grid_tools.h

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// ---------------------------------------------------------------------
//
// Copyright (C) 2001 - 2021 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.
//
// ---------------------------------------------------------------------
 
#ifndef dealii_grid_tools_h
#  define dealii_grid_tools_h
 
 
#  include <deal.II/base/config.h>
 
#  include <deal.II/base/bounding_box.h>
#  include <deal.II/base/geometry_info.h>
#  include <deal.II/base/std_cxx17/optional.h>
 
#  include <deal.II/boost_adaptors/bounding_box.h>
 
#  include <deal.II/distributed/shared_tria.h>
 
#  include <deal.II/dofs/dof_handler.h>
 
#  include <deal.II/fe/fe_values.h>
#  include <deal.II/fe/mapping.h>
#  include <deal.II/fe/mapping_q1.h>
 
#  include <deal.II/grid/manifold.h>
#  include <deal.II/grid/tria.h>
#  include <deal.II/grid/tria_accessor.h>
#  include <deal.II/grid/tria_iterator.h>
 
#  include <deal.II/hp/dof_handler.h>
 
#  include <deal.II/lac/la_parallel_vector.h>
#  include <deal.II/lac/la_vector.h>
#  include <deal.II/lac/petsc_vector.h>
#  include <deal.II/lac/sparsity_tools.h>
#  include <deal.II/lac/trilinos_vector.h>
 
#  include <deal.II/numerics/rtree.h>
 
DEAL_II_DISABLE_EXTRA_DIAGNOSTICS
#  include <boost/archive/binary_iarchive.hpp>
#  include <boost/archive/binary_oarchive.hpp>
#  include <boost/random/mersenne_twister.hpp>
#  include <boost/serialization/array.hpp>
#  include <boost/serialization/vector.hpp>
 
#  ifdef DEAL_II_WITH_ZLIB
#    include <boost/iostreams/device/back_inserter.hpp>
#    include <boost/iostreams/filter/gzip.hpp>
#    include <boost/iostreams/filtering_stream.hpp>
#    include <boost/iostreams/stream.hpp>
#  endif
DEAL_II_ENABLE_EXTRA_DIAGNOSTICS
 
#  include <bitset>
#  include <list>
#  include <set>
 
DEAL_II_NAMESPACE_OPEN
 
// Forward declarations
#  ifndef DOXYGEN
namespace parallel
{
  namespace distributed
  {
    template <int, int>
    class Triangulation;
  }
} // namespace parallel
 
namespace hp
{
  template <int, int>
  class MappingCollection;
}
 
class SparsityPattern;
#  endif
 
namespace internal
{
  template <int dim, int spacedim, class MeshType>
  class ActiveCellIterator
  {
  public:
#  ifndef _MSC_VER
    using type = typename MeshType::active_cell_iterator;
#  else
    using type = TriaActiveIterator<::CellAccessor<dim, spacedim>>;
#  endif
  };
 
#  ifdef _MSC_VER
  template <int dim, int spacedim>
  class ActiveCellIterator<dim, spacedim, ::DoFHandler<dim, spacedim>>
  {
  public:
    using type =
      TriaActiveIterator<::DoFCellAccessor<dim, spacedim, false>>;
  };
#  endif
 
 
#  ifdef _MSC_VER
  template <int dim, int spacedim>
  class ActiveCellIterator<dim, spacedim, ::hp::DoFHandler<dim, spacedim>>
  {
  public:
    using type =
      TriaActiveIterator<::DoFCellAccessor<dim, spacedim, false>>;
  };
#  endif
} // namespace internal
 
namespace GridTools
{
  template <int dim, int spacedim>
  class Cache;
 
 
  template <int dim, int spacedim>
  double
  diameter(const Triangulation<dim, spacedim> &tria);
 
  template <int dim, int spacedim>
  double
  volume(const Triangulation<dim, spacedim> &tria,
         const Mapping<dim, spacedim> &      mapping =
           (ReferenceCells::get_hypercube<dim>()
              .template get_default_linear_mapping<dim, spacedim>()));
 
  template <int dim, int spacedim>
  double
  minimal_cell_diameter(
    const Triangulation<dim, spacedim> &triangulation,
    const Mapping<dim, spacedim> &      mapping =
      (ReferenceCells::get_hypercube<dim>()
         .template get_default_linear_mapping<dim, spacedim>()));
 
  template <int dim, int spacedim>
  double
  maximal_cell_diameter(
    const Triangulation<dim, spacedim> &triangulation,
    const Mapping<dim, spacedim> &      mapping =
      (ReferenceCells::get_hypercube<dim>()
         .template get_default_linear_mapping<dim, spacedim>()));
 
  template <int dim>
  DEAL_II_DEPRECATED_EARLY double
  cell_measure(
    const std::vector<Point<dim>> &all_vertices,
    const unsigned int (&vertex_indices)[GeometryInfo<dim>::vertices_per_cell]);
 
  template <int dim>
  double
  cell_measure(const std::vector<Point<dim>> &      all_vertices,
               const ArrayView<const unsigned int> &vertex_indices);
 
  template <int dim, int spacedim>
  std::pair<DerivativeForm<1, dim, spacedim>, Tensor<1, spacedim>>
  affine_cell_approximation(const ArrayView<const Point<spacedim>> &vertices);
 
  template <int dim>
  Vector<double>
  compute_aspect_ratio_of_cells(const Mapping<dim> &      mapping,
                                const Triangulation<dim> &triangulation,
                                const Quadrature<dim> &   quadrature);
 
  template <int dim>
  double
  compute_maximum_aspect_ratio(const Mapping<dim> &      mapping,
                               const Triangulation<dim> &triangulation,
                               const Quadrature<dim> &   quadrature);
 
  template <int dim, int spacedim>
  BoundingBox<spacedim>
  compute_bounding_box(const Triangulation<dim, spacedim> &triangulation);
 
  template <typename Iterator>
  Point<Iterator::AccessorType::space_dimension>
  project_to_object(
    const Iterator &                                      object,
    const Point<Iterator::AccessorType::space_dimension> &trial_point);
 
  template <int dim, int spacedim>
  std::
    tuple<std::vector<Point<spacedim>>, std::vector<CellData<dim>>, SubCellData>
    get_coarse_mesh_description(const Triangulation<dim, spacedim> &tria);
 
 
  template <int dim, int spacedim>
  void
  delete_unused_vertices(std::vector<Point<spacedim>> &vertices,
                         std::vector<CellData<dim>> &  cells,
                         SubCellData &                 subcelldata);
 
  template <int dim, int spacedim>
  void
  delete_duplicated_vertices(std::vector<Point<spacedim>> &all_vertices,
                             std::vector<CellData<dim>> &  cells,
                             SubCellData &                 subcelldata,
                             std::vector<unsigned int> &   considered_vertices,
                             const double                  tol = 1e-12);
 
  template <int dim, int spacedim>
  void
  invert_all_negative_measure_cells(
    const std::vector<Point<spacedim>> &all_vertices,
    std::vector<CellData<dim>> &        cells);
 
  template <int dim>
  void
  consistently_order_cells(std::vector<CellData<dim>> &cells);
 
 
  template <int dim, typename Transformation, int spacedim>
  void
  transform(const Transformation &        transformation,
            Triangulation<dim, spacedim> &triangulation);
 
  template <int dim, int spacedim>
  void
  shift(const Tensor<1, spacedim> &   shift_vector,
        Triangulation<dim, spacedim> &triangulation);
 
 
  template <int dim>
  void
  rotate(const double angle, Triangulation<dim> &triangulation);
 
  template <int dim>
  void
  rotate(const double           angle,
         const unsigned int     axis,
         Triangulation<dim, 3> &triangulation);
 
  template <int dim>
  void
  laplace_transform(const std::map<unsigned int, Point<dim>> &new_points,
                    Triangulation<dim> &                      tria,
                    const Function<dim, double> *coefficient = nullptr,
                    const bool solve_for_absolute_positions  = false);
 
  template <int dim, int spacedim>
  std::map<unsigned int, Point<spacedim>>
  get_all_vertices_at_boundary(const Triangulation<dim, spacedim> &tria);
 
  template <int dim, int spacedim>
  void
  scale(const double                  scaling_factor,
        Triangulation<dim, spacedim> &triangulation);
 
