source/grid/grid_reordering.cc

// ---------------------------------------------------------------------
//
// Copyright (C) 2000 - 2020 by the deal.II authors
//
// This file is part of the deal.II library.
//
// The deal.II library is free software; you can use it, redistribute
// it, and/or modify it under the terms of the GNU Lesser General
// Public License as published by the Free Software Foundation; either
// version 2.1 of the License, or (at your option) any later version.
// The full text of the license can be found in the file LICENSE.md at
// the top level directory of deal.II.
//
// ---------------------------------------------------------------------


#include <deal.II/base/timer.h>
#include <deal.II/base/utilities.h>

#include <deal.II/grid/grid_reordering.h>
#include <deal.II/grid/grid_tools.h>

#include <algorithm>
#include <fstream>
#include <functional>
#include <iostream>
#include <set>

DEAL_II_NAMESPACE_OPEN


namespace
{
  /**
   * A simple data structure denoting an edge, i.e., the ordered pair
   * of its vertex indices. This is only used in the is_consistent()
   * function.
   */
  struct CheapEdge
  {
    /**
     * Construct an edge from the global indices of its two vertices.
     */
    CheapEdge(const unsigned int v0, const unsigned int v1)
      : v0(v0)
      , v1(v1)
    {}

    /**
     * Comparison operator for edges. It compares based on the
     * lexicographic ordering of the two vertex indices.
     */
    bool
    operator<(const CheapEdge &e) const
    {
      return ((v0 < e.v0) || ((v0 == e.v0) && (v1 < e.v1)));
    }

  private:
    /**
     * The global indices of the vertices that define the edge.
     */
    const unsigned int v0, v1;
  };


  /**
   * A function that determines whether the edges in a mesh are
   * already consistently oriented. It does so by adding all edges
   * of all cells into a set (which automatically eliminates
   * duplicates) but before that checks whether the reverse edge is
   * already in the set -- which would imply that a neighboring cell
   * is inconsistently oriented.
   */
  template <int dim>
  bool
  is_consistent(const std::vector<CellData<dim>> &cells)
  {
    std::set<CheapEdge> edges;

    for (typename std::vector<CellData<dim>>::const_iterator c = cells.begin();
         c != cells.end();
         ++c)
      {
        // construct the edges in reverse order. for each of them,
        // ensure that the reverse edge is not yet in the list of
        // edges (return false if the reverse edge already *is* in
        // the list) and then add the actual edge to it; std::set
        // eliminates duplicates automatically
        for (unsigned int l = 0; l < GeometryInfo<dim>::lines_per_cell; ++l)
          {
            const CheapEdge reverse_edge(
              c->vertices[GeometryInfo<dim>::line_to_cell_vertices(l, 1)],
              c->vertices[GeometryInfo<dim>::line_to_cell_vertices(l, 0)]);
            if (edges.find(reverse_edge) != edges.end())
              return false;


            // ok, not. insert edge in correct order
            const CheapEdge correct_edge(
              c->vertices[GeometryInfo<dim>::line_to_cell_vertices(l, 0)],
              c->vertices[GeometryInfo<dim>::line_to_cell_vertices(l, 1)]);
            edges.insert(correct_edge);
          }
      }

    // no conflicts found, so return true
    return true;
  }


  /**
   * A structure that describes some properties of parallel edges
   * such as what starter edges are (i.e., representative elements
   * of the sets of parallel edges within a cell) and what the set
   * of parallel edges to each edge is.
   */
  template <int dim>
  struct ParallelEdges
  {
    /**
     * An array that contains the indices of dim edges that can
     * serve as (arbitrarily chosen) starting points for the
     * dim sets of parallel edges within each cell.
     */
    static const unsigned int starter_edges[dim];

    /**
     * Number and indices of all of those edges parallel to each of the
     * edges in a cell.
     */
    static const unsigned int n_other_parallel_edges = (1 << (dim - 1)) - 1;
    static const unsigned int parallel_edges[GeometryInfo<dim>::lines_per_cell]
                                            [n_other_parallel_edges];
  };

  template <>
  const unsigned int ParallelEdges<2>::starter_edges[2] = {0, 2};

  template <>
  const unsigned int ParallelEdges<2>::parallel_edges[4][1] = {{1},
                                                               {0},
                                                               {3},
                                                               {2}};

  template <>
  const unsigned int ParallelEdges<3>::starter_edges[3] = {0, 2, 8};

  template <>
  const unsigned int ParallelEdges<3>::parallel_edges[12][3] = {
    {1, 4, 5},   // line 0
    {0, 4, 5},   // line 1
    {3, 6, 7},   // line 2
    {2, 6, 7},   // line 3
    {0, 1, 5},   // line 4
    {0, 1, 4},   // line 5
    {2, 3, 7},   // line 6
    {2, 3, 6},   // line 7
    {9, 10, 11}, // line 8
    {8, 10, 11}, // line 9
    {8, 9, 11},  // line 10
    {8, 9, 10}   // line 11
  };


  /**
   * A structure that store the index of a cell and, crucially, how a
   * given edge relates to this cell.
   */
  struct AdjacentCell
  {
    /**
     * Default constructor. Initialize the fields with invalid values.
     */
    AdjacentCell()
      : cell_index(numbers::invalid_unsigned_int)
      , edge_within_cell(numbers::invalid_unsigned_int)
    {}

