Utilities::fixed_power<2>(1 - p) *
                      ((1 - 2 * p) + 2 * p * x - Utilities::fixed_power<2>(x));
 
                const double dy_c =
                  (x <= p) ? 2 * m / Utilities::fixed_power<2>(p) * (p - x) :
                             2 * m / Utilities::fixed_power<2>(1 - p) * (p - x);
 
                const double y_t =
                  5 * t *
                  (0.2969 * std::sqrt(x) - 0.126 * x -
                   0.3516 * Utilities::fixed_power<2>(x) +
                   0.2843 * Utilities::fixed_power<3>(x) -
                   0.1036 * Utilities::fixed_power<4>(
                              x)); // half thickness at a position x
 
                const double theta = std::atan(dy_c);
 
                if (is_upper)
                  naca_points.emplace_back(x - y_t * std::sin(theta),
                                           y_c + y_t * std::cos(theta));
                else
                  naca_points.emplace_back(x + y_t * std::sin(theta),
                                           y_c - y_t * std::cos(theta));
              }
 
          return naca_points;
        }
 
 
 
        static std::array<std::vector<Point<2>>, 2>
        set_airfoil_length(const std::array<std::vector<Point<2>>, 2> &input,
                           const double desired_len)
        {
          std::array<std::vector<Point<2>>, 2> output;
          output[0] = set_airfoil_length(input[0], desired_len);
          output[1] = set_airfoil_length(input[1], desired_len);
 
          return output;
        }
 
        static std::vector<Point<2>>
        set_airfoil_length(const std::vector<Point<2>> &input,
                           const double                 desired_len)
        {
          std::vector<Point<2>> output = input;
 
          const double scale =
            desired_len / input.front().distance(input.back());
 
          for (auto &x : output)
            x *= scale;
 
          return output;
        }
 
        static std::vector<Point<2>>
        make_points_equidistant(
          const std::vector<Point<2>> &non_equidistant_points,
          const unsigned int           number_points)
        {
          const unsigned int n_points =
            non_equidistant_points
              .size(); // number provided airfoilpoints to interpolate
 
          // calculate arclength
          std::vector<double> arclength_L(non_equidistant_points.size(), 0);
          for (unsigned int i = 0; i < non_equidistant_points.size() - 1; ++i)
            arclength_L[i + 1] =
              arclength_L[i] +
              non_equidistant_points[i + 1].distance(non_equidistant_points[i]);
 
 
          const auto airfoil_length =
            arclength_L.back(); // arclength upper or lower side
          const auto deltaX = airfoil_length / (number_points - 1);
 
          // Create equidistant points: keep the first (and last) point
          // unchanged
          std::vector<Point<2>> equidist(
            number_points); // number_points is required points on each side for
                            // mesh
          equidist[0]                 = non_equidistant_points[0];
          equidist[number_points - 1] = non_equidistant_points[n_points - 1];
 
 
          // loop over all subsections
          for (unsigned int j = 0, i = 1; j < n_points - 1; ++j)
            {
              // get reference left and right end of this section
              const auto Lj  = arclength_L[j];
              const auto Ljp = arclength_L[j + 1];
 
              while (Lj <= i * deltaX && i * deltaX <= Ljp &&
                     i < number_points - 1)
                {
                  equidist[i] = Point<2>((i * deltaX - Lj) / (Ljp - Lj) *
                                           (non_equidistant_points[j + 1] -
                                            non_equidistant_points[j]) +
                                         non_equidistant_points[j]);
                  ++i;
                }
            }
          return equidist;
        }
 
 
 
        void
        make_coarse_grid(Triangulation<2> &tria) const
        {
          // create vector of serial triangulations for each block and
          // temporary storage for merging them
          std::vector<Triangulation<2>> trias(10);
 
          // helper function to create a subdivided quadrilateral
          auto make = [](Triangulation<2>                &tria,
                         const std::vector<Point<2>>     &corner_vertices,
                         const std::vector<unsigned int> &repetitions,
                         const unsigned int               material_id) {
            // create subdivided rectangle with corner points (-1,-1)
            // and (+1, +1). It serves as reference system
            GridGenerator::subdivided_hyper_rectangle(tria,
                                                      repetitions,
                                                      {-1, -1},
                                                      {+1, +1});
 
            // move all vertices to the correct position
            for (auto it = tria.begin_vertex(); it < tria.end_vertex(); ++it)
              {
                auto        &point = it->vertex();
                const double xi    = point[0];
                const double eta   = point[1];
 
                // bilinear mapping
                point = 0.25 * ((1 - xi) * (1 - eta) * corner_vertices[0] +
                                (1 + xi) * (1 - eta) * corner_vertices[1] +
                                (1 - xi) * (1 + eta) * corner_vertices[2] +
                                (1 + xi) * (1 + eta) * corner_vertices[3]);
              }
 
            // set material id of block
            for (auto cell : tria.active_cell_iterators())
              cell->set_material_id(material_id);
          };
 
          // create a subdivided quadrilateral for each block (see last number
          // of block id)
          make(trias[0],
               {A, B, D, C},
               {n_subdivision_y, n_subdivision_x_0},
               id_block_1);
          make(trias[1],
               {F, E, A, B},
               {n_subdivision_y, n_subdivision_x_0},
               id_block_4);
          make(trias[2],
               {C, H, D, G},
               {n_subdivision_x_1, n_subdivision_y},
               id_block_2);
          make(trias[3],
               {F, I, E, H},
               {n_subdivision_x_1, n_subdivision_y},
               id_block_5);
          make(trias[4],
               {H, J, G, K},
               {n_subdivision_x_2, n_subdivision_y},
               id_block_3);
          make(trias[5],
               {I, L, H, J},
               {n_subdivision_x_2, n_subdivision_y},
               id_block_6);
 
 
          // merge triangulation (warning: do not change the order here since
          // this might change the face ids)
          GridGenerator::merge_triangulations(trias[0], trias[1], trias[6]);
          GridGenerator::merge_triangulations(trias[2], trias[3], trias[7]);
          GridGenerator::merge_triangulations(trias[4], trias[5], trias[8]);
          GridGenerator::merge_triangulations(trias[6], trias[7], trias[9]);
          GridGenerator::merge_triangulations(trias[8], trias[9], tria);
        }
 
        /*
         * Loop over all (cells and) boundary faces of a given triangulation
         * and set the boundary_ids depending on the material_id of the cell and
         * the face number. The resulting boundary_ids are:
         * - 0: inlet
         * - 1: outlet
         * - 2: upper airfoil surface (aka. suction side)
         * - 3, lower airfoil surface (aka. pressure side),
         * - 4: upper far-field side
         * - 5: lower far-field side
         */
        static void
        set_boundary_ids(Triangulation<2> &tria)
        {
          for (auto cell : tria.active_cell_iterators())
            for (const unsigned int f : GeometryInfo<2>::face_indices())
              {
                if (cell->face(f)->at_boundary() == false)
                  continue;
 
                const auto mid = cell->material_id();
 
                if ((mid == id_block_1 && f == 0) ||
                    (mid == id_block_4 && f == 0))
                  cell->face(f)->set_boundary_id(0); // inlet
                else if ((mid == id_block_3 && f == 0) ||
                         (mid == id_block_6 && f == 2))
                  cell->face(f)->set_boundary_id(1); // outlet
                else if ((mid == id_block_1 && f == 1) ||
                         (mid == id_block_2 && f == 1))
                  cell->face(f)->set_boundary_id(2); // upper airfoil side
                else if ((mid == id_block_4 && f == 1) ||
                         (mid == id_block_5 && f == 3))
                  cell->face(f)->set_boundary_id(3); // lower airfoil side
                else if ((mid == id_block_2 && f == 0) ||
                         (mid == id_block_3 && f == 2))
                  cell->face(f)->set_boundary_id(4); // upper far-field side
                else if ((mid == id_block_5 && f == 2) ||
                         (mid == id_block_6 && f == 0))
                  cell->face(f)->set_boundary_id(5); // lower far-field side
                else
                  Assert(false, ExcIndexRange(mid, id_block_1, id_block_6));
              }
        }
 
        /*
         * Interpolate all vertices of the given triangulation onto the airfoil
         * geometry, depending on the material_id of the block.
         * Due to symmetry of coarse grid in respect to
         * x-axis (by definition of points A-L), blocks 1&4, 2&4 and 3&6 can be
         * interpolated with the same geometric computations Consider a
         * bias_factor and incline_factor during interpolation to obtain a more
         * dense mesh next to airfoil geometry and receive an inclined boundary
         * between block 2&3 and 5&6, respectively
         */
        void
        interpolate(Triangulation<2> &tria) const
        {
          // array storing the information if a vertex was processed
          std::vector<bool> vertex_processed(tria.n_vertices(), false);
 
          // rotation matrix for clockwise rotation of block 1 by angle gamma
          const Tensor<2, 2, double> rotation_matrix_1 =
            Physics::Transformations::Rotations::rotation_matrix_2d(-gamma);
          const Tensor<2, 2, double> rotation_matrix_2 =
            transpose(rotation_matrix_1);
 
          // horizontal offset in order to place coarse-grid node A in the
          // origin
          const Point<2, double> horizontal_offset(A[0], 0.0);
 
          // Move block 1 so that face BC coincides the x-axis
          const Point<2, double> trapeze_offset(0.0,
                                                std::sin(gamma) * edge_length);
 
          // loop over vertices of all cells
          for (const auto &cell : tria.cell_iterators())
            for (const unsigned int v : GeometryInfo<2>::vertex_indices())
              {
                // vertex has been already processed: nothing to do
                if (vertex_processed[cell->vertex_index(v)])
                  continue;
 
                // mark vertex as processed
                vertex_processed[cell->vertex_index(v)] = true;
 
                auto &node = cell->vertex(v);
 
                // distinguish blocks
                if (cell->material_id() == id_block_1 ||
                    cell->material_id() == id_block_4) // block 1 and 4
                  {
                    // step 1: rotate block 1 clockwise by gamma and move block
                    // 1 so that A[0] is on y-axis so that faces AD and BC are
                    // horizontal. This simplifies the computation of the
                    // required indices for interpolation (all x-nodes are
                    // positive) Move trapeze to be in first quadrant by adding
                    // trapeze_offset
                    Point<2, double> node_;
                    if (cell->material_id() == id_block_1)
                      {
                        node_ = Point<2, double>(rotation_matrix_1 *
                                                   (node - horizontal_offset) +
                                                 trapeze_offset);
                      }
                    // step 1: rotate block 4 counterclockwise and move down so
                    // that trapeze is located in fourth quadrant (subtracting
                    // trapeze_offset)
                    else if (cell->material_id() == id_block_4)
                      {
                        node_ = Point<2, double>(rotation_matrix_2 *
                                                   (node - horizontal_offset) -
                                                 trapeze_offset);
                      }
                    // step 2: compute indices ix and iy and interpolate
                    // trapezoid to a rectangle of length pi/2.
                    {
                      const double trapeze_height =
                        std::sin(gamma) * edge_length;
                      const double L   = height / std::sin(gamma);
                      const double l_a = std::cos(gamma) * edge_length;
                      const double l_b = trapeze_height * std::tan(gamma);
                      const double x1  = std::abs(node_[1]) / std::tan(gamma);
                      const double x2  = L - l_a - l_b;
                      const double x3  = std::abs(node_[1]) * std::tan(gamma);
                      const double Dx  = x1 + x2 + x3;
                      const double deltax =
                        (trapeze_height - std::abs(node_[1])) / std::tan(gamma);
                      const double dx = Dx / n_cells_x_0;
                      const double dy = trapeze_height / n_cells_y;
                      const int    ix =
                        static_cast<int>(std::round((node_[0] - deltax) / dx));
                      const int iy =
                        static_cast<int>(std::round(std::abs(node_[1]) / dy));
 
                      node_[0] = numbers::PI / 2 * (1.0 * ix) / n_cells_x_0;
                      node_[1] = height * (1.0 * iy) / n_cells_y;
                    }
 
                    // step 3: Interpolation between semicircle (of C-Mesh) and
                    // airfoil contour
                    {
                      const double dx = numbers::PI / 2 / n_cells_x_0;
                      const double dy = height / n_cells_y;
                      const int    ix =
                        static_cast<int>(std::round(node_[0] / dx));
                      const int iy =
                        static_cast<int>(std::round(node_[1] / dy));
                      const double alpha =
                        bias_alpha(1 - (1.0 * iy) / n_cells_y);
                      const double   theta = node_[0];
                      const Point<2> p(-height * std::cos(theta) + center_mesh,
                                       ((cell->material_id() == id_block_1) ?
                                          (height) :
                                          (-height)) *
                                         std::sin(theta));
                      node = airfoil_1D[(
                               (cell->material_id() == id_block_1) ? (0) : (1))]
                                       [ix] *
                               alpha +
                             p * (1 - alpha);
                    }
                  }
                else if (cell->material_id() == id_block_2 ||
                         cell->material_id() == id_block_5) // block 2 and 5
                  {
                    // geometric parameters and indices for interpolation
                    Assert(
                      (std::abs(D[1] - C[1]) == std::abs(F[1] - E[1])) &&
                        (std::abs(C[1]) == std::abs(E[1])) &&
                        (std::abs(G[1]) == std::abs(I[1])),
                      ExcMessage(
                        "Points D,C,G and E,F,I are not defined symmetric to "
                        "x-axis, which is required to interpolate block 2"
                        " and 5 with same geometric computations."));
                    const double l_y = D[1] - C[1];
                    const double l_h = D[1] - l_y;
                    const double by  = -l_h / length_b1_x * (node[0] - H[0]);
                    const double dy  = (height - by) / n_cells_y;
                    const int    iy  = static_cast<int>(
                      std::round((std::abs(node[1]) - by) / dy));
                    const double dx = length_b1_x / n_cells_x_1;
                    const int    ix = static_cast<int>(
                      std::round(std::abs(node[0] - center_mesh) / dx));
 
                    const double alpha = bias_alpha(1 - (1.0 * iy) / n_cells_y);
                    // define points on upper/lower horizontal far field side,
                    // i.e. face DG or FI. Incline factor to move points G and I
                    // to the right by distance incline_factor*length_b2
                    const Point<2> p(ix * dx + center_mesh +
                                       incline_factor * length_b2 * ix /
                                         n_cells_x_1,
                                     ((cell->material_id() == id_block_2) ?
                                        (height) :
                                        (-height)));
                    // interpolate between y = height and upper airfoil points
                    // (block2) or y = -height and lower airfoil points (block5)
                    node = airfoil_1D[(
                             (cell->material_id() == id_block_2) ? (0) : (1))]
                                     [n_cells_x_0 + ix] *
                             alpha +
                           p * (1 - alpha);
                  }
                else if (cell->material_id() == id_block_3 ||
                         cell->material_id() == id_block_6) // block 3 and 6
                  {
                    // compute indices ix and iy
                    const double dx = length_b2 / n_cells_x_2;
                    const double dy = height / n_cells_y;
                    const int    ix = static_cast<int>(
                      std::round(std::abs(node[0] - H[0]) / dx));
                    const int iy =
                      static_cast<int>(std::round(std::abs(node[1]) / dy));
 
                    const double alpha_y = bias_alpha(1 - 1.0 * iy / n_cells_y);
                    const double alpha_x =
                      bias_alpha(1 - (static_cast<double>(ix)) / n_cells_x_2);
                    // define on upper/lower horizontal far field side at y =
                    // +/- height, i.e. face GK or IL incline factor to move
                    // points G and H to the right
                    const Point<2> p1(J[0] - (1 - incline_factor) * length_b2 *
                                               (alpha_x),
                                      ((cell->material_id() == id_block_3) ?
                                         (height) :
                                         (-height)));
                    // define points on HJ but use tail_y as y-coordinate, in
                    // case last airfoil point has y =/= 0
                    const Point<2> p2(J[0] - alpha_x * length_b2, tail_y);
                    node = p1 * (1 - alpha_y) + p2 * alpha_y;
                  }
                else
                  {
                    Assert(false,
                           ExcIndexRange(cell->material_id(),
                                         id_block_1,
                                         id_block_6));
                  }
              }
        }
 
 
        /*
         * This function returns a bias factor 'alpha' which is used to make the
         * mesh more tight in close distance of the airfoil.
         * It is a bijective function mapping from [0,1] onto [0,1] where values
         * near 1 are made tighter.
         */
        double
        bias_alpha(double alpha) const
        {
          return std::tanh(bias_factor * alpha) / std::tanh(bias_factor);
        }
      };
    } // namespace
 
 
 
    void
    internal_create_triangulation(
      Triangulation<2, 2>                             &tria,
      std::vector<GridTools::PeriodicFacePair<
        typename Triangulation<2, 2>::cell_iterator>> *periodic_faces,
      const AdditionalData                            &additional_data)
    {
      MeshGenerator mesh_generator(additional_data);
      // Cast the triangulation to the right type so that the right
      // specialization of the function create_triangulation is picked up.
      if (auto *parallel_tria =
            dynamic_cast<::parallel::distributed::Triangulation<2, 2> *>(
              &tria))
        mesh_generator.create_triangulation(*parallel_tria, periodic_faces);
      else if (auto *parallel_tria = dynamic_cast<
                 ::parallel::fullydistributed::Triangulation<2, 2> *>(
                 &tria))
        mesh_generator.create_triangulation(*parallel_tria, periodic_faces);
      else
        mesh_generator.create_triangulation(tria, periodic_faces);
    }
 
    template <>
    void
    create_triangulation(Triangulation<1, 1> &, const AdditionalData &)
    {
      Assert(false, ExcMessage("Airfoils only exist for 2D and 3d!"));
    }
 
 
 
    template <>
    void
    create_triangulation(Triangulation<1, 1> &,
                         std::vector<GridTools::PeriodicFacePair<
                           typename Triangulation<1, 1>::cell_iterator>> &,
                         const AdditionalData &)
    {
      Assert(false, ExcMessage("Airfoils only exist for 2D and 3d!"));
    }
 
 
 
    template <>
    void
    create_triangulation(Triangulation<2, 2>  &tria,
                         const AdditionalData &additional_data)
    {
      internal_create_triangulation(tria, nullptr, additional_data);
    }
 
 
 
    template <>
    void
    create_triangulation(
      Triangulation<2, 2>                             &tria,
      std::vector<GridTools::PeriodicFacePair<
        typename Triangulation<2, 2>::cell_iterator>> &periodic_faces,
      const AdditionalData                            &additional_data)
    {
      internal_create_triangulation(tria, &periodic_faces, additional_data);
    }
 
 
 
    template <>
    void
    create_triangulation(
      Triangulation<3, 3>                             &tria,
      std::vector<GridTools::PeriodicFacePair<
        typename Triangulation<3, 3>::cell_iterator>> &periodic_faces,
      const AdditionalData                            &additional_data)
    {
      Assert(false, ExcMessage("3d airfoils are not implemented yet!"));
      (void)tria;
      (void)additional_data;
      (void)periodic_faces;
    }
  } // namespace Airfoil
 
 
  namespace
  {
    template <int dim, int spacedim>
    void
    colorize_hyper_rectangle(Triangulation<dim, spacedim> &tria)
    {
      // there is nothing to do in 1d
      if (dim > 1)
        {
          // there is only one cell, so
          // simple task
          const typename Triangulation<dim, spacedim>::cell_iterator cell =
            tria.begin();
          for (auto f : GeometryInfo<dim>::face_indices())
            cell->face(f)->set_boundary_id(f);
        }
    }
 
 
 
    template <int spacedim>
    void
    colorize_subdivided_hyper_rectangle(Triangulation<1, spacedim> &tria,
                                        const Point<spacedim> &,
                                        const Point<spacedim> &,
                                        const double)
    {
      for (typename Triangulation<1, spacedim>::cell_iterator cell =
             tria.begin();
           cell != tria.end();
           ++cell)
        if (cell->center()[0] > 0)
          cell->set_material_id(1);
      // boundary indicators are set to
      // 0 (left) and 1 (right) by default.
    }
 
 
 
    template <int dim, int spacedim>
    void
    colorize_subdivided_hyper_rectangle(Triangulation<dim, spacedim> &tria,
                                        const Point<spacedim>        &p1,
                                        const Point<spacedim>        &p2,
                                        const double                  epsilon)
    {
      // run through all faces and check
      // if one of their center coordinates matches
      // one of the corner points. Comparisons
      // are made using an epsilon which
      // should be smaller than the smallest cell
      // diameter.
 
      typename Triangulation<dim, spacedim>::face_iterator face =
                                                             tria.begin_face(),
                                                           endface =
                                                             tria.end_face();
      for (; face != endface; ++face)
        if (face->at_boundary())
          if (face->boundary_id() == 0)
            {
              const Point<spacedim> center(face->center());
 
              if (std::abs(center[0] - p1[0]) < epsilon)
                face->set_boundary_id(0);
              else if (std::abs(center[0] - p2[0]) < epsilon)
                face->set_boundary_id(1);
              else if (dim > 1 && std::abs(center[1] - p1[1]) < epsilon)
                face->set_boundary_id(2);
              else if (dim > 1 && std::abs(center[1] - p2[1]) < epsilon)
                face->set_boundary_id(3);
              else if (dim > 2 && std::abs(center[2] - p1[2]) < epsilon)
                face->set_boundary_id(4);
              else if (dim > 2 && std::abs(center[2] - p2[2]) < epsilon)
                face->set_boundary_id(5);
              else
                // triangulation says it
                // is on the boundary,
                // but we could not find
                // on which boundary.
                DEAL_II_ASSERT_UNREACHABLE();
            }
 
      for (const auto &cell : tria.cell_iterators())
        {
          types::material_id id = 0;
          for (unsigned int d = 0; d < dim; ++d)
            if (cell->center()[d] > 0)
              id += (1 << d);
          cell->set_material_id(id);
        }
    }
 
 
    template <int spacdim>
    void
    colorize_hyper_shell(Triangulation<2, spacdim> &tria,
                         const Point<spacdim> &,
                         const double,
                         const double)
    {
      // In spite of receiving geometrical
      // data, we do this only based on
      // topology.
 
      // For the mesh based on cube,
      // this is highly irregular
      for (auto cell = tria.begin(); cell != tria.end(); ++cell)
        {
          Assert(cell->face(2)->at_boundary(), ExcInternalError());
          cell->face(2)->set_all_boundary_ids(1);
        }
    }
 
 
    void
    colorize_hyper_shell(Triangulation<3> &tria,
                         const Point<3> &,
                         const double,
                         const double)
    {
      // the following uses a good amount
      // of knowledge about the
      // orientation of cells. this is
      // probably not good style...
      if (tria.n_cells() == 6)
        {
          Triangulation<3>::cell_iterator cell = tria.begin();
 
          Assert(cell->face(4)->at_boundary(), ExcInternalError());
          cell->face(4)->set_all_boundary_ids(1);
 
          ++cell;
          Assert(cell->face(2)->at_boundary(), ExcInternalError());
          cell->face(2)->set_all_boundary_ids(1);
 
          ++cell;
          Assert(cell->face(2)->at_boundary(), ExcInternalError());
          cell->face(2)->set_all_boundary_ids(1);
 
          ++cell;
          Assert(cell->face(0)->at_boundary(), ExcInternalError());
          cell->face(0)->set_all_boundary_ids(1);
 
          ++cell;
          Assert(cell->face(2)->at_boundary(), ExcInternalError());
          cell->face(2)->set_all_boundary_ids(1);
 
          ++cell;
          Assert(cell->face(0)->at_boundary(), ExcInternalError());
          cell->face(0)->set_all_boundary_ids(1);
        }
      else if (tria.n_cells() == 12)
        {
          // again use some internal
          // knowledge
          for (Triangulation<3>::cell_iterator cell = tria.begin();
               cell != tria.end();
               ++cell)
            {
              Assert(cell->face(5)->at_boundary(), ExcInternalError());
              cell->face(5)->set_all_boundary_ids(1);
            }
        }
      else if (tria.n_cells() == 96)
        {
          // the 96-cell hypershell is based on a once refined 12-cell
          // mesh. consequently, since the outer faces all are face_no==5
          // above, so they are here (unless they are in the interior). Use
          // this to assign boundary indicators, but also make sure that we
          // encounter exactly 48 such faces
          unsigned int count = 0;
          for (const auto &cell : tria.cell_iterators())
            if (cell->face(5)->at_boundary())
              {
                cell->face(5)->set_all_boundary_ids(1);
                ++count;
              }
          (void)count;
          Assert(count == 48, ExcInternalError());
        }
      else
        DEAL_II_NOT_IMPLEMENTED();
    }
 
 
 
    void
    colorize_quarter_hyper_shell(Triangulation<3> &tria,
                                 const Point<3>   &center,
                                 const double      inner_radius,
                                 const double      outer_radius)
    {
      if (tria.n_cells() != 3)
        AssertThrow(false, ExcNotImplemented());
 
      double middle = (outer_radius - inner_radius) / 2e0 + inner_radius;
      double eps    = 1e-3 * middle;
      Triangulation<3>::cell_iterator cell = tria.begin();
 
      for (; cell != tria.end(); ++cell)
        for (const unsigned int f : GeometryInfo<3>::face_indices())
          {
            if (!cell->face(f)->at_boundary())
              continue;
 
            double radius = cell->face(f)->center().norm() - center.norm();
            if (std::fabs(cell->face(f)->center()[0]) <
                eps) // x = 0 set boundary 2
              {
                cell->face(f)->set_boundary_id(2);
                for (unsigned int j = 0; j < GeometryInfo<3>::lines_per_face;
                     ++j)
                  if (cell->face(f)->line(j)->at_boundary())
                    if (std::fabs(cell->face(f)->line(j)->vertex(0).norm() -
                                  cell->face(f)->line(j)->vertex(1).norm()) >
                        eps)
                      cell->face(f)->line(j)->set_boundary_id(2);
              }
            else if (std::fabs(cell->face(f)->center()[1]) <
                     eps) // y = 0 set boundary 3
              {
                cell->face(f)->set_boundary_id(3);
                for (unsigned int j = 0; j < GeometryInfo<3>::lines_per_face;
                     ++j)
                  if (cell->face(f)->line(j)->at_boundary())
                    if (std::fabs(cell->face(f)->line(j)->vertex(0).norm() -
                                  cell->face(f)->line(j)->vertex(1).norm()) >
                        eps)
                      cell->face(f)->line(j)->set_boundary_id(3);
              }
            else if (std::fabs(cell->face(f)->center()[2]) <
                     eps) // z = 0 set boundary 4
              {
                cell->face(f)->set_boundary_id(4);
                for (unsigned int j = 0; j < GeometryInfo<3>::lines_per_face;
                     ++j)
                  if (cell->face(f)->line(j)->at_boundary())
                    if (std::fabs(cell->face(f)->line(j)->vertex(0).norm() -
                                  cell->face(f)->line(j)->vertex(1).norm()) >
                        eps)
                      cell->face(f)->line(j)->set_boundary_id(4);
              }
            else if (radius < middle) // inner radius set boundary 0
              {
                cell->face(f)->set_boundary_id(0);
                for (unsigned int j = 0; j < GeometryInfo<3>::lines_per_face;
                     ++j)
                  if (cell->face(f)->line(j)->at_boundary())
                    if (std::fabs(cell->face(f)->line(j)->vertex(0).norm() -
                                  cell->face(f)->line(j)->vertex(1).norm()) <
                        eps)
                      cell->face(f)->line(j)->set_boundary_id(0);
              }
            else if (radius > middle) // outer radius set boundary 1
              {
                cell->face(f)->set_boundary_id(1);
                for (unsigned int j = 0; j < GeometryInfo<3>::lines_per_face;
                     ++j)
                  if (cell->face(f)->line(j)->at_boundary())
                    if (std::fabs(cell->face(f)->line(j)->vertex(0).norm() -
                                  cell->face(f)->line(j)->vertex(1).norm()) <
                        eps)
                      cell->face(f)->line(j)->set_boundary_id(1);
              }
            else
              DEAL_II_ASSERT_UNREACHABLE();
          }
    }
 
  } // namespace
 
 
  template <int dim, int spacedim>
  void
  hyper_rectangle(Triangulation<dim, spacedim> &tria,
                  const Point<dim>             &p_1,
                  const Point<dim>             &p_2,
                  const bool                    colorize)
  {
    // First, extend dimensions from dim to spacedim and
    // normalize such that p1 is lower in all coordinate
    // directions. Additional entries will be 0.
    Point<spacedim> p1, p2;
    for (unsigned int i = 0; i < dim; ++i)
      {
        p1[i] = std::min(p_1[i], p_2[i]);
        p2[i] = std::max(p_1[i], p_2[i]);
      }
 
    std::vector<Point<spacedim>> vertices(GeometryInfo<dim>::vertices_per_cell);
    switch (dim)
      {
        case 1:
          vertices[0] = p1;
          vertices[1] = p2;
          break;
        case 2:
          vertices[0] = vertices[1] = p1;
          vertices[2] = vertices[3] = p2;
 
          vertices[1][0] = p2[0];
          vertices[2][0] = p1[0];
          break;
        case 3:
          vertices[0] = vertices[1] = vertices[2] = vertices[3] = p1;
          vertices[4] = vertices[5] = vertices[6] = vertices[7] = p2;
 
          vertices[1][0] = p2[0];
          vertices[2][1] = p2[1];
          vertices[3][0] = p2[0];
          vertices[3][1] = p2[1];
 
          vertices[4][0] = p1[0];
          vertices[4][1] = p1[1];
          vertices[5][1] = p1[1];
          vertices[6][0] = p1[0];
 
          break;
        default:
          DEAL_II_NOT_IMPLEMENTED();
      }
 
    // Prepare cell data
    std::vector<CellData<dim>> cells(1);
    for (const unsigned int i : GeometryInfo<dim>::vertex_indices())
      cells[0].vertices[i] = i;
    cells[0].material_id = 0;
 
    tria.create_triangulation(vertices, cells, SubCellData());
 
    // Assign boundary indicators
    if (colorize)
      colorize_hyper_rectangle(tria);
  }
 
 
 
  template <int dim, int spacedim>
  void
  hyper_cube(Triangulation<dim, spacedim> &tria,
             const double                  left,
             const double                  right,
             const bool                    colorize)
  {
    Assert(left < right,
           ExcMessage("Invalid left-to-right bounds of hypercube"));
 
    Point<dim> p1, p2;
    for (unsigned int i = 0; i < dim; ++i)
      {
        p1[i] = left;
        p2[i] = right;
      }
    hyper_rectangle(tria, p1, p2, colorize);
  }
 
 
 
  template <int dim>
  void
  simplex(Triangulation<dim> &tria, const std::vector<Point<dim>> &vertices)
  {
    AssertDimension(vertices.size(), dim + 1);
    Assert(dim > 1, ExcNotImplemented());
    Assert(dim < 4, ExcNotImplemented());
 
#  ifdef DEBUG
    Tensor<2, dim> vector_matrix;
    for (unsigned int d = 0; d < dim; ++d)
      for (unsigned int c = 1; c <= dim; ++c)
        vector_matrix[c - 1][d] = vertices[c][d] - vertices[0][d];
    Assert(determinant(vector_matrix) > 0.,
           ExcMessage("Vertices of simplex must form a right handed system"));
#  endif
 
    // Set up the vertices by first copying into points.
    std::vector<Point<dim>> points = vertices;
    Point<dim>              center;
    // Compute the edge midpoints and add up everything to compute the
    // center point.
    for (unsigned int i = 0; i <= dim; ++i)
      {
        points.push_back(0.5 * (points[i] + points[(i + 1) % (dim + 1)]));
        center += points[i];
      }
    if (dim > 2)
      {
        // In 3d, we have some more edges to deal with
        for (unsigned int i = 1; i < dim; ++i)
          points.push_back(0.5 * (points[i - 1] + points[i + 1]));
        // And we need face midpoints
        for (unsigned int i = 0; i <= dim; ++i)
          points.push_back(1. / 3. *
                           (points[i] + points[(i + 1) % (dim + 1)] +
                            points[(i + 2) % (dim + 1)]));
      }
    points.push_back((1. / (dim + 1)) * center);
 
    std::vector<CellData<dim>> cells(dim + 1);
    switch (dim)
      {
        case 2:
          AssertDimension(points.size(), 7);
          cells[0].vertices[0] = 0;
          cells[0].vertices[1] = 3;
          cells[0].vertices[2] = 5;
          cells[0].vertices[3] = 6;
          cells[0].material_id = 0;
 
          cells[1].vertices[0] = 3;
          cells[1].vertices[1] = 1;
          cells[1].vertices[2] = 6;
          cells[1].vertices[3] = 4;
          cells[1].material_id = 0;
 
          cells[2].vertices[0] = 5;
          cells[2].vertices[1] = 6;
          cells[2].vertices[2] = 2;
          cells[2].vertices[3] = 4;
          cells[2].material_id = 0;
          break;
        case 3:
          AssertDimension(points.size(), 15);
          cells[0].vertices[0] = 0;
          cells[0].vertices[1] = 4;
          cells[0].vertices[2] = 8;
          cells[0].vertices[3] = 10;
          cells[0].vertices[4] = 7;
          cells[0].vertices[5] = 13;
          cells[0].vertices[6] = 12;
          cells[0].vertices[7] = 14;
          cells[0].material_id = 0;
 
          cells[1].vertices[0] = 4;
          cells[1].vertices[1] = 1;
          cells[1].vertices[2] = 10;
          cells[1].vertices[3] = 5;
          cells[1].vertices[4] = 13;
          cells[1].vertices[5] = 9;
          cells[1].vertices[6] = 14;
          cells[1].vertices[7] = 11;
          cells[1].material_id = 0;
 
          cells[2].vertices[0] = 8;
          cells[2].vertices[1] = 10;
          cells[2].vertices[2] = 2;
          cells[2].vertices[3] = 5;
          cells[2].vertices[4] = 12;
          cells[2].vertices[5] = 14;
          cells[2].vertices[6] = 6;
          cells[2].vertices[7] = 11;
          cells[2].material_id = 0;
 
          cells[3].vertices[0] = 7;
          cells[3].vertices[1] = 13;
          cells[3].vertices[2] = 12;
          cells[3].vertices[3] = 14;
          cells[3].vertices[4] = 3;
          cells[3].vertices[5] = 9;
          cells[3].vertices[6] = 6;
          cells[3].vertices[7] = 11;
          cells[3].material_id = 0;
          break;
        default:
          DEAL_II_NOT_IMPLEMENTED();
      }
    tria.create_triangulation(points, cells, SubCellData());
  }
 
 
 
  template <int dim, int spacedim>
  void
  reference_cell(Triangulation<dim, spacedim> &tria,
                 const ReferenceCell          &reference_cell)
  {
    AssertDimension(dim, reference_cell.get_dimension());
 
    if (reference_cell == ReferenceCells::get_hypercube<dim>())
      {
        GridGenerator::hyper_cube(tria, 0, 1);
      }
    else
      {
        // Create an array that contains the vertices of the reference cell.
        // We can query these points from ReferenceCell, but then we have
        // to embed them into the spacedim-dimensional space.
        std::vector<Point<spacedim>> vertices(reference_cell.n_vertices());
        for (const unsigned int v : reference_cell.vertex_indices())
          {
            const Point<dim> this_vertex = reference_cell.vertex<dim>(v);
            for (unsigned int d = 0; d < dim; ++d)
              vertices[v][d] = this_vertex[d];
            // Point<spacedim> initializes everything to zero, so any remaining
            // elements are left at zero and we don't have to explicitly pad
            // from 'dim' to 'spacedim' here.
          }
 
        // Then make one cell out of these vertices. They are ordered correctly
        // already, so we just need to enumerate them
        std::vector<CellData<dim>> cells(1);
        cells[0].vertices.resize(reference_cell.n_vertices());
        for (const unsigned int v : reference_cell.vertex_indices())
          cells[0].vertices[v] = v;
 
        // Turn all of this into a triangulation
        tria.create_triangulation(vertices, cells, {});
      }
  }
 
  void
  moebius(Triangulation<3>  &tria,
          const unsigned int n_cells,
          const unsigned int n_rotations,
          const double       R,
          const double       r)
  {
    const unsigned int dim = 3;
    Assert(n_cells > 4,
           ExcMessage(
             "More than 4 cells are needed to create a moebius grid."));
    Assert(r > 0 && R > 0,
           ExcMessage("Outer and inner radius must be positive."));
    Assert(R > r,
           ExcMessage("Outer radius must be greater than inner radius."));
 
 
    std::vector<Point<dim>> vertices(4 * n_cells);
    double beta_step  = n_rotations * numbers::PI / 2.0 / n_cells;
    double alpha_step = 2.0 * numbers::PI / n_cells;
 
    for (unsigned int i = 0; i < n_cells; ++i)
      for (unsigned int j = 0; j < 4; ++j)
        {
          vertices[4 * i + j][0] =
            R * std::cos(i * alpha_step) +
            r * std::cos(i * beta_step + j * numbers::PI / 2.0) *
              std::cos(i * alpha_step);
          vertices[4 * i + j][1] =
            R * std::sin(i * alpha_step) +
            r * std::cos(i * beta_step + j * numbers::PI / 2.0) *
              std::sin(i * alpha_step);
          vertices[4 * i + j][2] =
            r * std::sin(i * beta_step + j * numbers::PI / 2.0);
        }
 
    unsigned int offset = 0;
 