  template <int dim, int spacedim>
  void
  distort_random(
    const double                  factor,
    Triangulation<dim, spacedim> &triangulation,
    const bool                    keep_boundary = true,
    const unsigned int            seed = boost::random::mt19937::default_seed);
 
  template <int dim, int spacedim>
  void
  remove_hanging_nodes(Triangulation<dim, spacedim> &tria,
                       const bool                    isotropic      = false,
                       const unsigned int            max_iterations = 100);
 
  template <int dim, int spacedim>
  void
  remove_anisotropy(Triangulation<dim, spacedim> &tria,
                    const double                  max_ratio      = 1.6180339887,
                    const unsigned int            max_iterations = 5);
 
  template <int dim, int spacedim>
  void
  regularize_corner_cells(Triangulation<dim, spacedim> &tria,
                          const double limit_angle_fraction = .75);
 
 
  template <int dim, int spacedim>
#  ifndef DOXYGEN
  std::tuple<
    std::vector<typename Triangulation<dim, spacedim>::active_cell_iterator>,
    std::vector<std::vector<Point<dim>>>,
    std::vector<std::vector<unsigned int>>>
#  else
  return_type
#  endif
  compute_point_locations(
    const Cache<dim, spacedim> &        cache,
    const std::vector<Point<spacedim>> &points,
    const typename Triangulation<dim, spacedim>::active_cell_iterator
      &cell_hint =
        typename Triangulation<dim, spacedim>::active_cell_iterator());
 
  template <int dim, int spacedim>
#  ifndef DOXYGEN
  std::tuple<
    std::vector<typename Triangulation<dim, spacedim>::active_cell_iterator>,
    std::vector<std::vector<Point<dim>>>,
    std::vector<std::vector<unsigned int>>,
    std::vector<unsigned int>>
#  else
  return_type
#  endif
  compute_point_locations_try_all(
    const Cache<dim, spacedim> &        cache,
    const std::vector<Point<spacedim>> &points,
    const typename Triangulation<dim, spacedim>::active_cell_iterator
      &cell_hint =
        typename Triangulation<dim, spacedim>::active_cell_iterator());
 
  template <int dim, int spacedim>
#  ifndef DOXYGEN
  std::tuple<
    std::vector<typename Triangulation<dim, spacedim>::active_cell_iterator>,
    std::vector<std::vector<Point<dim>>>,
    std::vector<std::vector<unsigned int>>,
    std::vector<std::vector<Point<spacedim>>>,
    std::vector<std::vector<unsigned int>>>
#  else
  return_type
#  endif
  distributed_compute_point_locations(
    const GridTools::Cache<dim, spacedim> &                cache,
    const std::vector<Point<spacedim>> &                   local_points,
    const std::vector<std::vector<BoundingBox<spacedim>>> &global_bboxes,
    const double                                           tolerance = 1e-10);
 
  namespace internal
  {
    template <int dim, int spacedim>
    struct DistributedComputePointLocationsInternal
    {
      std::vector<std::tuple<std::pair<int, int>,
                             unsigned int,
                             unsigned int,
                             Point<dim>,
                             Point<spacedim>,
                             unsigned int>>
        send_components;
 
      std::vector<unsigned int> send_ranks;
 
      std::vector<unsigned int> send_ptrs;
 
      std::vector<std::tuple<unsigned int, unsigned int, unsigned int>>
        recv_components;
 
      std::vector<unsigned int> recv_ranks;
 
      std::vector<unsigned int> recv_ptrs;
    };
 
    template <int dim, int spacedim>
    DistributedComputePointLocationsInternal<dim, spacedim>
    distributed_compute_point_locations(
      const GridTools::Cache<dim, spacedim> &                cache,
      const std::vector<Point<spacedim>> &                   points,
      const std::vector<std::vector<BoundingBox<spacedim>>> &global_bboxes,
      const double                                           tolerance,
      const bool                                             perform_handshake,
      const bool enforce_unique_mapping = false);
 
  } // namespace internal
 
  template <int dim, int spacedim>
  std::map<unsigned int, Point<spacedim>>
  extract_used_vertices(
    const Triangulation<dim, spacedim> &container,
    const Mapping<dim, spacedim> &      mapping =
      (ReferenceCells::get_hypercube<dim>()
         .template get_default_linear_mapping<dim, spacedim>()));
 
  template <int spacedim>
  unsigned int
  find_closest_vertex(const std::map<unsigned int, Point<spacedim>> &vertices,
                      const Point<spacedim> &                        p);
 
  template <int dim, template <int, int> class MeshType, int spacedim>
  unsigned int
  find_closest_vertex(const MeshType<dim, spacedim> &mesh,
                      const Point<spacedim> &        p,
                      const std::vector<bool> &      marked_vertices = {});
 
  template <int dim, template <int, int> class MeshType, int spacedim>
  unsigned int
  find_closest_vertex(const Mapping<dim, spacedim> & mapping,
                      const MeshType<dim, spacedim> &mesh,
                      const Point<spacedim> &        p,
                      const std::vector<bool> &      marked_vertices = {});
 
 
  template <int dim, template <int, int> class MeshType, int spacedim>
#  ifndef _MSC_VER
  std::vector<typename MeshType<dim, spacedim>::active_cell_iterator>
#  else
  std::vector<
    typename ::internal::
      ActiveCellIterator<dim, spacedim, MeshType<dim, spacedim>>::type>
#  endif
  find_cells_adjacent_to_vertex(const MeshType<dim, spacedim> &container,
                                const unsigned int             vertex_index);
 
  template <int dim, template <int, int> class MeshType, int spacedim>
#  ifndef _MSC_VER
  std::pair<typename MeshType<dim, spacedim>::active_cell_iterator, Point<dim>>
#  else
  std::pair<typename ::internal::
              ActiveCellIterator<dim, spacedim, MeshType<dim, spacedim>>::type,
            Point<dim>>
#  endif
  find_active_cell_around_point(const Mapping<dim, spacedim> & mapping,
                                const MeshType<dim, spacedim> &mesh,
                                const Point<spacedim> &        p,
                                const std::vector<bool> &marked_vertices = {},
                                const double             tolerance = 1.e-10);
 
  template <int dim, template <int, int> class MeshType, int spacedim>
#  ifndef _MSC_VER
  typename MeshType<dim, spacedim>::active_cell_iterator
#  else
  typename ::internal::
    ActiveCellIterator<dim, spacedim, MeshType<dim, spacedim>>::type
#  endif
  find_active_cell_around_point(const MeshType<dim, spacedim> &mesh,
                                const Point<spacedim> &        p,
                                const std::vector<bool> &marked_vertices = {},
                                const double             tolerance = 1.e-10);
 
  template <int dim, int spacedim>
  std::pair<typename DoFHandler<dim, spacedim>::active_cell_iterator,
            Point<dim>>
  find_active_cell_around_point(
    const hp::MappingCollection<dim, spacedim> &mapping,
    const DoFHandler<dim, spacedim> &           mesh,
    const Point<spacedim> &                     p,
    const double                                tolerance = 1.e-10);
 
  template <int dim, int spacedim>
  std::pair<typename Triangulation<dim, spacedim>::active_cell_iterator,
            Point<dim>>
  find_active_cell_around_point(
    const Cache<dim, spacedim> &cache,
    const Point<spacedim> &     p,
    const typename Triangulation<dim, spacedim>::active_cell_iterator &
                             cell_hint = typename Triangulation<dim, spacedim>::active_cell_iterator(),
    const std::vector<bool> &marked_vertices = {},
    const double             tolerance       = 1.e-10);
 
  template <int dim, template <int, int> class MeshType, int spacedim>
#  ifndef _MSC_VER
  std::pair<typename MeshType<dim, spacedim>::active_cell_iterator, Point<dim>>
#  else
  std::pair<typename ::internal::
              ActiveCellIterator<dim, spacedim, MeshType<dim, spacedim>>::type,
            Point<dim>>
#  endif
  find_active_cell_around_point(
    const Mapping<dim, spacedim> & mapping,
    const MeshType<dim, spacedim> &mesh,
    const Point<spacedim> &        p,
    const std::vector<
      std::set<typename MeshType<dim, spacedim>::active_cell_iterator>>
      &                                                  vertex_to_cell_map,
    const std::vector<std::vector<Tensor<1, spacedim>>> &vertex_to_cell_centers,
    const typename MeshType<dim, spacedim>::active_cell_iterator &cell_hint =
      typename MeshType<dim, spacedim>::active_cell_iterator(),
    const std::vector<bool> &                              marked_vertices = {},
    const RTree<std::pair<Point<spacedim>, unsigned int>> &used_vertices_rtree =
      RTree<std::pair<Point<spacedim>, unsigned int>>{},
    const double tolerance = 1.e-10);
 