    /**
     * Constructor. Initialize the fields with the given values.
     */
    AdjacentCell(const unsigned int cell_index,
                 const unsigned int edge_within_cell)
      : cell_index(cell_index)
      , edge_within_cell(edge_within_cell)
    {}


    unsigned int cell_index;
    unsigned int edge_within_cell;
  };


  template <int dim>
  class AdjacentCells;

  /**
   * A class that represents all of the cells adjacent to a given edge.
   * This class corresponds to the 2d case where each edge has at most
   * two adjacent cells.
   */
  template <>
  class AdjacentCells<2>
  {
  public:
    /**
     * An iterator that allows iterating over all cells adjacent
     * to the edge represented by the current object.
     */
    using const_iterator = const AdjacentCell *;

    /**
     * Add the given cell to the collection of cells adjacent to
     * the edge this object corresponds to. Since we are covering
     * the 2d case, the set of adjacent cells currently
     * represented by this object must have either zero or
     * one element already, since we can not add more than two
     * adjacent cells for each edge.
     */
    void
    push_back(const AdjacentCell &adjacent_cell)
    {
      if (adjacent_cells[0].cell_index == numbers::invalid_unsigned_int)
        adjacent_cells[0] = adjacent_cell;
      else
        {
          Assert(adjacent_cells[1].cell_index == numbers::invalid_unsigned_int,
                 ExcInternalError());
          adjacent_cells[1] = adjacent_cell;
        }
    }


    /**
     * Return an iterator to the first valid cell stored as adjacent to the
     * edge represented by the current object.
     */
    const_iterator
    begin() const
    {
      return adjacent_cells;
    }


    /**
     * Return an iterator to the element past the last valid cell stored
     * as adjacent to the edge represented by the current object.
     * @return
     */
    const_iterator
    end() const
    {
      // check whether the current object stores zero, one, or two
      // adjacent cells, and use this to point to the element past the
      // last valid one
      if (adjacent_cells[0].cell_index == numbers::invalid_unsigned_int)
        return adjacent_cells;
      else if (adjacent_cells[1].cell_index == numbers::invalid_unsigned_int)
        return adjacent_cells + 1;
      else
        return adjacent_cells + 2;
    }

  private:
    /**
     * References to the (at most) two cells that are adjacent to
     * the edge this object corresponds to. Unused elements are
     * default-initialized and have invalid values; in particular,
     * their cell_index field equals numbers::invalid_unsigned_int.
     */
    AdjacentCell adjacent_cells[2];
  };


  /**
   * A class that represents all of the cells adjacent to a given edge.
   * This class corresponds to the 3d case where each edge can have an
   * arbitrary number of adjacent cells. We represent this as a
   * std::vector<AdjacentCell>, from which class the current one is
   * derived and from which it inherits all of its member functions.
   */
  template <>
  class AdjacentCells<3> : public std::vector<AdjacentCell>
  {};


  /**
   * A class that describes all of the relevant properties of an
   * edge. For the purpose of what we do here, that includes the
   * indices of the two vertices, and the indices of the adjacent
   * cells (together with a description *where* in each of the
   * adjacent cells the edge is located). It also includes the
   * (global) direction of the edge: either from the first vertex to
   * the second, the other way around, or so far undetermined.
   */
  template <int dim>
  class Edge
  {
  public:
    /**
     * Constructor. Create the edge based on the information given
     * in @p cell, and selecting the edge with number @p edge_number
     * within this cell. Initialize the edge as unoriented.
     */
    Edge(const CellData<dim> &cell, const unsigned int edge_number)
      : orientation_status(not_oriented)
    {
      Assert(edge_number < GeometryInfo<dim>::lines_per_cell,
             ExcInternalError());

      // copy vertices for this particular line
      vertex_indices[0] =
        cell.vertices[GeometryInfo<dim>::line_to_cell_vertices(edge_number, 0)];
      vertex_indices[1] =
        cell.vertices[GeometryInfo<dim>::line_to_cell_vertices(edge_number, 1)];

      // bring them into standard orientation
      if (vertex_indices[0] > vertex_indices[1])
        std::swap(vertex_indices[0], vertex_indices[1]);
    }

    /**
     * Comparison operator for edges. It compares based on the
     * lexicographic ordering of the two vertex indices.
     */
    bool
    operator<(const Edge<dim> &e) const
    {
      return ((vertex_indices[0] < e.vertex_indices[0]) ||
              ((vertex_indices[0] == e.vertex_indices[0]) &&
               (vertex_indices[1] < e.vertex_indices[1])));
    }

    /**
     * Compare two edges for equality based on their vertex indices.
     */
    bool
    operator==(const Edge<dim> &e) const
    {
      return ((vertex_indices[0] == e.vertex_indices[0]) &&
              (vertex_indices[1] == e.vertex_indices[1]));
    }

    /**
     * The global indices of the two vertices that bound this edge. These
     * will be ordered so that the first index is less than the second.
     */
    unsigned int vertex_indices[2];

    /**
     * An enum that indicates the direction of this edge with
     * regard to the two vertices that bound it.
     */
    enum OrientationStatus
    {
      not_oriented,
      forward,
      backward
    };