    // This Triangulation is constructed using a numbering scheme in which
    // the front face is first and the back face is second,
    // which is more convenient for creating a Moebius loop
    static constexpr std::array<unsigned int, 8> local_vertex_numbering{
      {0, 1, 5, 4, 2, 3, 7, 6}};
    std::vector<CellData<dim>> cells(n_cells);
    for (unsigned int i = 0; i < n_cells; ++i)
      {
        for (unsigned int j = 0; j < 2; ++j)
          {
            cells[i].vertices[local_vertex_numbering[0 + 4 * j]] =
              offset + 0 + 4 * j;
            cells[i].vertices[local_vertex_numbering[1 + 4 * j]] =
              offset + 3 + 4 * j;
            cells[i].vertices[local_vertex_numbering[2 + 4 * j]] =
              offset + 2 + 4 * j;
            cells[i].vertices[local_vertex_numbering[3 + 4 * j]] =
              offset + 1 + 4 * j;
          }
        offset += 4;
        cells[i].material_id = 0;
      }
 
    // now correct the last four vertices
    cells[n_cells - 1].vertices[local_vertex_numbering[4]] =
      (0 + n_rotations) % 4;
    cells[n_cells - 1].vertices[local_vertex_numbering[5]] =
      (3 + n_rotations) % 4;
    cells[n_cells - 1].vertices[local_vertex_numbering[6]] =
      (2 + n_rotations) % 4;
    cells[n_cells - 1].vertices[local_vertex_numbering[7]] =
      (1 + n_rotations) % 4;
 
    GridTools::invert_all_negative_measure_cells(vertices, cells);
    tria.create_triangulation(vertices, cells, SubCellData());
  }
 
 
 
  template <>
  void
  torus<2, 3>(Triangulation<2, 3> &tria,
              const double         centerline_radius,
              const double         inner_radius,
              const unsigned int,
              const double)
  {
    Assert(centerline_radius > inner_radius,
           ExcMessage("The centerline radius must be greater than the "
                      "inner radius."));
    Assert(inner_radius > 0.0,
           ExcMessage("The inner radius must be positive."));
 
    const unsigned int           dim      = 2;
    const unsigned int           spacedim = 3;
    std::vector<Point<spacedim>> vertices(16);
 
    vertices[0]  = Point<spacedim>(centerline_radius - inner_radius, 0, 0);
    vertices[1]  = Point<spacedim>(centerline_radius, -inner_radius, 0);
    vertices[2]  = Point<spacedim>(centerline_radius + inner_radius, 0, 0);
    vertices[3]  = Point<spacedim>(centerline_radius, inner_radius, 0);
    vertices[4]  = Point<spacedim>(0, 0, centerline_radius - inner_radius);
    vertices[5]  = Point<spacedim>(0, -inner_radius, centerline_radius);
    vertices[6]  = Point<spacedim>(0, 0, centerline_radius + inner_radius);
    vertices[7]  = Point<spacedim>(0, inner_radius, centerline_radius);
    vertices[8]  = Point<spacedim>(-(centerline_radius - inner_radius), 0, 0);
    vertices[9]  = Point<spacedim>(-centerline_radius, -inner_radius, 0);
    vertices[10] = Point<spacedim>(-(centerline_radius + inner_radius), 0, 0);
    vertices[11] = Point<spacedim>(-centerline_radius, inner_radius, 0);
    vertices[12] = Point<spacedim>(0, 0, -(centerline_radius - inner_radius));
    vertices[13] = Point<spacedim>(0, -inner_radius, -centerline_radius);
    vertices[14] = Point<spacedim>(0, 0, -(centerline_radius + inner_radius));
    vertices[15] = Point<spacedim>(0, inner_radius, -centerline_radius);
 
    std::vector<CellData<dim>> cells(16);
    // Right Hand Orientation
    cells[0].vertices[0] = 0;
    cells[0].vertices[1] = 4;
    cells[0].vertices[2] = 3;
    cells[0].vertices[3] = 7;
    cells[0].material_id = 0;
 
    cells[1].vertices[0] = 1;
    cells[1].vertices[1] = 5;
    cells[1].vertices[2] = 0;
    cells[1].vertices[3] = 4;
    cells[1].material_id = 0;
 
    cells[2].vertices[0] = 2;
    cells[2].vertices[1] = 6;
    cells[2].vertices[2] = 1;
    cells[2].vertices[3] = 5;
    cells[2].material_id = 0;
 
    cells[3].vertices[0] = 3;
    cells[3].vertices[1] = 7;
    cells[3].vertices[2] = 2;
    cells[3].vertices[3] = 6;
    cells[3].material_id = 0;
 
    cells[4].vertices[0] = 4;
    cells[4].vertices[1] = 8;
    cells[4].vertices[2] = 7;
    cells[4].vertices[3] = 11;
    cells[4].material_id = 0;
 
    cells[5].vertices[0] = 5;
    cells[5].vertices[1] = 9;
    cells[5].vertices[2] = 4;
    cells[5].vertices[3] = 8;
    cells[5].material_id = 0;
 
    cells[6].vertices[0] = 6;
    cells[6].vertices[1] = 10;
    cells[6].vertices[2] = 5;
    cells[6].vertices[3] = 9;
    cells[6].material_id = 0;
 
    cells[7].vertices[0] = 7;
    cells[7].vertices[1] = 11;
    cells[7].vertices[2] = 6;
    cells[7].vertices[3] = 10;
    cells[7].material_id = 0;
 
    cells[8].vertices[0] = 8;
    cells[8].vertices[1] = 12;
    cells[8].vertices[2] = 11;
    cells[8].vertices[3] = 15;
    cells[8].material_id = 0;
 
    cells[9].vertices[0] = 9;
    cells[9].vertices[1] = 13;
    cells[9].vertices[2] = 8;
    cells[9].vertices[3] = 12;
    cells[9].material_id = 0;
 
    cells[10].vertices[0] = 10;
    cells[10].vertices[1] = 14;
    cells[10].vertices[2] = 9;
    cells[10].vertices[3] = 13;
    cells[10].material_id = 0;
 
    cells[11].vertices[0] = 11;
    cells[11].vertices[1] = 15;
    cells[11].vertices[2] = 10;
    cells[11].vertices[3] = 14;
    cells[11].material_id = 0;
 
    cells[12].vertices[0] = 12;
    cells[12].vertices[1] = 0;
    cells[12].vertices[2] = 15;
    cells[12].vertices[3] = 3;
    cells[12].material_id = 0;
 
    cells[13].vertices[0] = 13;
    cells[13].vertices[1] = 1;
    cells[13].vertices[2] = 12;
    cells[13].vertices[3] = 0;
    cells[13].material_id = 0;
 
    cells[14].vertices[0] = 14;
    cells[14].vertices[1] = 2;
    cells[14].vertices[2] = 13;
    cells[14].vertices[3] = 1;
    cells[14].material_id = 0;
 
    cells[15].vertices[0] = 15;
    cells[15].vertices[1] = 3;
    cells[15].vertices[2] = 14;
    cells[15].vertices[3] = 2;
    cells[15].material_id = 0;
 
    GridTools::consistently_order_cells(cells);
    tria.create_triangulation(vertices, cells, SubCellData());
 
    tria.set_all_manifold_ids(0);
    tria.set_manifold(0, TorusManifold<2>(centerline_radius, inner_radius));
  }
 
 
 
  namespace
  {
    static constexpr int circle_cell_vertices[5][4] = {{0, 1, 2, 3},
                                                       {0, 2, 6, 4},
                                                       {2, 3, 4, 5},
                                                       {1, 7, 3, 5},
                                                       {6, 4, 7, 5}};
  }
 
 
 
  template <>
  void
  torus<3, 3>(Triangulation<3, 3> &tria,
              const double         centerline_radius,
              const double         inner_radius,
              const unsigned int   n_cells_toroidal,
              const double         phi)
  {
    Assert(centerline_radius > inner_radius,
           ExcMessage("The centerline radius must be greater than the "
                      "inner radius."));
    Assert(inner_radius > 0.0,
           ExcMessage("The inner radius must be positive."));
    Assert(n_cells_toroidal > static_cast<unsigned int>(phi / numbers::PI),
           ExcMessage("Number of cells in toroidal direction has "
                      "to be at least 3 for a torus of polar extent 2*pi."));
    AssertThrow(phi > 0.0 && phi < 2.0 * numbers::PI + 1.0e-15,
                ExcMessage("Invalid angle phi specified."));
 
    // the first 8 vertices are in the x-y-plane
    const Point<3> p = Point<3>(centerline_radius, 0.0, 0.0);
    const double   a = 1. / (1 + std::sqrt(2.0));
    // A value of 1 indicates "open" torus with angle < 2*pi, which
    // means that we need an additional layer of vertices
    const unsigned int additional_layer =
      (phi < 2.0 * numbers::PI - 1.0e-15) ?
        1 :
        0; // torus is closed (angle of 2*pi)
    const unsigned int n_point_layers_toroidal =
      n_cells_toroidal + additional_layer;
    std::vector<Point<3>> vertices(8 * n_point_layers_toroidal);
    vertices[0] = p + Point<3>(-1, -1, 0) * (inner_radius / std::sqrt(2.0)),
    vertices[1] = p + Point<3>(+1, -1, 0) * (inner_radius / std::sqrt(2.0)),
    vertices[2] = p + Point<3>(-1, -1, 0) * (inner_radius / std::sqrt(2.0) * a),
    vertices[3] = p + Point<3>(+1, -1, 0) * (inner_radius / std::sqrt(2.0) * a),
    vertices[4] = p + Point<3>(-1, +1, 0) * (inner_radius / std::sqrt(2.0) * a),
    vertices[5] = p + Point<3>(+1, +1, 0) * (inner_radius / std::sqrt(2.0) * a),
    vertices[6] = p + Point<3>(-1, +1, 0) * (inner_radius / std::sqrt(2.0)),
    vertices[7] = p + Point<3>(+1, +1, 0) * (inner_radius / std::sqrt(2.0));
 
    // create remaining vertices by rotating around negative y-axis (the
    // direction is to ensure positive cell measures)
    const double phi_cell = phi / n_cells_toroidal;
    for (unsigned int c = 1; c < n_point_layers_toroidal; ++c)
      {
        for (unsigned int v = 0; v < 8; ++v)
          {
            const double inner_radius_2d = vertices[v][0];
            vertices[8 * c + v][0] = inner_radius_2d * std::cos(phi_cell * c);
            vertices[8 * c + v][1] = vertices[v][1];
            vertices[8 * c + v][2] = inner_radius_2d * std::sin(phi_cell * c);
          }
      }
 
    // cell connectivity
    std::vector<CellData<3>> cells(5 * n_cells_toroidal);
    for (unsigned int c = 0; c < n_cells_toroidal; ++c)
      {
        for (unsigned int j = 0; j < 2; ++j)
          {
            const unsigned int offset =
              (8 * (c + j)) % (8 * n_point_layers_toroidal);
 
            // cells in x-y-plane
            for (unsigned int c2 = 0; c2 < 5; ++c2)
              for (unsigned int i = 0; i < 4; ++i)
                cells[5 * c + c2].vertices[i + j * 4] =
                  offset + circle_cell_vertices[c2][i];
          }
 
        cells[5 * c].material_id = 0;
        // mark cell on torus centerline
        cells[5 * c + 1].material_id = 0;
        cells[5 * c + 2].material_id = 1;
        cells[5 * c + 3].material_id = 0;
        cells[5 * c + 4].material_id = 0;
      }
 
    tria.create_triangulation(vertices, cells, SubCellData());
 
    tria.reset_all_manifolds();
    tria.set_all_manifold_ids(0);
 
    for (const auto &cell : tria.cell_iterators())
      {
        // identify faces on torus surface and set manifold to 1
        for (const unsigned int f : GeometryInfo<3>::face_indices())
          {
            // faces 4 and 5 are those with normal vector aligned with torus
            // centerline
            if (cell->face(f)->at_boundary() && f != 4 && f != 5)
              {
                cell->face(f)->set_all_manifold_ids(1);
              }
          }
 
        // set manifold id to 2 for those cells that are on the torus centerline
        if (cell->material_id() == 1)
          {
            cell->set_all_manifold_ids(2);
            // reset to 0
            cell->set_material_id(0);
          }
      }
 
    tria.set_manifold(1, TorusManifold<3>(centerline_radius, inner_radius));
    tria.set_manifold(2,
                      CylindricalManifold<3>(Tensor<1, 3>({0., 1., 0.}),
                                             Point<3>()));
 
    tria.set_manifold(0, FlatManifold<3>());
    TransfiniteInterpolationManifold<3> transfinite;
    transfinite.initialize(tria);
    tria.set_manifold(0, transfinite);
  }
 
 
 
  template <int dim, int spacedim>
  void
  general_cell(Triangulation<dim, spacedim>       &tria,
               const std::vector<Point<spacedim>> &vertices,
               const bool                          colorize)
  {
    Assert(vertices.size() == ::GeometryInfo<dim>::vertices_per_cell,
           ExcMessage("Wrong number of vertices."));
 
    // First create a hyper_rectangle and then deform it.
    hyper_cube(tria, 0, 1, colorize);
 
    typename Triangulation<dim, spacedim>::active_cell_iterator cell =
      tria.begin_active();
    for (const unsigned int i : GeometryInfo<dim>::vertex_indices())
      cell->vertex(i) = vertices[i];
 
    // Check that the order of the vertices makes sense, i.e., the volume of the
    // cell is positive.
    Assert(GridTools::volume(tria, get_default_linear_mapping(tria)) > 0.,
           ExcMessage(
             "The volume of the cell is not greater than zero. "
             "This could be due to the wrong ordering of the vertices."));
  }
 
 
 
  template <>
  void
  parallelogram(Triangulation<3> &,
                const Point<3> (& /*corners*/)[3],
                const bool /*colorize*/)
  {
    DEAL_II_NOT_IMPLEMENTED();
  }
 
  template <>
  void
  parallelogram(Triangulation<1> &,
                const Point<1> (& /*corners*/)[1],
                const bool /*colorize*/)
  {
    DEAL_II_NOT_IMPLEMENTED();
  }
 
  // Implementation for 2d only
  template <>
  void
  parallelogram(Triangulation<2> &tria,
                const Point<2> (&corners)[2],
                const bool colorize)
  {
    Point<2>                    origin;
    std::array<Tensor<1, 2>, 2> edges;
    edges[0] = corners[0];
    edges[1] = corners[1];
    std::vector<unsigned int> subdivisions;
    subdivided_parallelepiped<2, 2>(
      tria, origin, edges, subdivisions, colorize);
  }
 
 
 
  template <int dim>
  void
  parallelepiped(Triangulation<dim> &tria,
                 const Point<dim> (&corners)[dim],
                 const bool colorize)
  {
    unsigned int n_subdivisions[dim];
    for (unsigned int i = 0; i < dim; ++i)
      n_subdivisions[i] = 1;
 
    // and call the function below
    subdivided_parallelepiped(tria, n_subdivisions, corners, colorize);
  }
 
  template <int dim>
  void
  subdivided_parallelepiped(Triangulation<dim> &tria,
                            const unsigned int  n_subdivisions,
                            const Point<dim> (&corners)[dim],
                            const bool colorize)
  {
    // Equalize number of subdivisions in each dim-direction, their
    // validity will be checked later
    unsigned int n_subdivisions_[dim];
    for (unsigned int i = 0; i < dim; ++i)
      n_subdivisions_[i] = n_subdivisions;
 
    // and call the function below
    subdivided_parallelepiped(tria, n_subdivisions_, corners, colorize);
  }
 
  template <int dim>
  void
  subdivided_parallelepiped(Triangulation<dim> &tria,
#  ifndef _MSC_VER
                            const unsigned int (&n_subdivisions)[dim],
#  else
                            const unsigned int *n_subdivisions,
#  endif
                            const Point<dim> (&corners)[dim],
                            const bool colorize)
  {
    Point<dim>                      origin;
    std::vector<unsigned int>       subdivisions;
    std::array<Tensor<1, dim>, dim> edges;
    for (unsigned int i = 0; i < dim; ++i)
      {
        subdivisions.push_back(n_subdivisions[i]);
        edges[i] = corners[i];
      }
 
    subdivided_parallelepiped<dim, dim>(
      tria, origin, edges, subdivisions, colorize);
  }
 
  // Parallelepiped implementation in 1d, 2d, and 3d. @note The
  // implementation in 1d is similar to hyper_rectangle(), and in 2d is
  // similar to parallelogram().
  template <int dim, int spacedim>
  void
  subdivided_parallelepiped(Triangulation<dim, spacedim>               &tria,
                            const Point<spacedim>                      &origin,
                            const std::array<Tensor<1, spacedim>, dim> &edges,
                            const std::vector<unsigned int> &subdivisions,
                            const bool                       colorize)
  {
    std::vector<unsigned int> compute_subdivisions = subdivisions;
    if (compute_subdivisions.empty())
      {
        compute_subdivisions.resize(dim, 1);
      }
 
    Assert(compute_subdivisions.size() == dim,
           ExcMessage("One subdivision must be provided for each dimension."));
    // check subdivisions
    for (unsigned int i = 0; i < dim; ++i)
      {
        Assert(compute_subdivisions[i] > 0,
               ExcInvalidRepetitions(subdivisions[i]));
        Assert(
          edges[i].norm() > 0,
          ExcMessage(
            "Edges in subdivided_parallelepiped() must not be degenerate."));
      }
 
    /*
     * Verify that the edge points to the right in 1d, vectors are oriented in
     * a counter clockwise direction in 2d, or form a right handed system in
     * 3d.
     */
    bool twisted_data = false;
    switch (dim)
      {
        case 1:
          {
            twisted_data = (edges[0][0] < 0);
            break;
          }
        case 2:
          {
            if (spacedim == 2) // this check does not make sense otherwise
              {
                const double plane_normal =
                  edges[0][0] * edges[1][1] - edges[0][1] * edges[1][0];
                twisted_data = (plane_normal < 0.0);
              }
            break;
          }
        case 3:
          {
            // Check that the first two vectors are not linear combinations to
            // avoid zero division later on.
            Assert(std::abs(edges[0] * edges[1] /
                              (edges[0].norm() * edges[1].norm()) -
                            1.0) > 1.0e-15,
                   ExcMessage(
                     "Edges in subdivided_parallelepiped() must point in"
                     " different directions."));
            const Tensor<1, spacedim> plane_normal =
              cross_product_3d(edges[0], edges[1]);
 
            /*
             * Ensure that edges 1, 2, and 3 form a right-handed set of
             * vectors. This works by applying the definition of the dot product
             *
             *     cos(theta) = dot(x, y)/(norm(x)*norm(y))
             *
             * and then, since the normal vector and third edge should both
             * point away from the plane formed by the first two edges, the
             * angle between them must be between 0 and pi/2; hence we just need
             *
             *     0 < dot(x, y).
             */
            twisted_data = (plane_normal * edges[2] < 0.0);
            break;
          }
        default:
          DEAL_II_ASSERT_UNREACHABLE();
      }
    (void)twisted_data; // make the static analyzer happy
    Assert(
      !twisted_data,
      ExcInvalidInputOrientation(
        "The triangulation you are trying to create will consist of cells"
        " with negative measures. This is usually the result of input data"
        " that does not define a right-handed coordinate system. The usual"
        " fix for this is to ensure that in 1d the given point is to the"
        " right of the origin (or the given edge tensor is positive), in 2d"
        " that the two edges (and their cross product) obey the right-hand"
        " rule (which may usually be done by switching the order of the"
        " points or edge tensors), or in 3d that the edges form a"
        " right-handed coordinate system (which may also be accomplished by"
        " switching the order of the first two points or edge tensors)."));
 
    // Check corners do not overlap (unique)
    for (unsigned int i = 0; i < dim; ++i)
      for (unsigned int j = i + 1; j < dim; ++j)
        Assert((edges[i] != edges[j]),
               ExcMessage(
                 "Degenerate edges of subdivided_parallelepiped encountered."));
 
    // Create a list of points
    std::vector<Point<spacedim>> points;
 
    switch (dim)
      {
        case 1:
          for (unsigned int x = 0; x <= compute_subdivisions[0]; ++x)
            points.push_back(origin + edges[0] / compute_subdivisions[0] * x);
          break;
 
        case 2:
          for (unsigned int y = 0; y <= compute_subdivisions[1]; ++y)
            for (unsigned int x = 0; x <= compute_subdivisions[0]; ++x)
              points.push_back(origin + edges[0] / compute_subdivisions[0] * x +
                               edges[1] / compute_subdivisions[1] * y);
          break;
 
        case 3:
          for (unsigned int z = 0; z <= compute_subdivisions[2]; ++z)
            for (unsigned int y = 0; y <= compute_subdivisions[1]; ++y)
              for (unsigned int x = 0; x <= compute_subdivisions[0]; ++x)
                points.push_back(origin +
                                 edges[0] / compute_subdivisions[0] * x +
                                 edges[1] / compute_subdivisions[1] * y +
                                 edges[2] / compute_subdivisions[2] * z);
          break;
 
        default:
          DEAL_II_NOT_IMPLEMENTED();
      }
 
    // Prepare cell data
    unsigned int n_cells = 1;
    for (unsigned int i = 0; i < dim; ++i)
      n_cells *= compute_subdivisions[i];
    std::vector<CellData<dim>> cells(n_cells);
 
    // Create fixed ordering of
    switch (dim)
      {
        case 1:
          for (unsigned int x = 0; x < compute_subdivisions[0]; ++x)
            {
              cells[x].vertices[0] = x;
              cells[x].vertices[1] = x + 1;
 
              // wipe material id
              cells[x].material_id = 0;
            }
          break;
 
        case 2:
          {
            // Shorthand
            const unsigned int n_dy = compute_subdivisions[1];
            const unsigned int n_dx = compute_subdivisions[0];
 
            for (unsigned int y = 0; y < n_dy; ++y)
              for (unsigned int x = 0; x < n_dx; ++x)
                {
                  const unsigned int c = y * n_dx + x;
                  cells[c].vertices[0] = y * (n_dx + 1) + x;
                  cells[c].vertices[1] = y * (n_dx + 1) + x + 1;
                  cells[c].vertices[2] = (y + 1) * (n_dx + 1) + x;
                  cells[c].vertices[3] = (y + 1) * (n_dx + 1) + x + 1;
 
                  // wipe material id
                  cells[c].material_id = 0;
                }
          }
          break;
 
        case 3:
          {
            // Shorthand
            const unsigned int n_dz = compute_subdivisions[2];
            const unsigned int n_dy = compute_subdivisions[1];
            const unsigned int n_dx = compute_subdivisions[0];
 
            for (unsigned int z = 0; z < n_dz; ++z)
              for (unsigned int y = 0; y < n_dy; ++y)
                for (unsigned int x = 0; x < n_dx; ++x)
                  {
                    const unsigned int c = z * n_dy * n_dx + y * n_dx + x;
 
                    cells[c].vertices[0] =
                      z * (n_dy + 1) * (n_dx + 1) + y * (n_dx + 1) + x;
                    cells[c].vertices[1] =
                      z * (n_dy + 1) * (n_dx + 1) + y * (n_dx + 1) + x + 1;
                    cells[c].vertices[2] =
                      z * (n_dy + 1) * (n_dx + 1) + (y + 1) * (n_dx + 1) + x;
                    cells[c].vertices[3] = z * (n_dy + 1) * (n_dx + 1) +
                                           (y + 1) * (n_dx + 1) + x + 1;
                    cells[c].vertices[4] =
                      (z + 1) * (n_dy + 1) * (n_dx + 1) + y * (n_dx + 1) + x;
                    cells[c].vertices[5] = (z + 1) * (n_dy + 1) * (n_dx + 1) +
                                           y * (n_dx + 1) + x + 1;
                    cells[c].vertices[6] = (z + 1) * (n_dy + 1) * (n_dx + 1) +
                                           (y + 1) * (n_dx + 1) + x;
                    cells[c].vertices[7] = (z + 1) * (n_dy + 1) * (n_dx + 1) +
                                           (y + 1) * (n_dx + 1) + x + 1;
 
                    // wipe material id
                    cells[c].material_id = 0;
                  }
            break;
          }
 
        default:
          DEAL_II_NOT_IMPLEMENTED();
      }
 
    // Create triangulation
    // reorder the cells to ensure that they satisfy the convention for
    // edge and face directions
    GridTools::consistently_order_cells(cells);
    tria.create_triangulation(points, cells, SubCellData());
 
    // Finally assign boundary indicators according to hyper_rectangle
    if (colorize)
      {
        typename Triangulation<dim>::active_cell_iterator cell =
                                                            tria.begin_active(),
                                                          endc = tria.end();
        for (; cell != endc; ++cell)
          {
            for (const unsigned int face : GeometryInfo<dim>::face_indices())
              {
                if (cell->face(face)->at_boundary())
                  cell->face(face)->set_boundary_id(face);
              }
          }
      }
  }
 
 
  template <int dim, int spacedim>
  void
  subdivided_hyper_cube(Triangulation<dim, spacedim> &tria,
                        const unsigned int            repetitions,
                        const double                  left,
                        const double                  right,
                        const bool                    colorize)
  {
    Assert(repetitions >= 1, ExcInvalidRepetitions(repetitions));
    Assert(left < right,
           ExcMessage("Invalid left-to-right bounds of hypercube"));
 
    Point<dim> p0, p1;
    for (unsigned int i = 0; i < dim; ++i)
      {
        p0[i] = left;
        p1[i] = right;
      }
 
    std::vector<unsigned int> reps(dim, repetitions);
    subdivided_hyper_rectangle(tria, reps, p0, p1, colorize);
  }
 
 
 
  template <int dim, int spacedim>
  void
  subdivided_hyper_rectangle(Triangulation<dim, spacedim>    &tria,
                             const std::vector<unsigned int> &repetitions,
                             const Point<dim>                &p_1,
                             const Point<dim>                &p_2,
                             const bool                       colorize)
  {
    Assert(repetitions.size() == dim, ExcInvalidRepetitionsDimension(dim));
 
    // First, extend dimensions from dim to spacedim and
    // normalize such that p1 is lower in all coordinate
    // directions. Additional entries will be 0.
    Point<spacedim> p1, p2;
    for (unsigned int i = 0; i < dim; ++i)
      {
        p1[i] = std::min(p_1[i], p_2[i]);
        p2[i] = std::max(p_1[i], p_2[i]);
      }
 
    // calculate deltas and validate input
    std::array<Point<spacedim>, dim> delta;
    for (unsigned int i = 0; i < dim; ++i)
      {
        Assert(repetitions[i] >= 1, ExcInvalidRepetitions(repetitions[i]));
 
        delta[i][i] = (p2[i] - p1[i]) / repetitions[i];
        Assert(
          delta[i][i] > 0.0,
          ExcMessage(
            "The first dim entries of coordinates of p1 and p2 need to be different."));
      }
 
    // then generate the points
    std::vector<Point<spacedim>> points;
    switch (dim)
      {
        case 1:
          for (unsigned int x = 0; x <= repetitions[0]; ++x)
            points.push_back(p1 + x * delta[0]);
          break;
 
        case 2:
          for (unsigned int y = 0; y <= repetitions[1]; ++y)
            for (unsigned int x = 0; x <= repetitions[0]; ++x)
              points.push_back(p1 + x * delta[0] + y * delta[1]);
          break;
 
        case 3:
          for (unsigned int z = 0; z <= repetitions[2]; ++z)
            for (unsigned int y = 0; y <= repetitions[1]; ++y)
              for (unsigned int x = 0; x <= repetitions[0]; ++x)
                points.push_back(p1 + x * delta[0] + y * delta[1] +
                                 z * delta[2]);
          break;
 
        default:
          DEAL_II_NOT_IMPLEMENTED();
      }
 
    // next create the cells
    std::vector<CellData<dim>> cells;
    switch (dim)
      {
        case 1:
          {
            cells.resize(repetitions[0]);
            for (unsigned int x = 0; x < repetitions[0]; ++x)
              {
                cells[x].vertices[0] = x;
                cells[x].vertices[1] = x + 1;
                cells[x].material_id = 0;
              }
            break;
          }
 
        case 2:
          {
            cells.resize(repetitions[1] * repetitions[0]);
            for (unsigned int y = 0; y < repetitions[1]; ++y)
              for (unsigned int x = 0; x < repetitions[0]; ++x)
                {
                  const unsigned int c = x + y * repetitions[0];
                  cells[c].vertices[0] = y * (repetitions[0] + 1) + x;
                  cells[c].vertices[1] = y * (repetitions[0] + 1) + x + 1;
                  cells[c].vertices[2] = (y + 1) * (repetitions[0] + 1) + x;
                  cells[c].vertices[3] = (y + 1) * (repetitions[0] + 1) + x + 1;
                  cells[c].material_id = 0;
                }
            break;
          }
 
        case 3:
          {
            const unsigned int n_x = (repetitions[0] + 1);
            const unsigned int n_xy =
              (repetitions[0] + 1) * (repetitions[1] + 1);
 
            cells.resize(repetitions[2] * repetitions[1] * repetitions[0]);
            for (unsigned int z = 0; z < repetitions[2]; ++z)
              for (unsigned int y = 0; y < repetitions[1]; ++y)
                for (unsigned int x = 0; x < repetitions[0]; ++x)
                  {
                    const unsigned int c = x + y * repetitions[0] +
                                           z * repetitions[0] * repetitions[1];
                    cells[c].vertices[0] = z * n_xy + y * n_x + x;
                    cells[c].vertices[1] = z * n_xy + y * n_x + x + 1;
                    cells[c].vertices[2] = z * n_xy + (y + 1) * n_x + x;
                    cells[c].vertices[3] = z * n_xy + (y + 1) * n_x + x + 1;
                    cells[c].vertices[4] = (z + 1) * n_xy + y * n_x + x;
                    cells[c].vertices[5] = (z + 1) * n_xy + y * n_x + x + 1;
                    cells[c].vertices[6] = (z + 1) * n_xy + (y + 1) * n_x + x;
                    cells[c].vertices[7] =
                      (z + 1) * n_xy + (y + 1) * n_x + x + 1;
                    cells[c].material_id = 0;
                  }
            break;
          }
 
        default:
          DEAL_II_NOT_IMPLEMENTED();
      }
 
    tria.create_triangulation(points, cells, SubCellData());
 
    if (colorize)
      {
        // to colorize, run through all
        // faces of all cells and set
        // boundary indicator to the
        // correct value if it was 0.
 
        // use a large epsilon to
        // compare numbers to avoid
        // roundoff problems.
        double epsilon = std::numeric_limits<double>::max();
        for (unsigned int i = 0; i < dim; ++i)
          epsilon = std::min(epsilon, 0.01 * delta[i][i]);
        Assert(epsilon > 0,
               ExcMessage(
                 "The distance between corner points must be positive."));
 
        // actual code is external since
        // 1-D is different from 2/3d.
        colorize_subdivided_hyper_rectangle(tria, p1, p2, epsilon);
      }
  }
 
 
 
  template <int dim>
  void
  subdivided_hyper_rectangle(Triangulation<dim>                     &tria,
                             const std::vector<std::vector<double>> &step_sz,
                             const Point<dim>                       &p_1,
                             const Point<dim>                       &p_2,
                             const bool                              colorize)
  {
    Assert(step_sz.size() == dim, ExcInvalidRepetitionsDimension(dim));
 
    // First, normalize input such that
    // p1 is lower in all coordinate
    // directions and check the consistency of
    // step sizes, i.e. that they all
    // add up to the sizes specified by
    // p_1 and p_2
    Point<dim>                       p1(p_1);
    Point<dim>                       p2(p_2);
    std::vector<std::vector<double>> step_sizes(step_sz);
 
    for (unsigned int i = 0; i < dim; ++i)
      {
        if (p1[i] > p2[i])
          {
            std::swap(p1[i], p2[i]);
            std::reverse(step_sizes[i].begin(), step_sizes[i].end());
          }
 
#  ifdef DEBUG
        double x = 0;
        for (unsigned int j = 0; j < step_sizes.at(i).size(); ++j)
          x += step_sizes[i][j];
        Assert(std::fabs(x - (p2[i] - p1[i])) <= 1e-12 * std::fabs(x),
               ExcMessage(
                 "The sequence of step sizes in coordinate direction " +
                 Utilities::int_to_string(i) +
                 " must be equal to the distance of the two given "
                 "points in this coordinate direction."));
#  endif
      }
 
 
    // then generate the necessary
    // points
    std::vector<Point<dim>> points;
    switch (dim)
      {
        case 1:
          {
            double x = 0;
            for (unsigned int i = 0;; ++i)
              {
                points.push_back(Point<dim>(p1[0] + x));
 
                // form partial sums. in
                // the last run through
                // avoid accessing
                // non-existent values
                // and exit early instead
                if (i == step_sizes[0].size())
                  break;
 
                x += step_sizes[0][i];
              }
            break;
          }
 
        case 2:
          {
            double y = 0;
            for (unsigned int j = 0;; ++j)
              {
                double x = 0;
                for (unsigned int i = 0;; ++i)
                  {
                    points.push_back(Point<dim>(p1[0] + x, p1[1] + y));
                    if (i == step_sizes[0].size())
                      break;
 
                    x += step_sizes[0][i];
                  }
 
                if (j == step_sizes[1].size())
                  break;
 
                y += step_sizes[1][j];
              }
            break;
          }
        case 3:
          {
            double z = 0;
            for (unsigned int k = 0;; ++k)
              {
                double y = 0;
                for (unsigned int j = 0;; ++j)
                  {
                    double x = 0;
                    for (unsigned int i = 0;; ++i)
                      {
                        points.push_back(
                          Point<dim>(p1[0] + x, p1[1] + y, p1[2] + z));
                        if (i == step_sizes[0].size())
                          break;
 
                        x += step_sizes[0][i];
                      }
 
                    if (j == step_sizes[1].size())
                      break;
 
                    y += step_sizes[1][j];
                  }
 
                if (k == step_sizes[2].size())
                  break;
 
                z += step_sizes[2][k];
              }
            break;
          }
 
        default:
          DEAL_II_NOT_IMPLEMENTED();
      }
 
    // next create the cells
    // Prepare cell data
    std::vector<CellData<dim>> cells;
    switch (dim)
      {
        case 1:
          {
            cells.resize(step_sizes[0].size());
            for (unsigned int x = 0; x < step_sizes[0].size(); ++x)
              {
                cells[x].vertices[0] = x;
                cells[x].vertices[1] = x + 1;
                cells[x].material_id = 0;
              }
            break;
          }
 
        case 2:
          {
            cells.resize(step_sizes[1].size() * step_sizes[0].size());
            for (unsigned int y = 0; y < step_sizes[1].size(); ++y)
              for (unsigned int x = 0; x < step_sizes[0].size(); ++x)
                {
                  const unsigned int c = x + y * step_sizes[0].size();
                  cells[c].vertices[0] = y * (step_sizes[0].size() + 1) + x;
                  cells[c].vertices[1] = y * (step_sizes[0].size() + 1) + x + 1;
                  cells[c].vertices[2] =
                    (y + 1) * (step_sizes[0].size() + 1) + x;
                  cells[c].vertices[3] =
                    (y + 1) * (step_sizes[0].size() + 1) + x + 1;
                  cells[c].material_id = 0;
                }
            break;
          }
 
        case 3:
          {
            const unsigned int n_x = (step_sizes[0].size() + 1);
            const unsigned int n_xy =
              (step_sizes[0].size() + 1) * (step_sizes[1].size() + 1);
 
            cells.resize(step_sizes[2].size() * step_sizes[1].size() *
                         step_sizes[0].size());
            for (unsigned int z = 0; z < step_sizes[2].size(); ++z)
              for (unsigned int y = 0; y < step_sizes[1].size(); ++y)
                for (unsigned int x = 0; x < step_sizes[0].size(); ++x)
                  {
                    const unsigned int c =
                      x + y * step_sizes[0].size() +
                      z * step_sizes[0].size() * step_sizes[1].size();
                    cells[c].vertices[0] = z * n_xy + y * n_x + x;
                    cells[c].vertices[1] = z * n_xy + y * n_x + x + 1;
                    cells[c].vertices[2] = z * n_xy + (y + 1) * n_x + x;
                    cells[c].vertices[3] = z * n_xy + (y + 1) * n_x + x + 1;
                    cells[c].vertices[4] = (z + 1) * n_xy + y * n_x + x;
                    cells[c].vertices[5] = (z + 1) * n_xy + y * n_x + x + 1;
                    cells[c].vertices[6] = (z + 1) * n_xy + (y + 1) * n_x + x;
                    cells[c].vertices[7] =
                      (z + 1) * n_xy + (y + 1) * n_x + x + 1;
                    cells[c].material_id = 0;
                  }
            break;
          }
 
        default:
          DEAL_II_NOT_IMPLEMENTED();
      }
 
    tria.create_triangulation(points, cells, SubCellData());
 
    if (colorize)
      {
        // to colorize, run through all
        // faces of all cells and set
        // boundary indicator to the
        // correct value if it was 0.
 