  template <int dim, template <int, int> class MeshType, int spacedim>
#  ifndef _MSC_VER
  std::vector<std::pair<typename MeshType<dim, spacedim>::active_cell_iterator,
                        Point<dim>>>
#  else
  std::vector<std::pair<
    typename ::internal::
      ActiveCellIterator<dim, spacedim, MeshType<dim, spacedim>>::type,
    Point<dim>>>
#  endif
  find_all_active_cells_around_point(
    const Mapping<dim, spacedim> & mapping,
    const MeshType<dim, spacedim> &mesh,
    const Point<spacedim> &        p,
    const double                   tolerance,
    const std::pair<typename MeshType<dim, spacedim>::active_cell_iterator,
                    Point<dim>> &  first_cell);
 
  template <int dim, template <int, int> class MeshType, int spacedim>
#  ifndef _MSC_VER
  std::vector<std::pair<typename MeshType<dim, spacedim>::active_cell_iterator,
                        Point<dim>>>
#  else
  std::vector<std::pair<
    typename ::internal::
      ActiveCellIterator<dim, spacedim, MeshType<dim, spacedim>>::type,
    Point<dim>>>
#  endif
  find_all_active_cells_around_point(
    const Mapping<dim, spacedim> & mapping,
    const MeshType<dim, spacedim> &mesh,
    const Point<spacedim> &        p,
    const double                   tolerance       = 1e-10,
    const std::vector<bool> &      marked_vertices = {});
 
  template <class MeshType>
  std::vector<typename MeshType::active_cell_iterator>
  get_active_child_cells(const typename MeshType::cell_iterator &cell);
 
  template <class MeshType>
  void
  get_active_neighbors(
    const typename MeshType::active_cell_iterator &       cell,
    std::vector<typename MeshType::active_cell_iterator> &active_neighbors);
 
  template <class MeshType>
  std::vector<typename MeshType::active_cell_iterator>
  compute_active_cell_halo_layer(
    const MeshType &mesh,
    const std::function<bool(const typename MeshType::active_cell_iterator &)>
      &predicate);
 
 
  template <class MeshType>
  std::vector<typename MeshType::cell_iterator>
  compute_cell_halo_layer_on_level(
    const MeshType &mesh,
    const std::function<bool(const typename MeshType::cell_iterator &)>
      &                predicate,
    const unsigned int level);
 
 
  template <class MeshType>
  std::vector<typename MeshType::active_cell_iterator>
  compute_ghost_cell_halo_layer(const MeshType &mesh);
 
  template <class MeshType>
  std::vector<typename MeshType::active_cell_iterator>
  compute_active_cell_layer_within_distance(
    const MeshType &mesh,
    const std::function<bool(const typename MeshType::active_cell_iterator &)>
      &          predicate,
    const double layer_thickness);
 
  template <class MeshType>
  std::vector<typename MeshType::active_cell_iterator>
  compute_ghost_cell_layer_within_distance(const MeshType &mesh,
                                           const double    layer_thickness);
 
  template <class MeshType>
  std::pair<Point<MeshType::space_dimension>, Point<MeshType::space_dimension>>
  compute_bounding_box(
    const MeshType &mesh,
    const std::function<bool(const typename MeshType::active_cell_iterator &)>
      &predicate);
 
  template <class MeshType>
  std::vector<BoundingBox<MeshType::space_dimension>>
  compute_mesh_predicate_bounding_box(
    const MeshType &mesh,
    const std::function<bool(const typename MeshType::active_cell_iterator &)>
      &                predicate,
    const unsigned int refinement_level = 0,
    const bool         allow_merge      = false,
    const unsigned int max_boxes        = numbers::invalid_unsigned_int);
 
  template <int spacedim>
#  ifndef DOXYGEN
  std::tuple<std::vector<std::vector<unsigned int>>,
             std::map<unsigned int, unsigned int>,
             std::map<unsigned int, std::vector<unsigned int>>>
#  else
  return_type
#  endif
  guess_point_owner(
    const std::vector<std::vector<BoundingBox<spacedim>>> &global_bboxes,
    const std::vector<Point<spacedim>> &                   points);
 
 
  template <int spacedim>
#  ifndef DOXYGEN
  std::tuple<std::map<unsigned int, std::vector<unsigned int>>,
             std::map<unsigned int, unsigned int>,
             std::map<unsigned int, std::vector<unsigned int>>>
#  else
  return_type
#  endif
  guess_point_owner(
    const RTree<std::pair<BoundingBox<spacedim>, unsigned int>> &covering_rtree,
    const std::vector<Point<spacedim>> &                         points);
 
 
  template <int dim, int spacedim>
  std::vector<
    std::set<typename Triangulation<dim, spacedim>::active_cell_iterator>>
  vertex_to_cell_map(const Triangulation<dim, spacedim> &triangulation);
 
  template <int dim, int spacedim>
  std::vector<std::vector<Tensor<1, spacedim>>>
  vertex_to_cell_centers_directions(
    const Triangulation<dim, spacedim> &mesh,
    const std::vector<
      std::set<typename Triangulation<dim, spacedim>::active_cell_iterator>>
      &vertex_to_cells);
 
 
  template <int dim, int spacedim>
  unsigned int
  find_closest_vertex_of_cell(
    const typename Triangulation<dim, spacedim>::active_cell_iterator &cell,
    const Point<spacedim> &                                            position,
    const Mapping<dim, spacedim> &                                     mapping =
      (ReferenceCells::get_hypercube<dim>()
         .template get_default_linear_mapping<dim, spacedim>()));
 
  template <int dim, int spacedim>
  std::map<unsigned int, types::global_vertex_index>
  compute_local_to_global_vertex_index_map(
    const parallel::distributed::Triangulation<dim, spacedim> &triangulation);
 
  template <int dim, int spacedim>
  std::pair<unsigned int, double>
  get_longest_direction(
    typename Triangulation<dim, spacedim>::active_cell_iterator cell);
 
 
  template <int dim, int spacedim>
  void
  get_face_connectivity_of_cells(
    const Triangulation<dim, spacedim> &triangulation,
    DynamicSparsityPattern &            connectivity);
 
  template <int dim, int spacedim>
  void
  get_vertex_connectivity_of_cells(
    const Triangulation<dim, spacedim> &triangulation,
    DynamicSparsityPattern &            connectivity);
 
  template <int dim, int spacedim>
  void
  get_vertex_connectivity_of_cells_on_level(
    const Triangulation<dim, spacedim> &triangulation,
    const unsigned int                  level,
    DynamicSparsityPattern &            connectivity);
 
  template <int dim, int spacedim>
  void
  partition_triangulation(const unsigned int               n_partitions,
                          Triangulation<dim, spacedim> &   triangulation,
                          const SparsityTools::Partitioner partitioner =
                            SparsityTools::Partitioner::metis);
 
  template <int dim, int spacedim>
  void
  partition_triangulation(const unsigned int               n_partitions,
                          const std::vector<unsigned int> &cell_weights,
                          Triangulation<dim, spacedim> &   triangulation,
                          const SparsityTools::Partitioner partitioner =
                            SparsityTools::Partitioner::metis);
 
  template <int dim, int spacedim>
  void
  partition_triangulation(const unsigned int            n_partitions,
                          const SparsityPattern &       cell_connection_graph,
                          Triangulation<dim, spacedim> &triangulation,
                          const SparsityTools::Partitioner partitioner =
                            SparsityTools::Partitioner::metis);
 
  template <int dim, int spacedim>
  void
  partition_triangulation(const unsigned int               n_partitions,
                          const std::vector<unsigned int> &cell_weights,
                          const SparsityPattern &       cell_connection_graph,
                          Triangulation<dim, spacedim> &triangulation,
                          const SparsityTools::Partitioner partitioner =
                            SparsityTools::Partitioner::metis);
 