    OrientationStatus orientation_status;

    /**
     * Store the set of cells adjacent to this edge (these cells then
     * also store *where* in the cell the edge is located).
     */
    AdjacentCells<dim> adjacent_cells;
  };


  /**
   * A data structure that represents a cell with all of its vertices
   * and edges.
   */
  template <int dim>
  struct Cell
  {
    /**
     * Construct a Cell object from a CellData object. Also take a
     * (sorted) list of edges and to point the edges of the current
     * object into this list of edges.
     */
    Cell(const CellData<dim> &c, const std::vector<Edge<dim>> &edge_list)
    {
      for (const unsigned int i : GeometryInfo<dim>::vertex_indices())
        vertex_indices[i] = c.vertices[i];

      // now for each of the edges of this cell, find the location inside the
      // given edge_list array and store than index
      for (unsigned int l = 0; l < GeometryInfo<dim>::lines_per_cell; ++l)
        {
          const Edge<dim> e(c, l);
          edge_indices[l] =
            (std::lower_bound(edge_list.begin(), edge_list.end(), e) -
             edge_list.begin());
          Assert(edge_indices[l] < edge_list.size(), ExcInternalError());
          Assert(edge_list[edge_indices[l]] == e, ExcInternalError())
        }
    }

    /**
     * A list of global indices for the vertices that bound this cell.
     */
    unsigned int vertex_indices[GeometryInfo<dim>::vertices_per_cell];

    /**
     * A list of indices into the 'edge_list' array passed to the constructor
     * for the edges of the current cell.
     */
    unsigned int edge_indices[GeometryInfo<dim>::lines_per_cell];
  };


  template <int dim>
  class EdgeDeltaSet;

  /**
   * A class that represents by how much the set of parallel edges
   * grows in each step. In the graph orientation paper, this set is
   * called $\Delta_k$, thus the name.
   *
   * In 2d, this set can only include zero, one, or two elements.
   * Consequently, the appropriate data structure is one in which
   * we store at most 2 elements in a fixed sized data structure.
   */
  template <>
  class EdgeDeltaSet<2>
  {
  public:
    /**
     * Iterator type for the elements of the set.
     */
    using const_iterator = const unsigned int *;

    /**
     * Default constructor. Initialize both slots as unused, corresponding
     * to an empty set.
     */
    EdgeDeltaSet()
    {
      edge_indices[0] = edge_indices[1] = numbers::invalid_unsigned_int;
    }


    /**
     * Delete the elements of the set by marking both slots as unused.
     */
    void
    clear()
    {
      edge_indices[0] = edge_indices[1] = numbers::invalid_unsigned_int;
    }

    /**
     * Insert one element into the set. This will fail if the set already
     * has two elements.
     */
    void
    insert(const unsigned int edge_index)
    {
      if (edge_indices[0] == numbers::invalid_unsigned_int)
        edge_indices[0] = edge_index;
      else
        {
          Assert(edge_indices[1] == numbers::invalid_unsigned_int,
                 ExcInternalError());
          edge_indices[1] = edge_index;
        }
    }


    /**
     * Return an iterator pointing to the first element of the set.
     */
    const_iterator
    begin() const
    {
      return edge_indices;
    }


    /**
     * Return an iterator pointing to the element past the last used one.
     */
    const_iterator
    end() const
    {
      // check whether the current object stores zero, one, or two
      // indices, and use this to point to the element past the
      // last valid one
      if (edge_indices[0] == numbers::invalid_unsigned_int)
        return edge_indices;
      else if (edge_indices[1] == numbers::invalid_unsigned_int)
        return edge_indices + 1;
      else
        return edge_indices + 2;
    }

  private:
    /**
     * Storage space to store the indices of at most two edges.
     */
    unsigned int edge_indices[2];
  };


  /**
   * A class that represents by how much the set of parallel edges
   * grows in each step. In the graph orientation paper, this set is
   * called $\Delta_k$, thus the name.
   *
   * In 3d, this set can have arbitrarily many elements, unlike the
   * 2d case specialized above. Consequently, we simply represent
   * the data structure with a std::set. Class derivation ensures
   * that we simply inherit all of the member functions of the
   * base class.
   */
  template <>
  class EdgeDeltaSet<3> : public std::set<unsigned int>
  {};


  /**
   * From a list of cells, build a sorted vector that contains all of the edges
   * that exist in the mesh.
   */
  template <int dim>
  std::vector<Edge<dim>>
  build_edges(const std::vector<CellData<dim>> &cells)
  {
    // build the edge list for all cells. because each cell has
    // GeometryInfo<dim>::lines_per_cell edges, the total number
    // of edges is this many times the number of cells. of course
    // some of them will be duplicates, and we throw them out below
    std::vector<Edge<dim>> edge_list;
    edge_list.reserve(cells.size() * GeometryInfo<dim>::lines_per_cell);
    for (unsigned int i = 0; i < cells.size(); ++i)
      for (unsigned int l = 0; l < GeometryInfo<dim>::lines_per_cell; ++l)
        edge_list.emplace_back(cells[i], l);