        // use a large epsilon to
        // compare numbers to avoid
        // roundoff problems.
        double min_size =
          *std::min_element(step_sizes[0].begin(), step_sizes[0].end());
        for (unsigned int i = 1; i < dim; ++i)
          min_size = std::min(min_size,
                              *std::min_element(step_sizes[i].begin(),
                                                step_sizes[i].end()));
        const double epsilon = 0.01 * min_size;
 
        // actual code is external since
        // 1-D is different from 2/3d.
        colorize_subdivided_hyper_rectangle(tria, p1, p2, epsilon);
      }
  }
 
 
 
  template <>
  void
  subdivided_hyper_rectangle(Triangulation<1>                       &tria,
                             const std::vector<std::vector<double>> &spacing,
                             const Point<1>                         &p,
                             const Table<1, types::material_id> &material_id,
                             const bool                          colorize)
  {
    Assert(spacing.size() == 1, ExcInvalidRepetitionsDimension(1));
 
    const unsigned int n_cells = material_id.size(0);
 
    Assert(spacing[0].size() == n_cells, ExcInvalidRepetitionsDimension(1));
 
    double delta = std::numeric_limits<double>::max();
    for (unsigned int i = 0; i < n_cells; ++i)
      {
        Assert(spacing[0][i] >= 0, ExcInvalidRepetitions(-1));
        delta = std::min(delta, spacing[0][i]);
      }
 
    // generate the necessary points
    std::vector<Point<1>> points;
    double                ax = p[0];
    for (unsigned int x = 0; x <= n_cells; ++x)
      {
        points.emplace_back(ax);
        if (x < n_cells)
          ax += spacing[0][x];
      }
    // create the cells
    unsigned int n_val_cells = 0;
    for (unsigned int i = 0; i < n_cells; ++i)
      if (material_id[i] != numbers::invalid_material_id)
        ++n_val_cells;
 
    std::vector<CellData<1>> cells(n_val_cells);
    unsigned int             id = 0;
    for (unsigned int x = 0; x < n_cells; ++x)
      if (material_id[x] != numbers::invalid_material_id)
        {
          cells[id].vertices[0] = x;
          cells[id].vertices[1] = x + 1;
          cells[id].material_id = material_id[x];
          ++id;
        }
    // create triangulation
    SubCellData t;
    GridTools::delete_unused_vertices(points, cells, t);
 
    tria.create_triangulation(points, cells, t);
 
    // set boundary indicator
    if (colorize)
      DEAL_II_NOT_IMPLEMENTED();
  }
 
 
  template <>
  void
  subdivided_hyper_rectangle(Triangulation<2>                       &tria,
                             const std::vector<std::vector<double>> &spacing,
                             const Point<2>                         &p,
                             const Table<2, types::material_id> &material_id,
                             const bool                          colorize)
  {
    Assert(spacing.size() == 2, ExcInvalidRepetitionsDimension(2));
 
    std::vector<unsigned int> repetitions(2);
    double                    delta = std::numeric_limits<double>::max();
    for (unsigned int i = 0; i < 2; ++i)
      {
        repetitions[i] = spacing[i].size();
        for (unsigned int j = 0; j < repetitions[i]; ++j)
          {
            Assert(spacing[i][j] >= 0, ExcInvalidRepetitions(-1));
            delta = std::min(delta, spacing[i][j]);
          }
        Assert(material_id.size(i) == repetitions[i],
               ExcInvalidRepetitionsDimension(i));
      }
 
    // generate the necessary points
    std::vector<Point<2>> points;
    double                ay = p[1];
    for (unsigned int y = 0; y <= repetitions[1]; ++y)
      {
        double ax = p[0];
        for (unsigned int x = 0; x <= repetitions[0]; ++x)
          {
            points.emplace_back(ax, ay);
            if (x < repetitions[0])
              ax += spacing[0][x];
          }
        if (y < repetitions[1])
          ay += spacing[1][y];
      }
 
    // create the cells
    unsigned int n_val_cells = 0;
    for (unsigned int i = 0; i < material_id.size(0); ++i)
      for (unsigned int j = 0; j < material_id.size(1); ++j)
        if (material_id[i][j] != numbers::invalid_material_id)
          ++n_val_cells;
 
    std::vector<CellData<2>> cells(n_val_cells);
    unsigned int             id = 0;
    for (unsigned int y = 0; y < repetitions[1]; ++y)
      for (unsigned int x = 0; x < repetitions[0]; ++x)
        if (material_id[x][y] != numbers::invalid_material_id)
          {
            cells[id].vertices[0] = y * (repetitions[0] + 1) + x;
            cells[id].vertices[1] = y * (repetitions[0] + 1) + x + 1;
            cells[id].vertices[2] = (y + 1) * (repetitions[0] + 1) + x;
            cells[id].vertices[3] = (y + 1) * (repetitions[0] + 1) + x + 1;
            cells[id].material_id = material_id[x][y];
            ++id;
          }
 
    // create triangulation
    SubCellData t;
    GridTools::delete_unused_vertices(points, cells, t);
 
    tria.create_triangulation(points, cells, t);
 
    // set boundary indicator
    if (colorize)
      {
        double                          eps  = 0.01 * delta;
        Triangulation<2>::cell_iterator cell = tria.begin(), endc = tria.end();
        for (; cell != endc; ++cell)
          {
            Point<2> cell_center = cell->center();
            for (const unsigned int f : GeometryInfo<2>::face_indices())
              if (cell->face(f)->boundary_id() == 0)
                {
                  Point<2> face_center = cell->face(f)->center();
                  for (unsigned int i = 0; i < 2; ++i)
                    {
                      if (face_center[i] < cell_center[i] - eps)
                        cell->face(f)->set_boundary_id(i * 2);
                      if (face_center[i] > cell_center[i] + eps)
                        cell->face(f)->set_boundary_id(i * 2 + 1);
                    }
                }
          }
      }
  }
 
 
  template <>
  void
  subdivided_hyper_rectangle(Triangulation<3>                       &tria,
                             const std::vector<std::vector<double>> &spacing,
                             const Point<3>                         &p,
                             const Table<3, types::material_id> &material_id,
                             const bool                          colorize)
  {
    const unsigned int dim = 3;
 
    Assert(spacing.size() == dim, ExcInvalidRepetitionsDimension(dim));
 
    std::array<unsigned int, dim> repetitions;
    double                        delta = std::numeric_limits<double>::max();
    for (unsigned int i = 0; i < dim; ++i)
      {
        repetitions[i] = spacing[i].size();
        for (unsigned int j = 0; j < repetitions[i]; ++j)
          {
            Assert(spacing[i][j] >= 0, ExcInvalidRepetitions(-1));
            delta = std::min(delta, spacing[i][j]);
          }
        Assert(material_id.size(i) == repetitions[i],
               ExcInvalidRepetitionsDimension(i));
      }
 
    // generate the necessary points
    std::vector<Point<dim>> points;
    double                  az = p[2];
    for (unsigned int z = 0; z <= repetitions[2]; ++z)
      {
        double ay = p[1];
        for (unsigned int y = 0; y <= repetitions[1]; ++y)
          {
            double ax = p[0];
            for (unsigned int x = 0; x <= repetitions[0]; ++x)
              {
                points.emplace_back(ax, ay, az);
                if (x < repetitions[0])
                  ax += spacing[0][x];
              }
            if (y < repetitions[1])
              ay += spacing[1][y];
          }
        if (z < repetitions[2])
          az += spacing[2][z];
      }
 
    // create the cells
    unsigned int n_val_cells = 0;
    for (unsigned int i = 0; i < material_id.size(0); ++i)
      for (unsigned int j = 0; j < material_id.size(1); ++j)
        for (unsigned int k = 0; k < material_id.size(2); ++k)
          if (material_id[i][j][k] != numbers::invalid_material_id)
            ++n_val_cells;
 
    std::vector<CellData<dim>> cells(n_val_cells);
    unsigned int               id  = 0;
    const unsigned int         n_x = (repetitions[0] + 1);
    const unsigned int n_xy = (repetitions[0] + 1) * (repetitions[1] + 1);
    for (unsigned int z = 0; z < repetitions[2]; ++z)
      for (unsigned int y = 0; y < repetitions[1]; ++y)
        for (unsigned int x = 0; x < repetitions[0]; ++x)
          if (material_id[x][y][z] != numbers::invalid_material_id)
            {
              cells[id].vertices[0] = z * n_xy + y * n_x + x;
              cells[id].vertices[1] = z * n_xy + y * n_x + x + 1;
              cells[id].vertices[2] = z * n_xy + (y + 1) * n_x + x;
              cells[id].vertices[3] = z * n_xy + (y + 1) * n_x + x + 1;
              cells[id].vertices[4] = (z + 1) * n_xy + y * n_x + x;
              cells[id].vertices[5] = (z + 1) * n_xy + y * n_x + x + 1;
              cells[id].vertices[6] = (z + 1) * n_xy + (y + 1) * n_x + x;
              cells[id].vertices[7] = (z + 1) * n_xy + (y + 1) * n_x + x + 1;
              cells[id].material_id = material_id[x][y][z];
              ++id;
            }
 
    // create triangulation
    SubCellData t;
    GridTools::delete_unused_vertices(points, cells, t);
 
    tria.create_triangulation(points, cells, t);
 
    // set boundary indicator
    if (colorize)
      {
        double                            eps  = 0.01 * delta;
        Triangulation<dim>::cell_iterator cell = tria.begin(),
                                          endc = tria.end();
        for (; cell != endc; ++cell)
          {
            Point<dim> cell_center = cell->center();
            for (auto f : GeometryInfo<dim>::face_indices())
              if (cell->face(f)->boundary_id() == 0)
                {
                  Point<dim> face_center = cell->face(f)->center();
                  for (unsigned int i = 0; i < dim; ++i)
                    {
                      if (face_center[i] < cell_center[i] - eps)
                        cell->face(f)->set_boundary_id(i * 2);
                      if (face_center[i] > cell_center[i] + eps)
                        cell->face(f)->set_boundary_id(i * 2 + 1);
                    }
                }
          }
      }
  }
 
  template <int dim, int spacedim>
  void
  cheese(Triangulation<dim, spacedim>    &tria,
         const std::vector<unsigned int> &holes)
  {
    AssertDimension(holes.size(), dim);
    // The corner points of the first cell. If there is a desire at
    // some point to change the geometry of the cells, they can be
    // made an argument to the function.
 
    Point<spacedim> p1;
    Point<spacedim> p2;
    for (unsigned int d = 0; d < dim; ++d)
      p2[d] = 1.;
 
    // then check that all repetitions
    // are >= 1, and calculate deltas
    // convert repetitions from double
    // to int by taking the ceiling.
    std::array<Point<spacedim>, dim> delta;
    std::array<unsigned int, dim>    repetitions;
    for (unsigned int i = 0; i < dim; ++i)
      {
        Assert(holes[i] >= 1,
               ExcMessage("At least one hole needed in each direction"));
        repetitions[i] = 2 * holes[i] + 1;
        delta[i][i]    = (p2[i] - p1[i]);
      }
 
    // then generate the necessary
    // points
    std::vector<Point<spacedim>> points;
    switch (dim)
      {
        case 1:
          for (unsigned int x = 0; x <= repetitions[0]; ++x)
            points.push_back(p1 + x * delta[0]);
          break;
 
        case 2:
          for (unsigned int y = 0; y <= repetitions[1]; ++y)
            for (unsigned int x = 0; x <= repetitions[0]; ++x)
              points.push_back(p1 + x * delta[0] + y * delta[1]);
          break;
 
        case 3:
          for (unsigned int z = 0; z <= repetitions[2]; ++z)
            for (unsigned int y = 0; y <= repetitions[1]; ++y)
              for (unsigned int x = 0; x <= repetitions[0]; ++x)
                points.push_back(p1 + x * delta[0] + y * delta[1] +
                                 z * delta[2]);
          break;
 
        default:
          DEAL_II_NOT_IMPLEMENTED();
      }
 
    // next create the cells
    // Prepare cell data
    std::vector<CellData<dim>> cells;
    switch (dim)
      {
        case 2:
          {
            cells.resize(repetitions[1] * repetitions[0] - holes[1] * holes[0]);
            unsigned int c = 0;
            for (unsigned int y = 0; y < repetitions[1]; ++y)
              for (unsigned int x = 0; x < repetitions[0]; ++x)
                {
                  if ((x % 2 == 1) && (y % 2 == 1))
                    continue;
                  Assert(c < cells.size(), ExcInternalError());
                  cells[c].vertices[0] = y * (repetitions[0] + 1) + x;
                  cells[c].vertices[1] = y * (repetitions[0] + 1) + x + 1;
                  cells[c].vertices[2] = (y + 1) * (repetitions[0] + 1) + x;
                  cells[c].vertices[3] = (y + 1) * (repetitions[0] + 1) + x + 1;
                  cells[c].material_id = 0;
                  ++c;
                }
            break;
          }
 
        case 3:
          {
            const unsigned int n_x = (repetitions[0] + 1);
            const unsigned int n_xy =
              (repetitions[0] + 1) * (repetitions[1] + 1);
 
            cells.resize(repetitions[2] * repetitions[1] * repetitions[0]);
 
            unsigned int c = 0;
            for (unsigned int z = 0; z < repetitions[2]; ++z)
              for (unsigned int y = 0; y < repetitions[1]; ++y)
                for (unsigned int x = 0; x < repetitions[0]; ++x)
                  {
                    Assert(c < cells.size(), ExcInternalError());
                    cells[c].vertices[0] = z * n_xy + y * n_x + x;
                    cells[c].vertices[1] = z * n_xy + y * n_x + x + 1;
                    cells[c].vertices[2] = z * n_xy + (y + 1) * n_x + x;
                    cells[c].vertices[3] = z * n_xy + (y + 1) * n_x + x + 1;
                    cells[c].vertices[4] = (z + 1) * n_xy + y * n_x + x;
                    cells[c].vertices[5] = (z + 1) * n_xy + y * n_x + x + 1;
                    cells[c].vertices[6] = (z + 1) * n_xy + (y + 1) * n_x + x;
                    cells[c].vertices[7] =
                      (z + 1) * n_xy + (y + 1) * n_x + x + 1;
                    cells[c].material_id = 0;
                    ++c;
                  }
            break;
          }
 
        default:
          DEAL_II_NOT_IMPLEMENTED();
      }
 
    tria.create_triangulation(points, cells, SubCellData());
  }
 
 
 
  template <>
  void
  plate_with_a_hole(Triangulation<1> & /*tria*/,
                    const double /*inner_radius*/,
                    const double /*outer_radius*/,
                    const double /*pad_bottom*/,
                    const double /*pad_top*/,
                    const double /*pad_left*/,
                    const double /*pad_right*/,
                    const Point<1> & /*center*/,
                    const types::manifold_id /*polar_manifold_id*/,
                    const types::manifold_id /*tfi_manifold_id*/,
                    const double /*L*/,
                    const unsigned int /*n_slices*/,
                    const bool /*colorize*/)
  {
    DEAL_II_NOT_IMPLEMENTED();
  }
 
 
 
  template <>
  void
  channel_with_cylinder(Triangulation<1> & /*tria*/,
                        const double /*shell_region_width*/,
                        const unsigned int /*n_shells*/,
                        const double /*skewness*/,
                        const bool /*colorize*/)
  {
    DEAL_II_NOT_IMPLEMENTED();
  }
 
 
 
  namespace internal
  {
    // helper function to check if point is in 2d box
    bool inline point_in_2d_box(const Point<2> &p,
                                const Point<2> &c,
                                const double    radius)
    {
      return (std::abs(p[0] - c[0]) < radius) &&
             (std::abs(p[1] - c[1]) < radius);
    }
 
 
 
    // Find the minimal distance between two vertices. This is useful for
    // computing a tolerance for merging vertices in
    // GridTools::merge_triangulations.
    template <int dim, int spacedim>
    double
    minimal_vertex_distance(const Triangulation<dim, spacedim> &triangulation)
    {
      double length = std::numeric_limits<double>::max();
      for (const auto &cell : triangulation.active_cell_iterators())
        for (unsigned int n = 0; n < GeometryInfo<dim>::lines_per_cell; ++n)
          length = std::min(length, cell->line(n)->diameter());
      return length;
    }
  } // namespace internal
 
 
 
  template <>
  void
  plate_with_a_hole(Triangulation<2>        &tria,
                    const double             inner_radius,
                    const double             outer_radius,
                    const double             pad_bottom,
                    const double             pad_top,
                    const double             pad_left,
                    const double             pad_right,
                    const Point<2>          &new_center,
                    const types::manifold_id polar_manifold_id,
                    const types::manifold_id tfi_manifold_id,
                    const double             L,
                    const unsigned int /*n_slices*/,
                    const bool colorize)
  {
    const bool with_padding =
      pad_bottom > 0 || pad_top > 0 || pad_left > 0 || pad_right > 0;
 
    Assert(pad_bottom >= 0., ExcMessage("Negative bottom padding."));
    Assert(pad_top >= 0., ExcMessage("Negative top padding."));
    Assert(pad_left >= 0., ExcMessage("Negative left padding."));
    Assert(pad_right >= 0., ExcMessage("Negative right padding."));
 
    const Point<2> center;
 
    auto min_line_length = [](const Triangulation<2> &tria) -> double {
      double length = std::numeric_limits<double>::max();
      for (const auto &cell : tria.active_cell_iterators())
        for (unsigned int n = 0; n < cell->n_lines(); ++n)
          length = std::min(length, cell->line(n)->diameter());
      return length;
    };
 
    // start by setting up the cylinder triangulation
    Triangulation<2>  cylinder_tria_maybe;
    Triangulation<2> &cylinder_tria = with_padding ? cylinder_tria_maybe : tria;
    GridGenerator::hyper_cube_with_cylindrical_hole(cylinder_tria,
                                                    inner_radius,
                                                    outer_radius,
                                                    L,
                                                    /*repetitions*/ 1,
                                                    colorize);
 
    // we will deal with face manifold ids after we merge triangulations
    for (const auto &cell : cylinder_tria.active_cell_iterators())
      cell->set_manifold_id(tfi_manifold_id);
 
    const Point<2> bl(-outer_radius - pad_left, -outer_radius - pad_bottom);
    const Point<2> tr(outer_radius + pad_right, outer_radius + pad_top);
    if (with_padding)
      {
        // hyper_cube_with_cylindrical_hole will have 2 cells along
        // each face, so the element size is outer_radius
 
        auto add_sizes = [](std::vector<double> &step_sizes,
                            const double         padding,
                            const double         h) -> void {
          // use std::round instead of std::ceil to improve aspect ratio
          // in case padding is only slightly larger than h.
          const auto rounded =
            static_cast<unsigned int>(std::round(padding / h));
          // in case padding is much smaller than h, make sure we
          // have at least 1 element
          const unsigned int num = (padding > 0. && rounded == 0) ? 1 : rounded;
          for (unsigned int i = 0; i < num; ++i)
            step_sizes.push_back(padding / num);
        };
 
        std::vector<std::vector<double>> step_sizes(2);
        // x-coord
        // left:
        add_sizes(step_sizes[0], pad_left, outer_radius);
        // center
        step_sizes[0].push_back(outer_radius);
        step_sizes[0].push_back(outer_radius);
        // right
        add_sizes(step_sizes[0], pad_right, outer_radius);
        // y-coord
        //   bottom
        add_sizes(step_sizes[1], pad_bottom, outer_radius);
        //   center
        step_sizes[1].push_back(outer_radius);
        step_sizes[1].push_back(outer_radius);
        //   top
        add_sizes(step_sizes[1], pad_top, outer_radius);
 
        // now create bulk
        Triangulation<2> bulk_tria;
        GridGenerator::subdivided_hyper_rectangle(
          bulk_tria, step_sizes, bl, tr, colorize);
 
        // now remove cells reserved from the cylindrical hole
        std::set<Triangulation<2>::active_cell_iterator> cells_to_remove;
        for (const auto &cell : bulk_tria.active_cell_iterators())
          if (internal::point_in_2d_box(cell->center(), center, outer_radius))
            cells_to_remove.insert(cell);
 
        Triangulation<2> tria_without_cylinder;
        GridGenerator::create_triangulation_with_removed_cells(
          bulk_tria, cells_to_remove, tria_without_cylinder);
 
        const double tolerance =
          std::min(min_line_length(tria_without_cylinder),
                   min_line_length(cylinder_tria)) /
          2.0;
 
        GridGenerator::merge_triangulations(
          tria_without_cylinder, cylinder_tria, tria, tolerance, true);
      }
 
    // now set manifold ids:
    for (const auto &cell : tria.active_cell_iterators())
      {
        // set all non-boundary manifold ids on the cells that came from the
        // grid around the cylinder to the new TFI manifold id.
        if (cell->manifold_id() == tfi_manifold_id)
          {
            for (const unsigned int face_n : cell->face_indices())
              {
                const auto &face = cell->face(face_n);
                if (face->at_boundary() &&
                    internal::point_in_2d_box(face->center(),
                                              center,
                                              outer_radius * (1. - 1e-12)))
                  face->set_manifold_id(polar_manifold_id);
                else
                  face->set_manifold_id(tfi_manifold_id);
              }
          }
        else
          {
            // ensure that all other manifold ids (including the faces
            // opposite the cylinder) are set to the flat id
            cell->set_all_manifold_ids(numbers::flat_manifold_id);
          }
      }
 
    static constexpr double tol =
      std::numeric_limits<double>::epsilon() * 10000;
    if (colorize)
      for (const auto &cell : tria.active_cell_iterators())
        for (const unsigned int face_n : cell->face_indices())
          {
            const auto face = cell->face(face_n);
            if (face->at_boundary())
              {
                const Point<2> center = face->center();
                // left side
                if (std::abs(center[0] - bl[0]) < tol * std::abs(bl[0]))
                  face->set_boundary_id(0);
                // right side
                else if (std::abs(center[0] - tr[0]) < tol * std::abs(tr[0]))
                  face->set_boundary_id(1);
                // bottom
                else if (std::abs(center[1] - bl[1]) < tol * std::abs(bl[1]))
                  face->set_boundary_id(2);
                // top
                else if (std::abs(center[1] - tr[1]) < tol * std::abs(tr[1]))
                  face->set_boundary_id(3);
                // cylinder boundary
                else
                  {
                    Assert(cell->manifold_id() == tfi_manifold_id,
                           ExcInternalError());
                    face->set_boundary_id(4);
                  }
              }
          }
 
    // move to the new center
    GridTools::shift(new_center, tria);
 
    PolarManifold<2> polar_manifold(new_center);
    tria.set_manifold(polar_manifold_id, polar_manifold);
    tria.set_manifold(tfi_manifold_id, FlatManifold<2>());
    TransfiniteInterpolationManifold<2> inner_manifold;
    inner_manifold.initialize(tria);
    tria.set_manifold(tfi_manifold_id, inner_manifold);
  }
 
 
 
  template <>
  void
  plate_with_a_hole(Triangulation<3>        &tria,
                    const double             inner_radius,
                    const double             outer_radius,
                    const double             pad_bottom,
                    const double             pad_top,
                    const double             pad_left,
                    const double             pad_right,
                    const Point<3>          &new_center,
                    const types::manifold_id polar_manifold_id,
                    const types::manifold_id tfi_manifold_id,
                    const double             L,
                    const unsigned int       n_slices,
                    const bool               colorize)
  {
    Triangulation<2> tria_2;
    plate_with_a_hole(tria_2,
                      inner_radius,
                      outer_radius,
                      pad_bottom,
                      pad_top,
                      pad_left,
                      pad_right,
                      Point<2>(new_center[0], new_center[1]),
                      polar_manifold_id,
                      tfi_manifold_id,
                      L,
                      n_slices,
                      colorize);
 
    // extrude to 3d
    extrude_triangulation(tria_2, n_slices, L, tria, true);
 
    // shift in Z direction to match specified center
    GridTools::shift(Point<3>(0, 0, new_center[2] - L / 2.), tria);
 
    // set up the new manifolds
    const Tensor<1, 3>           direction{{0.0, 0.0, 1.0}};
    const CylindricalManifold<3> cylindrical_manifold(
      direction,
      /*axial_point*/ new_center);
    tria.set_manifold(polar_manifold_id, FlatManifold<3>());
    tria.set_manifold(tfi_manifold_id, FlatManifold<3>());
    TransfiniteInterpolationManifold<3> inner_manifold;
    inner_manifold.initialize(tria);
 
    tria.set_manifold(polar_manifold_id, cylindrical_manifold);
    tria.set_manifold(tfi_manifold_id, inner_manifold);
  }
 
 
 
  template <>
  void
  channel_with_cylinder(Triangulation<2>  &tria,
                        const double       shell_region_width,
                        const unsigned int n_shells,
                        const double       skewness,
                        const bool         colorize)
  {
    Assert(0.0 <= shell_region_width && shell_region_width < 0.05,
           ExcMessage("The width of the shell region must be less than 0.05 "
                      "(and preferably close to 0.03)"));
    const types::manifold_id polar_manifold_id = 0;
    const types::manifold_id tfi_manifold_id   = 1;
 
    // We begin by setting up a grid that is 4 by 22 cells. While not
    // squares, these have pretty good aspect ratios.
    Triangulation<2> bulk_tria;
    GridGenerator::subdivided_hyper_rectangle(bulk_tria,
                                              {22u, 4u},
                                              Point<2>(0.0, 0.0),
                                              Point<2>(2.2, 0.41));
    // bulk_tria now looks like this:
    //
    //   +--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+
    //   |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |
    //   +--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+
    //   |  |XX|XX|  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |
    //   +--+--O--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+
    //   |  |XX|XX|  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |
    //   +--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+
    //   |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |
    //   +--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+
    //
    // Note that these cells are not quite squares: they are all 0.1 by
    // 0.1025.
    //
    // The next step is to remove the cells marked with XXs: we will place
    // the grid around the cylinder there later. The next loop does two
    // things:
    // 1. Determines which cells need to be removed from the Triangulation
    //    (i.e., find the cells marked with XX in the picture).
    // 2. Finds the location of the vertex marked with 'O' and uses that to
    //    calculate the shift vector for aligning cylinder_tria with
    //    tria_without_cylinder.
    std::set<Triangulation<2>::active_cell_iterator> cells_to_remove;
    Tensor<1, 2> cylinder_triangulation_offset;
    for (const auto &cell : bulk_tria.active_cell_iterators())
      {
        if ((cell->center() - Point<2>(0.2, 0.2)).norm() < 0.15)
          cells_to_remove.insert(cell);
 
        if (cylinder_triangulation_offset == Tensor<1, 2>())
          {
            for (const unsigned int vertex_n :
                 GeometryInfo<2>::vertex_indices())
              if (cell->vertex(vertex_n) == Point<2>())
                {
                  // cylinder_tria is centered at zero, so we need to
                  // shift it up and to the right by two cells:
                  cylinder_triangulation_offset =
                    2.0 * (cell->vertex(3) - Point<2>());
                  break;
                }
          }
      }
    Triangulation<2> tria_without_cylinder;
    GridGenerator::create_triangulation_with_removed_cells(
      bulk_tria, cells_to_remove, tria_without_cylinder);
 
    // set up the cylinder triangulation. Note that this function sets the
    // manifold ids of the interior boundary cells to 0
    // (polar_manifold_id).
    Triangulation<2> cylinder_tria;
    GridGenerator::hyper_cube_with_cylindrical_hole(cylinder_tria,
                                                    0.05 + shell_region_width,
                                                    0.41 / 4.0);
    // The bulk cells are not quite squares, so we need to move the left
    // and right sides of cylinder_tria inwards so that it fits in
    // bulk_tria:
    for (const auto &cell : cylinder_tria.active_cell_iterators())
      for (const unsigned int vertex_n : GeometryInfo<2>::vertex_indices())
        {
          if (std::abs(cell->vertex(vertex_n)[0] - -0.41 / 4.0) < 1e-10)
            cell->vertex(vertex_n)[0] = -0.1;
          else if (std::abs(cell->vertex(vertex_n)[0] - 0.41 / 4.0) < 1e-10)
            cell->vertex(vertex_n)[0] = 0.1;
        }
 
    // Assign interior manifold ids to be the TFI id.
    for (const auto &cell : cylinder_tria.active_cell_iterators())
      {
        cell->set_manifold_id(tfi_manifold_id);
        for (const unsigned int face_n : GeometryInfo<2>::face_indices())
          if (!cell->face(face_n)->at_boundary())
            cell->face(face_n)->set_manifold_id(tfi_manifold_id);
      }
    if (0.0 < shell_region_width)
      {
        Assert(0 < n_shells,
               ExcMessage("If the shell region has positive width then "
                          "there must be at least one shell."));
        Triangulation<2> shell_tria;
        GridGenerator::concentric_hyper_shells(shell_tria,
                                               Point<2>(),
                                               0.05,
                                               0.05 + shell_region_width,
                                               n_shells,
                                               skewness,
                                               8);
 
        // Make the tolerance as large as possible since these cells can
        // be quite close together
        const double vertex_tolerance =
          std::min(internal::minimal_vertex_distance(shell_tria),
                   internal::minimal_vertex_distance(cylinder_tria)) *
          0.5;
 
        shell_tria.set_all_manifold_ids(polar_manifold_id);
        Triangulation<2> temp;
        GridGenerator::merge_triangulations(
          shell_tria, cylinder_tria, temp, vertex_tolerance, true);
        cylinder_tria = std::move(temp);
      }
    GridTools::shift(cylinder_triangulation_offset, cylinder_tria);
 
    // Compute the tolerance again, since the shells may be very close to
    // each-other:
    const double vertex_tolerance =
      std::min(internal::minimal_vertex_distance(tria_without_cylinder),
               internal::minimal_vertex_distance(cylinder_tria)) /
      10;
    GridGenerator::merge_triangulations(
      tria_without_cylinder, cylinder_tria, tria, vertex_tolerance, true);
 
    // Move the vertices in the middle of the faces of cylinder_tria slightly
    // to give a better mesh quality. We have to balance the quality of these
    // cells with the quality of the outer cells (initially rectangles). For
    // constant radial distance, we would place them at the distance 0.1 *
    // sqrt(2.) from the center. In case the shell region width is more than
    // 0.1/6., we choose to place them at 0.1 * 4./3. from the center, which
    // ensures that the shortest edge of the outer cells is 2./3. of the
    // original length. If the shell region width is less, we make the edge
    // length of the inner part and outer part (in the shorter x direction)
    // the same.
    {
      const double shift =
        std::min(0.125 + shell_region_width * 0.5, 0.1 * 4. / 3.);
      for (const auto &cell : tria.active_cell_iterators())
        for (const unsigned int v : GeometryInfo<2>::vertex_indices())
          if (cell->vertex(v).distance(Point<2>(0.1, 0.205)) < 1e-10)
            cell->vertex(v) = Point<2>(0.2 - shift, 0.205);
          else if (cell->vertex(v).distance(Point<2>(0.3, 0.205)) < 1e-10)
            cell->vertex(v) = Point<2>(0.2 + shift, 0.205);
          else if (cell->vertex(v).distance(Point<2>(0.2, 0.1025)) < 1e-10)
            cell->vertex(v) = Point<2>(0.2, 0.2 - shift);
          else if (cell->vertex(v).distance(Point<2>(0.2, 0.3075)) < 1e-10)
            cell->vertex(v) = Point<2>(0.2, 0.2 + shift);
    }
 
    // Ensure that all manifold ids on a polar cell really are set to the
    // polar manifold id:
    for (const auto &cell : tria.active_cell_iterators())
      if (cell->manifold_id() == polar_manifold_id)
        cell->set_all_manifold_ids(polar_manifold_id);
 
    // Ensure that all other manifold ids (including the interior faces
    // opposite the cylinder) are set to the flat manifold id:
    for (const auto &cell : tria.active_cell_iterators())
      if (cell->manifold_id() != polar_manifold_id &&
          cell->manifold_id() != tfi_manifold_id)
        cell->set_all_manifold_ids(numbers::flat_manifold_id);
 
    // We need to calculate the current center so that we can move it later:
    // to start get a unique list of (points to) vertices on the cylinder
    std::vector<Point<2> *> cylinder_pointers;
    for (const auto &face : tria.active_face_iterators())
      if (face->manifold_id() == polar_manifold_id)
        {
          cylinder_pointers.push_back(&face->vertex(0));
          cylinder_pointers.push_back(&face->vertex(1));
        }
    // de-duplicate
    std::sort(cylinder_pointers.begin(), cylinder_pointers.end());
    cylinder_pointers.erase(std::unique(cylinder_pointers.begin(),
                                        cylinder_pointers.end()),
                            cylinder_pointers.end());
 
    // find the current center...
    Point<2> center;
    for (const Point<2> *const ptr : cylinder_pointers)
      center += *ptr / double(cylinder_pointers.size());
 
    // and recenter at (0.2, 0.2)
    for (Point<2> *const ptr : cylinder_pointers)
      *ptr += Point<2>(0.2, 0.2) - center;
 
    // attach manifolds
    PolarManifold<2> polar_manifold(Point<2>(0.2, 0.2));
    tria.set_manifold(polar_manifold_id, polar_manifold);
 
    tria.set_manifold(tfi_manifold_id, FlatManifold<2>());
    TransfiniteInterpolationManifold<2> inner_manifold;
    inner_manifold.initialize(tria);
    tria.set_manifold(tfi_manifold_id, inner_manifold);
 
    if (colorize)
      for (const auto &face : tria.active_face_iterators())
        if (face->at_boundary())
          {
            const Point<2> center = face->center();
            // left side
            if (std::abs(center[0] - 0.0) < 1e-10)
              face->set_boundary_id(0);
            // right side
            else if (std::abs(center[0] - 2.2) < 1e-10)
              face->set_boundary_id(1);
            // cylinder boundary
            else if (face->manifold_id() == polar_manifold_id)
              face->set_boundary_id(2);
            // sides of channel
            else
              {
                Assert(std::abs(center[1] - 0.00) < 1.0e-10 ||
                         std::abs(center[1] - 0.41) < 1.0e-10,
                       ExcInternalError());
                face->set_boundary_id(3);
              }
          }
  }
 
 
 
  template <>
  void
  channel_with_cylinder(Triangulation<3>  &tria,
                        const double       shell_region_width,
                        const unsigned int n_shells,
                        const double       skewness,
                        const bool         colorize)
  {
    Triangulation<2> tria_2;
    channel_with_cylinder(
      tria_2, shell_region_width, n_shells, skewness, colorize);
    extrude_triangulation(tria_2, 5, 0.41, tria, true);
 
    // set up the new 3d manifolds
    const types::manifold_id      cylindrical_manifold_id = 0;
    const types::manifold_id      tfi_manifold_id         = 1;
    const PolarManifold<2> *const m_ptr =
      dynamic_cast<const PolarManifold<2> *>(
        &tria_2.get_manifold(cylindrical_manifold_id));
    Assert(m_ptr != nullptr, ExcInternalError());
    const Point<3>     axial_point(m_ptr->center[0], m_ptr->center[1], 0.0);
    const Tensor<1, 3> direction{{0.0, 0.0, 1.0}};
 
    tria.set_manifold(cylindrical_manifold_id, FlatManifold<3>());
    tria.set_manifold(tfi_manifold_id, FlatManifold<3>());
    const CylindricalManifold<3> cylindrical_manifold(direction, axial_point);
    TransfiniteInterpolationManifold<3> inner_manifold;
    inner_manifold.initialize(tria);
    tria.set_manifold(cylindrical_manifold_id, cylindrical_manifold);
    tria.set_manifold(tfi_manifold_id, inner_manifold);
 
    // From extrude_triangulation: since the maximum boundary id of tria_2 was
    // 3, the bottom boundary id is 4 and the top is 5: both are walls, so set
    // them to 3
    if (colorize)
      for (const auto &face : tria.active_face_iterators())
        if (face->boundary_id() == 4 || face->boundary_id() == 5)
          face->set_boundary_id(3);
  }
 
 
 
  template <int dim, int spacedim>
  void
  hyper_cross(Triangulation<dim, spacedim>    &tria,
              const std::vector<unsigned int> &sizes,
              const bool                       colorize)
  {
    AssertDimension(sizes.size(), GeometryInfo<dim>::faces_per_cell);
    Assert(dim > 1, ExcNotImplemented());
    Assert(dim < 4, ExcNotImplemented());
 
    // If there is a desire at some point to change the geometry of
    // the cells, this tensor can be made an argument to the function.
    Tensor<1, dim> dimensions;
    for (unsigned int d = 0; d < dim; ++d)
      dimensions[d] = 1.;
 
    std::vector<Point<spacedim>> points;
    unsigned int                 n_cells = 1;
    for (const unsigned int i : GeometryInfo<dim>::face_indices())
      n_cells += sizes[i];
 
    std::vector<CellData<dim>> cells(n_cells);
    // Vertices of the center cell
    for (const unsigned int i : GeometryInfo<dim>::vertex_indices())
      {
        Point<spacedim> p;
        for (unsigned int d = 0; d < dim; ++d)
          p[d] = 0.5 * dimensions[d] *
                 GeometryInfo<dim>::unit_normal_orientation
                   [GeometryInfo<dim>::vertex_to_face[i][d]];
        points.push_back(p);
        cells[0].vertices[i] = i;
      }
    cells[0].material_id = 0;
 
    // The index of the first cell of the leg.
    unsigned int cell_index = 1;
    // The legs of the cross
    for (const unsigned int face : GeometryInfo<dim>::face_indices())
      {
        const unsigned int oface = GeometryInfo<dim>::opposite_face[face];
        const unsigned int dir = GeometryInfo<dim>::unit_normal_direction[face];
 
        // We are moving in the direction of face
        for (unsigned int j = 0; j < sizes[face]; ++j, ++cell_index)
          {
            const unsigned int last_cell = (j == 0) ? 0U : (cell_index - 1);
 
            for (unsigned int v = 0; v < GeometryInfo<dim>::vertices_per_face;
                 ++v)
              {
                const unsigned int cellv =
                  GeometryInfo<dim>::face_to_cell_vertices(face, v);
                const unsigned int ocellv =
                  GeometryInfo<dim>::face_to_cell_vertices(oface, v);
                // First the vertices which already exist
                cells[cell_index].vertices[ocellv] =
                  cells[last_cell].vertices[cellv];
 
                // Now the new vertices
                cells[cell_index].vertices[cellv] = points.size();
 