  template <int dim, int spacedim>
  void
  partition_triangulation_zorder(const unsigned int            n_partitions,
                                 Triangulation<dim, spacedim> &triangulation,
                                 const bool group_siblings = true);
 
  template <int dim, int spacedim>
  void
  partition_multigrid_levels(Triangulation<dim, spacedim> &triangulation);
 
  template <int dim, int spacedim>
  std::vector<types::subdomain_id>
  get_subdomain_association(const Triangulation<dim, spacedim> &triangulation,
                            const std::vector<CellId> &         cell_ids);
 
  template <int dim, int spacedim>
  void
  get_subdomain_association(const Triangulation<dim, spacedim> &triangulation,
                            std::vector<types::subdomain_id> &  subdomain);
 
  template <int dim, int spacedim>
  unsigned int
  count_cells_with_subdomain_association(
    const Triangulation<dim, spacedim> &triangulation,
    const types::subdomain_id           subdomain);
 
  template <int dim, int spacedim>
  std::vector<bool>
  get_locally_owned_vertices(const Triangulation<dim, spacedim> &triangulation);
 
 
  template <typename MeshType>
  std::list<std::pair<typename MeshType::cell_iterator,
                      typename MeshType::cell_iterator>>
  get_finest_common_cells(const MeshType &mesh_1, const MeshType &mesh_2);
 
  template <int dim, int spacedim>
  bool
  have_same_coarse_mesh(const Triangulation<dim, spacedim> &mesh_1,
                        const Triangulation<dim, spacedim> &mesh_2);
 
  template <typename MeshType>
  bool
  have_same_coarse_mesh(const MeshType &mesh_1, const MeshType &mesh_2);
 
 
  template <int dim, int spacedim>
  typename Triangulation<dim, spacedim>::DistortedCellList
  fix_up_distorted_child_cells(
    const typename Triangulation<dim, spacedim>::DistortedCellList
      &                           distorted_cells,
    Triangulation<dim, spacedim> &triangulation);
 
 
 
 
 
  template <class MeshType>
  std::vector<typename MeshType::active_cell_iterator>
  get_patch_around_cell(const typename MeshType::active_cell_iterator &cell);
 
 
  template <class Container>
  std::vector<typename Container::cell_iterator>
  get_cells_at_coarsest_common_level(
    const std::vector<typename Container::active_cell_iterator> &patch_cells);
 
  template <class Container>
  void
  build_triangulation_from_patch(
    const std::vector<typename Container::active_cell_iterator> &patch,
    Triangulation<Container::dimension, Container::space_dimension>
      &local_triangulation,
    std::map<
      typename Triangulation<Container::dimension,
                             Container::space_dimension>::active_cell_iterator,
      typename Container::active_cell_iterator> &patch_to_global_tria_map);
 
  template <int dim, int spacedim>
  std::map<
    types::global_dof_index,
    std::vector<typename DoFHandler<dim, spacedim>::active_cell_iterator>>
  get_dof_to_support_patch_map(DoFHandler<dim, spacedim> &dof_handler);
 
 
 
  template <typename CellIterator>
  struct PeriodicFacePair
  {
    CellIterator cell[2];
 
    unsigned int face_idx[2];
 
    std::bitset<3> orientation;
 
    FullMatrix<double> matrix;
  };
 
 
  template <typename FaceIterator>
  bool orthogonal_equality(
    std::bitset<3> &                                              orientation,
    const FaceIterator &                                          face1,
    const FaceIterator &                                          face2,
    const int                                                     direction,
    const Tensor<1, FaceIterator::AccessorType::space_dimension> &offset =
      Tensor<1, FaceIterator::AccessorType::space_dimension>(),
    const FullMatrix<double> &matrix = FullMatrix<double>());
 
 
  template <typename FaceIterator>
  bool
  orthogonal_equality(
    const FaceIterator &                                          face1,
    const FaceIterator &                                          face2,
    const int                                                     direction,
    const Tensor<1, FaceIterator::AccessorType::space_dimension> &offset =
      Tensor<1, FaceIterator::AccessorType::space_dimension>(),
    const FullMatrix<double> &matrix = FullMatrix<double>());
 
 
  template <typename MeshType>
  void
  collect_periodic_faces(
    const MeshType &         mesh,
    const types::boundary_id b_id1,
    const types::boundary_id b_id2,
    const int                direction,
    std::vector<PeriodicFacePair<typename MeshType::cell_iterator>>
      &                                         matched_pairs,
    const Tensor<1, MeshType::space_dimension> &offset =
      ::Tensor<1, MeshType::space_dimension>(),
    const FullMatrix<double> &matrix = FullMatrix<double>());
 
 
  template <typename MeshType>
  void
  collect_periodic_faces(
    const MeshType &         mesh,
    const types::boundary_id b_id,
    const int                direction,
    std::vector<PeriodicFacePair<typename MeshType::cell_iterator>>
      &                                                 matched_pairs,
    const ::Tensor<1, MeshType::space_dimension> &offset =
      ::Tensor<1, MeshType::space_dimension>(),
    const FullMatrix<double> &matrix = FullMatrix<double>());
 
 
  template <int dim, int spacedim>
  void
  copy_boundary_to_manifold_id(Triangulation<dim, spacedim> &tria,
                               const bool reset_boundary_ids = false);
 
  template <int dim, int spacedim>
  void
  map_boundary_to_manifold_ids(
    const std::vector<types::boundary_id> &src_boundary_ids,
    const std::vector<types::manifold_id> &dst_manifold_ids,
    Triangulation<dim, spacedim> &         tria,
    const std::vector<types::boundary_id> &reset_boundary_ids = {});
 
  template <int dim, int spacedim>
  void
  copy_material_to_manifold_id(Triangulation<dim, spacedim> &tria,
                               const bool compute_face_ids = false);
 
  template <int dim, int spacedim>
  void
  assign_co_dimensional_manifold_indicators(
    Triangulation<dim, spacedim> &            tria,
    const std::function<types::manifold_id(
      const std::set<types::manifold_id> &)> &disambiguation_function =
      [](const std::set<types::manifold_id> &manifold_ids) {
        if (manifold_ids.size() == 1)
          return *manifold_ids.begin();
        else
          return numbers::flat_manifold_id;
      },
    bool overwrite_only_flat_manifold_ids = true);
  template <typename DataType, typename MeshType>
  void
  exchange_cell_data_to_ghosts(
    const MeshType &                                     mesh,
    const std::function<std_cxx17::optional<DataType>(
      const typename MeshType::active_cell_iterator &)> &pack,
    const std::function<void(const typename MeshType::active_cell_iterator &,
                             const DataType &)> &        unpack,
    const std::function<bool(const typename MeshType::active_cell_iterator &)>
      &cell_filter =
        always_return<typename MeshType::active_cell_iterator, bool>{true});
 
  template <typename DataType, typename MeshType>
  void
  exchange_cell_data_to_level_ghosts(
    const MeshType &                                    mesh,
    const std::function<std_cxx17::optional<DataType>(
      const typename MeshType::level_cell_iterator &)> &pack,
    const std::function<void(const typename MeshType::level_cell_iterator &,
                             const DataType &)> &       unpack,
    const std::function<bool(const typename MeshType::level_cell_iterator &)> &
      cell_filter = always_return<typename MeshType::level_cell_iterator, bool>{
        true});
 
  /* Exchange with all processors of the MPI communicator @p mpi_communicator the vector of bounding
   * boxes @p local_bboxes.
   *
   * This function is meant to exchange bounding boxes describing the locally
   * owned cells in a distributed triangulation obtained with the function
   * GridTools::compute_mesh_predicate_bounding_box .
   *
   * The output vector's size is the number of processes of the MPI
   * communicator:
   * its i-th entry contains the vector @p local_bboxes of the i-th process.
   */
  template <int spacedim>
  std::vector<std::vector<BoundingBox<spacedim>>>
  exchange_local_bounding_boxes(
    const std::vector<BoundingBox<spacedim>> &local_bboxes,
    const MPI_Comm &                          mpi_communicator);
 
  template <int spacedim>
  RTree<std::pair<BoundingBox<spacedim>, unsigned int>>
  build_global_description_tree(
    const std::vector<BoundingBox<spacedim>> &local_description,
    const MPI_Comm &                          mpi_communicator);
 
  template <int dim, int spacedim>
  void
  collect_coinciding_vertices(
    const Triangulation<dim, spacedim> &               tria,
    std::map<unsigned int, std::vector<unsigned int>> &coinciding_vertex_groups,
    std::map<unsigned int, unsigned int> &vertex_to_coinciding_vertex_group);
 
  template <int dim, int spacedim>
  std::map<unsigned int, std::set<::types::subdomain_id>>
  compute_vertices_with_ghost_neighbors(
    const Triangulation<dim, spacedim> &tria);
 
  template <int dim, typename T>
  struct CellDataTransferBuffer
  {
    std::vector<CellId> cell_ids;
 
    std::vector<T> data;
 
    template <class Archive>
    void
    save(Archive &ar, const unsigned int version) const;
 
    template <class Archive>
    void
    load(Archive &ar, const unsigned int version);
 