    // next sort the edge list and then remove duplicates
    std::sort(edge_list.begin(), edge_list.end());
    edge_list.erase(std::unique(edge_list.begin(), edge_list.end()),
                    edge_list.end());

    return edge_list;
  }


  /**
   * Build the cell list. Update the edge array to let edges know
   * which cells are adjacent to them.
   */
  template <int dim>
  std::vector<Cell<dim>>
  build_cells_and_connect_edges(const std::vector<CellData<dim>> &cells,
                                std::vector<Edge<dim>> &          edges)
  {
    std::vector<Cell<dim>> cell_list;
    cell_list.reserve(cells.size());
    for (unsigned int i = 0; i < cells.size(); ++i)
      {
        // create our own data structure for the cells and let it
        // connect to the edges array
        cell_list.emplace_back(cells[i], edges);

        // then also inform the edges that they are adjacent
        // to the current cell, and where within this cell
        for (unsigned int l = 0; l < GeometryInfo<dim>::lines_per_cell; ++l)
          edges[cell_list.back().edge_indices[l]].adjacent_cells.push_back(
            AdjacentCell(i, l));
      }
    Assert(cell_list.size() == cells.size(), ExcInternalError());

    return cell_list;
  }


  /**
   * Return the index within 'cells' of the first cell that has at least one
   * edge that is not yet oriented.
   */
  template <int dim>
  unsigned int
  get_next_unoriented_cell(const std::vector<Cell<dim>> &cells,
                           const std::vector<Edge<dim>> &edges,
                           const unsigned int            current_cell)
  {
    for (unsigned int c = current_cell; c < cells.size(); ++c)
      for (unsigned int l = 0; l < GeometryInfo<dim>::lines_per_cell; ++l)
        if (edges[cells[c].edge_indices[l]].orientation_status ==
            Edge<dim>::not_oriented)
          return c;

    return numbers::invalid_unsigned_int;
  }


  /**
   * Given a set of cells and edges, orient all edges that are
   * (global) parallel to the one identified by the @p cell and
   * within it the one with index @p local_edge.
   */
  template <int dim>
  void
  orient_one_set_of_parallel_edges(const std::vector<Cell<dim>> &cells,
                                   std::vector<Edge<dim>> &      edges,
                                   const unsigned int            cell,
                                   const unsigned int            local_edge)
  {
    // choose the direction of the first edge. we have free choice
    // here and could simply choose "forward" if that's what pleases
    // us. however, for backward compatibility with the previous
    // implementation used till 2016, let us just choose the
    // direction so that it matches what we have in the given cell.
    //
    // in fact, in what can only be assumed to be a bug in the
    // original implementation, after orienting all edges, the code
    // that rotates the cells so that they match edge orientations
    // (see the rotate_cell() function below) rotated the cell two
    // more times by 90 degrees. this is ok -- it simply flips all
    // edge orientations, which leaves them valid. rather than do
    // the same in the current implementation, we can achieve the
    // same effect by modifying the rule above to choose the
    // direction of the starting edge of this parallel set
    // *opposite* to what it looks like in the current cell
    //
    // this bug only existed in the 2d implementation since there
    // were different implementations for 2d and 3d. consequently,
    // only replicate it for the 2d case and be "intuitive" in 3d.
    if (edges[cells[cell].edge_indices[local_edge]].vertex_indices[0] ==
        cells[cell].vertex_indices[GeometryInfo<dim>::line_to_cell_vertices(
          local_edge, 0)])
      // orient initial edge *opposite* to the way it is in the cell
      // (see above for the reason)
      edges[cells[cell].edge_indices[local_edge]].orientation_status =
        (dim == 2 ? Edge<dim>::backward : Edge<dim>::forward);
    else
      {
        Assert(
          edges[cells[cell].edge_indices[local_edge]].vertex_indices[0] ==
            cells[cell].vertex_indices[GeometryInfo<dim>::line_to_cell_vertices(
              local_edge, 1)],
          ExcInternalError());
        Assert(
          edges[cells[cell].edge_indices[local_edge]].vertex_indices[1] ==
            cells[cell].vertex_indices[GeometryInfo<dim>::line_to_cell_vertices(
              local_edge, 0)],
          ExcInternalError());

        // orient initial edge *opposite* to the way it is in the cell
        // (see above for the reason)
        edges[cells[cell].edge_indices[local_edge]].orientation_status =
          (dim == 2 ? Edge<dim>::forward : Edge<dim>::backward);
      }

    // walk outward from the given edge as described in
    // the algorithm in the paper that documents all of
    // this
    //
    // note that in 2d, each of the Deltas can at most
    // contain two elements, whereas in 3d it can be arbitrarily many
    EdgeDeltaSet<dim> Delta_k;
    EdgeDeltaSet<dim> Delta_k_minus_1;
    Delta_k_minus_1.insert(cells[cell].edge_indices[local_edge]);

    while (Delta_k_minus_1.begin() !=
           Delta_k_minus_1.end()) // while set is not empty
      {
        Delta_k.clear();

        for (typename EdgeDeltaSet<dim>::const_iterator delta =
               Delta_k_minus_1.begin();
             delta != Delta_k_minus_1.end();
             ++delta)
          {
            Assert(edges[*delta].orientation_status != Edge<dim>::not_oriented,
                   ExcInternalError());