                Point<spacedim> p = points[cells[cell_index].vertices[ocellv]];
                p[dir] += GeometryInfo<dim>::unit_normal_orientation[face] *
                          dimensions[dir];
                points.push_back(p);
              }
            cells[cell_index].material_id = (colorize) ? (face + 1U) : 0U;
          }
      }
    tria.create_triangulation(points, cells, SubCellData());
  }
 
 
  template <>
  void
  hyper_cube_slit(Triangulation<1> &, const double, const double, const bool)
  {
    DEAL_II_NOT_IMPLEMENTED();
  }
 
 
 
  template <>
  void
  enclosed_hyper_cube(Triangulation<1> &,
                      const double,
                      const double,
                      const double,
                      const bool)
  {
    DEAL_II_NOT_IMPLEMENTED();
  }
 
 
 
  template <>
  void
  hyper_L(Triangulation<1> &, const double, const double, const bool)
  {
    DEAL_II_NOT_IMPLEMENTED();
  }
 
 
 
  template <>
  void
  hyper_ball(Triangulation<1> &, const Point<1> &, const double, const bool)
  {
    DEAL_II_NOT_IMPLEMENTED();
  }
 
 
 
  template <>
  void
  hyper_ball_balanced(Triangulation<1> &, const Point<1> &, const double)
  {
    DEAL_II_NOT_IMPLEMENTED();
  }
 
 
 
  template <>
  void
  cylinder(Triangulation<1> &, const double, const double)
  {
    DEAL_II_NOT_IMPLEMENTED();
  }
 
 
  template <>
  void
  subdivided_cylinder(Triangulation<1> &,
                      const unsigned int,
                      const double,
                      const double)
  {
    DEAL_II_NOT_IMPLEMENTED();
  }
 
 
 
  template <>
  void
  truncated_cone(Triangulation<1> &, const double, const double, const double)
  {
    DEAL_II_NOT_IMPLEMENTED();
  }
 
 
 
  template <>
  void
  hyper_shell(Triangulation<1> &,
              const Point<1> &,
              const double,
              const double,
              const unsigned int,
              const bool)
  {
    DEAL_II_NOT_IMPLEMENTED();
  }
 
  template <>
  void
  cylinder_shell(Triangulation<1> &,
                 const double,
                 const double,
                 const double,
                 const unsigned int,
                 const unsigned int,
                 const bool)
  {
    DEAL_II_NOT_IMPLEMENTED();
  }
 
 
  template <>
  void
  quarter_hyper_ball(Triangulation<1> &, const Point<1> &, const double)
  {
    DEAL_II_NOT_IMPLEMENTED();
  }
 
 
  template <>
  void
  half_hyper_ball(Triangulation<1> &, const Point<1> &, const double)
  {
    DEAL_II_NOT_IMPLEMENTED();
  }
 
 
  template <>
  void
  half_hyper_shell(Triangulation<1> &,
                   const Point<1> &,
                   const double,
                   const double,
                   const unsigned int,
                   const bool)
  {
    DEAL_II_NOT_IMPLEMENTED();
  }
 
  template <>
  void
  quarter_hyper_shell(Triangulation<1> &,
                      const Point<1> &,
                      const double,
                      const double,
                      const unsigned int,
                      const bool)
  {
    DEAL_II_NOT_IMPLEMENTED();
  }
 
  template <>
  void
  enclosed_hyper_cube(Triangulation<2> &tria,
                      const double      left,
                      const double      right,
                      const double      thickness,
                      const bool        colorize)
  {
    Assert(left < right,
           ExcMessage("Invalid left-to-right bounds of enclosed hypercube"));
 
    std::vector<Point<2>> vertices(16);
    double                coords[4];
    coords[0] = left - thickness;
    coords[1] = left;
    coords[2] = right;
    coords[3] = right + thickness;
 
    unsigned int k = 0;
    for (const double y : coords)
      for (const double x : coords)
        vertices[k++] = Point<2>(x, y);
 
    const types::material_id materials[9] = {5, 4, 6, 1, 0, 2, 9, 8, 10};
 
    std::vector<CellData<2>> cells(9);
    k = 0;
    for (unsigned int i0 = 0; i0 < 3; ++i0)
      for (unsigned int i1 = 0; i1 < 3; ++i1)
        {
          cells[k].vertices[0] = i1 + 4 * i0;
          cells[k].vertices[1] = i1 + 4 * i0 + 1;
          cells[k].vertices[2] = i1 + 4 * i0 + 4;
          cells[k].vertices[3] = i1 + 4 * i0 + 5;
          if (colorize)
            cells[k].material_id = materials[k];
          ++k;
        }
    tria.create_triangulation(vertices,
                              cells,
                              SubCellData()); // no boundary information
  }
 
 
 
  // Implementation for 2d only
  template <>
  void
  hyper_cube_slit(Triangulation<2> &tria,
                  const double      left,
                  const double      right,
                  const bool        colorize)
  {
    const double             rl2                 = (right + left) / 2;
    const Point<2>           vertices[10]        = {Point<2>(left, left),
                                                    Point<2>(rl2, left),
                                                    Point<2>(rl2, rl2),
                                                    Point<2>(left, rl2),
                                                    Point<2>(right, left),
                                                    Point<2>(right, rl2),
                                                    Point<2>(rl2, right),
                                                    Point<2>(left, right),
                                                    Point<2>(right, right),
                                                    Point<2>(rl2, left)};
    const int                cell_vertices[4][4] = {{0, 1, 3, 2},
                                                    {9, 4, 2, 5},
                                                    {3, 2, 7, 6},
                                                    {2, 5, 6, 8}};
    std::vector<CellData<2>> cells(4, CellData<2>());
    for (unsigned int i = 0; i < 4; ++i)
      {
        for (unsigned int j = 0; j < 4; ++j)
          cells[i].vertices[j] = cell_vertices[i][j];
        cells[i].material_id = 0;
      }
    tria.create_triangulation(std::vector<Point<2>>(std::begin(vertices),
                                                    std::end(vertices)),
                              cells,
                              SubCellData()); // no boundary information
 
    if (colorize)
      {
        Triangulation<2>::cell_iterator cell = tria.begin();
        cell->face(1)->set_boundary_id(1);
        ++cell;
        cell->face(0)->set_boundary_id(2);
      }
  }
 
 
 
  template <>
  void
  truncated_cone(Triangulation<2> &triangulation,
                 const double      radius_0,
                 const double      radius_1,
                 const double      half_length)
  {
    Point<2> vertices_tmp[4];
 
    vertices_tmp[0] = Point<2>(-half_length, -radius_0);
    vertices_tmp[1] = Point<2>(half_length, -radius_1);
    vertices_tmp[2] = Point<2>(-half_length, radius_0);
    vertices_tmp[3] = Point<2>(half_length, radius_1);
 
    const std::vector<Point<2>> vertices(std::begin(vertices_tmp),
                                         std::end(vertices_tmp));
    unsigned int cell_vertices[1][GeometryInfo<2>::vertices_per_cell];
 
    for (const unsigned int i : GeometryInfo<2>::vertex_indices())
      cell_vertices[0][i] = i;
 
    std::vector<CellData<2>> cells(1, CellData<2>());
 
    for (const unsigned int i : GeometryInfo<2>::vertex_indices())
      cells[0].vertices[i] = cell_vertices[0][i];
 
    cells[0].material_id = 0;
    triangulation.create_triangulation(vertices, cells, SubCellData());
 
    Triangulation<2>::cell_iterator cell = triangulation.begin();
 
    cell->face(0)->set_boundary_id(1);
    cell->face(1)->set_boundary_id(2);
 
    for (unsigned int i = 2; i < 4; ++i)
      cell->face(i)->set_boundary_id(0);
  }
 
 
 
  // Implementation for 2d only
  template <>
  void
  hyper_L(Triangulation<2> &tria,
          const double      a,
          const double      b,
          const bool        colorize)
  {
    const Point<2> vertices[8]    = {Point<2>(a, a),
                                     Point<2>((a + b) / 2, a),
                                     Point<2>(b, a),
                                     Point<2>(a, (a + b) / 2),
                                     Point<2>((a + b) / 2, (a + b) / 2),
                                     Point<2>(b, (a + b) / 2),
                                     Point<2>(a, b),
                                     Point<2>((a + b) / 2, b)};
    const int cell_vertices[3][4] = {{0, 1, 3, 4}, {1, 2, 4, 5}, {3, 4, 6, 7}};
 
    std::vector<CellData<2>> cells(3, CellData<2>());
 
    for (unsigned int i = 0; i < 3; ++i)
      {
        for (unsigned int j = 0; j < 4; ++j)
          cells[i].vertices[j] = cell_vertices[i][j];
        cells[i].material_id = 0;
      }
 
    tria.create_triangulation(std::vector<Point<2>>(std::begin(vertices),
                                                    std::end(vertices)),
                              cells,
                              SubCellData());
 
    if (colorize)
      {
        Triangulation<2>::cell_iterator cell = tria.begin();
 
        cell->face(0)->set_boundary_id(0);
        cell->face(2)->set_boundary_id(1);
        ++cell;
 
        cell->face(1)->set_boundary_id(2);
        cell->face(2)->set_boundary_id(1);
        cell->face(3)->set_boundary_id(3);
        ++cell;
 
        cell->face(0)->set_boundary_id(0);
        cell->face(1)->set_boundary_id(4);
        cell->face(3)->set_boundary_id(5);
      }
  }
 
 
 
  template <int dim, int spacedim>
  void
  subdivided_hyper_L(Triangulation<dim, spacedim>    &tria,
                     const std::vector<unsigned int> &repetitions,
                     const Point<dim>                &bottom_left,
                     const Point<dim>                &top_right,
                     const std::vector<int>          &n_cells_to_remove)
  {
    Assert(dim > 1, ExcNotImplemented());
    // Check the consistency of the dimensions provided.
    AssertDimension(repetitions.size(), dim);
    AssertDimension(n_cells_to_remove.size(), dim);
    for (unsigned int d = 0; d < dim; ++d)
      {
        Assert(std::fabs(n_cells_to_remove[d]) <= repetitions[d],
               ExcMessage("Attempting to cut away too many cells."));
      }
    // Create the domain to be cut
    Triangulation<dim, spacedim> rectangle;
    GridGenerator::subdivided_hyper_rectangle(rectangle,
                                              repetitions,
                                              bottom_left,
                                              top_right);
    // compute the vertex of the cut step, we will cut according to the
    // location of the cartesian coordinates of the cell centers
    std::array<double, dim> h;
    Point<dim>              cut_step;
    for (unsigned int d = 0; d < dim; ++d)
      {
        // mesh spacing in each direction in cartesian coordinates
        h[d] = (top_right[d] - bottom_left[d]) / repetitions[d];
        // left to right, bottom to top, front to back
        if (n_cells_to_remove[d] >= 0)
          {
            // cartesian coordinates of vertex location
            cut_step[d] =
              h[d] * std::fabs(n_cells_to_remove[d]) + bottom_left[d];
          }
        // right to left, top to bottom, back to front
        else
          {
            cut_step[d] = top_right[d] - h[d] * std::fabs(n_cells_to_remove[d]);
          }
      }
 
 
    // compute cells to remove
    std::set<typename Triangulation<dim, spacedim>::active_cell_iterator>
      cells_to_remove;
    for (const auto &cell : rectangle.active_cell_iterators())
      {
        bool remove_cell = true;
        for (unsigned int d = 0; d < dim && remove_cell; ++d)
          if ((n_cells_to_remove[d] > 0 && cell->center()[d] >= cut_step[d]) ||
              (n_cells_to_remove[d] < 0 && cell->center()[d] <= cut_step[d]))
            remove_cell = false;
        if (remove_cell)
          cells_to_remove.insert(cell);
      }
 
    GridGenerator::create_triangulation_with_removed_cells(rectangle,
                                                           cells_to_remove,
                                                           tria);
  }
 
 
 
  // Implementation for 2d only
  template <>
  void
  hyper_ball(Triangulation<2> &tria,
             const Point<2>   &p,
             const double      radius,
             const bool        internal_manifolds)
  {
    // equilibrate cell sizes at
    // transition from the inner part
    // to the radial cells
    const double   a           = 1. / (1 + std::sqrt(2.0));
    const Point<2> vertices[8] = {
      p + Point<2>(-1, -1) * (radius / std::sqrt(2.0)),
      p + Point<2>(+1, -1) * (radius / std::sqrt(2.0)),
      p + Point<2>(-1, -1) * (radius / std::sqrt(2.0) * a),
      p + Point<2>(+1, -1) * (radius / std::sqrt(2.0) * a),
      p + Point<2>(-1, +1) * (radius / std::sqrt(2.0) * a),
      p + Point<2>(+1, +1) * (radius / std::sqrt(2.0) * a),
      p + Point<2>(-1, +1) * (radius / std::sqrt(2.0)),
      p + Point<2>(+1, +1) * (radius / std::sqrt(2.0))};
 
    std::vector<CellData<2>> cells(5, CellData<2>());
 
    for (unsigned int i = 0; i < 5; ++i)
      {
        for (unsigned int j = 0; j < 4; ++j)
          cells[i].vertices[j] = circle_cell_vertices[i][j];
        cells[i].material_id = 0;
        cells[i].manifold_id = i == 2 ? numbers::flat_manifold_id : 1;
      }
 
    tria.create_triangulation(std::vector<Point<2>>(std::begin(vertices),
                                                    std::end(vertices)),
                              cells,
                              SubCellData()); // no boundary information
    tria.set_all_manifold_ids_on_boundary(0);
    tria.set_manifold(0, SphericalManifold<2>(p));
    if (internal_manifolds)
      tria.set_manifold(1, SphericalManifold<2>(p));
    else
      tria.set_manifold(1, FlatManifold<2>());
  }
 
 
 
  template <int spacedim>
  void
  hyper_shell_2D(Triangulation<2, spacedim> &tria,
                 const Point<spacedim>      &center,
                 const double                inner_radius,
                 const double                outer_radius,
                 const unsigned int          n_cells,
                 const bool                  colorize)
  {
    Assert((inner_radius > 0) && (inner_radius < outer_radius),
           ExcInvalidRadii());
 
    const double pi = numbers::PI;
 
    // determine the number of cells
    // for the grid. if not provided by
    // the user determine it such that
    // the length of each cell on the
    // median (in the middle between
    // the two circles) is equal to its
    // radial extent (which is the
    // difference between the two
    // radii)
    const unsigned int N =
      (n_cells == 0 ? static_cast<unsigned int>(
                        std::ceil((2 * pi * (outer_radius + inner_radius) / 2) /
                                  (outer_radius - inner_radius))) :
                      n_cells);
 
    // set up N vertices on the
    // outer and N vertices on
    // the inner circle. the
    // first N ones are on the
    // outer one, and all are
    // numbered counter-clockwise
    std::vector<Point<spacedim>> vertices(2 * N);
    for (unsigned int i = 0; i < N; ++i)
      {
        Point<spacedim> point;
        point[0] = std::cos(2 * pi * i / N);
        point[1] = std::sin(2 * pi * i / N);
 
        vertices[i]     = point * outer_radius;
        vertices[i + N] = vertices[i] * (inner_radius / outer_radius);
 
        vertices[i] += center;
        vertices[i + N] += center;
      }
 
    std::vector<CellData<2>> cells(N, CellData<2>());
 
    for (unsigned int i = 0; i < N; ++i)
      {
        cells[i].vertices[0] = i;
        cells[i].vertices[1] = (i + 1) % N;
        cells[i].vertices[2] = N + i;
        cells[i].vertices[3] = N + ((i + 1) % N);
 
        cells[i].material_id = 0;
      }
 
    tria.create_triangulation(vertices, cells, SubCellData());
 
    if (colorize)
      colorize_hyper_shell(tria, center, inner_radius, outer_radius);
 
    tria.set_all_manifold_ids(0);
    tria.set_manifold(0, SphericalManifold<2, spacedim>(center));
  }
 
 
 
  template <>
  void
  hyper_shell(Triangulation<2>  &tria,
              const Point<2>    &center,
              const double       inner_radius,
              const double       outer_radius,
              const unsigned int n_cells,
              const bool         colorize)
  {
    hyper_shell_2D(tria, center, inner_radius, outer_radius, n_cells, colorize);
  }
 
 
 
  template <>
  void
  hyper_shell(Triangulation<2, 3> &tria,
              const Point<3>      &center,
              const double         inner_radius,
              const double         outer_radius,
              const unsigned int   n_cells,
              const bool           colorize)
  {
    hyper_shell_2D(tria, center, inner_radius, outer_radius, n_cells, colorize);
  }
 
 
 
  template <int dim>
  void
  eccentric_hyper_shell(Triangulation<dim> &tria,
                        const Point<dim>   &inner_center,
                        const Point<dim>   &outer_center,
                        const double        inner_radius,
                        const double        outer_radius,
                        const unsigned int  n_cells)
  {
    GridGenerator::hyper_shell(
      tria, outer_center, inner_radius, outer_radius, n_cells, true);
 
    // check the consistency of the dimensions provided
    Assert(
      outer_radius - inner_radius > outer_center.distance(inner_center),
      ExcInternalError(
        "The inner radius is greater than or equal to the outer radius plus eccentricity."));
 
    // shift nodes along the inner boundary according to the position of
    // inner_circle
    std::set<Point<dim> *> vertices_to_move;
 
    for (const auto &face : tria.active_face_iterators())
      if (face->boundary_id() == 0)
        for (unsigned int v = 0; v < GeometryInfo<dim>::vertices_per_face; ++v)
          vertices_to_move.insert(&face->vertex(v));
 
    const auto shift = inner_center - outer_center;
    for (const auto &p : vertices_to_move)
      (*p) += shift;
 
    // the original hyper_shell function assigns the same manifold id
    // to all cells and faces. Set all manifolds ids to a different
    // value (2), then use boundary ids to assign different manifolds to
    // the inner (0) and outer manifolds (1). Use a transfinite manifold
    // for all faces and cells aside from the boundaries.
    tria.set_all_manifold_ids(2);
    GridTools::copy_boundary_to_manifold_id(tria);
 
    SphericalManifold<dim> inner_manifold(inner_center);
    SphericalManifold<dim> outer_manifold(outer_center);
 
    tria.set_manifold(0, FlatManifold<dim>());
    tria.set_manifold(1, FlatManifold<dim>());
    tria.set_manifold(2, FlatManifold<dim>());
    TransfiniteInterpolationManifold<dim> transfinite;
    transfinite.initialize(tria);
 
    tria.set_manifold(0, inner_manifold);
    tria.set_manifold(1, outer_manifold);
    tria.set_manifold(2, transfinite);
  }
 
 
 
  // Implementation for 2d only
  template <>
  void
  cylinder(Triangulation<2> &tria,
           const double      radius,
           const double      half_length)
  {
    Point<2> p1(-half_length, -radius);
    Point<2> p2(half_length, radius);
 
    hyper_rectangle(tria, p1, p2, true);
 
    Triangulation<2>::face_iterator f   = tria.begin_face();
    Triangulation<2>::face_iterator end = tria.end_face();
    while (f != end)
      {
        switch (f->boundary_id())
          {
            case 0:
              f->set_boundary_id(1);
              break;
            case 1:
              f->set_boundary_id(2);
              break;
            default:
              f->set_boundary_id(0);
              break;
          }
        ++f;
      }
  }
 
  template <>
  void
  subdivided_cylinder(Triangulation<2> &,
                      const unsigned int,
                      const double,
                      const double)
  {
    DEAL_II_NOT_IMPLEMENTED();
  }
 
 
 
  // Implementation for 2d only
  template <>
  void
  cylinder_shell(Triangulation<2> &,
                 const double,
                 const double,
                 const double,
                 const unsigned int,
                 const unsigned int,
                 const bool)
  {
    DEAL_II_NOT_IMPLEMENTED();
  }
 
 
  template <>
  void
  quarter_hyper_ball(Triangulation<2> &tria,
                     const Point<2>   &p,
                     const double      radius)
  {
    const unsigned int dim = 2;
 
    // the numbers 0.55647 and 0.42883 have been found by a search for the
    // best aspect ratio (defined as the maximal between the minimal singular
    // value of the Jacobian)
    const Point<dim> vertices[7] = {p + Point<dim>(0, 0) * radius,
                                    p + Point<dim>(+1, 0) * radius,
                                    p + Point<dim>(+1, 0) * (radius * 0.55647),
                                    p + Point<dim>(0, +1) * (radius * 0.55647),
                                    p + Point<dim>(+1, +1) * (radius * 0.42883),
                                    p + Point<dim>(0, +1) * radius,
                                    p + Point<dim>(+1, +1) *
                                          (radius / std::sqrt(2.0))};
 
    const int cell_vertices[3][4] = {{0, 2, 3, 4}, {1, 6, 2, 4}, {5, 3, 6, 4}};
 
    std::vector<CellData<dim>> cells(3, CellData<dim>());
 
    for (unsigned int i = 0; i < 3; ++i)
      {
        for (unsigned int j = 0; j < 4; ++j)
          cells[i].vertices[j] = cell_vertices[i][j];
        cells[i].material_id = 0;
      }
 
    tria.create_triangulation(std::vector<Point<dim>>(std::begin(vertices),
                                                      std::end(vertices)),
                              cells,
                              SubCellData()); // no boundary information
 
    Triangulation<dim>::cell_iterator cell = tria.begin();
    Triangulation<dim>::cell_iterator end  = tria.end();
 
    tria.set_all_manifold_ids_on_boundary(0);
 
    while (cell != end)
      {
        for (const unsigned int i : GeometryInfo<dim>::face_indices())
          {
            if (cell->face(i)->boundary_id() ==
                numbers::internal_face_boundary_id)
              continue;
 
            // If one the components is the same as the respective
            // component of the center, then this is part of the plane
            if (cell->face(i)->center()[0] < p[0] + 1.e-5 * radius ||
                cell->face(i)->center()[1] < p[1] + 1.e-5 * radius)
              {
                cell->face(i)->set_boundary_id(1);
                cell->face(i)->set_manifold_id(numbers::flat_manifold_id);
              }
          }
        ++cell;
      }
    tria.set_manifold(0, SphericalManifold<2>(p));
  }
 
 
  template <>
  void
  half_hyper_ball(Triangulation<2> &tria,
                  const Point<2>   &p,
                  const double      radius)
  {
    // equilibrate cell sizes at
    // transition from the inner part
    // to the radial cells
    const double   a           = 1. / (1 + std::sqrt(2.0));
    const Point<2> vertices[8] = {
      p + Point<2>(0, -1) * radius,
      p + Point<2>(+1, -1) * (radius / std::sqrt(2.0)),
      p + Point<2>(0, -1) * (radius / std::sqrt(2.0) * a),
      p + Point<2>(+1, -1) * (radius / std::sqrt(2.0) * a),
      p + Point<2>(0, +1) * (radius / std::sqrt(2.0) * a),
      p + Point<2>(+1, +1) * (radius / std::sqrt(2.0) * a),
      p + Point<2>(0, +1) * radius,
      p + Point<2>(+1, +1) * (radius / std::sqrt(2.0))};
 
    const int cell_vertices[5][4] = {{0, 1, 2, 3},
                                     {2, 3, 4, 5},
                                     {1, 7, 3, 5},
                                     {6, 4, 7, 5}};
 
    std::vector<CellData<2>> cells(4, CellData<2>());
 
    for (unsigned int i = 0; i < 4; ++i)
      {
        for (unsigned int j = 0; j < 4; ++j)
          cells[i].vertices[j] = cell_vertices[i][j];
        cells[i].material_id = 0;
      }
 
    tria.create_triangulation(std::vector<Point<2>>(std::begin(vertices),
                                                    std::end(vertices)),
                              cells,
                              SubCellData()); // no boundary information
 
    Triangulation<2>::cell_iterator cell = tria.begin();
    Triangulation<2>::cell_iterator end  = tria.end();
 
    tria.set_all_manifold_ids_on_boundary(0);
 
    while (cell != end)
      {
        for (const unsigned int i : GeometryInfo<2>::face_indices())
          {
            if (cell->face(i)->boundary_id() ==
                numbers::internal_face_boundary_id)
              continue;
 
            // If x is zero, then this is part of the plane
            if (cell->face(i)->center()[0] < p[0] + 1.e-5 * radius)
              {
                cell->face(i)->set_boundary_id(1);
                cell->face(i)->set_manifold_id(numbers::flat_manifold_id);
              }
          }
        ++cell;
      }
    tria.set_manifold(0, SphericalManifold<2>(p));
  }
 
 
 
  // Implementation for 2d only
  template <>
  void
  half_hyper_shell(Triangulation<2>  &tria,
                   const Point<2>    &center,
                   const double       inner_radius,
                   const double       outer_radius,
                   const unsigned int n_cells,
                   const bool         colorize)
  {
    Assert((inner_radius > 0) && (inner_radius < outer_radius),
           ExcInvalidRadii());
 
    const double pi = numbers::PI;
    // determine the number of cells
    // for the grid. if not provided by
    // the user determine it such that
    // the length of each cell on the
    // median (in the middle between
    // the two circles) is equal to its
    // radial extent (which is the
    // difference between the two
    // radii)
    const unsigned int N =
      (n_cells == 0 ? static_cast<unsigned int>(
                        std::ceil((pi * (outer_radius + inner_radius) / 2) /
                                  (outer_radius - inner_radius))) :
                      n_cells);
 
    // set up N+1 vertices on the
    // outer and N+1 vertices on
    // the inner circle. the
    // first N+1 ones are on the
    // outer one, and all are
    // numbered counter-clockwise
    std::vector<Point<2>> vertices(2 * (N + 1));
    for (unsigned int i = 0; i <= N; ++i)
      {
        // enforce that the x-coordinates
        // of the first and last point of
        // each half-circle are exactly
        // zero (contrary to what we may
        // compute using the imprecise
        // value of pi)
        vertices[i] =
          Point<2>(((i == 0) || (i == N) ? 0 : std::cos(pi * i / N - pi / 2)),
                   std::sin(pi * i / N - pi / 2)) *
          outer_radius;
        vertices[i + N + 1] = vertices[i] * (inner_radius / outer_radius);
 
        vertices[i] += center;
        vertices[i + N + 1] += center;
      }
 
 
    std::vector<CellData<2>> cells(N, CellData<2>());
 
    for (unsigned int i = 0; i < N; ++i)
      {
        cells[i].vertices[0] = i;
        cells[i].vertices[1] = (i + 1) % (N + 1);
        cells[i].vertices[2] = N + 1 + i;
        cells[i].vertices[3] = N + 1 + ((i + 1) % (N + 1));
 
        cells[i].material_id = 0;
      }
 
    tria.create_triangulation(vertices, cells, SubCellData());
 
    if (colorize)
      {
        Triangulation<2>::cell_iterator cell = tria.begin();
        for (; cell != tria.end(); ++cell)
          {
            cell->face(2)->set_boundary_id(1);
          }
        tria.begin()->face(0)->set_boundary_id(3);
 
        tria.last()->face(1)->set_boundary_id(2);
      }
    tria.set_all_manifold_ids(0);
    tria.set_manifold(0, SphericalManifold<2>(center));
  }
 
 
  template <>
  void
  quarter_hyper_shell(Triangulation<2>  &tria,
                      const Point<2>    &center,
                      const double       inner_radius,
                      const double       outer_radius,
                      const unsigned int n_cells,
                      const bool         colorize)
  {
    Assert((inner_radius > 0) && (inner_radius < outer_radius),
           ExcInvalidRadii());
 
    const double pi = numbers::PI;
    // determine the number of cells
    // for the grid. if not provided by
    // the user determine it such that
    // the length of each cell on the
    // median (in the middle between
    // the two circles) is equal to its
    // radial extent (which is the
    // difference between the two
    // radii)
    const unsigned int N =
      (n_cells == 0 ? static_cast<unsigned int>(
                        std::ceil((pi * (outer_radius + inner_radius) / 4) /
                                  (outer_radius - inner_radius))) :
                      n_cells);
 
    // set up N+1 vertices on the
    // outer and N+1 vertices on
    // the inner circle. the
    // first N+1 ones are on the
    // outer one, and all are
    // numbered counter-clockwise
    std::vector<Point<2>> vertices(2 * (N + 1));
    for (unsigned int i = 0; i <= N; ++i)
      {
        // enforce that the x-coordinates
        // of the last point is exactly
        // zero (contrary to what we may
        // compute using the imprecise
        // value of pi)
        vertices[i] = Point<2>(((i == N) ? 0 : std::cos(pi * i / N / 2)),
                               std::sin(pi * i / N / 2)) *
                      outer_radius;
        vertices[i + N + 1] = vertices[i] * (inner_radius / outer_radius);
 
        vertices[i] += center;
        vertices[i + N + 1] += center;
      }
 
 
    std::vector<CellData<2>> cells(N, CellData<2>());
 
    for (unsigned int i = 0; i < N; ++i)
      {
        cells[i].vertices[0] = i;
        cells[i].vertices[1] = (i + 1) % (N + 1);
        cells[i].vertices[2] = N + 1 + i;
        cells[i].vertices[3] = N + 1 + ((i + 1) % (N + 1));
 
        cells[i].material_id = 0;
      }
 
    tria.create_triangulation(vertices, cells, SubCellData());
 
    if (colorize)
      {
        Triangulation<2>::cell_iterator cell = tria.begin();
        for (; cell != tria.end(); ++cell)
          {
            cell->face(2)->set_boundary_id(1);
          }
        tria.begin()->face(0)->set_boundary_id(3);
 
        tria.last()->face(1)->set_boundary_id(2);
      }
 
    tria.set_all_manifold_ids(0);
    tria.set_manifold(0, SphericalManifold<2>(center));
  }
 
 
 
  // Implementation for 3d only
  template <>
  void
  hyper_cube_slit(Triangulation<3> &tria,
                  const double      left,
                  const double      right,
                  const bool        colorize)
  {
    const double rl2 = (right + left) / 2;
    const double len = (right - left) / 2.;
 
    const Point<3> vertices[20] = {
      Point<3>(left, left, -len / 2.),   Point<3>(rl2, left, -len / 2.),
      Point<3>(rl2, rl2, -len / 2.),     Point<3>(left, rl2, -len / 2.),
      Point<3>(right, left, -len / 2.),  Point<3>(right, rl2, -len / 2.),
      Point<3>(rl2, right, -len / 2.),   Point<3>(left, right, -len / 2.),
      Point<3>(right, right, -len / 2.), Point<3>(rl2, left, -len / 2.),
      Point<3>(left, left, len / 2.),    Point<3>(rl2, left, len / 2.),
      Point<3>(rl2, rl2, len / 2.),      Point<3>(left, rl2, len / 2.),
      Point<3>(right, left, len / 2.),   Point<3>(right, rl2, len / 2.),
      Point<3>(rl2, right, len / 2.),    Point<3>(left, right, len / 2.),
      Point<3>(right, right, len / 2.),  Point<3>(rl2, left, len / 2.)};
    const int cell_vertices[4][8] = {{0, 1, 3, 2, 10, 11, 13, 12},
                                     {9, 4, 2, 5, 19, 14, 12, 15},
                                     {3, 2, 7, 6, 13, 12, 17, 16},
                                     {2, 5, 6, 8, 12, 15, 16, 18}};
    std::vector<CellData<3>> cells(4, CellData<3>());
    for (unsigned int i = 0; i < 4; ++i)
      {
        for (unsigned int j = 0; j < 8; ++j)
          cells[i].vertices[j] = cell_vertices[i][j];
        cells[i].material_id = 0;
      }
    tria.create_triangulation(std::vector<Point<3>>(std::begin(vertices),
                                                    std::end(vertices)),
                              cells,
                              SubCellData()); // no boundary information
 
    if (colorize)
      {
        Triangulation<3>::cell_iterator cell = tria.begin();
        cell->face(1)->set_boundary_id(1);
        ++cell;
        cell->face(0)->set_boundary_id(2);
      }
  }
 
 
 
  // Implementation for 3d only
  template <>
  void
  enclosed_hyper_cube(Triangulation<3> &tria,
                      const double      left,
                      const double      right,
                      const double      thickness,
                      const bool        colorize)
  {
    Assert(left < right,
           ExcMessage("Invalid left-to-right bounds of enclosed hypercube"));
 
    std::vector<Point<3>> vertices(64);
    double                coords[4];
    coords[0] = left - thickness;
    coords[1] = left;
    coords[2] = right;
    coords[3] = right + thickness;
 
    unsigned int k = 0;
    for (const double z : coords)
      for (const double y : coords)
        for (const double x : coords)
          vertices[k++] = Point<3>(x, y, z);
 
    const types::material_id materials[27] = {21, 20, 22, 17, 16, 18, 25,
                                              24, 26, 5,  4,  6,  1,  0,
                                              2,  9,  8,  10, 37, 36, 38,
                                              33, 32, 34, 41, 40, 42};
 
    std::vector<CellData<3>> cells(27);
    k = 0;
    for (unsigned int z = 0; z < 3; ++z)
      for (unsigned int y = 0; y < 3; ++y)
        for (unsigned int x = 0; x < 3; ++x)
          {
            cells[k].vertices[0] = x + 4 * y + 16 * z;
            cells[k].vertices[1] = x + 4 * y + 16 * z + 1;
            cells[k].vertices[2] = x + 4 * y + 16 * z + 4;
            cells[k].vertices[3] = x + 4 * y + 16 * z + 5;
            cells[k].vertices[4] = x + 4 * y + 16 * z + 16;
            cells[k].vertices[5] = x + 4 * y + 16 * z + 17;
            cells[k].vertices[6] = x + 4 * y + 16 * z + 20;
            cells[k].vertices[7] = x + 4 * y + 16 * z + 21;
            if (colorize)
              cells[k].material_id = materials[k];
            ++k;
          }
    tria.create_triangulation(vertices,
                              cells,
                              SubCellData()); // no boundary information
  }
 
 
 
  template <>
  void
  truncated_cone(Triangulation<3> &triangulation,
                 const double      radius_0,
                 const double      radius_1,
                 const double      half_length)
  {
    Assert(triangulation.n_cells() == 0,
           ExcMessage("The output triangulation object needs to be empty."));
    Assert(0 < radius_0, ExcMessage("The radii must be positive."));
    Assert(0 < radius_1, ExcMessage("The radii must be positive."));
    Assert(0 < half_length, ExcMessage("The half length must be positive."));
 
    const auto n_slices = 1 + static_cast<unsigned int>(std::ceil(
                                half_length / std::max(radius_0, radius_1)));
 
    Triangulation<2> triangulation_2;
    GridGenerator::hyper_ball(triangulation_2, Point<2>(), radius_0);
    GridGenerator::extrude_triangulation(triangulation_2,
                                         n_slices,
                                         2 * half_length,
                                         triangulation);
    GridTools::rotate(Tensor<1, 3>({0., 1., 0.}), numbers::PI_2, triangulation);
    GridTools::shift(Tensor<1, 3>({-half_length, 0.0, 0.0}), triangulation);
    // At this point we have a cylinder. Multiply the y and z coordinates by a
    // factor that scales (with x) linearly between radius_0 and radius_1 to fix
    // the circle radii and interior points:
    auto shift_radii = [=](const Point<3> &p) {
      const double slope  = (radius_1 / radius_0 - 1.0) / (2.0 * half_length);
      const double factor = slope * (p[0] - -half_length) + 1.0;
      return Point<3>(p[0], factor * p[1], factor * p[2]);
    };
    GridTools::transform(shift_radii, triangulation);
 
    // Set boundary ids at -half_length to 1 and at half_length to 2. Set the
    // manifold id on hull faces (i.e., faces not on either end) to 0.
    for (const auto &face : triangulation.active_face_iterators())
      if (face->at_boundary())
        {
          if (std::abs(face->center()[0] - -half_length) < 1e-8 * half_length)
            face->set_boundary_id(1);
          else if (std::abs(face->center()[0] - half_length) <
                   1e-8 * half_length)
            face->set_boundary_id(2);
          else
            face->set_all_manifold_ids(0);
        }
 
    triangulation.set_manifold(0, CylindricalManifold<3>());
  }
 
 
  // Implementation for 3d only
  template <>
  void
  hyper_L(Triangulation<3> &tria,
          const double      a,
          const double      b,
          const bool        colorize)
  {
    // we slice out the top back right
    // part of the cube
    const Point<3> vertices[26] = {
      // front face of the big cube
      Point<3>(a, a, a),
      Point<3>((a + b) / 2, a, a),
      Point<3>(b, a, a),
      Point<3>(a, a, (a + b) / 2),
      Point<3>((a + b) / 2, a, (a + b) / 2),
      Point<3>(b, a, (a + b) / 2),
      Point<3>(a, a, b),
      Point<3>((a + b) / 2, a, b),
      Point<3>(b, a, b),
      // middle face of the big cube
      Point<3>(a, (a + b) / 2, a),
      Point<3>((a + b) / 2, (a + b) / 2, a),
      Point<3>(b, (a + b) / 2, a),
      Point<3>(a, (a + b) / 2, (a + b) / 2),
      Point<3>((a + b) / 2, (a + b) / 2, (a + b) / 2),
      Point<3>(b, (a + b) / 2, (a + b) / 2),
      Point<3>(a, (a + b) / 2, b),
      Point<3>((a + b) / 2, (a + b) / 2, b),
      Point<3>(b, (a + b) / 2, b),
      // back face of the big cube
      // last (top right) point is missing
      Point<3>(a, b, a),
      Point<3>((a + b) / 2, b, a),
      Point<3>(b, b, a),
      Point<3>(a, b, (a + b) / 2),
      Point<3>((a + b) / 2, b, (a + b) / 2),
      Point<3>(b, b, (a + b) / 2),
      Point<3>(a, b, b),
      Point<3>((a + b) / 2, b, b)};
    const int cell_vertices[7][8] = {{0, 1, 9, 10, 3, 4, 12, 13},
                                     {1, 2, 10, 11, 4, 5, 13, 14},
                                     {3, 4, 12, 13, 6, 7, 15, 16},
                                     {4, 5, 13, 14, 7, 8, 16, 17},
                                     {9, 10, 18, 19, 12, 13, 21, 22},
                                     {10, 11, 19, 20, 13, 14, 22, 23},
                                     {12, 13, 21, 22, 15, 16, 24, 25}};
 
    std::vector<CellData<3>> cells(7, CellData<3>());
 
    for (unsigned int i = 0; i < 7; ++i)
      {
        for (unsigned int j = 0; j < 8; ++j)
          cells[i].vertices[j] = cell_vertices[i][j];
        cells[i].material_id = 0;
      }
 
    tria.create_triangulation(std::vector<Point<3>>(std::begin(vertices),
                                                    std::end(vertices)),
                              cells,
                              SubCellData()); // no boundary information
 
    if (colorize)
      {
        DEAL_II_NOT_IMPLEMENTED();
      }
  }
 
 
 