#  ifdef DOXYGEN
    template <class Archive>
    void
    serialize(Archive &archive, const unsigned int version);
#  else
    // This macro defines the serialize() method that is compatible with
    // the templated save() and load() method that have been implemented.
    BOOST_SERIALIZATION_SPLIT_MEMBER()
#  endif
  };
 
 
 
  template <int dim, typename VectorType>
  class MarchingCubeAlgorithm
  {
  public:
    using value_type = typename VectorType::value_type;
 
    MarchingCubeAlgorithm(const Mapping<dim, dim> &      mapping,
                          const FiniteElement<dim, dim> &fe,
                          const unsigned int             n_subdivisions = 1,
                          const double                   tolerance = 1e-10);
 
    void
    process(const DoFHandler<dim> &         background_dof_handler,
            const VectorType &              ls_vector,
            const double                    iso_level,
            std::vector<Point<dim>> &       vertices,
            std::vector<CellData<dim - 1>> &cells) const;
 
    void
    process_cell(const typename DoFHandler<dim>::active_cell_iterator &cell,
                 const VectorType &              ls_vector,
                 const double                    iso_level,
                 std::vector<Point<dim>> &       vertices,
                 std::vector<CellData<dim - 1>> &cells) const;
 
  private:
    static Quadrature<dim>
    create_quadrature_rule(const unsigned int n_subdivisions);
 
    void
    process_cell(std::vector<value_type> &       ls_values,
                 const std::vector<Point<dim>> & points,
                 const double                    iso_level,
                 std::vector<Point<dim>> &       vertices,
                 std::vector<CellData<dim - 1>> &cells) const;
 
    void
    process_sub_cell(const std::vector<value_type> & ls_values,
                     const std::vector<Point<2>> &   points,
                     const std::vector<unsigned int> mask,
                     const double                    iso_level,
                     std::vector<Point<2>> &         vertices,
                     std::vector<CellData<1>> &      cells) const;
 
    void
    process_sub_cell(const std::vector<value_type> & ls_values,
                     const std::vector<Point<3>> &   points,
                     const std::vector<unsigned int> mask,
                     const double                    iso_level,
                     std::vector<Point<3>> &         vertices,
                     std::vector<CellData<2>> &      cells) const;
 
    const unsigned int n_subdivisions;
 
    const double tolerance;
 
    mutable FEValues<dim> fe_values;
  };
 
 
 
 
  DeclException1(ExcInvalidNumberOfPartitions,
                 int,
                 << "The number of partitions you gave is " << arg1
                 << ", but must be greater than zero.");
  DeclException1(ExcNonExistentSubdomain,
                 int,
                 << "The subdomain id " << arg1
                 << " has no cells associated with it.");
  DeclException0(ExcTriangulationHasBeenRefined);
 
  DeclException1(ExcScalingFactorNotPositive,
                 double,
                 << "The scaling factor must be positive, but it is " << arg1
                 << ".");
 
  DeclException1(ExcVertexNotUsed,
                 unsigned int,
                 << "The given vertex with index " << arg1
                 << " is not used in the given triangulation.");
 
} /*namespace GridTools*/
 
 
DeclExceptionMsg(ExcMeshNotOrientable,
                 "The edges of the mesh are not consistently orientable.");
 
 
 
/* ----------------- Template function --------------- */
 
#  ifndef DOXYGEN
 
namespace GridTools
{
  template <int dim>
  double
  cell_measure(
    const std::vector<Point<dim>> &all_vertices,
    const unsigned int (&indices)[GeometryInfo<dim>::vertices_per_cell])
  {
    // We forward call to the ArrayView version:
    const ArrayView<const unsigned int> view(
      indices, GeometryInfo<dim>::vertices_per_cell);
    return cell_measure(all_vertices, view);
  }
 
 
 
  // This specialization is defined here so that the general template in the
  // source file doesn't need to have further 1D overloads for the internal
  // functions it calls.
  template <>
  inline Triangulation<1, 1>::DistortedCellList
  fix_up_distorted_child_cells(const Triangulation<1, 1>::DistortedCellList &,
                               Triangulation<1, 1> &)
  {
    return {};
  }
 
 
 
  template <int dim, typename Predicate, int spacedim>
  void
  transform(const Predicate &             predicate,
            Triangulation<dim, spacedim> &triangulation)
  {
    std::vector<bool> treated_vertices(triangulation.n_vertices(), false);
 
    // loop over all active cells, and
    // transform those vertices that
    // have not yet been touched. note
    // that we get to all vertices in
    // the triangulation by only
    // visiting the active cells.
    typename Triangulation<dim, spacedim>::active_cell_iterator
      cell = triangulation.begin_active(),
      endc = triangulation.end();
    for (; cell != endc; ++cell)
      for (const unsigned int v : cell->vertex_indices())
        if (treated_vertices[cell->vertex_index(v)] == false)
          {
            // transform this vertex
            cell->vertex(v) = predicate(cell->vertex(v));
            // and mark it as treated
            treated_vertices[cell->vertex_index(v)] = true;
          };
 
 
    // now fix any vertices on hanging nodes so that we don't create any holes
    if (dim == 2)
      {
        typename Triangulation<dim, spacedim>::active_cell_iterator
          cell = triangulation.begin_active(),
          endc = triangulation.end();
        for (; cell != endc; ++cell)
          for (const unsigned int face : cell->face_indices())
            if (cell->face(face)->has_children() &&
                !cell->face(face)->at_boundary())
              {
                Assert(cell->reference_cell() ==
                         ReferenceCells::get_hypercube<dim>(),
                       ExcNotImplemented());
 
                // this line has children
                cell->face(face)->child(0)->vertex(1) =
                  (cell->face(face)->vertex(0) + cell->face(face)->vertex(1)) /
                  2;
              }
      }
    else if (dim == 3)
      {
        typename Triangulation<dim, spacedim>::active_cell_iterator
          cell = triangulation.begin_active(),
          endc = triangulation.end();
        for (; cell != endc; ++cell)
          for (const unsigned int face : cell->face_indices())
            if (cell->face(face)->has_children() &&
                !cell->face(face)->at_boundary())
              {
                Assert(cell->reference_cell() ==
                         ReferenceCells::get_hypercube<dim>(),
                       ExcNotImplemented());
 
                // this face has hanging nodes
                cell->face(face)->child(0)->vertex(1) =
                  (cell->face(face)->vertex(0) + cell->face(face)->vertex(1)) /
                  2.0;
                cell->face(face)->child(0)->vertex(2) =
                  (cell->face(face)->vertex(0) + cell->face(face)->vertex(2)) /
                  2.0;
                cell->face(face)->child(1)->vertex(3) =
                  (cell->face(face)->vertex(1) + cell->face(face)->vertex(3)) /
                  2.0;
                cell->face(face)->child(2)->vertex(3) =
                  (cell->face(face)->vertex(2) + cell->face(face)->vertex(3)) /
                  2.0;
 
                // center of the face
                cell->face(face)->child(0)->vertex(3) =
                  (cell->face(face)->vertex(0) + cell->face(face)->vertex(1) +
                   cell->face(face)->vertex(2) + cell->face(face)->vertex(3)) /
                  4.0;
              }
      }
 
    // Make sure FEValues notices that the mesh has changed
    triangulation.signals.mesh_movement();
  }
 