            // now go through the cells adjacent to this edge
            for (typename AdjacentCells<dim>::const_iterator adjacent_cell =
                   edges[*delta].adjacent_cells.begin();
                 adjacent_cell != edges[*delta].adjacent_cells.end();
                 ++adjacent_cell)
              {
                const unsigned int K = adjacent_cell->cell_index;
                const unsigned int delta_is_edge_in_K =
                  adjacent_cell->edge_within_cell;

                // figure out the direction of delta with respect to the cell K
                // (in the orientation in which the user has given it to us)
                const unsigned int first_edge_vertex =
                  (edges[*delta].orientation_status == Edge<dim>::forward ?
                     edges[*delta].vertex_indices[0] :
                     edges[*delta].vertex_indices[1]);
                const unsigned int first_edge_vertex_in_K =
                  cells[K]
                    .vertex_indices[GeometryInfo<dim>::line_to_cell_vertices(
                      delta_is_edge_in_K, 0)];
                Assert(
                  first_edge_vertex == first_edge_vertex_in_K ||
                    first_edge_vertex ==
                      cells[K].vertex_indices[GeometryInfo<
                        dim>::line_to_cell_vertices(delta_is_edge_in_K, 1)],
                  ExcInternalError());

                // now figure out which direction the each of the "opposite"
                // edges needs to be oriented into.
                for (unsigned int o_e = 0;
                     o_e < ParallelEdges<dim>::n_other_parallel_edges;
                     ++o_e)
                  {
                    // get the index of the opposite edge and select which its
                    // first vertex needs to be based on how the current edge is
                    // oriented in the current cell
                    const unsigned int opposite_edge =
                      cells[K].edge_indices[ParallelEdges<
                        dim>::parallel_edges[delta_is_edge_in_K][o_e]];
                    const unsigned int first_opposite_edge_vertex =
                      cells[K].vertex_indices
                        [GeometryInfo<dim>::line_to_cell_vertices(
                          ParallelEdges<dim>::parallel_edges[delta_is_edge_in_K]
                                                            [o_e],
                          (first_edge_vertex == first_edge_vertex_in_K ? 0 :
                                                                         1))];

                    // then determine the orientation of the edge based on
                    // whether the vertex we want to be the edge's first
                    // vertex is already the first vertex of the edge, or
                    // whether it points in the opposite direction
                    const typename Edge<dim>::OrientationStatus
                      opposite_edge_orientation =
                        (edges[opposite_edge].vertex_indices[0] ==
                             first_opposite_edge_vertex ?
                           Edge<dim>::forward :
                           Edge<dim>::backward);

                    // see if the opposite edge (there is only one in 2d) has
                    // already been oriented.
                    if (edges[opposite_edge].orientation_status ==
                        Edge<dim>::not_oriented)
                      {
                        // the opposite edge is not yet oriented. do orient it
                        // and add it to Delta_k
                        edges[opposite_edge].orientation_status =
                          opposite_edge_orientation;
                        Delta_k.insert(opposite_edge);
                      }
                    else
                      {
                        // this opposite edge has already been oriented. it
                        // should be consistent with the current one in 2d,
                        // while in 3d it may in fact be mis-oriented, and in
                        // that case the mesh will not be orientable. indicate
                        // this by throwing an exception that we can catch
                        // further up; this has the advantage that we can
                        // propagate through a couple of functions without
                        // having to do error checking and without modifying the
                        // 'cells' array that the user gave us
                        if (dim == 2)
                          {
                            Assert(edges[opposite_edge].orientation_status ==
                                     opposite_edge_orientation,
                                   ExcMeshNotOrientable());
                          }
                        else if (dim == 3)
                          {
                            if (edges[opposite_edge].orientation_status !=
                                opposite_edge_orientation)
                              throw ExcMeshNotOrientable();
                          }
                        else
                          Assert(false, ExcNotImplemented());
                      }
                  }
              }
          }

        // finally copy the new set to the previous one
        // (corresponding to increasing 'k' by one in the
        // algorithm)
        Delta_k_minus_1 = Delta_k;
      }
  }


  /**
   * Given data structures @p cell_list and @p edge_list, where
   * all edges are already oriented, rotate the cell with
   * index @p cell_index in such a way that its local coordinate
   * system matches the ones of the adjacent edges. Store the
   * rotated order of vertices in <code>raw_cells[cell_index]</code>.
   */
  template <int dim>
  void
  rotate_cell(const std::vector<Cell<dim>> &cell_list,
              const std::vector<Edge<dim>> &edge_list,
              const unsigned int            cell_index,
              std::vector<CellData<dim>> &  raw_cells)
  {
    // find the first vertex of the cell. this is the vertex where dim edges
    // originate, so for each of the edges record which the starting vertex is
    unsigned int starting_vertex_of_edge[GeometryInfo<dim>::lines_per_cell];
    for (unsigned int e = 0; e < GeometryInfo<dim>::lines_per_cell; ++e)
      {
        Assert(edge_list[cell_list[cell_index].edge_indices[e]]
                   .orientation_status != Edge<dim>::not_oriented,
               ExcInternalError());
        if (edge_list[cell_list[cell_index].edge_indices[e]]
              .orientation_status == Edge<dim>::forward)
          starting_vertex_of_edge[e] =
            edge_list[cell_list[cell_index].edge_indices[e]].vertex_indices[0];
        else
          starting_vertex_of_edge[e] =
            edge_list[cell_list[cell_index].edge_indices[e]].vertex_indices[1];
      }