  // Implementation for 3d only
  template <>
  void
  hyper_ball(Triangulation<3> &tria,
             const Point<3>   &p,
             const double      radius,
             const bool        internal_manifold)
  {
    const double a =
      1. / (1 + std::sqrt(3.0)); // equilibrate cell sizes at transition
    // from the inner part to the radial
    // cells
    const unsigned int n_vertices           = 16;
    const Point<3>     vertices[n_vertices] = {
      // first the vertices of the inner
      // cell
      p + Point<3>(-1, -1, -1) * (radius / std::sqrt(3.0) * a),
      p + Point<3>(+1, -1, -1) * (radius / std::sqrt(3.0) * a),
      p + Point<3>(+1, -1, +1) * (radius / std::sqrt(3.0) * a),
      p + Point<3>(-1, -1, +1) * (radius / std::sqrt(3.0) * a),
      p + Point<3>(-1, +1, -1) * (radius / std::sqrt(3.0) * a),
      p + Point<3>(+1, +1, -1) * (radius / std::sqrt(3.0) * a),
      p + Point<3>(+1, +1, +1) * (radius / std::sqrt(3.0) * a),
      p + Point<3>(-1, +1, +1) * (radius / std::sqrt(3.0) * a),
      // now the eight vertices at
      // the outer sphere
      p + Point<3>(-1, -1, -1) * (radius / std::sqrt(3.0)),
      p + Point<3>(+1, -1, -1) * (radius / std::sqrt(3.0)),
      p + Point<3>(+1, -1, +1) * (radius / std::sqrt(3.0)),
      p + Point<3>(-1, -1, +1) * (radius / std::sqrt(3.0)),
      p + Point<3>(-1, +1, -1) * (radius / std::sqrt(3.0)),
      p + Point<3>(+1, +1, -1) * (radius / std::sqrt(3.0)),
      p + Point<3>(+1, +1, +1) * (radius / std::sqrt(3.0)),
      p + Point<3>(-1, +1, +1) * (radius / std::sqrt(3.0)),
    };
 
    // one needs to draw the seven cubes to
    // understand what's going on here
    const unsigned int n_cells                   = 7;
    const int          cell_vertices[n_cells][8] = {
      {0, 1, 4, 5, 3, 2, 7, 6},      // center
      {8, 9, 12, 13, 0, 1, 4, 5},    // bottom
      {9, 13, 1, 5, 10, 14, 2, 6},   // right
      {11, 10, 3, 2, 15, 14, 7, 6},  // top
      {8, 0, 12, 4, 11, 3, 15, 7},   // left
      {8, 9, 0, 1, 11, 10, 3, 2},    // front
      {12, 4, 13, 5, 15, 7, 14, 6}}; // back
 
    std::vector<CellData<3>> cells(n_cells, CellData<3>());
 
    for (unsigned int i = 0; i < n_cells; ++i)
      {
        for (const unsigned int j : GeometryInfo<3>::vertex_indices())
          cells[i].vertices[j] = cell_vertices[i][j];
        cells[i].material_id = 0;
        cells[i].manifold_id = i == 0 ? numbers::flat_manifold_id : 1;
      }
 
    tria.create_triangulation(std::vector<Point<3>>(std::begin(vertices),
                                                    std::end(vertices)),
                              cells,
                              SubCellData()); // no boundary information
    tria.set_all_manifold_ids_on_boundary(0);
    tria.set_manifold(0, SphericalManifold<3>(p));
    if (internal_manifold)
      tria.set_manifold(1, SphericalManifold<3>(p));
    else
      tria.set_manifold(1, FlatManifold<3>());
  }
 
 
 
  void
  non_standard_orientation_mesh(Triangulation<2>  &tria,
                                const unsigned int n_rotate_middle_square)
  {
    AssertThrow(n_rotate_middle_square < 4,
                ExcMessage("The number of rotation by pi/2 of the right square "
                           "must be in the half-open range [0,4)."));
 
    constexpr unsigned int dim = 2;
 
    const unsigned int         n_cells = 5;
    std::vector<CellData<dim>> cells(n_cells);
 
    // Corner points of the cube [0,1]^2
    const std::vector<Point<dim>> vertices = {Point<dim>(0, 0),  // 0
                                              Point<dim>(1, 0),  // 1
                                              Point<dim>(0, 1),  // 2
                                              Point<dim>(1, 1),  // 3
                                              Point<dim>(2, 0),  // 4
                                              Point<dim>(2, 1),  // 5
                                              Point<dim>(3, 0),  // 6
                                              Point<dim>(3, 1),  // 7
                                              Point<dim>(1, -1), // 8
                                              Point<dim>(2, -1), // 9
                                              Point<dim>(1, 2),  // 10
                                              Point<dim>(2, 2)}; // 11
 
 
    // consistent orientation
    unsigned int cell_vertices[n_cells][4] = {{0, 1, 2, 3},
                                              {1, 4, 3, 5}, // rotating cube
                                              {8, 9, 1, 4},
                                              {4, 6, 5, 7},
                                              {3, 5, 10, 11}};
 
    switch (n_rotate_middle_square)
      {
        case /* rotate right square */ 1:
          {
            cell_vertices[1][0] = 4;
            cell_vertices[1][1] = 5;
            cell_vertices[1][2] = 1;
            cell_vertices[1][3] = 3;
            break;
          }
 
        case /* rotate right square */ 2:
          {
            cell_vertices[1][0] = 5;
            cell_vertices[1][1] = 3;
            cell_vertices[1][2] = 4;
            cell_vertices[1][3] = 1;
            break;
          }
 
        case /* rotate right square */ 3:
          {
            cell_vertices[1][0] = 3;
            cell_vertices[1][1] = 1;
            cell_vertices[1][2] = 5;
            cell_vertices[1][3] = 4;
            break;
          }
 
        default /* 0 */:
          break;
      } // switch
 
    cells.resize(n_cells, CellData<dim>());
 
    for (unsigned int cell_index = 0; cell_index < n_cells; ++cell_index)
      {
        for (const unsigned int vertex_index :
             GeometryInfo<dim>::vertex_indices())
          {
            cells[cell_index].vertices[vertex_index] =
              cell_vertices[cell_index][vertex_index];
            cells[cell_index].material_id = 0;
          }
      }
 
    tria.create_triangulation(vertices, cells, SubCellData());
  }
 
 
  void
  non_standard_orientation_mesh(Triangulation<3> &tria,
                                const bool        face_orientation,
                                const bool        face_flip,
                                const bool        face_rotation,
                                const bool        manipulate_left_cube)
  {
    constexpr unsigned int dim = 3;
 
    const unsigned int         n_cells = 2;
    std::vector<CellData<dim>> cells(n_cells);
 
    // Corner points of the cube [0,1]^3
    const std::vector<Point<dim>> vertices = {Point<dim>(0, 0, 0),  // 0
                                              Point<dim>(1, 0, 0),  // 1
                                              Point<dim>(0, 1, 0),  // 2
                                              Point<dim>(1, 1, 0),  // 3
                                              Point<dim>(0, 0, 1),  // 4
                                              Point<dim>(1, 0, 1),  // 5
                                              Point<dim>(0, 1, 1),  // 6
                                              Point<dim>(1, 1, 1),  // 7
                                              Point<dim>(2, 0, 0),  // 8
                                              Point<dim>(2, 1, 0),  // 9
                                              Point<dim>(2, 0, 1),  // 10
                                              Point<dim>(2, 1, 1)}; // 11
 
    unsigned int cell_vertices[n_cells][8] = {
      {0, 1, 2, 3, 4, 5, 6, 7},    // unit cube
      {1, 8, 3, 9, 5, 10, 7, 11}}; // shifted cube
 
    // binary to case number
    const unsigned int this_case = 4 * static_cast<int>(face_orientation) +
                                   2 * static_cast<int>(face_flip) +
                                   static_cast<int>(face_rotation);
 
    if (manipulate_left_cube)
      {
        switch (this_case)
          {
            case 0:
              {
                cell_vertices[0][0] = 1;
                cell_vertices[0][1] = 0;
                cell_vertices[0][2] = 5;
                cell_vertices[0][3] = 4;
                cell_vertices[0][4] = 3;
                cell_vertices[0][5] = 2;
                cell_vertices[0][6] = 7;
                cell_vertices[0][7] = 6;
                break;
              }
 
            case 1:
              {
                cell_vertices[0][0] = 5;
                cell_vertices[0][1] = 4;
                cell_vertices[0][2] = 7;
                cell_vertices[0][3] = 6;
                cell_vertices[0][4] = 1;
                cell_vertices[0][5] = 0;
                cell_vertices[0][6] = 3;
                cell_vertices[0][7] = 2;
                break;
              }
 
            case 2:
              {
                cell_vertices[0][0] = 7;
                cell_vertices[0][1] = 6;
                cell_vertices[0][2] = 3;
                cell_vertices[0][3] = 2;
                cell_vertices[0][4] = 5;
                cell_vertices[0][5] = 4;
                cell_vertices[0][6] = 1;
                cell_vertices[0][7] = 0;
                break;
              }
            case 3:
              {
                cell_vertices[0][0] = 3;
                cell_vertices[0][1] = 2;
                cell_vertices[0][2] = 1;
                cell_vertices[0][3] = 0;
                cell_vertices[0][4] = 7;
                cell_vertices[0][5] = 6;
                cell_vertices[0][6] = 5;
                cell_vertices[0][7] = 4;
                break;
              }
 
            case 4:
              {
                cell_vertices[0][0] = 0;
                cell_vertices[0][1] = 1;
                cell_vertices[0][2] = 2;
                cell_vertices[0][3] = 3;
                cell_vertices[0][4] = 4;
                cell_vertices[0][5] = 5;
                cell_vertices[0][6] = 6;
                cell_vertices[0][7] = 7;
                break;
              }
 
            case 5:
              {
                cell_vertices[0][0] = 2;
                cell_vertices[0][1] = 3;
                cell_vertices[0][2] = 6;
                cell_vertices[0][3] = 7;
                cell_vertices[0][4] = 0;
                cell_vertices[0][5] = 1;
                cell_vertices[0][6] = 4;
                cell_vertices[0][7] = 5;
                break;
              }
 
            case 6:
              {
                cell_vertices[0][0] = 6;
                cell_vertices[0][1] = 7;
                cell_vertices[0][2] = 4;
                cell_vertices[0][3] = 5;
                cell_vertices[0][4] = 2;
                cell_vertices[0][5] = 3;
                cell_vertices[0][6] = 0;
                cell_vertices[0][7] = 1;
                break;
              }
 
            case 7:
              {
                cell_vertices[0][0] = 4;
                cell_vertices[0][1] = 5;
                cell_vertices[0][2] = 0;
                cell_vertices[0][3] = 1;
                cell_vertices[0][4] = 6;
                cell_vertices[0][5] = 7;
                cell_vertices[0][6] = 2;
                cell_vertices[0][7] = 3;
                break;
              }
          } // switch
      }
    else
      {
        switch (this_case)
          {
            case 0:
              {
                cell_vertices[1][0] = 8;
                cell_vertices[1][1] = 1;
                cell_vertices[1][2] = 10;
                cell_vertices[1][3] = 5;
                cell_vertices[1][4] = 9;
                cell_vertices[1][5] = 3;
                cell_vertices[1][6] = 11;
                cell_vertices[1][7] = 7;
                break;
              }
 
            case 1:
              {
                cell_vertices[1][0] = 10;
                cell_vertices[1][1] = 5;
                cell_vertices[1][2] = 11;
                cell_vertices[1][3] = 7;
                cell_vertices[1][4] = 8;
                cell_vertices[1][5] = 1;
                cell_vertices[1][6] = 9;
                cell_vertices[1][7] = 3;
                break;
              }
 
            case 2:
              {
                cell_vertices[1][0] = 11;
                cell_vertices[1][1] = 7;
                cell_vertices[1][2] = 9;
                cell_vertices[1][3] = 3;
                cell_vertices[1][4] = 10;
                cell_vertices[1][5] = 5;
                cell_vertices[1][6] = 8;
                cell_vertices[1][7] = 1;
                break;
              }
 
            case 3:
              {
                cell_vertices[1][0] = 9;
                cell_vertices[1][1] = 3;
                cell_vertices[1][2] = 8;
                cell_vertices[1][3] = 1;
                cell_vertices[1][4] = 11;
                cell_vertices[1][5] = 7;
                cell_vertices[1][6] = 10;
                cell_vertices[1][7] = 5;
                break;
              }
 
            case 4:
              {
                cell_vertices[1][0] = 1;
                cell_vertices[1][1] = 8;
                cell_vertices[1][2] = 3;
                cell_vertices[1][3] = 9;
                cell_vertices[1][4] = 5;
                cell_vertices[1][5] = 10;
                cell_vertices[1][6] = 7;
                cell_vertices[1][7] = 11;
                break;
              }
 
            case 5:
              {
                cell_vertices[1][0] = 5;
                cell_vertices[1][1] = 10;
                cell_vertices[1][2] = 1;
                cell_vertices[1][3] = 8;
                cell_vertices[1][4] = 7;
                cell_vertices[1][5] = 11;
                cell_vertices[1][6] = 3;
                cell_vertices[1][7] = 9;
                break;
              }
 
            case 6:
              {
                cell_vertices[1][0] = 7;
                cell_vertices[1][1] = 11;
                cell_vertices[1][2] = 5;
                cell_vertices[1][3] = 10;
                cell_vertices[1][4] = 3;
                cell_vertices[1][5] = 9;
                cell_vertices[1][6] = 1;
                cell_vertices[1][7] = 8;
                break;
              }
 
            case 7:
              {
                cell_vertices[1][0] = 3;
                cell_vertices[1][1] = 9;
                cell_vertices[1][2] = 7;
                cell_vertices[1][3] = 11;
                cell_vertices[1][4] = 1;
                cell_vertices[1][5] = 8;
                cell_vertices[1][6] = 5;
                cell_vertices[1][7] = 10;
                break;
              }
          } // switch
      }
 
    cells.resize(n_cells, CellData<dim>());
 
    for (unsigned int cell_index = 0; cell_index < n_cells; ++cell_index)
      {
        for (const unsigned int vertex_index :
             GeometryInfo<dim>::vertex_indices())
          {
            cells[cell_index].vertices[vertex_index] =
              cell_vertices[cell_index][vertex_index];
            cells[cell_index].material_id = 0;
          }
      }
 
    tria.create_triangulation(vertices, cells, SubCellData());
  }
 
 
 
  template <int spacedim>
  void
  hyper_sphere(Triangulation<spacedim - 1, spacedim> &tria,
               const Point<spacedim>                 &p,
               const double                           radius)
  {
    Triangulation<spacedim> volume_mesh;
    GridGenerator::hyper_ball(volume_mesh, p, radius);
    const std::set<types::boundary_id> boundary_ids = {0};
    GridGenerator::extract_boundary_mesh(volume_mesh, tria, boundary_ids);
    tria.set_all_manifold_ids(0);
    tria.set_manifold(0, SphericalManifold<spacedim - 1, spacedim>(p));
  }
 
 
 
  // Implementation for 3d only
  template <>
  void
  subdivided_cylinder(Triangulation<3>  &tria,
                      const unsigned int x_subdivisions,
                      const double       radius,
                      const double       half_length)
  {
    // Copy the base from hyper_ball<3>
    // and transform it to yz
    const double d = radius / std::sqrt(2.0);
    const double a = d / (1 + std::sqrt(2.0));
 
    std::vector<Point<3>> vertices;
    const double          initial_height   = -half_length;
    const double          height_increment = 2. * half_length / x_subdivisions;
 
    for (unsigned int rep = 0; rep < (x_subdivisions + 1); ++rep)
      {
        const double height = initial_height + height_increment * rep;
 
        vertices.emplace_back(-d, height, -d);
        vertices.emplace_back(d, height, -d);
        vertices.emplace_back(-a, height, -a);
        vertices.emplace_back(a, height, -a);
        vertices.emplace_back(-a, height, a);
        vertices.emplace_back(a, height, a);
        vertices.emplace_back(-d, height, d);
        vertices.emplace_back(d, height, d);
      }
 
    // Turn cylinder such that y->x
    for (auto &vertex : vertices)
      {
        const double h = vertex[1];
        vertex[1]      = -vertex[0];
        vertex[0]      = h;
      }
 
    std::vector<std::vector<int>> cell_vertices;
    cell_vertices.push_back({0, 1, 8, 9, 2, 3, 10, 11});
    cell_vertices.push_back({0, 2, 8, 10, 6, 4, 14, 12});
    cell_vertices.push_back({2, 3, 10, 11, 4, 5, 12, 13});
    cell_vertices.push_back({1, 7, 9, 15, 3, 5, 11, 13});
    cell_vertices.push_back({6, 4, 14, 12, 7, 5, 15, 13});
 
    for (unsigned int rep = 1; rep < x_subdivisions; ++rep)
      {
        for (unsigned int i = 0; i < 5; ++i)
          {
            std::vector<int> new_cell_vertices(8);
            for (unsigned int j = 0; j < 8; ++j)
              new_cell_vertices[j] = cell_vertices[i][j] + 8 * rep;
            cell_vertices.push_back(new_cell_vertices);
          }
      }
 
    unsigned int n_cells = x_subdivisions * 5;
 
    std::vector<CellData<3>> cells(n_cells, CellData<3>());
 
    for (unsigned int i = 0; i < n_cells; ++i)
      {
        for (unsigned int j = 0; j < 8; ++j)
          cells[i].vertices[j] = cell_vertices[i][j];
        cells[i].material_id = 0;
      }
 
    tria.create_triangulation(std::vector<Point<3>>(std::begin(vertices),
                                                    std::end(vertices)),
                              cells,
                              SubCellData()); // no boundary information
 
    // set boundary indicators for the
    // faces at the ends to 1 and 2,
    // respectively. note that we also
    // have to deal with those lines
    // that are purely in the interior
    // of the ends. we determine whether
    // an edge is purely in the
    // interior if one of its vertices
    // is at coordinates '+-a' as set
    // above
    tria.set_all_manifold_ids_on_boundary(0);
 
    // Tolerance is calculated using the minimal length defining
    // the cylinder
    const double tolerance = 1e-5 * std::min(radius, half_length);
 
    for (const auto &cell : tria.cell_iterators())
      for (const unsigned int i : GeometryInfo<3>::face_indices())
        if (cell->at_boundary(i))
          {
            if (cell->face(i)->center()[0] > half_length - tolerance)
              {
                cell->face(i)->set_boundary_id(2);
                cell->face(i)->set_manifold_id(numbers::flat_manifold_id);
 
                for (unsigned int e = 0; e < GeometryInfo<3>::lines_per_face;
                     ++e)
                  if ((std::fabs(cell->face(i)->line(e)->vertex(0)[1]) == a) ||
                      (std::fabs(cell->face(i)->line(e)->vertex(0)[2]) == a) ||
                      (std::fabs(cell->face(i)->line(e)->vertex(1)[1]) == a) ||
                      (std::fabs(cell->face(i)->line(e)->vertex(1)[2]) == a))
                    {
                      cell->face(i)->line(e)->set_boundary_id(2);
                      cell->face(i)->line(e)->set_manifold_id(
                        numbers::flat_manifold_id);
                    }
              }
            else if (cell->face(i)->center()[0] < -half_length + tolerance)
              {
                cell->face(i)->set_boundary_id(1);
                cell->face(i)->set_manifold_id(numbers::flat_manifold_id);
 
                for (unsigned int e = 0; e < GeometryInfo<3>::lines_per_face;
                     ++e)
                  if ((std::fabs(cell->face(i)->line(e)->vertex(0)[1]) == a) ||
                      (std::fabs(cell->face(i)->line(e)->vertex(0)[2]) == a) ||
                      (std::fabs(cell->face(i)->line(e)->vertex(1)[1]) == a) ||
                      (std::fabs(cell->face(i)->line(e)->vertex(1)[2]) == a))
                    {
                      cell->face(i)->line(e)->set_boundary_id(1);
                      cell->face(i)->line(e)->set_manifold_id(
                        numbers::flat_manifold_id);
                    }
              }
          }
    tria.set_manifold(0, CylindricalManifold<3>());
  }
 
  // Implementation for 3d only
  template <>
  void
  cylinder(Triangulation<3> &tria,
           const double      radius,
           const double      half_length)
  {
    subdivided_cylinder(tria, 2, radius, half_length);
  }
 
  template <>
  void
  quarter_hyper_ball(Triangulation<3> &tria,
                     const Point<3>   &center,
                     const double      radius)
  {
    const unsigned int dim = 3;
 
    // the parameters a (intersection on the octant lines from center), b
    // (intersection within the octant faces) and c (position inside the
    // octant) have been derived by equilibrating the minimal singular value
    // of the Jacobian of the four cells around the center point c and, as a
    // secondary measure, to minimize the aspect ratios defined as the maximal
    // divided by the minimal singular values throughout cells
    const double     a            = 0.528;
    const double     b            = 0.4533;
    const double     c            = 0.3752;
    const Point<dim> vertices[15] = {
      center + Point<dim>(0, 0, 0) * radius,
      center + Point<dim>(+1, 0, 0) * radius,
      center + Point<dim>(+1, 0, 0) * (radius * a),
      center + Point<dim>(0, +1, 0) * (radius * a),
      center + Point<dim>(+1, +1, 0) * (radius * b),
      center + Point<dim>(0, +1, 0) * radius,
      center + Point<dim>(+1, +1, 0) * radius / std::sqrt(2.0),
      center + Point<dim>(0, 0, 1) * radius * a,
      center + Point<dim>(+1, 0, 1) * radius / std::sqrt(2.0),
      center + Point<dim>(+1, 0, 1) * (radius * b),
      center + Point<dim>(0, +1, 1) * (radius * b),
      center + Point<dim>(+1, +1, 1) * (radius * c),
      center + Point<dim>(0, +1, 1) * radius / std::sqrt(2.0),
      center + Point<dim>(+1, +1, 1) * (radius / (std::sqrt(3.0))),
      center + Point<dim>(0, 0, 1) * radius};
    const int cell_vertices[4][8] = {{0, 2, 3, 4, 7, 9, 10, 11},
                                     {1, 6, 2, 4, 8, 13, 9, 11},
                                     {5, 3, 6, 4, 12, 10, 13, 11},
                                     {7, 9, 10, 11, 14, 8, 12, 13}};
 
    std::vector<CellData<dim>> cells(4, CellData<dim>());
 
    for (unsigned int i = 0; i < 4; ++i)
      {
        for (unsigned int j = 0; j < 8; ++j)
          cells[i].vertices[j] = cell_vertices[i][j];
        cells[i].material_id = 0;
      }
 
    tria.create_triangulation(std::vector<Point<dim>>(std::begin(vertices),
                                                      std::end(vertices)),
                              cells,
                              SubCellData()); // no boundary information
 
    Triangulation<dim>::cell_iterator cell = tria.begin();
    Triangulation<dim>::cell_iterator end  = tria.end();
 
    tria.set_all_manifold_ids_on_boundary(0);
    while (cell != end)
      {
        for (const unsigned int i : GeometryInfo<dim>::face_indices())
          {
            if (cell->face(i)->boundary_id() ==
                numbers::internal_face_boundary_id)
              continue;
 
            // If x,y or z is zero, then this is part of the plane
            if (cell->face(i)->center()[0] < center[0] + 1.e-5 * radius ||
                cell->face(i)->center()[1] < center[1] + 1.e-5 * radius ||
                cell->face(i)->center()[2] < center[2] + 1.e-5 * radius)
              {
                cell->face(i)->set_boundary_id(1);
                cell->face(i)->set_manifold_id(numbers::flat_manifold_id);
                // also set the boundary indicators of the bounding lines,
                // unless both vertices are on the perimeter
                for (unsigned int j = 0; j < GeometryInfo<3>::lines_per_face;
                     ++j)
                  {
                    const Point<3> line_vertices[2] = {
                      cell->face(i)->line(j)->vertex(0),
                      cell->face(i)->line(j)->vertex(1)};
                    if ((std::fabs(line_vertices[0].distance(center) - radius) >
                         1e-5 * radius) ||
                        (std::fabs(line_vertices[1].distance(center) - radius) >
                         1e-5 * radius))
                      {
                        cell->face(i)->line(j)->set_boundary_id(1);
                        cell->face(i)->line(j)->set_manifold_id(
                          numbers::flat_manifold_id);
                      }
                  }
              }
          }
        ++cell;
      }
    tria.set_manifold(0, SphericalManifold<3>(center));
  }
 
 
 
  // Implementation for 3d only
  template <>
  void
  half_hyper_ball(Triangulation<3> &tria,
                  const Point<3>   &center,
                  const double      radius)
  {
    // These are for the two lower squares
    const double d = radius / std::sqrt(2.0);
    const double a = d / (1 + std::sqrt(2.0));
    // These are for the two upper square
    const double b = a / 2.0;
    const double c = d / 2.0;
    // And so are these
    const double hb = radius * std::sqrt(3.0) / 4.0;
    const double hc = radius * std::sqrt(3.0) / 2.0;
 
    Point<3> vertices[16] = {
      center + Point<3>(0, d, -d),
      center + Point<3>(0, -d, -d),
      center + Point<3>(0, a, -a),
      center + Point<3>(0, -a, -a),
      center + Point<3>(0, a, a),
      center + Point<3>(0, -a, a),
      center + Point<3>(0, d, d),
      center + Point<3>(0, -d, d),
 
      center + Point<3>(hc, c, -c),
      center + Point<3>(hc, -c, -c),
      center + Point<3>(hb, b, -b),
      center + Point<3>(hb, -b, -b),
      center + Point<3>(hb, b, b),
      center + Point<3>(hb, -b, b),
      center + Point<3>(hc, c, c),
      center + Point<3>(hc, -c, c),
    };
 
    int cell_vertices[6][8] = {{0, 1, 8, 9, 2, 3, 10, 11},
                               {0, 2, 8, 10, 6, 4, 14, 12},
                               {2, 3, 10, 11, 4, 5, 12, 13},
                               {1, 7, 9, 15, 3, 5, 11, 13},
                               {6, 4, 14, 12, 7, 5, 15, 13},
                               {8, 10, 9, 11, 14, 12, 15, 13}};
 
    std::vector<CellData<3>> cells(6, CellData<3>());
 
    for (unsigned int i = 0; i < 6; ++i)
      {
        for (unsigned int j = 0; j < 8; ++j)
          cells[i].vertices[j] = cell_vertices[i][j];
        cells[i].material_id = 0;
      }
 
    tria.create_triangulation(std::vector<Point<3>>(std::begin(vertices),
                                                    std::end(vertices)),
                              cells,
                              SubCellData()); // no boundary information
 
    Triangulation<3>::cell_iterator cell = tria.begin();
    Triangulation<3>::cell_iterator end  = tria.end();
 
    tria.set_all_manifold_ids_on_boundary(0);
 
    // go over all faces. for the ones on the flat face, set boundary
    // indicator for face and edges to one; the rest will remain at
    // zero but we have to pay attention to those edges that are
    // at the perimeter of the flat face since they should not be
    // set to one
    while (cell != end)
      {
        for (const unsigned int i : GeometryInfo<3>::face_indices())
          {
            if (!cell->at_boundary(i))
              continue;
 
            // If the center is on the plane x=0, this is a planar element. set
            // its boundary indicator. also set the boundary indicators of the
            // bounding faces unless both vertices are on the perimeter
            if (cell->face(i)->center()[0] < center[0] + 1.e-5 * radius)
              {
                cell->face(i)->set_boundary_id(1);
                cell->face(i)->set_manifold_id(numbers::flat_manifold_id);
                for (unsigned int j = 0; j < GeometryInfo<3>::lines_per_face;
                     ++j)
                  {
                    const Point<3> line_vertices[2] = {
                      cell->face(i)->line(j)->vertex(0),
                      cell->face(i)->line(j)->vertex(1)};
                    if ((std::fabs(line_vertices[0].distance(center) - radius) >
                         1e-5 * radius) ||
                        (std::fabs(line_vertices[1].distance(center) - radius) >
                         1e-5 * radius))
                      {
                        cell->face(i)->line(j)->set_boundary_id(1);
                        cell->face(i)->line(j)->set_manifold_id(
                          numbers::flat_manifold_id);
                      }
                  }
              }
          }
        ++cell;
      }
    tria.set_manifold(0, SphericalManifold<3>(center));
  }
 
 
 
  template <int dim>
  void
  hyper_ball_balanced(Triangulation<dim> &tria,
                      const Point<dim>   &p,
                      const double        radius)
  {
    // We create the ball by duplicating the information in each dimension at
    // a time by appropriate rotations, starting from the quarter ball. The
    // rotations make sure we do not generate inverted cells that would appear
    // if we tried the slightly simpler approach to simply mirror the cells.
    //
    // Make the rotations easy by centering at the origin now and shifting by p
    // later.
 
    Triangulation<dim> tria_piece;
    GridGenerator::quarter_hyper_ball(tria_piece, Point<dim>(), radius);
 
    for (unsigned int round = 0; round < dim; ++round)
      {
        Triangulation<dim> tria_copy;
        tria_copy.copy_triangulation(tria_piece);
        tria_piece.clear();
        std::vector<Point<dim>> new_points(tria_copy.n_vertices());
        if (round == 0)
          for (unsigned int v = 0; v < tria_copy.n_vertices(); ++v)
            {
              // rotate by 90 degrees counterclockwise
              new_points[v][0] = -tria_copy.get_vertices()[v][1];
              new_points[v][1] = tria_copy.get_vertices()[v][0];
              if (dim == 3)
                new_points[v][2] = tria_copy.get_vertices()[v][2];
            }
        else if (round == 1)
          {
            for (unsigned int v = 0; v < tria_copy.n_vertices(); ++v)
              {
                // rotate by 180 degrees along the xy plane
                new_points[v][0] = -tria_copy.get_vertices()[v][0];
                new_points[v][1] = -tria_copy.get_vertices()[v][1];
                if (dim == 3)
                  new_points[v][2] = tria_copy.get_vertices()[v][2];
              }
          }
        else if (round == 2)
          for (unsigned int v = 0; v < tria_copy.n_vertices(); ++v)
            {
              // rotate by 180 degrees along the xz plane
              Assert(dim == 3, ExcInternalError());
              new_points[v][0] = -tria_copy.get_vertices()[v][0];
              new_points[v][1] = tria_copy.get_vertices()[v][1];
              new_points[v][2] = -tria_copy.get_vertices()[v][2];
            }
        else
          DEAL_II_ASSERT_UNREACHABLE();
 
 
        // the cell data is exactly the same as before
        std::vector<CellData<dim>> cells;
        cells.reserve(tria_copy.n_cells());
        for (const auto &cell : tria_copy.cell_iterators())
          {
            CellData<dim> data;
            for (const unsigned int v : GeometryInfo<dim>::vertex_indices())
              data.vertices[v] = cell->vertex_index(v);
            data.material_id = cell->material_id();
            data.manifold_id = cell->manifold_id();
            cells.push_back(data);
          }
 
        Triangulation<dim> rotated_tria;
        rotated_tria.create_triangulation(new_points, cells, SubCellData());
 
        // merge the triangulations - this will make sure that the duplicate
        // vertices in the interior are absorbed
        if (round == dim - 1)
          merge_triangulations(tria_copy, rotated_tria, tria, 1e-12 * radius);
        else
          merge_triangulations(tria_copy,
                               rotated_tria,
                               tria_piece,
                               1e-12 * radius);
      }
 
    for (const auto &cell : tria.cell_iterators())
      if (cell->center().norm_square() > 0.4 * radius)
        cell->set_manifold_id(1);
      else
        cell->set_all_manifold_ids(numbers::flat_manifold_id);
    GridTools::shift(p, tria);
 
    tria.set_manifold(1, FlatManifold<dim>());
 
    tria.set_all_manifold_ids_on_boundary(0);
    tria.set_manifold(0, SphericalManifold<dim>(p));
  }
 
  // To work around an internal clang-13 error we need to split up the
  // individual hyper shell functions. This has the added bonus of making the
  // control flow easier to follow - some hyper shell functions call others.
  namespace internal
  {
    namespace
    {
      void
      hyper_shell_6(Triangulation<3> &tria,
                    const Point<3>   &p,
                    const double      inner_radius,
                    const double      outer_radius)
      {
        std::vector<Point<3>>    vertices;
        std::vector<CellData<3>> cells;
 
        const double irad = inner_radius / std::sqrt(3.0);
        const double orad = outer_radius / std::sqrt(3.0);
 
        // Corner points of the cube [-1,1]^3
        static const std::array<Point<3>, 8> hexahedron = {{{-1, -1, -1}, //
                                                            {+1, -1, -1}, //
                                                            {-1, +1, -1}, //
                                                            {+1, +1, -1}, //
                                                            {-1, -1, +1}, //
                                                            {+1, -1, +1}, //
                                                            {-1, +1, +1}, //
                                                            {+1, +1, +1}}};
 
        // Start with the shell bounded by two nested cubes
        for (unsigned int i = 0; i < 8; ++i)
          vertices.push_back(p + hexahedron[i] * irad);
        for (unsigned int i = 0; i < 8; ++i)
          vertices.push_back(p + hexahedron[i] * orad);
 
        const unsigned int n_cells                   = 6;
        const int          cell_vertices[n_cells][8] = {
          {8, 9, 10, 11, 0, 1, 2, 3},    // bottom
          {9, 11, 1, 3, 13, 15, 5, 7},   // right
          {12, 13, 4, 5, 14, 15, 6, 7},  // top
          {8, 0, 10, 2, 12, 4, 14, 6},   // left
          {8, 9, 0, 1, 12, 13, 4, 5},    // front
          {10, 2, 11, 3, 14, 6, 15, 7}}; // back
 
        cells.resize(n_cells, CellData<3>());
 
        for (unsigned int i = 0; i < n_cells; ++i)
          {
            for (const unsigned int j : GeometryInfo<3>::vertex_indices())
              cells[i].vertices[j] = cell_vertices[i][j];
            cells[i].material_id = 0;
          }
 
        tria.create_triangulation(vertices, cells, SubCellData());
        tria.set_all_manifold_ids(0);
        tria.set_manifold(0, SphericalManifold<3>(p));
      }
 
      void
      hyper_shell_12(Triangulation<3> &tria,
                     const Point<3>   &p,
                     const double      inner_radius,
                     const double      outer_radius)
      {
        std::vector<Point<3>>    vertices;
        std::vector<CellData<3>> cells;
 
        const double irad = inner_radius / std::sqrt(3.0);
        const double orad = outer_radius / std::sqrt(3.0);
 
        // A more regular subdivision can be obtained by two nested rhombic
        // dodecahedra
        //
        // Octahedron inscribed in the cube [-1,1]^3
        static const std::array<Point<3>, 6> octahedron = {{{-1, 0, 0}, //
                                                            {1, 0, 0},  //
                                                            {0, -1, 0}, //
                                                            {0, 1, 0},  //
                                                            {0, 0, -1}, //
                                                            {0, 0, 1}}};
 