 
 
  template <class MeshType>
  std::vector<typename MeshType::active_cell_iterator>
  get_active_child_cells(const typename MeshType::cell_iterator &cell)
  {
    std::vector<typename MeshType::active_cell_iterator> child_cells;
 
    if (cell->has_children())
      {
        for (unsigned int child = 0; child < cell->n_children(); ++child)
          if (cell->child(child)->has_children())
            {
              const std::vector<typename MeshType::active_cell_iterator>
                children = get_active_child_cells<MeshType>(cell->child(child));
              child_cells.insert(child_cells.end(),
                                 children.begin(),
                                 children.end());
            }
          else
            child_cells.push_back(cell->child(child));
      }
 
    return child_cells;
  }
 
 
 
  template <class MeshType>
  void
  get_active_neighbors(
    const typename MeshType::active_cell_iterator &       cell,
    std::vector<typename MeshType::active_cell_iterator> &active_neighbors)
  {
    active_neighbors.clear();
    for (const unsigned int n : cell->face_indices())
      if (!cell->at_boundary(n))
        {
          if (MeshType::dimension == 1)
            {
              // check children of neighbor. note
              // that in 1d children of the neighbor
              // may be further refined. In 1d the
              // case is simple since we know what
              // children bound to the present cell
              typename MeshType::cell_iterator neighbor_child =
                cell->neighbor(n);
              if (!neighbor_child->is_active())
                {
                  while (neighbor_child->has_children())
                    neighbor_child = neighbor_child->child(n == 0 ? 1 : 0);
 
                  Assert(neighbor_child->neighbor(n == 0 ? 1 : 0) == cell,
                         ExcInternalError());
                }
              active_neighbors.push_back(neighbor_child);
            }
          else
            {
              if (cell->face(n)->has_children())
                // this neighbor has children. find
                // out which border to the present
                // cell
                for (unsigned int c = 0;
                     c < cell->face(n)->n_active_descendants();
                     ++c)
                  active_neighbors.push_back(
                    cell->neighbor_child_on_subface(n, c));
              else
                {
                  // the neighbor must be active
                  // himself
                  Assert(cell->neighbor(n)->is_active(), ExcInternalError());
                  active_neighbors.push_back(cell->neighbor(n));
                }
            }
        }
  }
 
 
 
  namespace internal
  {
    namespace ProjectToObject
    {
      struct CrossDerivative
      {
        const unsigned int direction_0;
        const unsigned int direction_1;
 
        CrossDerivative(const unsigned int d0, const unsigned int d1);
      };
 
      inline CrossDerivative::CrossDerivative(const unsigned int d0,
                                              const unsigned int d1)
        : direction_0(d0)
        , direction_1(d1)
      {}
 
 
 
      template <typename F>
      inline auto
      centered_first_difference(const double center,
                                const double step,
                                const F &f) -> decltype(f(center) - f(center))
      {
        return (f(center + step) - f(center - step)) / (2.0 * step);
      }
 
 
 
      template <typename F>
      inline auto
      centered_second_difference(const double center,
                                 const double step,
                                 const F &f) -> decltype(f(center) - f(center))
      {
        return (f(center + step) - 2.0 * f(center) + f(center - step)) /
               (step * step);
      }
 
 
 
      template <int structdim, typename F>
      inline auto
      cross_stencil(
        const CrossDerivative cross_derivative,
        const Tensor<1, GeometryInfo<structdim>::vertices_per_cell> &center,
        const double                                                 step,
        const F &f) -> decltype(f(center) - f(center))
      {
        Tensor<1, GeometryInfo<structdim>::vertices_per_cell> simplex_vector;
        simplex_vector[cross_derivative.direction_0] = 0.5 * step;
        simplex_vector[cross_derivative.direction_1] = -0.5 * step;
        return (-4.0 * f(center) - 1.0 * f(center + simplex_vector) -
                1.0 / 3.0 * f(center - simplex_vector) +
                16.0 / 3.0 * f(center + 0.5 * simplex_vector)) /
               step;
      }
 
 
 
      template <int spacedim, int structdim, typename F>
      inline double
      gradient_entry(
        const unsigned int     row_n,
        const unsigned int     dependent_direction,
        const Point<spacedim> &p0,
        const Tensor<1, GeometryInfo<structdim>::vertices_per_cell> &center,
        const double                                                 step,
        const F &                                                    f)
      {
        Assert(row_n < GeometryInfo<structdim>::vertices_per_cell &&
                 dependent_direction <
                   GeometryInfo<structdim>::vertices_per_cell,
               ExcMessage("This function assumes that the last weight is a "
                          "dependent variable (and hence we cannot take its "
                          "derivative directly)."));
        Assert(row_n != dependent_direction,
               ExcMessage(
                 "We cannot differentiate with respect to the variable "
                 "that is assumed to be dependent."));
 
        const Point<spacedim>     manifold_point = f(center);
        const Tensor<1, spacedim> stencil_value  = cross_stencil<structdim>(
          {row_n, dependent_direction}, center, step, f);
        double entry = 0.0;
        for (unsigned int dim_n = 0; dim_n < spacedim; ++dim_n)
          entry +=
            -2.0 * (p0[dim_n] - manifold_point[dim_n]) * stencil_value[dim_n];
        return entry;
      }
 
      template <typename Iterator, int spacedim, int structdim>
      Point<spacedim>
      project_to_d_linear_object(const Iterator &       object,
                                 const Point<spacedim> &trial_point)
      {
        // let's look at this for simplicity for a quad (structdim==2) in a
        // space with spacedim>2 (notate trial_point by y): all points on the
        // surface are given by
        //   x(\xi) = sum_i v_i phi_x(\xi)
        // where v_i are the vertices of the quad, and \xi=(\xi_1,\xi_2) are the
        // reference coordinates of the quad. so what we are trying to do is
        // find a point x on the surface that is closest to the point y. there
        // are different ways to solve this problem, but in the end it's a
        // nonlinear problem and we have to find reference coordinates \xi so
        // that J(\xi) = 1/2 || x(\xi)-y ||^2 is minimal. x(\xi) is a function
        // that is structdim-linear in \xi, so J(\xi) is a polynomial of degree
        // 2*structdim that we'd like to minimize. unless structdim==1, we'll
        // have to use a Newton method to find the answer. This leads to the
        // following formulation of Newton steps:
        //
        // Given \xi_k, find \delta\xi_k so that
        //   H_k \delta\xi_k = - F_k
        // where H_k is an approximation to the second derivatives of J at
        // \xi_k, and F_k is the first derivative of J.  We'll iterate this a
        // number of times until the right hand side is small enough. As a
        // stopping criterion, we terminate if ||\delta\xi||<eps.
        //
        // As for the Hessian, the best choice would be
        //   H_k = J''(\xi_k)
        // but we'll opt for the simpler Gauss-Newton form
        //   H_k = A^T A
        // i.e.
        //   (H_k)_{nm} = \sum_{i,j} v_i*v_j *
        //                   \partial_n phi_i *
        //                   \partial_m phi_j
        // we start at xi=(0.5, 0.5).
        Point<structdim> xi;
        for (unsigned int d = 0; d < structdim; ++d)
          xi[d] = 0.5;
 
        Point<spacedim> x_k;
        for (const unsigned int i : GeometryInfo<structdim>::vertex_indices())
          x_k += object->vertex(i) *
                 GeometryInfo<structdim>::d_linear_shape_function(xi, i);
 
        do
          {
            Tensor<1, structdim> F_k;
            for (const unsigned int i :
                 GeometryInfo<structdim>::vertex_indices())
              F_k +=
                (x_k - trial_point) * object->vertex(i) *
                GeometryInfo<structdim>::d_linear_shape_function_gradient(xi,
                                                                          i);
 
            Tensor<2, structdim> H_k;
            for (const unsigned int i :
                 GeometryInfo<structdim>::vertex_indices())
              for (const unsigned int j :
                   GeometryInfo<structdim>::vertex_indices())
                {
                  Tensor<2, structdim> tmp = outer_product(
                    GeometryInfo<structdim>::d_linear_shape_function_gradient(
                      xi, i),
                    GeometryInfo<structdim>::d_linear_shape_function_gradient(
                      xi, j));
                  H_k += (object->vertex(i) * object->vertex(j)) * tmp;
                }
 
            const Tensor<1, structdim> delta_xi = -invert(H_k) * F_k;
            xi += delta_xi;
 
            x_k = Point<spacedim>();
            for (const unsigned int i :
                 GeometryInfo<structdim>::vertex_indices())
              x_k += object->vertex(i) *
                     GeometryInfo<structdim>::d_linear_shape_function(xi, i);
 
            if (delta_xi.norm() < 1e-7)
              break;
          }
        while (true);
 
        return x_k;
      }
    } // namespace ProjectToObject
  }   // namespace internal
 