    // find the vertex number that appears dim times. this will then be
    // the vertex at which we want to locate the origin of the cell's
    // coordinate system (i.e., vertex 0)
    unsigned int origin_vertex_of_cell = numbers::invalid_unsigned_int;
    switch (dim)
      {
        case 2:
          {
            // in 2d, we can simply enumerate the possibilities where the
            // origin may be located because edges zero and one don't share
            // any vertices, and the same for edges two and three
            if ((starting_vertex_of_edge[0] == starting_vertex_of_edge[2]) ||
                (starting_vertex_of_edge[0] == starting_vertex_of_edge[3]))
              origin_vertex_of_cell = starting_vertex_of_edge[0];
            else if ((starting_vertex_of_edge[1] ==
                      starting_vertex_of_edge[2]) ||
                     (starting_vertex_of_edge[1] == starting_vertex_of_edge[3]))
              origin_vertex_of_cell = starting_vertex_of_edge[1];
            else
              Assert(false, ExcInternalError());

            break;
          }

        case 3:
          {
            // one could probably do something similar in 3d, but that seems
            // more complicated than one wants to write down. just go
            // through the list of possible starting vertices and check
            for (origin_vertex_of_cell = 0;
                 origin_vertex_of_cell < GeometryInfo<dim>::vertices_per_cell;
                 ++origin_vertex_of_cell)
              if (std::count(starting_vertex_of_edge,
                             starting_vertex_of_edge +
                               GeometryInfo<dim>::lines_per_cell,
                             cell_list[cell_index]
                               .vertex_indices[origin_vertex_of_cell]) == dim)
                break;
            Assert(origin_vertex_of_cell < GeometryInfo<dim>::vertices_per_cell,
                   ExcInternalError());

            break;
          }

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

    // now rotate raw_cells[cell_index] in such a way that its orientation
    // matches that of cell_list[cell_index]
    switch (dim)
      {
        case 2:
          {
            // in 2d, we can literally rotate the cell until its origin
            // matches the one that we have determined above should be
            // the origin vertex
            //
            // when doing a rotation, take into account the ordering of
            // vertices (not in clockwise or counter-clockwise sense)
            while (raw_cells[cell_index].vertices[0] != origin_vertex_of_cell)
              {
                const unsigned int tmp = raw_cells[cell_index].vertices[0];
                raw_cells[cell_index].vertices[0] =
                  raw_cells[cell_index].vertices[1];
                raw_cells[cell_index].vertices[1] =
                  raw_cells[cell_index].vertices[3];
                raw_cells[cell_index].vertices[3] =
                  raw_cells[cell_index].vertices[2];
                raw_cells[cell_index].vertices[2] = tmp;
              }
            break;
          }

        case 3:
          {
            // in 3d, the situation is a bit more complicated. from above, we
            // now know which vertex is at the origin (because 3 edges originate
            // from it), but that still leaves 3 possible rotations of the cube.
            // the important realization is that we can choose any of them:
            // in all 3 rotations, all edges originate from the one vertex,
            // and that fixes the directions of all 12 edges in the cube because
            // these 3 cover all 3 equivalence classes! consequently, we can
            // select an arbitrary one among the permutations -- for
            // example the following ones:
            static const unsigned int cube_permutations[8][8] = {
              {0, 1, 2, 3, 4, 5, 6, 7},
              {1, 5, 3, 7, 0, 4, 2, 6},
              {2, 6, 0, 4, 3, 7, 1, 5},
              {3, 2, 1, 0, 7, 6, 5, 4},
              {4, 0, 6, 2, 5, 1, 7, 3},
              {5, 4, 7, 6, 1, 0, 3, 2},
              {6, 7, 4, 5, 2, 3, 0, 1},
              {7, 3, 5, 1, 6, 2, 4, 0}};

            unsigned int
              temp_vertex_indices[GeometryInfo<dim>::vertices_per_cell];
            for (const unsigned int v : GeometryInfo<dim>::vertex_indices())
              temp_vertex_indices[v] =
                raw_cells[cell_index]
                  .vertices[cube_permutations[origin_vertex_of_cell][v]];
            for (const unsigned int v : GeometryInfo<dim>::vertex_indices())
              raw_cells[cell_index].vertices[v] = temp_vertex_indices[v];

            break;
          }

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


  /**
   * Given a set of cells, find globally unique edge orientations
   * and then rotate cells so that the coordinate system of the cell
   * coincides with the coordinate systems of the adjacent edges.
   */
  template <int dim>
  void
  reorient(std::vector<CellData<dim>> &cells)
  {
    // first build the arrays that connect cells to edges and the other
    // way around
    std::vector<Edge<dim>> edge_list = build_edges(cells);
    std::vector<Cell<dim>> cell_list =
      build_cells_and_connect_edges(cells, edge_list);