        // Corner points of the cube [-1,1]^3
        static const std::array<Point<3>, 8> hexahedron = {{{-1, -1, -1}, //
                                                            {+1, -1, -1}, //
                                                            {-1, +1, -1}, //
                                                            {+1, +1, -1}, //
                                                            {-1, -1, +1}, //
                                                            {+1, -1, +1}, //
                                                            {-1, +1, +1}, //
                                                            {+1, +1, +1}}};
 
        for (unsigned int i = 0; i < 8; ++i)
          vertices.push_back(p + hexahedron[i] * irad);
        for (unsigned int i = 0; i < 6; ++i)
          vertices.push_back(p + octahedron[i] * inner_radius);
        for (unsigned int i = 0; i < 8; ++i)
          vertices.push_back(p + hexahedron[i] * orad);
        for (unsigned int i = 0; i < 6; ++i)
          vertices.push_back(p + octahedron[i] * outer_radius);
 
        const unsigned int n_cells            = 12;
        const unsigned int rhombi[n_cells][4] = {{10, 4, 0, 8},
                                                 {4, 13, 8, 6},
                                                 {10, 5, 4, 13},
                                                 {1, 9, 10, 5},
                                                 {9, 7, 5, 13},
                                                 {7, 11, 13, 6},
                                                 {9, 3, 7, 11},
                                                 {1, 12, 9, 3},
                                                 {12, 2, 3, 11},
                                                 {2, 8, 11, 6},
                                                 {12, 0, 2, 8},
                                                 {1, 10, 12, 0}};
 
        cells.resize(n_cells, CellData<3>());
 
        for (unsigned int i = 0; i < n_cells; ++i)
          {
            for (unsigned int j = 0; j < 4; ++j)
              {
                cells[i].vertices[j]     = rhombi[i][j];
                cells[i].vertices[j + 4] = rhombi[i][j] + 14;
              }
            cells[i].material_id = 0;
          }
 
        tria.create_triangulation(vertices, cells, SubCellData());
        tria.set_all_manifold_ids(0);
        tria.set_manifold(0, SphericalManifold<3>(p));
      }
 
      void
      hyper_shell_24_48(Triangulation<3>  &tria,
                        const unsigned int n,
                        const unsigned int n_refinement_steps,
                        const Point<3>    &p,
                        const double       inner_radius,
                        const double       outer_radius)
      {
        // These two meshes are created by first creating a mesh of the
        // 6-cell/12-cell version, refining globally, and removing the outer
        // half of the cells. For 192 and more cells, we do this iteratively
        // several times, always refining and removing the outer half. Thus, the
        // outer radius for the start is larger and set as 2^n_refinement_steps
        // such that it exactly gives the desired radius in the end. It would
        // have been slightly less code to treat refinement steps recursively
        // for 192 cells or beyond, but unfortunately we could end up with the
        // 96 cell case which is not what we want. Thus, we need to implement a
        // loop manually here.
        Triangulation<3>   tmp;
        const unsigned int outer_radius_factor = 1 << n_refinement_steps;
        if (n == 24)
          hyper_shell_6(tmp,
                        p,
                        inner_radius,
                        outer_radius_factor * outer_radius -
                          (outer_radius_factor - 1) * inner_radius);
        else if (n == 48)
          hyper_shell_12(tmp,
                         p,
                         inner_radius,
                         outer_radius_factor * outer_radius -
                           (outer_radius_factor - 1) * inner_radius);
        else
          Assert(n == 24 || n == 48, ExcInternalError());
        for (unsigned int r = 0; r < n_refinement_steps; ++r)
          {
            tmp.refine_global(1);
            std::set<Triangulation<3>::active_cell_iterator> cells_to_remove;
 
            // We remove all cells which do not have exactly four vertices
            // at the inner radius (plus some tolerance).
            for (const auto &cell : tmp.active_cell_iterators())
              {
                unsigned int n_vertices_inside = 0;
                for (const auto v : GeometryInfo<3>::vertex_indices())
                  if ((cell->vertex(v) - p).norm_square() <
                      inner_radius * inner_radius * (1 + 1e-12))
                    ++n_vertices_inside;
                if (n_vertices_inside < 4)
                  cells_to_remove.insert(cell);
              }
 
            AssertDimension(cells_to_remove.size(), tmp.n_active_cells() / 2);
            if (r == n_refinement_steps - 1)
              create_triangulation_with_removed_cells(tmp,
                                                      cells_to_remove,
                                                      tria);
            else
              {
                Triangulation<3> copy;
                create_triangulation_with_removed_cells(tmp,
                                                        cells_to_remove,
                                                        copy);
                tmp = std::move(copy);
                tmp.set_all_manifold_ids(0);
                tmp.set_manifold(0, SphericalManifold<3>(p));
              }
          }
        tria.set_all_manifold_ids(0);
        tria.set_manifold(0, SphericalManifold<3>(p));
      }
 
    } // namespace
  }   // namespace internal
 
 
 
  template <>
  void
  hyper_shell(Triangulation<3>  &tria,
              const Point<3>    &p,
              const double       inner_radius,
              const double       outer_radius,
              const unsigned int n_cells,
              const bool         colorize)
  {
    Assert((inner_radius > 0) && (inner_radius < outer_radius),
           ExcInvalidRadii());
 
    unsigned int n_refinement_steps = 0;
    unsigned int n_cells_coarsened  = n_cells;
    if (n_cells != 96 && n_cells > 12)
      while (n_cells_coarsened > 12 && n_cells_coarsened % 4 == 0)
        {
          ++n_refinement_steps;
          n_cells_coarsened /= 4;
        }
    Assert(n_cells == 0 || n_cells == 6 || n_cells == 12 || n_cells == 96 ||
             (n_refinement_steps > 0 &&
              (n_cells_coarsened == 6 || n_cells_coarsened == 12)),
           ExcMessage("Invalid number of coarse mesh cells"));
 
    const unsigned int n = n_refinement_steps > 0 ?
                             4 * n_cells_coarsened :
                             ((n_cells == 0) ? 6 : n_cells);
 
    switch (n)
      {
        case 6:
          internal::hyper_shell_6(tria, p, inner_radius, outer_radius);
          break;
        case 12:
          internal::hyper_shell_12(tria, p, inner_radius, outer_radius);
          break;
        case 24:
        case 48:
          internal::hyper_shell_24_48(
            tria, n, n_refinement_steps, p, inner_radius, outer_radius);
          break;
        case 96:
          {
            // create a triangulation based on the 12-cell version. This
            // function was needed before SphericalManifold was written: it
            // manually adjusted the interior vertices to lie along concentric
            // spheres. Nowadays we can just refine globally:
            Triangulation<3> tmp;
            internal::hyper_shell_12(tmp, p, inner_radius, outer_radius);
            tmp.refine_global(1);
            flatten_triangulation(tmp, tria);
            tria.set_all_manifold_ids(0);
            tria.set_manifold(0, SphericalManifold<3>(p));
            break;
          }
        default:
          {
            Assert(false, ExcMessage("Invalid number of coarse mesh cells."));
          }
      }
 
    if (n_cells > 0)
      AssertDimension(tria.n_global_active_cells(), n_cells);
 
    if (colorize)
      colorize_hyper_shell(tria, p, inner_radius, outer_radius);
  }
 
 
 
  // Implementation for 3d only
  template <>
  void
  half_hyper_shell(Triangulation<3> &tria,
                   const Point<3>   &center,
                   const double      inner_radius,
                   const double      outer_radius,
                   const unsigned int /*n_cells*/,
                   const bool colorize)
  {
    Assert((inner_radius > 0) && (inner_radius < outer_radius),
           ExcInvalidRadii());
 
    // These are for the two lower squares
    const double d = outer_radius / std::sqrt(2.0);
    const double a = inner_radius / std::sqrt(2.0);
    // These are for the two upper square
    const double b = a / 2.0;
    const double c = d / 2.0;
    // And so are these
    const double hb = inner_radius * std::sqrt(3.0) / 2.0;
    const double hc = outer_radius * std::sqrt(3.0) / 2.0;
 
    Point<3> vertices[16] = {
      center + Point<3>(0, d, -d),
      center + Point<3>(0, -d, -d),
      center + Point<3>(0, a, -a),
      center + Point<3>(0, -a, -a),
      center + Point<3>(0, a, a),
      center + Point<3>(0, -a, a),
      center + Point<3>(0, d, d),
      center + Point<3>(0, -d, d),
 
      center + Point<3>(hc, c, -c),
      center + Point<3>(hc, -c, -c),
      center + Point<3>(hb, b, -b),
      center + Point<3>(hb, -b, -b),
      center + Point<3>(hb, b, b),
      center + Point<3>(hb, -b, b),
      center + Point<3>(hc, c, c),
      center + Point<3>(hc, -c, c),
    };
 
    int cell_vertices[5][8] = {{0, 1, 8, 9, 2, 3, 10, 11},
                               {0, 2, 8, 10, 6, 4, 14, 12},
                               {1, 7, 9, 15, 3, 5, 11, 13},
                               {6, 4, 14, 12, 7, 5, 15, 13},
                               {8, 10, 9, 11, 14, 12, 15, 13}};
 
    std::vector<CellData<3>> cells(5, CellData<3>());
 
    for (unsigned int i = 0; i < 5; ++i)
      {
        for (unsigned int j = 0; j < 8; ++j)
          cells[i].vertices[j] = cell_vertices[i][j];
        cells[i].material_id = 0;
      }
 
    tria.create_triangulation(std::vector<Point<3>>(std::begin(vertices),
                                                    std::end(vertices)),
                              cells,
                              SubCellData()); // no boundary information
 
    if (colorize)
      {
        // We want to use a standard boundary description where
        // the boundary is not curved. Hence set boundary id 2 to
        // to all faces in a first step.
        Triangulation<3>::cell_iterator cell = tria.begin();
        for (; cell != tria.end(); ++cell)
          for (const unsigned int i : GeometryInfo<3>::face_indices())
            if (cell->at_boundary(i))
              cell->face(i)->set_all_boundary_ids(2);
 
        // Next look for the curved boundaries. If the x value of the
        // center of the face is not equal to center(0), we're on a curved
        // boundary. Then decide whether the center is nearer to the inner
        // or outer boundary to set the correct boundary id.
        for (cell = tria.begin(); cell != tria.end(); ++cell)
          for (const unsigned int i : GeometryInfo<3>::face_indices())
            if (cell->at_boundary(i))
              {
                const Triangulation<3>::face_iterator face = cell->face(i);
 
                const Point<3> face_center(face->center());
                if (std::abs(face_center[0] - center[0]) >
                    1.e-6 * face_center.norm())
                  {
                    if (std::abs((face_center - center).norm() - inner_radius) <
                        std::abs((face_center - center).norm() - outer_radius))
                      face->set_all_boundary_ids(0);
                    else
                      face->set_all_boundary_ids(1);
                  }
              }
      }
    tria.set_all_manifold_ids(0);
    tria.set_manifold(0, SphericalManifold<3>(center));
  }
 
 
  // Implementation for 3d only
  template <>
  void
  quarter_hyper_shell(Triangulation<3>  &tria,
                      const Point<3>    &center,
                      const double       inner_radius,
                      const double       outer_radius,
                      const unsigned int n,
                      const bool         colorize)
  {
    Assert((inner_radius > 0) && (inner_radius < outer_radius),
           ExcInvalidRadii());
    if (n == 0 || n == 3)
      {
        const double a = inner_radius * std::sqrt(2.0) / 2e0;
        const double b = outer_radius * std::sqrt(2.0) / 2e0;
        const double c = a * std::sqrt(3.0) / 2e0;
        const double d = b * std::sqrt(3.0) / 2e0;
        const double e = outer_radius / 2e0;
        const double h = inner_radius / 2e0;
 
        std::vector<Point<3>> vertices;
 
        vertices.push_back(center + Point<3>(0, inner_radius, 0)); // 0
        vertices.push_back(center + Point<3>(a, a, 0));            // 1
        vertices.push_back(center + Point<3>(b, b, 0));            // 2
        vertices.push_back(center + Point<3>(0, outer_radius, 0)); // 3
        vertices.push_back(center + Point<3>(0, a, a));            // 4
        vertices.push_back(center + Point<3>(c, c, h));            // 5
        vertices.push_back(center + Point<3>(d, d, e));            // 6
        vertices.push_back(center + Point<3>(0, b, b));            // 7
        vertices.push_back(center + Point<3>(inner_radius, 0, 0)); // 8
        vertices.push_back(center + Point<3>(outer_radius, 0, 0)); // 9
        vertices.push_back(center + Point<3>(a, 0, a));            // 10
        vertices.push_back(center + Point<3>(b, 0, b));            // 11
        vertices.push_back(center + Point<3>(0, 0, inner_radius)); // 12
        vertices.push_back(center + Point<3>(0, 0, outer_radius)); // 13
 
        const int cell_vertices[3][8] = {
          {0, 1, 3, 2, 4, 5, 7, 6},
          {1, 8, 2, 9, 5, 10, 6, 11},
          {4, 5, 7, 6, 12, 10, 13, 11},
        };
        std::vector<CellData<3>> cells(3);
 
        for (unsigned int i = 0; i < 3; ++i)
          {
            for (unsigned int j = 0; j < 8; ++j)
              cells[i].vertices[j] = cell_vertices[i][j];
            cells[i].material_id = 0;
          }
 
        tria.create_triangulation(vertices,
                                  cells,
                                  SubCellData()); // no boundary information
      }
    else
      {
        AssertThrow(false, ExcNotImplemented());
      }
 
    if (colorize)
      colorize_quarter_hyper_shell(tria, center, inner_radius, outer_radius);
 
    tria.set_all_manifold_ids(0);
    tria.set_manifold(0, SphericalManifold<3>(center));
  }
 
 
  // Implementation for 3d only
  template <>
  void
  cylinder_shell(Triangulation<3>  &tria,
                 const double       length,
                 const double       inner_radius,
                 const double       outer_radius,
                 const unsigned int n_radial_cells,
                 const unsigned int n_axial_cells,
                 const bool         colorize)
  {
    Assert((inner_radius > 0) && (inner_radius < outer_radius),
           ExcInvalidRadii());
 
    const double pi = numbers::PI;
 
    // determine the number of cells
    // for the grid. if not provided by
    // the user determine it such that
    // the length of each cell on the
    // median (in the middle between
    // the two circles) is equal to its
    // radial extent (which is the
    // difference between the two
    // radii)
    const unsigned int N_r =
      (n_radial_cells == 0 ? static_cast<unsigned int>(std::ceil(
                               (2 * pi * (outer_radius + inner_radius) / 2) /
                               (outer_radius - inner_radius))) :
                             n_radial_cells);
    const unsigned int N_z =
      (n_axial_cells == 0 ?
         static_cast<unsigned int>(std::ceil(
           length / (2 * pi * (outer_radius + inner_radius) / 2 / N_r))) :
         n_axial_cells);
 
    // set up N vertices on the
    // outer and N vertices on
    // the inner circle. the
    // first N ones are on the
    // outer one, and all are
    // numbered counter-clockwise
    std::vector<Point<2>> vertices_2d(2 * N_r);
    for (unsigned int i = 0; i < N_r; ++i)
      {
        vertices_2d[i] =
          Point<2>(std::cos(2 * pi * i / N_r), std::sin(2 * pi * i / N_r)) *
          outer_radius;
        vertices_2d[i + N_r] = vertices_2d[i] * (inner_radius / outer_radius);
      }
 
    std::vector<Point<3>> vertices_3d;
    vertices_3d.reserve(2 * N_r * (N_z + 1));
    for (unsigned int j = 0; j <= N_z; ++j)
      for (unsigned int i = 0; i < 2 * N_r; ++i)
        {
          const Point<3> v(vertices_2d[i][0],
                           vertices_2d[i][1],
                           j * length / N_z);
          vertices_3d.push_back(v);
        }
 
    std::vector<CellData<3>> cells(N_r * N_z, CellData<3>());
 
    for (unsigned int j = 0; j < N_z; ++j)
      for (unsigned int i = 0; i < N_r; ++i)
        {
          cells[i + j * N_r].vertices[0] = i + (j + 1) * 2 * N_r;
          cells[i + j * N_r].vertices[1] = (i + 1) % N_r + (j + 1) * 2 * N_r;
          cells[i + j * N_r].vertices[2] = i + j * 2 * N_r;
          cells[i + j * N_r].vertices[3] = (i + 1) % N_r + j * 2 * N_r;
 
          cells[i + j * N_r].vertices[4] = N_r + i + (j + 1) * 2 * N_r;
          cells[i + j * N_r].vertices[5] =
            N_r + ((i + 1) % N_r) + (j + 1) * 2 * N_r;
          cells[i + j * N_r].vertices[6] = N_r + i + j * 2 * N_r;
          cells[i + j * N_r].vertices[7] = N_r + ((i + 1) % N_r) + j * 2 * N_r;
 
          cells[i + j * N_r].material_id = 0;
        }
 
    tria.create_triangulation(vertices_3d, cells, SubCellData());
    tria.set_all_manifold_ids(0);
    tria.set_manifold(0, CylindricalManifold<3>(2));
 
    if (!colorize)
      return;
 
    // If colorize, set boundary id on the triangualtion.
    // Inner cylinder has boundary id 0
    // Outer cylinder has boundary id 1
    // Bottom (Z-) part of the cylinder has boundary id 2
    // Top (Z+) part of the cylinder has boundary id 3
 
    // Define tolerance to help detect boundary conditions
    // First we define the tolerance along the z axis to identify
    // bottom and top cells.
    double eps_z = 1e-6 * length;
 
    // Gather the inner radius from the faces instead of the argument, this is
    // more robust for some aspect ratios. First initialize the outer to 0 and
    // the inner to a large value
    double face_inner_radius = std::numeric_limits<double>::max();
    double face_outer_radius = 0.;
 
    // Loop over the cells once to acquire the min and max radius at the face
    // centers. Otherwise, for some cell ratio, the center of the faces can be
    // at a radius which is significantly different from the one prescribed.
    for (const auto &cell : tria.active_cell_iterators())
      for (const unsigned int f : GeometryInfo<3>::face_indices())
        {
          if (!cell->face(f)->at_boundary())
            continue;
 
          const auto   face_center = cell->face(f)->center();
          const double z           = face_center[2];
 
          if ((std::fabs(z) > eps_z) &&
              (std::fabs(z - length) > eps_z)) // Not a zmin or zmax boundary
            {
              const double radius = std::sqrt(face_center[0] * face_center[0] +
                                              face_center[1] * face_center[1]);
              face_inner_radius   = std::min(face_inner_radius, radius);
              face_outer_radius   = std::max(face_outer_radius, radius);
            }
        }
 
    double mid_radial_distance = 0.5 * (face_outer_radius - face_inner_radius);
 
    for (const auto &cell : tria.active_cell_iterators())
      for (const unsigned int f : GeometryInfo<3>::face_indices())
        {
          if (cell->face(f)->at_boundary())
            {
              const auto face_center = cell->face(f)->center();
 
              const double radius = std::sqrt(face_center[0] * face_center[0] +
                                              face_center[1] * face_center[1]);
 
              const double z = face_center[2];
 
              if (std::fabs(z) < eps_z) // z = 0 set boundary 2
                {
                  cell->face(f)->set_boundary_id(2);
                }
              else if (std::fabs(z - length) <
                       eps_z) // z = length set boundary 3
                {
                  cell->face(f)->set_boundary_id(3);
                }
              else if (std::fabs(radius - face_inner_radius) >
                       mid_radial_distance) // r =  outer_radius set boundary 1
                {
                  cell->face(f)->set_boundary_id(1);
                }
              else if (std::fabs(radius - face_inner_radius) <
                       mid_radial_distance) // r =  inner_radius set boundary 0
                {
                  cell->face(f)->set_boundary_id(0);
                }
              else
                DEAL_II_ASSERT_UNREACHABLE();
            }
        }
  }
 
 
  template <int dim, int spacedim>
  void
  merge_triangulations(
    const std::vector<const Triangulation<dim, spacedim> *> &triangulations,
    Triangulation<dim, spacedim>                            &result,
    const double duplicated_vertex_tolerance,
    const bool   copy_manifold_ids,
    const bool   copy_boundary_ids)
  {
    std::vector<Point<spacedim>> vertices;
    std::vector<CellData<dim>>   cells;
    SubCellData                  subcell_data;
 
    unsigned int n_accumulated_vertices = 0;
    for (const auto triangulation : triangulations)
      {
        Assert(triangulation->n_levels() == 1,
               ExcMessage("The input triangulations must be non-empty "
                          "and must not be refined."));
 
        auto [tria_vertices, tria_cells, tria_subcell_data] =
          GridTools::get_coarse_mesh_description(*triangulation);
        Assert(tria_vertices.size() == triangulation->n_vertices(),
               ExcInternalError());
        Assert(tria_cells.size() == triangulation->n_cells(),
               ExcInternalError());
 
        // Copy the vertices of the current triangulation into the merged list,
        // and then let the vertex indices of the cells refer to those in
        // the merged list:
        vertices.insert(vertices.end(),
                        tria_vertices.begin(),
                        tria_vertices.end());
        for (CellData<dim> &cell_data : tria_cells)
          {
            for (unsigned int &vertex_n : cell_data.vertices)
              vertex_n += n_accumulated_vertices;
            cells.push_back(cell_data);
          }
 
        // Skip copying lines with no manifold information.
        if (copy_manifold_ids)
          {
            for (CellData<1> &line_data : tria_subcell_data.boundary_lines)
              {
                if (line_data.manifold_id == numbers::flat_manifold_id)
                  continue;
                for (unsigned int &vertex_n : line_data.vertices)
                  vertex_n += n_accumulated_vertices;
                line_data.boundary_id =
                  numbers::internal_face_boundary_id; // default
                subcell_data.boundary_lines.push_back(line_data);
              }
 
            for (CellData<2> &quad_data : tria_subcell_data.boundary_quads)
              {
                if (quad_data.manifold_id == numbers::flat_manifold_id)
                  continue;
                for (unsigned int &vertex_n : quad_data.vertices)
                  vertex_n += n_accumulated_vertices;
                quad_data.boundary_id =
                  numbers::internal_face_boundary_id; // default
                subcell_data.boundary_quads.push_back(quad_data);
              }
          }
 
        n_accumulated_vertices += triangulation->n_vertices();
      }
 
    // throw out duplicated vertices
    std::vector<unsigned int> considered_vertices;
    GridTools::delete_duplicated_vertices(vertices,
                                          cells,
                                          subcell_data,
                                          considered_vertices,
                                          duplicated_vertex_tolerance);
 
    // reorder the cells to ensure that they satisfy the convention for
    // edge and face directions
    if (std::all_of(cells.begin(), cells.end(), [](const auto &cell) {
          return cell.vertices.size() ==
                 ReferenceCells::get_hypercube<dim>().n_vertices();
        }))
      GridTools::consistently_order_cells(cells);
    result.clear();
    result.create_triangulation(vertices, cells, subcell_data);
 
    if (!copy_manifold_ids)
      result.set_all_manifold_ids(numbers::flat_manifold_id);
 
    if (copy_boundary_ids)
      {
        auto result_cell = result.begin();
        for (const auto &tria : triangulations)
          {
            for (const auto &cell : tria->cell_iterators())
              {
                for (const auto &f : cell->face_indices())
                  if (result_cell->face(f)->at_boundary())
                    result_cell->face(f)->set_boundary_id(
                      cell->face(f)->boundary_id());
                ++result_cell;
              }
          }
      }
 
    Assert(duplicated_vertex_tolerance > 0.0 ||
             n_accumulated_vertices == result.n_vertices(),
           ExcInternalError());
  }
 
 
 
  template <int dim, int spacedim>
  void
  merge_triangulations(const Triangulation<dim, spacedim> &triangulation_1,
                       const Triangulation<dim, spacedim> &triangulation_2,
                       Triangulation<dim, spacedim>       &result,
                       const double duplicated_vertex_tolerance,
                       const bool   copy_manifold_ids,
                       const bool   copy_boundary_ids)
  {
    // if either Triangulation is empty then merging is just a copy.
    if (triangulation_1.n_cells() == 0)
      {
        if (&result != &triangulation_2)
          result.copy_triangulation(triangulation_2);
      }
    else if (triangulation_2.n_cells() == 0)
      {
        if (&result != &triangulation_1)
          result.copy_triangulation(triangulation_1);
      }
    else
      merge_triangulations({&triangulation_1, &triangulation_2},
                           result,
                           duplicated_vertex_tolerance,
                           copy_manifold_ids,
                           copy_boundary_ids);
  }
 
 
 
  namespace
  {
    template <int structdim>
    void
    delete_duplicated_objects(std::vector<CellData<structdim>> &subcell_data)
    {
      static_assert(structdim == 1 || structdim == 2,
                    "This function is only implemented for lines and "
                    "quadrilaterals.");
      // start by making sure that all objects representing the same vertices
      // are numbered in the same way by canonicalizing the numberings. This
      // makes it possible to detect duplicates.
      for (CellData<structdim> &cell_data : subcell_data)
        {
          if (structdim == 1)
            std::sort(std::begin(cell_data.vertices),
                      std::end(cell_data.vertices));
          else if (structdim == 2)
            {
              // rotate the vertex numbers so that the lowest one is first
              std::array<unsigned int, 4> renumbering{};
              std::copy(std::begin(cell_data.vertices),
                        std::end(cell_data.vertices),
                        renumbering.begin());
 
              // convert to old style vertex numbering. This makes the
              // permutations easy since the valid configurations are
              //
              // 3  2   2  1   1  0   0  3
              // 0  1   3  0   2  3   1  2
              // (0123) (3012) (2310) (1230)
              //
              // rather than the lexical ordering which is harder to permute
              // by rotation.
              std::swap(renumbering[2], renumbering[3]);
              std::rotate(renumbering.begin(),
                          std::min_element(renumbering.begin(),
                                           renumbering.end()),
                          renumbering.end());
              // convert to new style
              std::swap(renumbering[2], renumbering[3]);
              // deal with cases where we might have
              //
              // 3 2   1 2
              // 0 1   0 3
              //
              // by forcing the second vertex (in lexical ordering) to be
              // smaller than the third
              if (renumbering[1] > renumbering[2])
                std::swap(renumbering[1], renumbering[2]);
              std::copy(renumbering.begin(),
                        renumbering.end(),
                        std::begin(cell_data.vertices));
            }
        }
 
      // Now that all cell objects have been canonicalized they can be sorted:
      auto compare = [](const CellData<structdim> &a,
                        const CellData<structdim> &b) {
        return std::lexicographical_compare(std::begin(a.vertices),
                                            std::end(a.vertices),
                                            std::begin(b.vertices),
                                            std::end(b.vertices));
      };
      std::sort(subcell_data.begin(), subcell_data.end(), compare);
 
      // Finally, determine which objects are duplicates. Duplicates are
      // assumed to be interior objects, so delete all but one and change the
      // boundary id:
      auto left = subcell_data.begin();
      while (left != subcell_data.end())
        {
          const auto right =
            std::upper_bound(left, subcell_data.end(), *left, compare);
          // if the range has more than one item, then there are duplicates -
          // set all boundary ids in the range to the internal boundary id
          if (left + 1 != right)
            for (auto it = left; it != right; ++it)
              {
                it->boundary_id = numbers::internal_face_boundary_id;
                Assert(it->manifold_id == left->manifold_id,
                       ExcMessage(
                         "In the process of grid generation a single "
                         "line or quadrilateral has been assigned two "
                         "different manifold ids. This can happen when "
                         "a Triangulation is copied, e.g., via "
                         "GridGenerator::replicate_triangulation() and "
                         "not all external boundary faces have the same "
                         "manifold id. Double check that all faces "
                         "which you expect to be merged together have "
                         "the same manifold id."));
              }
          left = right;
        }
 
      subcell_data.erase(std::unique(subcell_data.begin(), subcell_data.end()),
                         subcell_data.end());
    }
  } // namespace
 
 
 
  template <int dim, int spacedim>
  void
  replicate_triangulation(const Triangulation<dim, spacedim> &input,
                          const std::vector<unsigned int>    &extents,
                          Triangulation<dim, spacedim>       &result)
  {
    AssertDimension(dim, extents.size());
#  ifdef DEBUG
    for (const auto &extent : extents)
      Assert(0 < extent,
             ExcMessage("The Triangulation must be copied at least one time in "
                        "each coordinate dimension."));
#  endif
    const BoundingBox<spacedim> bbox(input.get_vertices());
    const auto                 &min = bbox.get_boundary_points().first;
    const auto                 &max = bbox.get_boundary_points().second;
 
    std::array<Tensor<1, spacedim>, dim> offsets;
    for (unsigned int d = 0; d < dim; ++d)
      offsets[d][d] = max[d] - min[d];
 
    Triangulation<dim, spacedim> tria_to_replicate;
    tria_to_replicate.copy_triangulation(input);
    for (unsigned int d = 0; d < dim; ++d)
      {
        auto [input_vertices, input_cell_data, input_subcell_data] =
          GridTools::get_coarse_mesh_description(tria_to_replicate);
 
        std::vector<Point<spacedim>> output_vertices     = input_vertices;
        std::vector<CellData<dim>>   output_cell_data    = input_cell_data;
        SubCellData                  output_subcell_data = input_subcell_data;
 
        for (unsigned int k = 1; k < extents[d]; ++k)
          {
            const std::size_t vertex_offset = k * input_vertices.size();
            // vertices
            for (const Point<spacedim> &point : input_vertices)
              output_vertices.push_back(point + double(k) * offsets[d]);
            // cell data
            for (const CellData<dim> &cell_data : input_cell_data)
              {
                output_cell_data.push_back(cell_data);
                for (unsigned int &vertex : output_cell_data.back().vertices)
                  vertex += vertex_offset;
              }
            // subcell data
            for (const CellData<1> &boundary_line :
                 input_subcell_data.boundary_lines)
              {
                output_subcell_data.boundary_lines.push_back(boundary_line);
                for (unsigned int &vertex :
                     output_subcell_data.boundary_lines.back().vertices)
                  vertex += vertex_offset;
              }
            for (const CellData<2> &boundary_quad :
                 input_subcell_data.boundary_quads)
              {
                output_subcell_data.boundary_quads.push_back(boundary_quad);
                for (unsigned int &vertex :
                     output_subcell_data.boundary_quads.back().vertices)
                  vertex += vertex_offset;
              }
          }
        // check all vertices: since the grid is coarse, most will be on the
        // boundary anyway
        std::vector<unsigned int> boundary_vertices;
        GridTools::delete_duplicated_vertices(
          output_vertices,
          output_cell_data,
          output_subcell_data,
          boundary_vertices,
          1e-6 * input.begin_active()->diameter());
        // delete_duplicated_vertices also deletes any unused vertices
        // deal with any reordering issues created by delete_duplicated_vertices
        GridTools::consistently_order_cells(output_cell_data);
        // clean up the boundary ids of the boundary objects: note that we
        // have to do this after delete_duplicated_vertices so that boundary
        // objects are actually duplicated at this point
        if (dim == 2)
          delete_duplicated_objects(output_subcell_data.boundary_lines);
        else if (dim == 3)
          {
            delete_duplicated_objects(output_subcell_data.boundary_quads);
            for (CellData<1> &boundary_line :
                 output_subcell_data.boundary_lines)
              // set boundary lines to the default value - let
              // create_triangulation figure out the rest.
              boundary_line.boundary_id = numbers::internal_face_boundary_id;
          }
 
        tria_to_replicate.clear();
        tria_to_replicate.create_triangulation(output_vertices,
                                               output_cell_data,
                                               output_subcell_data);
      }
 
    result.copy_triangulation(tria_to_replicate);
  }
 
 
 
  template <int dim, int spacedim>
  void
  create_union_triangulation(
    const Triangulation<dim, spacedim> &triangulation_1,
    const Triangulation<dim, spacedim> &triangulation_2,
    Triangulation<dim, spacedim>       &result)
  {
    Assert(GridTools::have_same_coarse_mesh(triangulation_1, triangulation_2),
           ExcMessage("The two input triangulations are not derived from "
                      "the same coarse mesh as required."));
    Assert((dynamic_cast<
              const parallel::distributed::Triangulation<dim, spacedim> *>(
              &triangulation_1) == nullptr) &&
             (dynamic_cast<
                const parallel::distributed::Triangulation<dim, spacedim> *>(
                &triangulation_2) == nullptr),
           ExcMessage("The source triangulations for this function must both "
                      "be available entirely locally, and not be distributed "
                      "triangulations."));
 
    // first copy triangulation_1, and
    // then do as many iterations as
    // there are levels in
    // triangulation_2 to refine
    // additional cells. since this is
    // the maximum number of
    // refinements to get from the
    // coarse grid to triangulation_2,
    // it is clear that this is also
    // the maximum number of
    // refinements to get from any cell
    // on triangulation_1 to
    // triangulation_2
    result.clear();
    result.copy_triangulation(triangulation_1);
    for (unsigned int iteration = 0; iteration < triangulation_2.n_levels();
         ++iteration)
      {
        InterGridMap<Triangulation<dim, spacedim>> intergrid_map;
        intergrid_map.make_mapping(result, triangulation_2);
 
        bool any_cell_flagged = false;
        for (const auto &result_cell : result.active_cell_iterators())
          if (intergrid_map[result_cell]->has_children())
            {
              any_cell_flagged = true;
              result_cell->set_refine_flag();
            }
 
        if (any_cell_flagged == false)
          break;
        else
          result.execute_coarsening_and_refinement();
      }
  }
 
 
 
  template <int dim, int spacedim>
  void
  create_triangulation_with_removed_cells(
    const Triangulation<dim, spacedim> &input_triangulation,
    const std::set<typename Triangulation<dim, spacedim>::active_cell_iterator>
                                 &cells_to_remove,
    Triangulation<dim, spacedim> &result)
  {
    // simply copy the vertices; we will later strip those
    // that turn out to be unused
    std::vector<Point<spacedim>> vertices = input_triangulation.get_vertices();
 
    // the loop through the cells and copy stuff, excluding
    // the ones we are to remove
    std::vector<CellData<dim>> cells;
    for (const auto &cell : input_triangulation.active_cell_iterators())
      if (cells_to_remove.find(cell) == cells_to_remove.end())
        {
          Assert(static_cast<unsigned int>(cell->level()) ==
                   input_triangulation.n_levels() - 1,
                 ExcMessage(
                   "Your input triangulation appears to have "
                   "adaptively refined cells. This is not allowed. You can "
                   "only call this function on a triangulation in which "
                   "all cells are on the same refinement level."));
 
          CellData<dim> this_cell;
          for (const unsigned int v : GeometryInfo<dim>::vertex_indices())
            this_cell.vertices[v] = cell->vertex_index(v);
          this_cell.material_id = cell->material_id();
          cells.push_back(this_cell);
        }
 
    // throw out duplicated vertices from the two meshes, reorder vertices as
    // necessary and create the triangulation
    SubCellData               subcell_data;
    std::vector<unsigned int> considered_vertices;
    GridTools::delete_duplicated_vertices(vertices,
                                          cells,
                                          subcell_data,
                                          considered_vertices);
 
    // then clear the old triangulation and create the new one
    result.clear();
    result.create_triangulation(vertices, cells, subcell_data);
  }
 
 
 
  void
  extrude_triangulation(
    const Triangulation<2, 2>             &input,
    const unsigned int                     n_slices,
    const double                           height,
    Triangulation<3, 3>                   &result,
    const bool                             copy_manifold_ids,
    const std::vector<types::manifold_id> &manifold_priorities)
  {
    Assert(input.n_levels() == 1,
           ExcMessage(
             "The input triangulation must be a coarse mesh, i.e., it must "
             "not have been refined."));
    Assert(result.n_cells() == 0,
           ExcMessage("The output triangulation object needs to be empty."));
    Assert(height > 0,
           ExcMessage("The given height for extrusion must be positive."));
    Assert(n_slices >= 2,
           ExcMessage(
             "The number of slices for extrusion must be at least 2."));
 
    const double        delta_h = height / (n_slices - 1);
    std::vector<double> slices_z_values;
    for (unsigned int i = 0; i < n_slices; ++i)
      slices_z_values.push_back(i * delta_h);
    extrude_triangulation(
      input, slices_z_values, result, copy_manifold_ids, manifold_priorities);
  }
 
 
 
  void
  extrude_triangulation(
    const Triangulation<2, 2>             &input,
    const unsigned int                     n_slices,
    const double                           height,
    Triangulation<2, 2>                   &result,
    const bool                             copy_manifold_ids,
    const std::vector<types::manifold_id> &manifold_priorities)
  {
    (void)input;
    (void)n_slices;
    (void)height;
    (void)result;
    (void)copy_manifold_ids;
    (void)manifold_priorities;
 
    AssertThrow(false,
                ExcMessage(
                  "GridTools::extrude_triangulation() is only available "
                  "for Triangulation<3, 3> as output triangulation."));
  }
 
 
 
  void
  extrude_triangulation(
    const Triangulation<2, 2>             &input,
    const std::vector<double>             &slice_coordinates,
    Triangulation<3, 3>                   &result,
    const bool                             copy_manifold_ids,
    const std::vector<types::manifold_id> &manifold_priorities)
  {
    Assert(input.n_levels() == 1,
           ExcMessage(
             "The input triangulation must be a coarse mesh, i.e., it must "
             "not have been refined."));
    Assert(result.n_cells() == 0,
           ExcMessage("The output triangulation object needs to be empty."));
    Assert(slice_coordinates.size() >= 2,
           ExcMessage(
             "The number of slices for extrusion must be at least 2."));
    Assert(std::is_sorted(slice_coordinates.begin(), slice_coordinates.end()),
           ExcMessage("Slice z-coordinates should be in ascending order"));
    Assert(input.all_reference_cells_are_hyper_cube(),
           ExcMessage(
             "This function is only implemented for quadrilateral meshes."));
 
    const auto priorities = [&]() -> std::vector<types::manifold_id> {
      // if a non-empty (i.e., not the default) vector is given for
      // manifold_priorities then use it (but check its validity in debug
      // mode)
      if (0 < manifold_priorities.size())
        {
#  ifdef DEBUG
          // check that the provided manifold_priorities is valid
          std::vector<types::manifold_id> sorted_manifold_priorities =
            manifold_priorities;
          std::sort(sorted_manifold_priorities.begin(),
                    sorted_manifold_priorities.end());
          Assert(std::unique(sorted_manifold_priorities.begin(),
                             sorted_manifold_priorities.end()) ==
                   sorted_manifold_priorities.end(),
                 ExcMessage(
                   "The given vector of manifold ids may not contain any "
                   "duplicated entries."));
          std::vector<types::manifold_id> sorted_manifold_ids =
            input.get_manifold_ids();
          std::sort(sorted_manifold_ids.begin(), sorted_manifold_ids.end());
          if (sorted_manifold_priorities != sorted_manifold_ids)
            {
              std::ostringstream message;
              message << "The given triangulation has manifold ids {";
              for (const types::manifold_id manifold_id : sorted_manifold_ids)
                if (manifold_id != sorted_manifold_ids.back())
                  message << manifold_id << ", ";
              message << sorted_manifold_ids.back() << "}, but \n"
                      << "    the given vector of manifold ids is {";
              for (const types::manifold_id manifold_id : manifold_priorities)
                if (manifold_id != manifold_priorities.back())
                  message << manifold_id << ", ";
              message
                << manifold_priorities.back() << "}.\n"
                << "    These vectors should contain the same elements.\n";
              const std::string m = message.str();
              Assert(false, ExcMessage(m));
            }
#  endif
          return manifold_priorities;
        }
      // otherwise use the default ranking: ascending order, but TFI manifolds
      // are at the end.
      std::vector<types::manifold_id> default_priorities =
        input.get_manifold_ids();
      const auto first_tfi_it = std::partition(
        default_priorities.begin(),
        default_priorities.end(),
        [&input](const types::manifold_id &id) {
          return dynamic_cast<const TransfiniteInterpolationManifold<2, 2> *>(
                   &input.get_manifold(id)) == nullptr;
        });
      std::sort(default_priorities.begin(), first_tfi_it);
      std::sort(first_tfi_it, default_priorities.end());
 
      return default_priorities;
    }();
 
    const std::size_t        n_slices = slice_coordinates.size();
    std::vector<Point<3>>    points(n_slices * input.n_vertices());
    std::vector<CellData<3>> cells;
    cells.reserve((n_slices - 1) * input.n_active_cells());
 