 
 
  namespace internal
  {
    // We hit an internal compiler error in ICC 15 if we define this as a lambda
    // inside the project_to_object function below.
    template <int structdim>
    inline bool
    weights_are_ok(
      const Tensor<1, GeometryInfo<structdim>::vertices_per_cell> &v)
    {
      // clang has trouble figuring out structdim here, so define it
      // again:
      static const std::size_t n_vertices_per_cell =
        Tensor<1, GeometryInfo<structdim>::vertices_per_cell>::
          n_independent_components;
      std::array<double, n_vertices_per_cell> copied_weights;
      for (unsigned int i = 0; i < n_vertices_per_cell; ++i)
        {
          copied_weights[i] = v[i];
          if (v[i] < 0.0 || v[i] > 1.0)
            return false;
        }
 
      // check the sum: try to avoid some roundoff errors by summing in order
      std::sort(copied_weights.begin(), copied_weights.end());
      const double sum =
        std::accumulate(copied_weights.begin(), copied_weights.end(), 0.0);
      return std::abs(sum - 1.0) < 1e-10; // same tolerance used in manifold.cc
    }
  } // namespace internal
 
  template <typename Iterator>
  Point<Iterator::AccessorType::space_dimension>
  project_to_object(
    const Iterator &                                      object,
    const Point<Iterator::AccessorType::space_dimension> &trial_point)
  {
    const int spacedim  = Iterator::AccessorType::space_dimension;
    const int structdim = Iterator::AccessorType::structure_dimension;
 
    Point<spacedim> projected_point = trial_point;
 
    if (structdim >= spacedim)
      return projected_point;
    else if (structdim == 1 || structdim == 2)
      {
        using namespace internal::ProjectToObject;
        // Try to use the special flat algorithm for quads (this is better
        // than the general algorithm in 3D). This does not take into account
        // whether projected_point is outside the quad, but we optimize along
        // lines below anyway:
        const int                      dim = Iterator::AccessorType::dimension;
        const Manifold<dim, spacedim> &manifold = object->get_manifold();
        if (structdim == 2 && dynamic_cast<const FlatManifold<dim, spacedim> *>(
                                &manifold) != nullptr)
          {
            projected_point =
              project_to_d_linear_object<Iterator, spacedim, structdim>(
                object, trial_point);
          }
        else
          {
            // We want to find a point on the convex hull (defined by the
            // vertices of the object and the manifold description) that is
            // relatively close to the trial point. This has a few issues:
            //
            // 1. For a general convex hull we are not guaranteed that a unique
            //    minimum exists.
            // 2. The independent variables in the optimization process are the
            //    weights given to Manifold::get_new_point, which must sum to 1,
            //    so we cannot use standard finite differences to approximate a
            //    gradient.
            //
            // There is not much we can do about 1., but for 2. we can derive
            // finite difference stencils that work on a structdim-dimensional
            // simplex and rewrite the optimization problem to use those
            // instead. Consider the structdim 2 case and let
            //
            // F(c0, c1, c2, c3) = Manifold::get_new_point(vertices, {c0, c1,
            // c2, c3})
            //
            // where {c0, c1, c2, c3} are the weights for the four vertices on
            // the quadrilateral. We seek to minimize the Euclidean distance
            // between F(...) and trial_point. We can solve for c3 in terms of
            // the other weights and get, for one coordinate direction
            //
            // d/dc0 ((x0 - F(c0, c1, c2, 1 - c0 - c1 - c2))^2)
            //      = -2(x0 - F(...)) (d/dc0 F(...) - d/dc3 F(...))
            //
            // where we substitute back in for c3 after taking the
            // derivative. We can compute a stencil for the cross derivative
            // d/dc0 - d/dc3: this is exactly what cross_stencil approximates
            // (and gradient_entry computes the sum over the independent
            // variables). Below, we somewhat arbitrarily pick the last
            // component as the dependent one.
            //
            // Since we can now calculate derivatives of the objective
            // function we can use gradient descent to minimize it.
            //
            // Of course, this is much simpler in the structdim = 1 case (we
            // could rewrite the projection as a 1D optimization problem), but
            // to reduce the potential for bugs we use the same code in both
            // cases.
            const double step_size = object->diameter() / 64.0;
 
            constexpr unsigned int n_vertices_per_cell =
              GeometryInfo<structdim>::vertices_per_cell;
 
            std::array<Point<spacedim>, n_vertices_per_cell> vertices;
            for (unsigned int vertex_n = 0; vertex_n < n_vertices_per_cell;
                 ++vertex_n)
              vertices[vertex_n] = object->vertex(vertex_n);
 
            auto get_point_from_weights =
              [&](const Tensor<1, n_vertices_per_cell> &weights)
              -> Point<spacedim> {
              return object->get_manifold().get_new_point(
                make_array_view(vertices.begin(), vertices.end()),
                make_array_view(weights.begin_raw(), weights.end_raw()));
            };
 
            // pick the initial weights as (normalized) inverse distances from
            // the trial point:
            Tensor<1, n_vertices_per_cell> guess_weights;
            double                         guess_weights_sum = 0.0;
            for (unsigned int vertex_n = 0; vertex_n < n_vertices_per_cell;
                 ++vertex_n)
              {
                const double distance =
                  vertices[vertex_n].distance(trial_point);
                if (distance == 0.0)
                  {
                    guess_weights           = 0.0;
                    guess_weights[vertex_n] = 1.0;
                    guess_weights_sum       = 1.0;
                    break;
                  }
                else
                  {
                    guess_weights[vertex_n] = 1.0 / distance;
                    guess_weights_sum += guess_weights[vertex_n];
                  }
              }
            guess_weights /= guess_weights_sum;
            Assert(internal::weights_are_ok<structdim>(guess_weights),
                   ExcInternalError());
 
            // The optimization algorithm consists of two parts:
            //
            // 1. An outer loop where we apply the gradient descent algorithm.
            // 2. An inner loop where we do a line search to find the optimal
            //    length of the step one should take in the gradient direction.
            //
            for (unsigned int outer_n = 0; outer_n < 40; ++outer_n)
              {
                const unsigned int dependent_direction =
                  n_vertices_per_cell - 1;
                Tensor<1, n_vertices_per_cell> current_gradient;
                for (unsigned int row_n = 0; row_n < n_vertices_per_cell;
                     ++row_n)
                  {
                    if (row_n != dependent_direction)
                      {
                        current_gradient[row_n] =
                          gradient_entry<spacedim, structdim>(
                            row_n,
                            dependent_direction,
                            trial_point,
                            guess_weights,
                            step_size,
                            get_point_from_weights);
 
                        current_gradient[dependent_direction] -=
                          current_gradient[row_n];
                      }
                  }
 
                // We need to travel in the -gradient direction, as noted
                // above, but we may not want to take a full step in that
                // direction; instead, guess that we will go -0.5*gradient and
                // do quasi-Newton iteration to pick the best multiplier. The
                // goal is to find a scalar alpha such that
                //
                // F(x - alpha g)
                //
                // is minimized, where g is the gradient and F is the
                // objective function. To find the optimal value we find roots
                // of the derivative of the objective function with respect to
                // alpha by Newton iteration, where we approximate the first
                // and second derivatives of F(x - alpha g) with centered
                // finite differences.
                double gradient_weight = -0.5;
                auto   gradient_weight_objective_function =
                  [&](const double gradient_weight_guess) -> double {
                  return (trial_point -
                          get_point_from_weights(guess_weights +
                                                 gradient_weight_guess *
                                                   current_gradient))
                    .norm_square();
                };
 
                for (unsigned int inner_n = 0; inner_n < 10; ++inner_n)
                  {
                    const double update_numerator = centered_first_difference(
                      gradient_weight,
                      step_size,
                      gradient_weight_objective_function);
                    const double update_denominator =
                      centered_second_difference(
                        gradient_weight,
                        step_size,
                        gradient_weight_objective_function);
 