    // then loop over all cells and start orienting parallel edge sets
    // of cells that still have non-oriented edges
    unsigned int next_cell_with_unoriented_edge = 0;
    while ((next_cell_with_unoriented_edge = get_next_unoriented_cell(
              cell_list, edge_list, next_cell_with_unoriented_edge)) !=
           numbers::invalid_unsigned_int)
      {
        // see which edge sets are still not oriented
        //
        // we do not need to look at each edge because if we orient edge
        // 0, we will end up with edge 1 also oriented (in 2d; in 3d, there
        // will be 3 other edges that are also oriented). there are only
        // dim independent sets of edges, so loop over these.
        //
        // we need to check whether each one of these starter edges may
        // already be oriented because the line (sheet) that connects
        // globally parallel edges may be self-intersecting in the
        // current cell
        for (unsigned int l = 0; l < dim; ++l)
          if (edge_list[cell_list[next_cell_with_unoriented_edge]
                          .edge_indices[ParallelEdges<dim>::starter_edges[l]]]
                .orientation_status == Edge<dim>::not_oriented)
            orient_one_set_of_parallel_edges(
              cell_list,
              edge_list,
              next_cell_with_unoriented_edge,
              ParallelEdges<dim>::starter_edges[l]);

        // ensure that we have really oriented all edges now, not just
        // the starter edges
        for (unsigned int l = 0; l < GeometryInfo<dim>::lines_per_cell; ++l)
          Assert(
            edge_list[cell_list[next_cell_with_unoriented_edge].edge_indices[l]]
                .orientation_status != Edge<dim>::not_oriented,
            ExcInternalError());
      }

    // now that we have oriented all edges, we need to rotate cells
    // so that the edges point in the right direction with the now
    // rotated coordinate system
    for (unsigned int c = 0; c < cells.size(); ++c)
      rotate_cell(cell_list, edge_list, c, cells);
  }


  // overload of the function above for 1d -- there is nothing
  // to orient in that case
  void reorient(std::vector<CellData<1>> &)
  {}
} // namespace


// anonymous namespace for internal helper functions
namespace
{
  /**
   * A set of functions that
   * reorder the data from the
   * "current" to the "classic"
   * format of vertex numbering of
   * cells and faces. These functions
   * do the reordering of their
   * arguments in-place.
   */
  void reorder_new_to_old_style(std::vector<CellData<1>> &)
  {}


  void reorder_new_to_old_style(std::vector<CellData<2>> &cells)
  {
    for (auto &cell : cells)
      std::swap(cell.vertices[2], cell.vertices[3]);
  }


  void reorder_new_to_old_style(std::vector<CellData<3>> &cells)
  {
    unsigned int tmp[GeometryInfo<3>::vertices_per_cell];
    for (auto &cell : cells)
      {
        for (const unsigned int i : GeometryInfo<3>::vertex_indices())
          tmp[i] = cell.vertices[i];
        for (const unsigned int i : GeometryInfo<3>::vertex_indices())
          cell.vertices[i] = tmp[GeometryInfo<3>::ucd_to_deal[i]];
      }
  }


  /**
   * And now also in the opposite direction.
   */
  void reorder_old_to_new_style(std::vector<CellData<1>> &)
  {}


  void reorder_old_to_new_style(std::vector<CellData<2>> &cells)
  {
    // just invert the permutation:
    reorder_new_to_old_style(cells);
  }


  void reorder_old_to_new_style(std::vector<CellData<3>> &cells)
  {
    // undo the ordering above
    unsigned int tmp[GeometryInfo<3>::vertices_per_cell];
    for (auto &cell : cells)
      {
        for (const unsigned int i : GeometryInfo<3>::vertex_indices())
          tmp[i] = cell.vertices[i];
        for (const unsigned int i : GeometryInfo<3>::vertex_indices())
          cell.vertices[GeometryInfo<3>::ucd_to_deal[i]] = tmp[i];
      }
  }
} // namespace


template <int dim, int spacedim>
void
GridReordering<dim, spacedim>::reorder_cells(std::vector<CellData<dim>> &cells,
                                             const bool use_new_style_ordering)
{
  Assert(cells.size() != 0,
         ExcMessage("List of elements to orient must have at least one cell"));

  // there is nothing for us to do in 1d
  if (dim == 1)
    return;

  // if necessary, convert to new-style format
  if (use_new_style_ordering == false)
    reorder_old_to_new_style(cells);

  // check if grids are already consistent. if so, do
  // nothing. if not, then do the reordering
  if (!is_consistent(cells))
    try
      {
        reorient(cells);
      }
    catch (const ExcMeshNotOrientable &)
      {
        // the mesh is not orientable. this is acceptable if we are in 3d,
        // as class Triangulation knows how to handle this, but it is
        // not in 2d; in that case, re-throw the exception
        if (dim < 3)
          throw;
      }