    // copy the array of points as many times as there will be slices,
    // one slice at a time. The z-axis value are defined in slices_coordinates
    for (std::size_t slice_n = 0; slice_n < n_slices; ++slice_n)
      {
        for (std::size_t vertex_n = 0; vertex_n < input.n_vertices();
             ++vertex_n)
          {
            const Point<2> vertex = input.get_vertices()[vertex_n];
            points[slice_n * input.n_vertices() + vertex_n] =
              Point<3>(vertex[0], vertex[1], slice_coordinates[slice_n]);
          }
      }
 
    // then create the cells of each of the slices, one stack at a
    // time
    for (const auto &cell : input.active_cell_iterators())
      {
        for (std::size_t slice_n = 0; slice_n < n_slices - 1; ++slice_n)
          {
            CellData<3> this_cell;
            for (const unsigned int vertex_n :
                 GeometryInfo<2>::vertex_indices())
              {
                this_cell.vertices[vertex_n] =
                  cell->vertex_index(vertex_n) + slice_n * input.n_vertices();
                this_cell
                  .vertices[vertex_n + GeometryInfo<2>::vertices_per_cell] =
                  cell->vertex_index(vertex_n) +
                  (slice_n + 1) * input.n_vertices();
              }
 
            this_cell.material_id = cell->material_id();
            if (copy_manifold_ids)
              this_cell.manifold_id = cell->manifold_id();
            cells.push_back(this_cell);
          }
      }
 
    // Next, create face data for all faces that are orthogonal to the x-y
    // plane
    SubCellData               subcell_data;
    std::vector<CellData<2>> &quads           = subcell_data.boundary_quads;
    types::boundary_id        max_boundary_id = 0;
    quads.reserve(input.n_active_lines() * (n_slices - 1) +
                  input.n_active_cells() * 2);
    for (const auto &face : input.active_face_iterators())
      {
        CellData<2> quad;
        quad.boundary_id = face->boundary_id();
        if (face->at_boundary())
          max_boundary_id = std::max(max_boundary_id, quad.boundary_id);
        if (copy_manifold_ids)
          quad.manifold_id = face->manifold_id();
        for (std::size_t slice_n = 0; slice_n < n_slices - 1; ++slice_n)
          {
            quad.vertices[0] =
              face->vertex_index(0) + slice_n * input.n_vertices();
            quad.vertices[1] =
              face->vertex_index(1) + slice_n * input.n_vertices();
            quad.vertices[2] =
              face->vertex_index(0) + (slice_n + 1) * input.n_vertices();
            quad.vertices[3] =
              face->vertex_index(1) + (slice_n + 1) * input.n_vertices();
            quads.push_back(quad);
          }
      }
 
    // if necessary, create face data for faces parallel to the x-y
    // plane. This is only necessary if we need to set manifolds.
    if (copy_manifold_ids)
      for (const auto &cell : input.active_cell_iterators())
        {
          CellData<2> quad;
          quad.boundary_id = numbers::internal_face_boundary_id;
          quad.manifold_id = cell->manifold_id(); // check is outside loop
          for (std::size_t slice_n = 1; slice_n < n_slices - 1; ++slice_n)
            {
              quad.vertices[0] =
                cell->vertex_index(0) + slice_n * input.n_vertices();
              quad.vertices[1] =
                cell->vertex_index(1) + slice_n * input.n_vertices();
              quad.vertices[2] =
                cell->vertex_index(2) + slice_n * input.n_vertices();
              quad.vertices[3] =
                cell->vertex_index(3) + slice_n * input.n_vertices();
              quads.push_back(quad);
            }
        }
 
    // then mark the bottom and top boundaries of the extruded mesh
    // with max_boundary_id+1 and max_boundary_id+2. check that this
    // remains valid
    Assert((max_boundary_id != numbers::invalid_boundary_id) &&
             (max_boundary_id + 1 != numbers::invalid_boundary_id) &&
             (max_boundary_id + 2 != numbers::invalid_boundary_id),
           ExcMessage(
             "The input triangulation to this function is using boundary "
             "indicators in a range that do not allow using "
             "max_boundary_id+1 and max_boundary_id+2 as boundary "
             "indicators for the bottom and top faces of the "
             "extruded triangulation."));
    const types::boundary_id bottom_boundary_id = max_boundary_id + 1;
    const types::boundary_id top_boundary_id    = max_boundary_id + 2;
    for (const auto &cell : input.active_cell_iterators())
      {
        CellData<2> quad;
        quad.boundary_id = bottom_boundary_id;
        quad.vertices[0] = cell->vertex_index(0);
        quad.vertices[1] = cell->vertex_index(1);
        quad.vertices[2] = cell->vertex_index(2);
        quad.vertices[3] = cell->vertex_index(3);
        if (copy_manifold_ids)
          quad.manifold_id = cell->manifold_id();
        quads.push_back(quad);
 
        quad.boundary_id = top_boundary_id;
        for (unsigned int &vertex : quad.vertices)
          vertex += (n_slices - 1) * input.n_vertices();
        if (copy_manifold_ids)
          quad.manifold_id = cell->manifold_id();
        quads.push_back(quad);
      }
 
    // use all of this to finally create the extruded 3d
    // triangulation.  it is not necessary to call
    // GridTools::consistently_order_cells() because the cells we have
    // constructed above are automatically correctly oriented. this is
    // because the 2d base mesh is always correctly oriented, and
    // extruding it automatically yields a correctly oriented 3d mesh,
    // as discussed in the edge orientation paper mentioned in the
    // introduction to the @ref reordering "reordering module".
    result.create_triangulation(points, cells, subcell_data);
 
    for (auto manifold_id_it = priorities.rbegin();
         manifold_id_it != priorities.rend();
         ++manifold_id_it)
      for (const auto &face : result.active_face_iterators())
        if (face->manifold_id() == *manifold_id_it)
          for (unsigned int line_n = 0;
               line_n < GeometryInfo<3>::lines_per_face;
               ++line_n)
            face->line(line_n)->set_manifold_id(*manifold_id_it);
  }
 
 
 
  void
  extrude_triangulation(
    const Triangulation<2, 2>             &input,
    const std::vector<double>             &slice_coordinates,
    Triangulation<2, 2>                   &result,
    const bool                             copy_manifold_ids,
    const std::vector<types::manifold_id> &manifold_priorities)
  {
    (void)input;
    (void)slice_coordinates;
    (void)result;
    (void)copy_manifold_ids;
    (void)manifold_priorities;
 
    AssertThrow(false,
                ExcMessage(
                  "GridTools::extrude_triangulation() is only available "
                  "for Triangulation<3, 3> as output triangulation."));
  }
 
 
 
  template <>
  void
  hyper_cube_with_cylindrical_hole(Triangulation<1> &,
                                   const double,
                                   const double,
                                   const double,
                                   const unsigned int,
                                   const bool)
  {
    DEAL_II_NOT_IMPLEMENTED();
  }
 
 
 
  template <int spacedim>
  void
  hyper_cube_with_cylindrical_hole_2D(Triangulation<2, spacedim> &triangulation,
                                      const double                inner_radius,
                                      const double                outer_radius,
                                      const double,       // width,
                                      const unsigned int, // width_repetition,
                                      const bool colorize)
  {
    const int dim = 2;
 
    Assert(inner_radius < outer_radius,
           ExcMessage("outer_radius has to be bigger than inner_radius."));
 
    const Point<spacedim> center;
 
    // We create a hyper_shell (i.e., an annulus) in two dimensions, and then we
    // modify it by pulling the vertices on the diagonals out to where the
    // corners of a square would be:
    hyper_shell(triangulation, center, inner_radius, outer_radius, 8);
    triangulation.set_all_manifold_ids(numbers::flat_manifold_id);
    std::vector<bool> treated_vertices(triangulation.n_vertices(), false);
    for (const auto &cell : triangulation.active_cell_iterators())
      {
        for (auto f : GeometryInfo<dim>::face_indices())
          if (cell->face(f)->at_boundary())
            for (const unsigned int v : cell->face(f)->vertex_indices())
              if (/* is the vertex on the outer ring? */
                  (std::fabs(cell->face(f)->vertex(v).norm() - outer_radius) <
                   1e-12 * outer_radius)
                  /* and */
                  &&
                  /* is the vertex on one of the two diagonals? */
                  (std::fabs(std::fabs(cell->face(f)->vertex(v)[0]) -
                             std::fabs(cell->face(f)->vertex(v)[1])) <
                   1e-12 * outer_radius))
                cell->face(f)->vertex(v) *= std::sqrt(2.);
      }
    const double eps = 1e-3 * outer_radius;
    for (const auto &cell : triangulation.active_cell_iterators())
      {
        for (const unsigned int f : cell->face_indices())
          if (cell->face(f)->at_boundary())
            {
              const double dx = cell->face(f)->center()[0] - center[0];
              const double dy = cell->face(f)->center()[1] - center[1];
              if (colorize)
                {
                  if (std::abs(dx + outer_radius) < eps)
                    cell->face(f)->set_boundary_id(0);
                  else if (std::abs(dx - outer_radius) < eps)
                    cell->face(f)->set_boundary_id(1);
                  else if (std::abs(dy + outer_radius) < eps)
                    cell->face(f)->set_boundary_id(2);
                  else if (std::abs(dy - outer_radius) < eps)
                    cell->face(f)->set_boundary_id(3);
                  else
                    {
                      cell->face(f)->set_boundary_id(4);
                      cell->face(f)->set_manifold_id(0);
                    }
                }
              else
                {
                  const double d = (cell->face(f)->center() - center).norm();
                  if (d - inner_radius < 0)
                    {
                      cell->face(f)->set_boundary_id(1);
                      cell->face(f)->set_manifold_id(0);
                    }
                  else
                    cell->face(f)->set_boundary_id(0);
                }
            }
      }
    triangulation.set_manifold(0, PolarManifold<2, spacedim>(center));
  }
 
 
 
  template <>
  void
  hyper_cube_with_cylindrical_hole(Triangulation<2, 2> &triangulation,
                                   const double         inner_radius,
                                   const double         outer_radius,
                                   const double         width,
                                   const unsigned int   width_repetition,
                                   const bool           colorize)
  {
    hyper_cube_with_cylindrical_hole_2D(triangulation,
                                        inner_radius,
                                        outer_radius,
                                        width,
                                        width_repetition,
                                        colorize);
  }
 
 
 
  template <>
  void
  hyper_cube_with_cylindrical_hole(Triangulation<2, 3> &triangulation,
                                   const double         inner_radius,
                                   const double         outer_radius,
                                   const double         width,
                                   const unsigned int   width_repetition,
                                   const bool           colorize)
  {
    hyper_cube_with_cylindrical_hole_2D(triangulation,
                                        inner_radius,
                                        outer_radius,
                                        width,
                                        width_repetition,
                                        colorize);
  }
 
 
 
  template <int dim>
  void
  concentric_hyper_shells(Triangulation<dim> &triangulation,
                          const Point<dim>   &center,
                          const double        inner_radius,
                          const double        outer_radius,
                          const unsigned int  n_shells,
                          const double        skewness,
                          const unsigned int  n_cells,
                          const bool          colorize)
  {
    Assert(dim == 2 || dim == 3, ExcNotImplemented());
    (void)colorize;
    (void)n_cells;
    Assert(inner_radius < outer_radius,
           ExcMessage("outer_radius has to be bigger than inner_radius."));
    if (n_shells == 0)
      return; // empty Triangulation
 
    std::vector<double> radii;
    radii.push_back(inner_radius);
    for (unsigned int shell_n = 1; shell_n < n_shells; ++shell_n)
      if (skewness == 0.0)
        // same as below, but works in the limiting case of zero skewness
        radii.push_back(inner_radius +
                        (outer_radius - inner_radius) *
                          (1.0 - (1.0 - double(shell_n) / n_shells)));
      else
        radii.push_back(
          inner_radius +
          (outer_radius - inner_radius) *
            (1.0 - std::tanh(skewness * (1.0 - double(shell_n) / n_shells)) /
                     std::tanh(skewness)));
    radii.push_back(outer_radius);
 
    double grid_vertex_tolerance = 0.0;
    for (unsigned int shell_n = 0; shell_n < radii.size() - 1; ++shell_n)
      {
        Triangulation<dim> current_shell;
        GridGenerator::hyper_shell(current_shell,
                                   center,
                                   radii[shell_n],
                                   radii[shell_n + 1],
                                   n_cells == 0 ? (dim == 2 ? 8 : 12) :
                                                  n_cells);
 
        // The innermost shell has the smallest cells: use that to set the
        // vertex merging tolerance
        if (grid_vertex_tolerance == 0.0)
          grid_vertex_tolerance =
            0.5 * internal::minimal_vertex_distance(current_shell);
 
        Triangulation<dim> temp(std::move(triangulation));
        triangulation.clear();
        GridGenerator::merge_triangulations(current_shell,
                                            temp,
                                            triangulation,
                                            grid_vertex_tolerance);
      }
 
    const types::manifold_id manifold_id = 0;
    triangulation.set_all_manifold_ids(manifold_id);
    if (dim == 2)
      triangulation.set_manifold(manifold_id, PolarManifold<dim>(center));
    else if (dim == 3)
      triangulation.set_manifold(manifold_id, SphericalManifold<dim>(center));
 
    // We use boundary vertex positions to see if things are on the inner or
    // outer boundary.
    constexpr double radial_vertex_tolerance =
      100.0 * std::numeric_limits<double>::epsilon();
    auto assert_vertex_distance_within_tolerance =
      [center, radial_vertex_tolerance](
        const TriaIterator<TriaAccessor<dim - 1, dim, dim>> face,
        const double                                        radius) {
        (void)center;
        (void)radial_vertex_tolerance;
        (void)face;
        (void)radius;
        for (unsigned int vertex_n = 0;
             vertex_n < GeometryInfo<dim>::vertices_per_face;
             ++vertex_n)
          {
            Assert(std::abs((face->vertex(vertex_n) - center).norm() - radius) <
                     (center.norm() + radius) * radial_vertex_tolerance,
                   ExcInternalError());
          }
      };
    if (colorize)
      for (const auto &cell : triangulation.active_cell_iterators())
        for (const unsigned int face_n : GeometryInfo<dim>::face_indices())
          {
            auto face = cell->face(face_n);
            if (face->at_boundary())
              {
                if (((face->vertex(0) - center).norm() - inner_radius) <
                    (center.norm() + inner_radius) * radial_vertex_tolerance)
                  {
                    // we must be at an inner face, but check
                    assert_vertex_distance_within_tolerance(face, inner_radius);
                    face->set_all_boundary_ids(0);
                  }
                else
                  {
                    // we must be at an outer face, but check
                    assert_vertex_distance_within_tolerance(face, outer_radius);
                    face->set_all_boundary_ids(1);
                  }
              }
          }
  }
 
 
 
  template <>
  void
  hyper_cube_with_cylindrical_hole(Triangulation<3>  &triangulation,
                                   const double       inner_radius,
                                   const double       outer_radius,
                                   const double       L,
                                   const unsigned int Nz,
                                   const bool         colorize)
  {
    const int dim = 3;
 
    Assert(inner_radius < outer_radius,
           ExcMessage("outer_radius has to be bigger than inner_radius."));
    Assert(L > 0, ExcMessage("Must give positive extension L"));
    Assert(Nz >= 1, ExcLowerRange(1, Nz));
 
    // Start with a cylinder shell with the correct inner and outer radius
    // and as many layers as requested
    cylinder_shell(triangulation, L, inner_radius, outer_radius, 8, Nz, false);
    triangulation.set_all_manifold_ids(numbers::flat_manifold_id);
 
    // Then loop over all vertices that are at the boundary (by looping
    // over all cells, their faces, and if the face is at the boundary,
    // their vertices. If we haven't touched that vertex yet, see if
    // we need to move it from its cylinder mantle position to the
    // outer boundary of the box.
    std::vector<bool> treated_vertices(triangulation.n_vertices(), false);
    for (const auto &cell : triangulation.active_cell_iterators())
      {
        for (const auto f : cell->face_indices())
          if (cell->face(f)->at_boundary())
            {
              for (const unsigned int v : cell->face(f)->vertex_indices())
                {
                  const unsigned int vv = cell->face(f)->vertex_index(v);
                  if (treated_vertices[vv] == false)
                    {
                      treated_vertices[vv] = true;
 
                      // The vertices we have to treat are the ones that
                      // have x=y or x=-y and are at the outer ring -- that is,
                      // they are on the diagonal in the x-y plane and radius
                      // equal to outer_radius. These need to be pulled out to
                      // the corner point of the square, i.e., their x and y
                      // coordinates need to be multiplied by sqrt(2),
                      // whereas the z coordinate remains unchanged:
                      const Point<dim> vertex_location =
                        cell->face(f)->vertex(v);
                      if ((std::fabs(std::fabs(vertex_location[0]) -
                                     std::fabs(vertex_location[1])) <
                           1e-12 * outer_radius) &&
                          (std::fabs(vertex_location[0] * vertex_location[0] +
                                     vertex_location[1] * vertex_location[1] -
                                     outer_radius * outer_radius) <
                           1e-12 * outer_radius))
                        cell->face(f)->vertex(v) =
                          Point<3>(vertex_location[0] * std::sqrt(2.0),
                                   vertex_location[1] * std::sqrt(2.0),
                                   vertex_location[2]);
                    }
                }
            }
      }
    double eps = 1e-3 * outer_radius;
    for (const auto &cell : triangulation.active_cell_iterators())
      {
        for (const unsigned int f : cell->face_indices())
          if (cell->face(f)->at_boundary())
            {
              const double dx = cell->face(f)->center()[0];
              const double dy = cell->face(f)->center()[1];
              const double dz = cell->face(f)->center()[2];
 
              if (colorize)
                {
                  if (std::abs(dx + outer_radius) < eps)
                    cell->face(f)->set_boundary_id(0);
 
                  else if (std::abs(dx - outer_radius) < eps)
                    cell->face(f)->set_boundary_id(1);
 
                  else if (std::abs(dy + outer_radius) < eps)
                    cell->face(f)->set_boundary_id(2);
 
                  else if (std::abs(dy - outer_radius) < eps)
                    cell->face(f)->set_boundary_id(3);
 
                  else if (std::abs(dz) < eps)
                    cell->face(f)->set_boundary_id(4);
 
                  else if (std::abs(dz - L) < eps)
                    cell->face(f)->set_boundary_id(5);
 
                  else
                    {
                      cell->face(f)->set_all_boundary_ids(6);
                      cell->face(f)->set_all_manifold_ids(0);
                    }
                }
              else
                {
                  Point<dim> c   = cell->face(f)->center();
                  c[2]           = 0;
                  const double d = c.norm();
                  if (d - inner_radius < 0)
                    {
                      cell->face(f)->set_all_boundary_ids(1);
                      cell->face(f)->set_all_manifold_ids(0);
                    }
                  else
                    cell->face(f)->set_boundary_id(0);
                }
            }
      }
    triangulation.set_manifold(0, CylindricalManifold<3>(2));
  }
 
 
 
  template <int dim, int spacedim1, int spacedim2>
  void
  flatten_triangulation(const Triangulation<dim, spacedim1> &in_tria,
                        Triangulation<dim, spacedim2>       &out_tria)
  {
    Assert((dynamic_cast<
              const parallel::distributed::Triangulation<dim, spacedim1> *>(
              &in_tria) == nullptr),
           ExcMessage(
             "This function cannot be used on "
             "parallel::distributed::Triangulation objects as inputs."));
    Assert(in_tria.has_hanging_nodes() == false,
           ExcMessage("This function does not work for meshes that have "
                      "hanging nodes."));
 
 
    const unsigned int spacedim = std::min(spacedim1, spacedim2);
    const std::vector<Point<spacedim1>> &in_vertices = in_tria.get_vertices();
 
    // Create an array of vertices, with components either truncated
    // or extended by zeroes.
    std::vector<Point<spacedim2>> v(in_vertices.size());
    for (unsigned int i = 0; i < in_vertices.size(); ++i)
      for (unsigned int d = 0; d < spacedim; ++d)
        v[i][d] = in_vertices[i][d];
 
    std::vector<CellData<dim>> cells(in_tria.n_active_cells());
    for (const auto &cell : in_tria.active_cell_iterators())
      {
        const unsigned int id = cell->active_cell_index();
 
        cells[id].vertices.resize(cell->n_vertices());
        for (const auto i : cell->vertex_indices())
          cells[id].vertices[i] = cell->vertex_index(i);
        cells[id].material_id = cell->material_id();
        cells[id].manifold_id = cell->manifold_id();
      }
 
    SubCellData subcelldata;
    switch (dim)
      {
        case 1:
          {
            // Nothing to do in 1d
            break;
          }
 
        case 2:
          {
            std::vector<bool> user_flags_line;
            in_tria.save_user_flags_line(user_flags_line);
            const_cast<Triangulation<dim, spacedim1> &>(in_tria)
              .clear_user_flags_line();
 
            // Loop over all the faces of the triangulation and create
            // objects that describe their boundary and manifold ids.
            for (const auto &face : in_tria.active_face_iterators())
              {
                if (face->at_boundary())
                  {
                    CellData<1> boundary_line;
 
                    boundary_line.vertices.resize(face->n_vertices());
                    for (const auto i : face->vertex_indices())
                      boundary_line.vertices[i] = face->vertex_index(i);
                    boundary_line.boundary_id = face->boundary_id();
                    boundary_line.manifold_id = face->manifold_id();
 
                    subcelldata.boundary_lines.emplace_back(
                      std::move(boundary_line));
                  }
                else
                  // The face is not at the boundary. We won't have to set
                  // boundary_ids (that is not possible for interior faces), but
                  // we need to do something if the manifold-id is not the
                  // default.
                  //
                  // We keep track via the user flags whether we have already
                  // dealt with a face or not. (We need to do that here because
                  // we will return to interior faces twice, once for each
                  // neighbor, whereas we only touch each of the boundary faces
                  // above once.)
                  if ((face->user_flag_set() == false) &&
                      (face->manifold_id() != numbers::flat_manifold_id))
                    {
                      CellData<1> boundary_line;
 
                      boundary_line.vertices.resize(face->n_vertices());
                      for (const auto i : face->vertex_indices())
                        boundary_line.vertices[i] = face->vertex_index(i);
                      boundary_line.boundary_id =
                        numbers::internal_face_boundary_id;
                      boundary_line.manifold_id = face->manifold_id();
 
                      subcelldata.boundary_lines.emplace_back(
                        std::move(boundary_line));
 
                      face->set_user_flag();
                    }
              }
 
            // Reset the user flags to their previous values:
            const_cast<Triangulation<dim, spacedim1> &>(in_tria)
              .load_user_flags_line(user_flags_line);
 
            break;
          }
 
        case 3:
          {
            std::vector<bool> user_flags_line;
            in_tria.save_user_flags_line(user_flags_line);
            const_cast<Triangulation<dim, spacedim1> &>(in_tria)
              .clear_user_flags_line();
 
            std::vector<bool> user_flags_quad;
            in_tria.save_user_flags_quad(user_flags_quad);
            const_cast<Triangulation<dim, spacedim1> &>(in_tria)
              .clear_user_flags_quad();
 
            // Loop over all the faces of the triangulation and create
            // objects that describe their boundary and manifold ids.
            for (const auto &face : in_tria.active_face_iterators())
              {
                if (face->at_boundary())
                  {
                    CellData<2> boundary_face;
 
                    boundary_face.vertices.resize(face->n_vertices());
                    for (const auto i : face->vertex_indices())
                      boundary_face.vertices[i] = face->vertex_index(i);
                    boundary_face.boundary_id = face->boundary_id();
                    boundary_face.manifold_id = face->manifold_id();
 
                    subcelldata.boundary_quads.emplace_back(
                      std::move(boundary_face));
 
                    // Then also loop over the edges and do the same. We would
                    // accidentally create duplicates for edges that are part of
                    // two boundary faces. To avoid this, use the user_flag on
                    // edges to mark those that we have already visited. (Note
                    // how we save and restore those above and below.)
                    for (unsigned int e = 0; e < face->n_lines(); ++e)
                      if (face->line(e)->user_flag_set() == false)
                        {
                          const typename Triangulation<dim,
                                                       spacedim1>::line_iterator
                                      edge = face->line(e);
                          CellData<1> boundary_edge;
 
                          boundary_edge.vertices.resize(edge->n_vertices());
                          for (const auto i : edge->vertex_indices())
                            boundary_edge.vertices[i] = edge->vertex_index(i);
                          boundary_edge.boundary_id = edge->boundary_id();
                          boundary_edge.manifold_id = edge->manifold_id();
 
                          subcelldata.boundary_lines.emplace_back(
                            std::move(boundary_edge));
 
                          edge->set_user_flag();
                        }
                  }
                else
                  // The face is not at the boundary. We won't have to set
                  // boundary_ids (that is not possible for interior faces), but
                  // we need to do something if the manifold-id is not the
                  // default.
                  //
                  // We keep track via the user flags whether we have already
                  // dealt with a face or not. (We need to do that here because
                  // we will return to interior faces twice, once for each
                  // neighbor, whereas we only touch each of the boundary faces
                  // above once.)
                  //
                  // Note that if we have already dealt with a face, then we
                  // have also already dealt with the edges and don't have
                  // to worry about that any more separately.
                  if (face->user_flag_set() == false)
                    {
                      if (face->manifold_id() != numbers::flat_manifold_id)
                        {
                          CellData<2> boundary_face;
 
                          boundary_face.vertices.resize(face->n_vertices());
                          for (const auto i : face->vertex_indices())
                            boundary_face.vertices[i] = face->vertex_index(i);
                          boundary_face.boundary_id =
                            numbers::internal_face_boundary_id;
                          boundary_face.manifold_id = face->manifold_id();
 
                          subcelldata.boundary_quads.emplace_back(
                            std::move(boundary_face));
 
                          face->set_user_flag();
                        }
 
                      // Then also loop over the edges of this face. Because
                      // every boundary edge must also be a part of a boundary
                      // face, we can ignore these. But it is possible that we
                      // have already encountered an interior edge through a
                      // previous face, and in that case we have to just ignore
                      // it
                      for (unsigned int e = 0; e < face->n_lines(); ++e)
                        if (face->line(e)->at_boundary() == false)
                          if (face->line(e)->user_flag_set() == false)
                            {
                              const typename Triangulation<dim, spacedim1>::
                                line_iterator edge = face->line(e);
                              CellData<1>     boundary_edge;
 
                              boundary_edge.vertices.resize(edge->n_vertices());
                              for (const auto i : edge->vertex_indices())
                                boundary_edge.vertices[i] =
                                  edge->vertex_index(i);
                              boundary_edge.boundary_id =
                                numbers::internal_face_boundary_id;
                              boundary_edge.manifold_id = edge->manifold_id();
 
                              subcelldata.boundary_lines.emplace_back(
                                std::move(boundary_edge));
 
                              edge->set_user_flag();
                            }
                    }
              }
 
            // Reset the user flags to their previous values:
            const_cast<Triangulation<dim, spacedim1> &>(in_tria)
              .load_user_flags_line(user_flags_line);
            const_cast<Triangulation<dim, spacedim1> &>(in_tria)
              .load_user_flags_quad(user_flags_quad);
 
            break;
          }
        default:
          DEAL_II_ASSERT_UNREACHABLE();
      }
 
    out_tria.create_triangulation(v, cells, subcelldata);
 
    for (const auto i : out_tria.get_manifold_ids())
      if (i != numbers::flat_manifold_id)
        out_tria.set_manifold(i, FlatManifold<dim, spacedim2>());
  }
 
 
 
  template <int dim, int spacedim>
  void
  convert_hypercube_to_simplex_mesh(const Triangulation<dim, spacedim> &in_tria,
                                    Triangulation<dim, spacedim> &out_tria)
  {
    Assert(dim > 1, ExcNotImplemented());
    AssertThrow(
      in_tria.all_reference_cells_are_hyper_cube(),
      ExcMessage(
        "GridGenerator::convert_hypercube_to_simplex_mesh() expects a mesh that consists only of quads/hexes."));
 
    Triangulation<dim, spacedim> temp_tria;
    if (in_tria.n_global_levels() > 1)
      {
        AssertThrow(!in_tria.has_hanging_nodes(), ExcNotImplemented());
        flatten_triangulation(in_tria, temp_tria);
      }
    const Triangulation<dim, spacedim> &ref_tria =
      in_tria.n_global_levels() > 1 ? temp_tria : in_tria;
 
    // static tables with the definitions of cells, faces and edges by its
    // vertices for 2d and 3d. For the inheritance of the manifold_id,
    // definitions of inner-faces and boundary-faces are required. In case of
    // 3d, also inner-edges and boundary-edges need to be defined.
 
    // Cell definition 2d:
    // A quadrilateral element is converted to 8 simplices elements. Each
    // triangle is defined by 3 vertices.
    static const ndarray<unsigned int, 8, 3> table_2D_cell = {{{{0, 6, 4}},
                                                               {{8, 4, 6}},
                                                               {{8, 6, 5}},
                                                               {{1, 5, 6}},
                                                               {{2, 4, 7}},
                                                               {{8, 7, 4}},
                                                               {{8, 5, 7}},
                                                               {{3, 7, 5}}}};
 
    // Cell definition 3d:
    // A hexahedron element is converted to 24 tetrahedron elements. Each
    // tetrahedron is defined by 4 vertices.
    static const ndarray<unsigned int, 24, 4> vertex_ids_for_cells_3d = {
      {{{0, 1, 12, 10}},  {{2, 3, 11, 12}},  {{7, 6, 11, 13}},
       {{5, 4, 13, 10}},  {{0, 2, 8, 12}},   {{4, 6, 13, 8}},
       {{5, 13, 7, 9}},   {{1, 9, 3, 12}},   {{0, 8, 4, 10}},
       {{1, 5, 9, 10}},   {{3, 7, 11, 9}},   {{2, 6, 8, 11}},
       {{12, 13, 10, 9}}, {{12, 13, 9, 11}}, {{12, 13, 11, 8}},
       {{12, 13, 8, 10}}, {{13, 8, 10, 4}},  {{13, 10, 9, 5}},
       {{13, 9, 11, 7}},  {{13, 11, 8, 6}},  {{10, 12, 9, 1}},
       {{9, 12, 11, 3}},  {{11, 12, 8, 2}},  {{8, 12, 10, 0}}}};
 
    // Boundary-faces 2d:
    // After converting, each of the 4 quadrilateral faces is defined by faces
    // of 2 different triangles, i.e., lines. Note that lines are defined by 2
    // vertices.
    static const ndarray<unsigned int, 4, 2, 2>
      vertex_ids_for_boundary_faces_2d = {{{{{{0, 4}}, {{4, 2}}}},
                                           {{{{1, 5}}, {{5, 3}}}},
                                           {{{{0, 6}}, {{6, 1}}}},
                                           {{{{2, 7}}, {{7, 3}}}}}};
 
    // Boundary-faces 3d:
    // After converting, each of the 6 hexahedron faces corresponds to faces of
    // 4 different tetrahedron faces, i.e., triangles. Note that a triangle is
    // defined by 3 vertices.
    static const ndarray<unsigned int, 6, 4, 3>
      vertex_ids_for_boundary_faces_3d = {
        {{{{{0, 4, 8}}, {{4, 8, 6}}, {{8, 6, 2}}, {{0, 2, 8}}}},
         {{{{1, 3, 9}}, {{3, 9, 7}}, {{9, 7, 5}}, {{1, 9, 5}}}},
         {{{{0, 1, 10}}, {{1, 10, 5}}, {{10, 5, 4}}, {{0, 10, 4}}}},
         {{{{2, 3, 11}}, {{3, 11, 7}}, {{11, 7, 6}}, {{2, 11, 6}}}},
         {{{{0, 1, 12}}, {{1, 12, 3}}, {{12, 3, 2}}, {{0, 12, 2}}}},
         {{{{4, 5, 13}}, {{5, 13, 7}}, {{13, 7, 6}}, {{4, 13, 6}}}}}};
 
    // Inner-faces 2d:
    // The converted triangulation based on simplices has 8 faces that do not
    // form the boundary, i.e. inner-faces, each defined by 2 vertices.
    static const ndarray<unsigned int, 8, 2> vertex_ids_for_inner_faces_2d = {
      {{{6, 4}},
       {{6, 8}},
       {{6, 5}},
       {{4, 8}},
       {{8, 5}},
       {{7, 4}},
       {{7, 8}},
       {{7, 5}}}};
 
    // Inner-faces 3d:
    // The converted triangulation based on simplices has 72 faces that do not
    // form the boundary, i.e. inner-faces, each defined by 3 vertices.
    static const ndarray<unsigned int, 72, 3> vertex_ids_for_inner_faces_3d = {
      {{{0, 12, 10}},  {{12, 1, 10}},  {{12, 1, 9}},  {{12, 3, 9}},
       {{12, 2, 11}},  {{12, 3, 11}},  {{12, 0, 8}},  {{12, 2, 8}},
       {{9, 13, 5}},   {{13, 7, 9}},   {{11, 7, 13}}, {{11, 6, 13}},
       {{4, 8, 13}},   {{6, 8, 13}},   {{4, 13, 10}}, {{13, 5, 10}},
       {{10, 9, 5}},   {{10, 9, 1}},   {{11, 9, 7}},  {{11, 9, 3}},
       {{8, 11, 2}},   {{8, 11, 6}},   {{8, 10, 0}},  {{8, 10, 4}},
       {{12, 3, 9}},   {{12, 9, 11}},  {{12, 3, 11}}, {{3, 9, 11}},
       {{2, 12, 8}},   {{2, 12, 11}},  {{2, 11, 8}},  {{8, 12, 11}},
       {{0, 12, 10}},  {{0, 12, 8}},   {{0, 8, 10}},  {{8, 10, 12}},
       {{12, 1, 10}},  {{12, 1, 9}},   {{1, 10, 9}},  {{10, 9, 12}},
       {{10, 8, 4}},   {{10, 8, 13}},  {{4, 13, 8}},  {{4, 13, 10}},
       {{10, 9, 13}},  {{10, 9, 5}},   {{13, 5, 10}}, {{13, 5, 9}},
       {{13, 7, 9}},   {{13, 7, 11}},  {{9, 11, 13}}, {{9, 11, 7}},
       {{8, 11, 13}},  {{8, 11, 6}},   {{6, 13, 8}},  {{6, 13, 11}},
       {{12, 13, 10}}, {{12, 13, 8}},  {{8, 10, 13}}, {{8, 10, 12}},
       {{12, 13, 10}}, {{12, 13, 9}},  {{10, 9, 13}}, {{10, 9, 12}},
       {{12, 13, 9}},  {{12, 13, 11}}, {{9, 11, 13}}, {{9, 11, 12}},
       {{12, 13, 11}}, {{12, 13, 8}},  {{8, 11, 13}}, {{8, 11, 12}}}};
 
    // Inner-edges 3d:
    // The converted triangulation based on simplices has 60 edges that do not
    // coincide with the boundary, i.e. inner-edges, each defined by 2 vertices.
    static const ndarray<unsigned int, 60, 2> vertex_ids_for_inner_edges_3d = {
      {{{12, 10}}, {{12, 9}},  {{12, 11}}, {{12, 8}},  {{9, 13}},  {{11, 13}},
       {{8, 13}},  {{10, 13}}, {{10, 9}},  {{9, 11}},  {{11, 8}},  {{8, 10}},
       {{12, 9}},  {{12, 11}}, {{11, 9}},  {{12, 8}},  {{12, 11}}, {{11, 8}},
       {{12, 8}},  {{12, 10}}, {{10, 8}},  {{12, 10}}, {{12, 9}},  {{9, 10}},
       {{13, 10}}, {{13, 8}},  {{8, 10}},  {{13, 10}}, {{13, 9}},  {{9, 10}},
       {{13, 11}}, {{13, 9}},  {{11, 9}},  {{13, 11}}, {{13, 8}},  {{11, 8}},
       {{12, 13}}, {{8, 10}},  {{8, 13}},  {{10, 13}}, {{8, 12}},  {{10, 12}},
       {{12, 13}}, {{10, 9}},  {{10, 13}}, {{9, 13}},  {{10, 12}}, {{9, 12}},
       {{12, 13}}, {{9, 11}},  {{9, 13}},  {{11, 13}}, {{9, 12}},  {{11, 12}},
       {{12, 13}}, {{11, 8}},  {{11, 13}}, {{8, 13}},  {{11, 12}}, {{8, 12}}}};
 