  // and convert back if necessary
  if (use_new_style_ordering == false)
    reorder_new_to_old_style(cells);
}


template <>
void
GridReordering<1>::invert_all_cells_of_negative_grid(
  const std::vector<Point<1>> &,
  std::vector<CellData<1>> &)
{
  // nothing to be done in 1d
}


template <>
void
GridReordering<1, 2>::invert_all_cells_of_negative_grid(
  const std::vector<Point<2>> &,
  std::vector<CellData<1>> &)
{
  // nothing to be done in 1d
}


template <>
void
GridReordering<1, 3>::invert_all_cells_of_negative_grid(
  const std::vector<Point<3>> &,
  std::vector<CellData<1>> &)
{
  // nothing to be done in 1d
}


template <>
void
GridReordering<2>::invert_all_cells_of_negative_grid(
  const std::vector<Point<2>> &all_vertices,
  std::vector<CellData<2>> &   cells)
{
  unsigned int vertices_lex[GeometryInfo<2>::vertices_per_cell];
  unsigned int n_negative_cells = 0;
  for (auto &cell : cells)
    {
      // GridTools::cell_measure
      // requires the vertices to be
      // in lexicographic ordering
      for (const unsigned int i : GeometryInfo<2>::vertex_indices())
        vertices_lex[GeometryInfo<2>::ucd_to_deal[i]] = cell.vertices[i];
      if (GridTools::cell_measure<2>(all_vertices, vertices_lex) < 0)
        {
          ++n_negative_cells;
          std::swap(cell.vertices[1], cell.vertices[3]);

          // Check whether the resulting cell is now ok.
          // If not, then the grid is seriously broken and
          // we just give up.
          for (const unsigned int i : GeometryInfo<2>::vertex_indices())
            vertices_lex[GeometryInfo<2>::ucd_to_deal[i]] = cell.vertices[i];
          AssertThrow(GridTools::cell_measure<2>(all_vertices, vertices_lex) >
                        0,
                      ExcInternalError());
        }
    }

  // We assume that all cells of a grid have
  // either positive or negative volumes but
  // not both mixed. Although above reordering
  // might work also on single cells, grids
  // with both kind of cells are very likely to
  // be broken. Check for this here.
  AssertThrow(
    n_negative_cells == 0 || n_negative_cells == cells.size(),
    ExcMessage(
      std::string("This class assumes that either all cells have positive "
                  "volume, or that all cells have been specified in an "
                  "inverted vertex order so that their volume is negative. "
                  "(In the latter case, this class automatically inverts "
                  "every cell.) However, the mesh you have specified "
                  "appears to have both cells with positive and cells with "
                  "negative volume. You need to check your mesh which "
                  "cells these are and how they got there.\n"
                  "As a hint, of the total ") +
      std::to_string(cells.size()) + " cells in the mesh, " +
      std::to_string(n_negative_cells) + " appear to have a negative volume."));
}


template <>
void
GridReordering<2, 3>::invert_all_cells_of_negative_grid(
  const std::vector<Point<3>> &,
  std::vector<CellData<2>> &)
{
  Assert(false, ExcNotImplemented());
}


template <>
void
GridReordering<3>::invert_all_cells_of_negative_grid(
  const std::vector<Point<3>> &all_vertices,
  std::vector<CellData<3>> &   cells)
{
  unsigned int vertices_lex[GeometryInfo<3>::vertices_per_cell];
  unsigned int n_negative_cells = 0;
  for (auto &cell : cells)
    {
      // GridTools::cell_measure
      // requires the vertices to be
      // in lexicographic ordering
      for (const unsigned int i : GeometryInfo<3>::vertex_indices())
        vertices_lex[GeometryInfo<3>::ucd_to_deal[i]] = cell.vertices[i];
      if (GridTools::cell_measure<3>(all_vertices, vertices_lex) < 0)
        {
          ++n_negative_cells;
          // reorder vertices: swap front and back face
          for (unsigned int i = 0; i < 4; ++i)
            std::swap(cell.vertices[i], cell.vertices[i + 4]);

          // Check whether the resulting cell is now ok.
          // If not, then the grid is seriously broken and
          // we just give up.
          for (const unsigned int i : GeometryInfo<3>::vertex_indices())
            vertices_lex[GeometryInfo<3>::ucd_to_deal[i]] = cell.vertices[i];
          AssertThrow(GridTools::cell_measure<3>(all_vertices, vertices_lex) >
                        0,
                      ExcInternalError());
        }
    }

  // We assume that all cells of a
  // grid have either positive or
  // negative volumes but not both
  // mixed. Although above reordering
  // might work also on single cells,
  // grids with both kind of cells
  // are very likely to be
  // broken. Check for this here.
  AssertThrow(n_negative_cells == 0 || n_negative_cells == cells.size(),
              ExcMessage("While sorting the cells that will be passed for "
                         "creating a Triangulation object, deal.II found that "
                         "some but not all cells have a negative volume. (If "
                         "all cells had a negative volume, they would simply "
                         "all have been inverted.) This usually happens in "
                         "hand-generated meshes if one accidentally uses an "
                         "incorrect convention for ordering the vertices in "
                         "one or more cells; in that case, you may want to "
                         "double check that you specified the vertex indices "
                         "in their correct order. If you are reading a mesh "
                         "that was created by a mesh generator, then this "
                         "exception indicates that some of the cells created "
                         "are so badly distorted that their volume becomes "
                         "negative; this commonly occurs at complex geometric "
                         "features, and you may see if the problem can be "
                         "fixed by playing with the parameters that control "
                         "mesh properties in your mesh generator, such as "
                         "the number of cells, the mesh density, etc."));
}


/* ------------------------ explicit instantiations ------------------- */
template class GridReordering<1, 1>;
template class GridReordering<1, 2>;
template class GridReordering<1, 3>;
template class GridReordering<2, 2>;
template class GridReordering<2, 3>;
template class GridReordering<3, 3>;

DEAL_II_NAMESPACE_CLOSE