    // Boundary-edges 3d:
    // For each of the 6 boundary-faces of the hexahedron, there are 8 edges (of
    // different tetrahedrons) that coincide with the boundary, i.e.
    // boundary-edges. Each boundary-edge is defined by 2 vertices. 4 of these
    // edges are new (they are placed in the middle of a presently existing
    // face); the other 4 coincide with edges present in the hexahedral
    // triangulation. The new 4 edges inherit the manifold id of the relevant
    // face, but the other 4 need to be copied from the input and thus do not
    // require a lookup table.
    static const ndarray<unsigned int, 6, 4, 2>
      vertex_ids_for_new_boundary_edges_3d = {
        {{{{{4, 8}}, {{6, 8}}, {{0, 8}}, {{2, 8}}}},
         {{{{5, 9}}, {{7, 9}}, {{1, 9}}, {{3, 9}}}},
         {{{{4, 10}}, {{5, 10}}, {{0, 10}}, {{1, 10}}}},
         {{{{6, 11}}, {{7, 11}}, {{2, 11}}, {{3, 11}}}},
         {{{{2, 12}}, {{3, 12}}, {{0, 12}}, {{1, 12}}}},
         {{{{6, 13}}, {{7, 13}}, {{4, 13}}, {{5, 13}}}}}};
 
    std::vector<Point<spacedim>> vertices;
    std::vector<CellData<dim>>   cells;
    SubCellData                  subcell_data;
 
    // store for each vertex and face the assigned index so that we only
    // assign them a value once
    std::vector<unsigned int> old_to_new_vertex_indices(
      ref_tria.n_vertices(), numbers::invalid_unsigned_int);
    std::vector<unsigned int> face_to_new_vertex_indices(
      ref_tria.n_faces(), numbers::invalid_unsigned_int);
 
    // We first have to create all of the new vertices. To do this, we loop over
    // all cells and on each cell
    // (i) copy the existing vertex locations (and record their new indices in
    // the 'old_to_new_vertex_indices' vector),
    // (ii) create new midpoint vertex locations for each face (and record their
    // new indices in the 'face_to_new_vertex_indices' vector),
    // (iii) create new midpoint vertex locations for each cell (dim = 2 only)
    for (const auto &cell : ref_tria.cell_iterators())
      {
        // temporary array storing the global indices of each cell entity in the
        // sequence: vertices, edges/faces, cell
        std::array<unsigned int, dim == 2 ? 9 : 14> local_vertex_indices;
 
        // (i) copy the existing vertex locations
        for (const auto v : cell->vertex_indices())
          {
            const auto v_global = cell->vertex_index(v);
 
            if (old_to_new_vertex_indices[v_global] ==
                numbers::invalid_unsigned_int)
              {
                old_to_new_vertex_indices[v_global] = vertices.size();
                vertices.push_back(cell->vertex(v));
              }
 
            AssertIndexRange(v, local_vertex_indices.size());
            local_vertex_indices[v] = old_to_new_vertex_indices[v_global];
          }
 
        // (ii) create new midpoint vertex locations for each face
        for (const auto f : cell->face_indices())
          {
            const auto f_global = cell->face_index(f);
 
            if (face_to_new_vertex_indices[f_global] ==
                numbers::invalid_unsigned_int)
              {
                face_to_new_vertex_indices[f_global] = vertices.size();
                vertices.push_back(
                  cell->face(f)->center(/*respect_manifold*/ true));
              }
 
            AssertIndexRange(cell->n_vertices() + f,
                             local_vertex_indices.size());
            local_vertex_indices[cell->n_vertices() + f] =
              face_to_new_vertex_indices[f_global];
          }
 
        // (iii) create new midpoint vertex locations for each cell
        if (dim == 2)
          {
            AssertIndexRange(cell->n_vertices() + cell->n_faces(),
                             local_vertex_indices.size());
            local_vertex_indices[cell->n_vertices() + cell->n_faces()] =
              vertices.size();
            vertices.push_back(cell->center(/*respect_manifold*/ true));
          }
 
        // helper function for creating cells and subcells
        const auto add_cell = [&](const unsigned int struct_dim,
                                  const auto        &index_vertices,
                                  const unsigned int material_or_boundary_id,
                                  const unsigned int manifold_id = 0) {
          // sub-cell data only has to be stored if the information differs
          // from the default
          if (struct_dim < dim &&
              (material_or_boundary_id == numbers::internal_face_boundary_id &&
               manifold_id == numbers::flat_manifold_id))
            return;
 
          if (struct_dim == dim) // cells
            {
              AssertDimension(index_vertices.size(), dim + 1);
 
              CellData<dim> cell_data(index_vertices.size());
              cell_data.material_id =
                material_or_boundary_id;           // inherit material id
              cell_data.manifold_id = manifold_id; // inherit cell-manifold id
              for (unsigned int i = 0; i < index_vertices.size(); ++i)
                {
                  AssertIndexRange(index_vertices[i],
                                   local_vertex_indices.size());
                  cell_data.vertices[i] =
                    local_vertex_indices[index_vertices[i]];
                }
              cells.push_back(cell_data);
            }
          else if (dim == 2 && struct_dim == 1) // an edge of a simplex
            {
              Assert(index_vertices.size() == 2, ExcInternalError());
              CellData<1> boundary_line(2);
              boundary_line.boundary_id = material_or_boundary_id;
              boundary_line.manifold_id = manifold_id;
              for (unsigned int i = 0; i < index_vertices.size(); ++i)
                {
                  AssertIndexRange(index_vertices[i],
                                   local_vertex_indices.size());
                  boundary_line.vertices[i] =
                    local_vertex_indices[index_vertices[i]];
                }
              subcell_data.boundary_lines.push_back(boundary_line);
            }
          else if (dim == 3 && struct_dim == 2) // a face of a tetrahedron
            {
              Assert(index_vertices.size() == 3, ExcInternalError());
              CellData<2> boundary_quad(3);
              boundary_quad.material_id = material_or_boundary_id;
              boundary_quad.manifold_id = manifold_id;
              for (unsigned int i = 0; i < index_vertices.size(); ++i)
                {
                  AssertIndexRange(index_vertices[i],
                                   local_vertex_indices.size());
                  boundary_quad.vertices[i] =
                    local_vertex_indices[index_vertices[i]];
                }
              subcell_data.boundary_quads.push_back(boundary_quad);
            }
          else if (dim == 3 && struct_dim == 1) // an edge of a tetrahedron
            {
              Assert(index_vertices.size() == 2, ExcInternalError());
              CellData<1> boundary_line(2);
              boundary_line.boundary_id = material_or_boundary_id;
              boundary_line.manifold_id = manifold_id;
              for (unsigned int i = 0; i < index_vertices.size(); ++i)
                {
                  AssertIndexRange(index_vertices[i],
                                   local_vertex_indices.size());
                  boundary_line.vertices[i] =
                    local_vertex_indices[index_vertices[i]];
                }
              subcell_data.boundary_lines.push_back(boundary_line);
            }
          else
            {
              DEAL_II_NOT_IMPLEMENTED();
            }
        };
 
        const auto material_id_cell = cell->material_id();
 
        // create cells one by one
        if (dim == 2)
          {
            // get cell-manifold id from current quad cell
            const auto manifold_id_cell = cell->manifold_id();
            // inherit cell manifold
            for (const auto &cell_vertices : table_2D_cell)
              add_cell(dim, cell_vertices, material_id_cell, manifold_id_cell);
 
            // inherit inner manifold (faces)
            for (const auto &face_vertices : vertex_ids_for_inner_faces_2d)
              // set inner tri-faces according to cell-manifold of quad
              // element, set invalid b_id, since we do not want to modify
              // b_id inside
              add_cell(1,
                       face_vertices,
                       numbers::internal_face_boundary_id,
                       manifold_id_cell);
          }
        else if (dim == 3)
          {
            // get cell-manifold id from current quad cell
            const auto manifold_id_cell = cell->manifold_id();
            // inherit cell manifold
            for (const auto &cell_vertices : vertex_ids_for_cells_3d)
              add_cell(dim, cell_vertices, material_id_cell, manifold_id_cell);
 
            // set manifold of inner FACES of tets according to
            // hex-cell-manifold
            for (const auto &face_vertices : vertex_ids_for_inner_faces_3d)
              add_cell(2,
                       face_vertices,
                       numbers::internal_face_boundary_id,
                       manifold_id_cell);
 
            // set manifold of inner EDGES of tets according to
            // hex-cell-manifold
            for (const auto &edge_vertices : vertex_ids_for_inner_edges_3d)
              add_cell(1,
                       edge_vertices,
                       numbers::internal_face_boundary_id,
                       manifold_id_cell);
          }
        else
          DEAL_II_NOT_IMPLEMENTED();
 
        // Set up sub-cell data.
        for (const auto f : cell->face_indices())
          {
            const auto bid = cell->face(f)->boundary_id();
            const auto mid = cell->face(f)->manifold_id();
 
            // process boundary-faces: set boundary and manifold ids
            if (dim == 2) // 2d boundary-faces
              {
                for (const auto &face_vertices :
                     vertex_ids_for_boundary_faces_2d[f])
                  add_cell(1, face_vertices, bid, mid);
              }
            else if (dim == 3) // 3d boundary-faces
              {
                // set manifold ids of tet-boundary-faces according to
                // hex-boundary-faces
                for (const auto &face_vertices :
                     vertex_ids_for_boundary_faces_3d[f])
                  add_cell(2, face_vertices, bid, mid);
                // set manifold ids of new tet-boundary-edges according to
                // hex-boundary-faces
                for (const auto &edge_vertices :
                     vertex_ids_for_new_boundary_edges_3d[f])
                  add_cell(1, edge_vertices, bid, mid);
              }
            else
              DEAL_II_NOT_IMPLEMENTED();
          }
 
        // set manifold ids of edges that were already present in the
        // triangulation.
        if (dim == 3)
          {
            for (const auto e : cell->line_indices())
              {
                auto edge = cell->line(e);
                // Rather than use add_cell(), which does additional index
                // translation, just add edges directly into subcell_data since
                // we already know the correct global vertex indices.
                CellData<1> edge_data;
                edge_data.vertices[0] =
                  old_to_new_vertex_indices[edge->vertex_index(0)];
                edge_data.vertices[1] =
                  old_to_new_vertex_indices[edge->vertex_index(1)];
                edge_data.boundary_id = edge->boundary_id();
                edge_data.manifold_id = edge->manifold_id();
 
                subcell_data.boundary_lines.push_back(std::move(edge_data));
              }
          }
      }
 
    out_tria.clear();
    out_tria.create_triangulation(vertices, cells, subcell_data);
 
    for (const auto i : out_tria.get_manifold_ids())
      if (i != numbers::flat_manifold_id)
        out_tria.set_manifold(i, FlatManifold<dim, spacedim>());
  }
 
 
 
  template <int spacedim>
  void
  convert_hypercube_to_simplex_mesh(const Triangulation<1, spacedim> &in_tria,
                                    Triangulation<1, spacedim>       &out_tria)
  {
    out_tria.copy_triangulation(in_tria);
    return;
  }
 
 
 
  template <int dim, int spacedim>
  void
  alfeld_split_of_simplex_mesh(const Triangulation<dim, spacedim> &in_tria,
                               Triangulation<dim, spacedim>       &out_tria)
  {
    Assert(dim == 2, ExcNotImplemented());
 
    Triangulation<dim, spacedim> temp_tria;
    if (in_tria.n_global_levels() > 1)
      {
        AssertThrow(!in_tria.has_hanging_nodes(), ExcNotImplemented());
        GridGenerator::flatten_triangulation(in_tria, temp_tria);
      }
    const Triangulation<dim, spacedim> &ref_tria =
      in_tria.n_global_levels() > 1 ? temp_tria : in_tria;
 
    // Three triangles connecting to barycenter with vertex index 3:
    static const ndarray<unsigned int, 3, 3> table_2D_cell = {
      {{{0, 1, 3}}, {{1, 2, 3}}, {{2, 0, 3}}}};
 
    // Boundary-faces 2d:
    // Each face of the original simplex is defined by the following vertices:
    static const ndarray<unsigned int, 4, 2, 2>
      vertex_ids_for_boundary_faces_2d = {
        {{{{{0, 1}}}}, {{{{1, 2}}}}, {{{{2, 0}}}}}};
 
    std::vector<Point<spacedim>> vertices;
    std::vector<CellData<dim>>   cells;
    SubCellData                  subcell_data;
 
    // for each vertex we store the assigned index so that we only
    // assign them a value once
    std::vector<unsigned int> old_to_new_vertex_indices(
      ref_tria.n_vertices(), numbers::invalid_unsigned_int);
 
    // We first have to create all of the new vertices. To do this, we loop over
    // all cells and on each cell
    // (i) copy the existing vertex locations (and record their new indices in
    // the 'old_to_new_vertex_indices' vector),
    // (ii) create new barycenter vertex location
    for (const auto &cell : ref_tria.cell_iterators())
      {
        AssertThrow(
          cell->reference_cell().is_simplex(),
          ExcMessage(
            "Cell with invalid ReferenceCell encountered. GridGenerator::alfeld_split_of_simplex_mesh() "
            "only supports simplex meshes as input."));
 
        // temporary array storing the global indices of each cell entity in the
        // sequence: vertices, edges/faces, cell
        std::array<unsigned int, 4> local_vertex_indices;
 
        // (i) copy the existing vertex locations
        Point<spacedim> barycenter;
        for (const auto v : cell->vertex_indices())
          {
            const auto v_global = cell->vertex_index(v);
 
            if (old_to_new_vertex_indices[v_global] ==
                numbers::invalid_unsigned_int)
              {
                old_to_new_vertex_indices[v_global] = vertices.size();
                vertices.push_back(cell->vertex(v));
              }
 
            AssertIndexRange(v, local_vertex_indices.size());
            local_vertex_indices[v] = old_to_new_vertex_indices[v_global];
 
            barycenter += vertices[local_vertex_indices[v]] - Point<spacedim>();
          }
 
        // (ii) barycenter:
        local_vertex_indices[3] = vertices.size();
        vertices.push_back(barycenter / 3.);
 
        // helper function for creating cells and subcells
        const auto add_cell = [&](const unsigned int struct_dim,
                                  const auto        &index_vertices,
                                  const unsigned int material_or_boundary_id,
                                  const unsigned int manifold_id = 0) {
          // sub-cell data only has to be stored if the information differs
          // from the default
          if (struct_dim < dim &&
              (material_or_boundary_id == numbers::internal_face_boundary_id &&
               manifold_id == numbers::flat_manifold_id))
            return;
 
          if (struct_dim == dim) // cells
            {
              AssertDimension(index_vertices.size(), dim + 1);
 
              CellData<dim> cell_data(index_vertices.size());
              for (unsigned int i = 0; i < index_vertices.size(); ++i)
                {
                  AssertIndexRange(index_vertices[i],
                                   local_vertex_indices.size());
                  cell_data.vertices[i] =
                    local_vertex_indices[index_vertices[i]];
                }
              cell_data.material_id =
                material_or_boundary_id;           // inherit material id
              cell_data.manifold_id = manifold_id; // inherit cell-manifold id
              cells.push_back(cell_data);
            }
          else if (dim == 2 && struct_dim == 1) // an edge of a simplex
            {
              Assert(index_vertices.size() == 2, ExcInternalError());
              CellData<1> boundary_line(2);
              boundary_line.boundary_id = material_or_boundary_id;
              boundary_line.manifold_id = manifold_id;
              for (unsigned int i = 0; i < index_vertices.size(); ++i)
                {
                  AssertIndexRange(index_vertices[i],
                                   local_vertex_indices.size());
                  boundary_line.vertices[i] =
                    local_vertex_indices[index_vertices[i]];
                }
              subcell_data.boundary_lines.push_back(boundary_line);
            }
          else if (dim == 3 && struct_dim == 2) // a face of a tetrahedron
            {
              Assert(index_vertices.size() == 3, ExcInternalError());
              CellData<2> boundary_quad(3);
              boundary_quad.material_id = material_or_boundary_id;
              boundary_quad.manifold_id = manifold_id;
              for (unsigned int i = 0; i < index_vertices.size(); ++i)
                {
                  AssertIndexRange(index_vertices[i],
                                   local_vertex_indices.size());
                  boundary_quad.vertices[i] =
                    local_vertex_indices[index_vertices[i]];
                }
              subcell_data.boundary_quads.push_back(boundary_quad);
            }
          else if (dim == 3 && struct_dim == 1) // an edge of a tetrahedron
            {
              Assert(index_vertices.size() == 2, ExcInternalError());
              CellData<1> boundary_line(2);
              boundary_line.boundary_id = material_or_boundary_id;
              boundary_line.manifold_id = manifold_id;
              for (unsigned int i = 0; i < index_vertices.size(); ++i)
                {
                  AssertIndexRange(index_vertices[i],
                                   local_vertex_indices.size());
                  boundary_line.vertices[i] =
                    local_vertex_indices[index_vertices[i]];
                }
              subcell_data.boundary_lines.push_back(boundary_line);
            }
          else
            {
              DEAL_II_NOT_IMPLEMENTED();
            }
        };
 
        const auto material_id_cell = cell->material_id();
 
        // create cells one by one
        if (dim == 2)
          {
            // get cell-manifold id from current quad cell
            const auto manifold_id_cell = cell->manifold_id();
 
            for (const auto &cell_vertices : table_2D_cell)
              add_cell(dim, cell_vertices, material_id_cell, manifold_id_cell);
          }
        else
          DEAL_II_NOT_IMPLEMENTED();
 
        // Set up sub-cell data.
        for (const auto f : cell->face_indices())
          {
            const auto bid = cell->face(f)->boundary_id();
            const auto mid = cell->face(f)->manifold_id();
 
            // process boundary-faces: set boundary and manifold ids
            if (dim == 2) // 2d boundary-faces
              {
                for (const auto &face_vertices :
                     vertex_ids_for_boundary_faces_2d[f])
                  add_cell(1, face_vertices, bid, mid);
              }
            else
              DEAL_II_NOT_IMPLEMENTED();
          }
      }
 
    out_tria.clear();
    out_tria.create_triangulation(vertices, cells, subcell_data);
 
    for (const auto i : out_tria.get_manifold_ids())
      if (i != numbers::flat_manifold_id)
        out_tria.set_manifold(i, FlatManifold<dim, spacedim>());
  }
 
 
 
  template <template <int, int> class MeshType, int dim, int spacedim>
  DEAL_II_CXX20_REQUIRES(
    (concepts::is_triangulation_or_dof_handler<MeshType<dim, spacedim>>))
#  ifndef _MSC_VER
  std::map<typename MeshType<dim - 1, spacedim>::cell_iterator,
           typename MeshType<dim, spacedim>::face_iterator>
#  else
  typename ExtractBoundaryMesh<MeshType, dim, spacedim>::return_type
#  endif
    extract_boundary_mesh(const MeshType<dim, spacedim>      &volume_mesh,
                          MeshType<dim - 1, spacedim>        &surface_mesh,
                          const std::set<types::boundary_id> &boundary_ids)
  {
    Assert((dynamic_cast<
              const parallel::distributed::Triangulation<dim, spacedim> *>(
              &volume_mesh.get_triangulation()) == nullptr),
           ExcNotImplemented());
 
    // This function works using the following assumption:
    //    Triangulation::create_triangulation(...) will create cells that
    //    preserve the order of cells passed in using the CellData argument;
    //    also, that it will not reorder the vertices.
 
    // dimension of the boundary mesh
    const unsigned int boundary_dim = dim - 1;
 
    // temporary map for level==0
    // iterator to face is stored along with face number
    // (this is required by the algorithm to adjust the normals of the
    // cells of the boundary mesh)
    std::vector<
      std::pair<typename MeshType<dim, spacedim>::face_iterator, unsigned int>>
      temporary_mapping_level0;
 
    // vector indicating whether a vertex of the volume mesh has
    // already been visited (necessary to avoid duplicate vertices in
    // boundary mesh)
    std::vector<bool> touched(volume_mesh.get_triangulation().n_vertices(),
                              false);
 
    // data structures required for creation of boundary mesh
    std::vector<CellData<boundary_dim>> cells;
    SubCellData                         subcell_data;
    std::vector<Point<spacedim>>        vertices;
 
    // volume vertex indices to surf ones
    std::map<unsigned int, unsigned int> map_vert_index;
 
    // define swapping of vertices to get proper normal orientation of boundary
    // mesh;
    // the entry (i,j) of swap_matrix stores the index of the vertex of
    // the boundary cell corresponding to the j-th vertex on the i-th face
    // of the underlying volume cell
    // if e.g. face 3 of a volume cell is considered and vertices 1 and 2 of the
    // corresponding boundary cell are swapped to get
    // proper normal orientation, swap_matrix[3]=( 0, 2, 1, 3 )
    Table<2, unsigned int> swap_matrix(
      GeometryInfo<spacedim>::faces_per_cell,
      GeometryInfo<dim - 1>::vertices_per_cell);
    for (unsigned int i1 = 0; i1 < GeometryInfo<spacedim>::faces_per_cell; ++i1)
      {
        for (unsigned int i2 = 0; i2 < GeometryInfo<dim - 1>::vertices_per_cell;
             i2++)
          swap_matrix[i1][i2] = i2;
      }
    // vertex swapping such that normals on the surface mesh point out of the
    // underlying volume
    if (dim == 3)
      {
        std::swap(swap_matrix[0][1], swap_matrix[0][2]);
        std::swap(swap_matrix[2][1], swap_matrix[2][2]);
        std::swap(swap_matrix[4][1], swap_matrix[4][2]);
      }
    else if (dim == 2)
      {
        std::swap(swap_matrix[1][0], swap_matrix[1][1]);
        std::swap(swap_matrix[2][0], swap_matrix[2][1]);
      }
 
    // Create boundary mesh and mapping
    // from only level(0) cells of volume_mesh
    for (typename MeshType<dim, spacedim>::cell_iterator cell =
           volume_mesh.begin(0);
         cell != volume_mesh.end(0);
         ++cell)
      for (const unsigned int i : GeometryInfo<dim>::face_indices())
        {
          const typename MeshType<dim, spacedim>::face_iterator face =
            cell->face(i);
 
          if (face->at_boundary() &&
              (boundary_ids.empty() ||
               (boundary_ids.find(face->boundary_id()) != boundary_ids.end())))
            {
              CellData<boundary_dim> c_data;
 
              for (const unsigned int j :
                   GeometryInfo<boundary_dim>::vertex_indices())
                {
                  const unsigned int v_index = face->vertex_index(j);
 
                  if (!touched[v_index])
                    {
                      vertices.push_back(face->vertex(j));
                      map_vert_index[v_index] = vertices.size() - 1;
                      touched[v_index]        = true;
                    }
 
                  c_data.vertices[swap_matrix[i][j]] = map_vert_index[v_index];
                }
              c_data.material_id =
                static_cast<types::material_id>(face->boundary_id());
              c_data.manifold_id = face->manifold_id();
 
 
              // in 3d, we need to make sure we copy the manifold
              // indicators from the edges of the volume mesh to the
              // edges of the surface mesh
              //
              // we set default boundary ids for boundary lines
              // and numbers::internal_face_boundary_id for internal lines
              if (dim == 3)
                for (unsigned int e = 0; e < 4; ++e)
                  {
                    // see if we already saw this edge from a
                    // neighboring face, either in this or the reverse
                    // orientation. if so, skip it.
                    {
                      bool edge_found = false;
                      for (auto &boundary_line : subcell_data.boundary_lines)
                        if (((boundary_line.vertices[0] ==
                              map_vert_index[face->line(e)->vertex_index(0)]) &&
                             (boundary_line.vertices[1] ==
                              map_vert_index[face->line(e)->vertex_index(
                                1)])) ||
                            ((boundary_line.vertices[0] ==
                              map_vert_index[face->line(e)->vertex_index(1)]) &&
                             (boundary_line.vertices[1] ==
                              map_vert_index[face->line(e)->vertex_index(0)])))
                          {
                            boundary_line.boundary_id =
                              numbers::internal_face_boundary_id;
                            edge_found = true;
                            break;
                          }
                      if (edge_found == true)
                        // try next edge of current face
                        continue;
                    }
 
                    CellData<1> edge;
                    edge.vertices[0] =
                      map_vert_index[face->line(e)->vertex_index(0)];
                    edge.vertices[1] =
                      map_vert_index[face->line(e)->vertex_index(1)];
                    edge.boundary_id = 0;
                    edge.manifold_id = face->line(e)->manifold_id();
 
                    subcell_data.boundary_lines.push_back(edge);
                  }
 
              cells.push_back(c_data);
              temporary_mapping_level0.push_back(std::make_pair(face, i));
            }
        }
 
    // create level 0 surface triangulation
    Assert(cells.size() > 0, ExcMessage("No boundary faces selected"));
    const_cast<Triangulation<dim - 1, spacedim> &>(
      surface_mesh.get_triangulation())
      .create_triangulation(vertices, cells, subcell_data);
 
    // in 2d: set default boundary ids for "boundary vertices"
    if (dim == 2)
      {
        for (const auto &cell : surface_mesh.active_cell_iterators())
          for (unsigned int vertex = 0; vertex < 2; ++vertex)
            if (cell->face(vertex)->at_boundary())
              cell->face(vertex)->set_boundary_id(0);
      }
 
    // Make mapping for level 0
 
    // temporary map between cells on the boundary and corresponding faces of
    // domain mesh (each face is characterized by an iterator to the face and
    // the face number within the underlying cell)
    std::vector<std::pair<
      const typename MeshType<dim - 1, spacedim>::cell_iterator,
      std::pair<typename MeshType<dim, spacedim>::face_iterator, unsigned int>>>
      temporary_map_boundary_cell_face;
    for (const auto &cell : surface_mesh.active_cell_iterators())
      temporary_map_boundary_cell_face.push_back(
        std::make_pair(cell, temporary_mapping_level0.at(cell->index())));
 
 
    // refine the boundary mesh according to the refinement of the underlying
    // volume mesh,
    // algorithm:
    //   (1) check which cells on refinement level i need to be refined
    //   (2) do refinement (yields cells on level i+1)
    //   (3) repeat for the next level (i+1->i) until refinement is completed
 
    // stores the index into temporary_map_boundary_cell_face at which
    // presently deepest refinement level of boundary mesh begins
    unsigned int index_cells_deepest_level = 0;
    do
      {
        bool changed = false;
 
        // vector storing cells which have been marked for
        // refinement
        std::vector<unsigned int> cells_refined;
 
        // loop over cells of presently deepest level of boundary triangulation
        for (unsigned int cell_n = index_cells_deepest_level;
             cell_n < temporary_map_boundary_cell_face.size();
             cell_n++)
          {
            // mark boundary cell for refinement if underlying volume face has
            // children
            if (temporary_map_boundary_cell_face[cell_n]
                  .second.first->has_children())
              {
                // algorithm only works for
                // isotropic refinement!
                Assert(temporary_map_boundary_cell_face[cell_n]
                           .second.first->refinement_case() ==
                         RefinementCase<dim - 1>::isotropic_refinement,
                       ExcNotImplemented());
                temporary_map_boundary_cell_face[cell_n]
                  .first->set_refine_flag();
                cells_refined.push_back(cell_n);
                changed = true;
              }
          }
 
        // if cells have been marked for refinement (i.e., presently deepest
        // level is not the deepest level of the volume mesh)
        if (changed)
          {
            // do actual refinement
            const_cast<Triangulation<dim - 1, spacedim> &>(
              surface_mesh.get_triangulation())
              .execute_coarsening_and_refinement();
 
            // add new level of cells to temporary_map_boundary_cell_face
            index_cells_deepest_level = temporary_map_boundary_cell_face.size();
            for (const auto &refined_cell_n : cells_refined)
              {
                const typename MeshType<dim - 1, spacedim>::cell_iterator
                  refined_cell =
                    temporary_map_boundary_cell_face[refined_cell_n].first;
                const typename MeshType<dim,
                                        spacedim>::face_iterator refined_face =
                  temporary_map_boundary_cell_face[refined_cell_n].second.first;
                const unsigned int refined_face_number =
                  temporary_map_boundary_cell_face[refined_cell_n]
                    .second.second;
                for (unsigned int child_n = 0;
                     child_n < refined_cell->n_children();
                     ++child_n)
                  // at this point, the swapping of vertices done earlier must
                  // be taken into account to get the right association between
                  // volume faces and boundary cells!
                  temporary_map_boundary_cell_face.push_back(
                    std::make_pair(refined_cell->child(
                                     swap_matrix[refined_face_number][child_n]),
                                   std::make_pair(refined_face->child(child_n),
                                                  refined_face_number)));
              }
          }
        // we are at the deepest level of refinement of the volume mesh
        else
          break;
      }
    while (true);
 
    // generate the final mapping from the temporary mapping
    std::map<typename MeshType<dim - 1, spacedim>::cell_iterator,
             typename MeshType<dim, spacedim>::face_iterator>
      surface_to_volume_mapping;
    for (unsigned int i = 0; i < temporary_map_boundary_cell_face.size(); ++i)
      surface_to_volume_mapping[temporary_map_boundary_cell_face[i].first] =
        temporary_map_boundary_cell_face[i].second.first;
 
    // TODO: we attach flat manifolds here; one should attach submanifolds here
    const auto attached_mids =
      surface_mesh.get_triangulation().get_manifold_ids();
    for (const auto i : volume_mesh.get_triangulation().get_manifold_ids())
      if (i != numbers::flat_manifold_id &&
          std::find(attached_mids.begin(), attached_mids.end(), i) ==
            attached_mids.end())
        const_cast<Triangulation<dim - 1, spacedim> &>(
          surface_mesh.get_triangulation())
          .set_manifold(i, FlatManifold<dim - 1, spacedim>());
 
    return surface_to_volume_mapping;
  }
 
 
 
  template <int dim, int spacedim>
  void
  subdivided_hyper_rectangle_with_simplices(
    Triangulation<dim, spacedim>    &tria,
    const std::vector<unsigned int> &repetitions,
    const Point<dim>                &p1,
    const Point<dim>                &p2,
    const bool                       colorize)
  {
    AssertDimension(dim, spacedim);
 
    std::vector<Point<spacedim>> vertices;
    std::vector<CellData<dim>>   cells;
 
    if (dim == 2)
      {
        // determine cell sizes
        const Point<dim> dx((p2[0] - p1[0]) / repetitions[0],
                            (p2[1] - p1[1]) / repetitions[1]);
 
        // create vertices
        for (unsigned int j = 0; j <= repetitions[1]; ++j)
          for (unsigned int i = 0; i <= repetitions[0]; ++i)
            vertices.push_back(
              Point<spacedim>(p1[0] + dx[0] * i, p1[1] + dx[1] * j));
 
        // create cells
        for (unsigned int j = 0; j < repetitions[1]; ++j)
          for (unsigned int i = 0; i < repetitions[0]; ++i)
            {
              // create reference QUAD cell
              std::array<unsigned int, 4> quad{{
                (j + 0) * (repetitions[0] + 1) + i + 0, //
                (j + 0) * (repetitions[0] + 1) + i + 1, //
                (j + 1) * (repetitions[0] + 1) + i + 0, //
                (j + 1) * (repetitions[0] + 1) + i + 1  //
              }};                                       //
 
              // TRI cell 0
              {
                CellData<dim> tri;
                tri.vertices = {quad[0], quad[1], quad[2]};
                cells.push_back(tri);
              }
 
              // TRI cell 1
              {
                CellData<dim> tri;
                tri.vertices = {quad[3], quad[2], quad[1]};
                cells.push_back(tri);
              }
            }
      }
    else if (dim == 3)
      {
        // determine cell sizes
        const Point<dim> dx((p2[0] - p1[0]) / repetitions[0],
                            (p2[1] - p1[1]) / repetitions[1],
                            (p2[2] - p1[2]) / repetitions[2]);
 
        // create vertices
        for (unsigned int k = 0; k <= repetitions[2]; ++k)
          for (unsigned int j = 0; j <= repetitions[1]; ++j)
            for (unsigned int i = 0; i <= repetitions[0]; ++i)
              vertices.push_back(Point<spacedim>(p1[0] + dx[0] * i,
                                                 p1[1] + dx[1] * j,
                                                 p1[2] + dx[2] * k));
 
        // create cells
        for (unsigned int k = 0; k < repetitions[2]; ++k)
          for (unsigned int j = 0; j < repetitions[1]; ++j)
            for (unsigned int i = 0; i < repetitions[0]; ++i)
              {
                // create reference HEX cell
                std::array<unsigned int, 8> quad{
                  {(k + 0) * (repetitions[0] + 1) * (repetitions[1] + 1) +
                     (j + 0) * (repetitions[0] + 1) + i + 0,
                   (k + 0) * (repetitions[0] + 1) * (repetitions[1] + 1) +
                     (j + 0) * (repetitions[0] + 1) + i + 1,
                   (k + 0) * (repetitions[0] + 1) * (repetitions[1] + 1) +
                     (j + 1) * (repetitions[0] + 1) + i + 0,
                   (k + 0) * (repetitions[0] + 1) * (repetitions[1] + 1) +
                     (j + 1) * (repetitions[0] + 1) + i + 1,
                   (k + 1) * (repetitions[0] + 1) * (repetitions[1] + 1) +
                     (j + 0) * (repetitions[0] + 1) + i + 0,
                   (k + 1) * (repetitions[0] + 1) * (repetitions[1] + 1) +
                     (j + 0) * (repetitions[0] + 1) + i + 1,
                   (k + 1) * (repetitions[0] + 1) * (repetitions[1] + 1) +
                     (j + 1) * (repetitions[0] + 1) + i + 0,
                   (k + 1) * (repetitions[0] + 1) * (repetitions[1] + 1) +
                     (j + 1) * (repetitions[0] + 1) + i + 1}};
 
                // TET cell 0
                {
                  CellData<dim> cell;
                  if (((i % 2) + (j % 2) + (k % 2)) % 2 == 0)
                    cell.vertices = {{quad[0], quad[1], quad[2], quad[4]}};
                  else
                    cell.vertices = {{quad[0], quad[1], quad[3], quad[5]}};
 
                  cells.push_back(cell);
                }
 
                // TET cell 1
                {
                  CellData<dim> cell;
                  if (((i % 2) + (j % 2) + (k % 2)) % 2 == 0)
                    cell.vertices = {{quad[2], quad[1], quad[3], quad[7]}};
                  else
                    cell.vertices = {{quad[0], quad[3], quad[2], quad[6]}};
                  cells.push_back(cell);
                }
 
                // TET cell 2
                {
                  CellData<dim> cell;
                  if (((i % 2) + (j % 2) + (k % 2)) % 2 == 0)
                    cell.vertices = {{quad[1], quad[4], quad[5], quad[7]}};
                  else
                    cell.vertices = {{quad[0], quad[4], quad[5], quad[6]}};
                  cells.push_back(cell);
                }
 
                // TET cell 3
                {
                  CellData<dim> cell;
                  if (((i % 2) + (j % 2) + (k % 2)) % 2 == 0)
                    cell.vertices = {{quad[2], quad[4], quad[7], quad[6]}};
                  else
                    cell.vertices = {{quad[3], quad[5], quad[7], quad[6]}};
                  cells.push_back(cell);
                }
 
                // TET cell 4
                {
                  CellData<dim> cell;
                  if (((i % 2) + (j % 2) + (k % 2)) % 2 == 0)
                    cell.vertices = {{quad[1], quad[2], quad[4], quad[7]}};
                  else
                    cell.vertices = {{quad[0], quad[3], quad[6], quad[5]}};
                  cells.push_back(cell);
                }
              }
      }
    else
      {
        AssertThrow(false, ExcNotImplemented());
      }
 
    // actually create triangulation
    tria.create_triangulation(vertices, cells, SubCellData());
 
    if (colorize)
      {
        // to colorize, run through all
        // faces of all cells and set
        // boundary indicator to the
        // correct value if it was 0.
 
        // use a large epsilon to
        // compare numbers to avoid
        // roundoff problems.
        double epsilon = std::numeric_limits<double>::max();
        for (unsigned int i = 0; i < dim; ++i)
          epsilon = std::min(epsilon,
                             0.01 * (std::abs(p2[i] - p1[i]) / repetitions[i]));
        Assert(epsilon > 0,
               ExcMessage(
                 "The distance between corner points must be positive."));
 
        // actual code is external since
        // 1-D is different from 2/3d.
        colorize_subdivided_hyper_rectangle(tria, p1, p2, epsilon);
      }
  }
 
 
 
  template <int dim, int spacedim>
  void
  subdivided_hyper_cube_with_simplices(Triangulation<dim, spacedim> &tria,
                                       const unsigned int repetitions,
                                       const double       p1,
                                       const double       p2,
                                       const bool         colorize)
  {
    if (dim == 2)
      {
        subdivided_hyper_rectangle_with_simplices(
          tria, {{repetitions, repetitions}}, {p1, p1}, {p2, p2}, colorize);
      }
    else if (dim == 3)
      {
        subdivided_hyper_rectangle_with_simplices(
          tria,
          {{repetitions, repetitions, repetitions}},
          {p1, p1, p1},
          {p2, p2, p2},
          colorize);
      }
    else
      {
        AssertThrow(false, ExcNotImplemented());
      }
  }
} // namespace GridGenerator
 
// explicit instantiations
#  include "grid_generator.inst"
 
#endif // DOXYGEN
 
DEAL_II_NAMESPACE_CLOSE