                    // avoid division by zero. Note that we limit the gradient
                    // weight below
                    if (std::abs(update_denominator) == 0.0)
                      break;
                    gradient_weight =
                      gradient_weight - update_numerator / update_denominator;
 
                    // Put a fairly lenient bound on the largest possible
                    // gradient (things tend to be locally flat, so the gradient
                    // itself is usually small)
                    if (std::abs(gradient_weight) > 10)
                      {
                        gradient_weight = -10.0;
                        break;
                      }
                  }
 
                // It only makes sense to take convex combinations with weights
                // between zero and one. If the update takes us outside of this
                // region then rescale the update to stay within the region and
                // try again
                Tensor<1, n_vertices_per_cell> tentative_weights =
                  guess_weights + gradient_weight * current_gradient;
 
                double new_gradient_weight = gradient_weight;
                for (unsigned int iteration_count = 0; iteration_count < 40;
                     ++iteration_count)
                  {
                    if (internal::weights_are_ok<structdim>(tentative_weights))
                      break;
 
                    for (unsigned int i = 0; i < n_vertices_per_cell; ++i)
                      {
                        if (tentative_weights[i] < 0.0)
                          {
                            tentative_weights -=
                              (tentative_weights[i] / current_gradient[i]) *
                              current_gradient;
                          }
                        if (tentative_weights[i] < 0.0 ||
                            1.0 < tentative_weights[i])
                          {
                            new_gradient_weight /= 2.0;
                            tentative_weights =
                              guess_weights +
                              new_gradient_weight * current_gradient;
                          }
                      }
                  }
 
                // the update might still send us outside the valid region, so
                // check again and quit if the update is still not valid
                if (!internal::weights_are_ok<structdim>(tentative_weights))
                  break;
 
                // if we cannot get closer by traveling in the gradient
                // direction then quit
                if (get_point_from_weights(tentative_weights)
                      .distance(trial_point) <
                    get_point_from_weights(guess_weights).distance(trial_point))
                  guess_weights = tentative_weights;
                else
                  break;
                Assert(internal::weights_are_ok<structdim>(guess_weights),
                       ExcInternalError());
              }
            Assert(internal::weights_are_ok<structdim>(guess_weights),
                   ExcInternalError());
            projected_point = get_point_from_weights(guess_weights);
          }
 
        // if structdim == 2 and the optimal point is not on the interior then
        // we may be able to get a more accurate result by projecting onto the
        // lines.
        if (structdim == 2)
          {
            std::array<Point<spacedim>, GeometryInfo<structdim>::lines_per_cell>
              line_projections;
            for (unsigned int line_n = 0;
                 line_n < GeometryInfo<structdim>::lines_per_cell;
                 ++line_n)
              {
                line_projections[line_n] =
                  project_to_object(object->line(line_n), trial_point);
              }
            std::sort(line_projections.begin(),
                      line_projections.end(),
                      [&](const Point<spacedim> &a, const Point<spacedim> &b) {
                        return a.distance(trial_point) <
                               b.distance(trial_point);
                      });
            if (line_projections[0].distance(trial_point) <
                projected_point.distance(trial_point))
              projected_point = line_projections[0];
          }
      }
    else
      {
        Assert(false, ExcNotImplemented());
        return projected_point;
      }
 
    return projected_point;
  }
 
 
 
  template <int dim, typename T>
  template <class Archive>
  void
  CellDataTransferBuffer<dim, T>::save(Archive &ar,
                                       const unsigned int /*version*/) const
  {
    Assert(cell_ids.size() == data.size(),
           ExcDimensionMismatch(cell_ids.size(), data.size()));
    // archive the cellids in an efficient binary format
    const std::size_t n_cells = cell_ids.size();
    ar &              n_cells;
    for (const auto &id : cell_ids)
      {
        CellId::binary_type binary_cell_id = id.template to_binary<dim>();
        ar &                binary_cell_id;
      }
 
    ar &data;
  }
 
 
 
  template <int dim, typename T>
  template <class Archive>
  void
  CellDataTransferBuffer<dim, T>::load(Archive &ar,
                                       const unsigned int /*version*/)
  {
    std::size_t n_cells;
    ar &        n_cells;
    cell_ids.clear();
    cell_ids.reserve(n_cells);
    for (unsigned int c = 0; c < n_cells; ++c)
      {
        CellId::binary_type value;
        ar &                value;
        cell_ids.emplace_back(value);
      }
    ar &data;
  }
 
 
  namespace internal
  {
    template <typename DataType,
              typename MeshType,
              typename MeshCellIteratorType>
    inline void
    exchange_cell_data(
      const MeshType &mesh,
      const std::function<
        std_cxx17::optional<DataType>(const MeshCellIteratorType &)> &pack,
      const std::function<void(const MeshCellIteratorType &, const DataType &)>
        &                                                         unpack,
      const std::function<bool(const MeshCellIteratorType &)> &   cell_filter,
      const std::function<void(
        const std::function<void(const MeshCellIteratorType &,
                                 const types::subdomain_id)> &)> &process_cells,
      const std::function<std::set<types::subdomain_id>(
        const parallel::TriangulationBase<MeshType::dimension,
                                          MeshType::space_dimension> &)>
        &compute_ghost_owners)
    {
#    ifndef DEAL_II_WITH_MPI
      (void)mesh;
      (void)pack;
      (void)unpack;
      (void)cell_filter;
      (void)process_cells;
      (void)compute_ghost_owners;
      Assert(false, ExcNeedsMPI());
#    else
      constexpr int dim      = MeshType::dimension;
      constexpr int spacedim = MeshType::space_dimension;
      auto          tria =
        dynamic_cast<const parallel::TriangulationBase<dim, spacedim> *>(
          &mesh.get_triangulation());
      Assert(
        tria != nullptr,
        ExcMessage(
          "The function exchange_cell_data_to_ghosts() only works with parallel triangulations."));
 
      if (const auto tria = dynamic_cast<
            const parallel::shared::Triangulation<dim, spacedim> *>(
            &mesh.get_triangulation()))
        {
          Assert(
            tria->with_artificial_cells(),
            ExcMessage(
              "The functions GridTools::exchange_cell_data_to_ghosts() and "
              "GridTools::exchange_cell_data_to_level_ghosts() can only "
              "operate on a single layer ghost cells. However, you have "
              "given a Triangulation object of type "
              "parallel::shared::Triangulation without artificial cells "
              "resulting in arbitrary numbers of ghost layers."));
        }
 
      // build list of cells to request for each neighbor
      std::set<::types::subdomain_id> ghost_owners =
        compute_ghost_owners(*tria);
      std::map<::types::subdomain_id,
               std::vector<typename CellId::binary_type>>
        neighbor_cell_list;
 
      for (const auto ghost_owner : ghost_owners)
        neighbor_cell_list[ghost_owner] = {};
 
      process_cells([&](const auto &cell, const auto key) {
        if (cell_filter(cell))
          neighbor_cell_list[key].emplace_back(
            cell->id().template to_binary<spacedim>());
      });
 
      Assert(ghost_owners.size() == neighbor_cell_list.size(),
             ExcInternalError());
 
 
      // Before sending & receiving, make sure we protect this section with
      // a mutex:
      static Utilities::MPI::CollectiveMutex      mutex;
      Utilities::MPI::CollectiveMutex::ScopedLock lock(
        mutex, tria->get_communicator());
 
      const int mpi_tag =
        Utilities::MPI::internal::Tags::exchange_cell_data_request;
      const int mpi_tag_reply =
        Utilities::MPI::internal::Tags::exchange_cell_data_reply;
 
      // send our requests:
      std::vector<MPI_Request> requests(ghost_owners.size());
      {
        unsigned int idx = 0;
        for (const auto &it : neighbor_cell_list)
          {
            // send the data about the relevant cells
            const int ierr = MPI_Isend(it.second.data(),
                                       it.second.size() * sizeof(it.second[0]),
                                       MPI_BYTE,
                                       it.first,
                                       mpi_tag,
                                       tria->get_communicator(),
                                       &requests[idx]);
            AssertThrowMPI(ierr);
            ++idx;
          }
      }
 
      using DestinationToBufferMap =
        std::map<::types::subdomain_id,
                 GridTools::CellDataTransferBuffer<dim, DataType>>;
      DestinationToBufferMap destination_to_data_buffer_map;
 
      // receive requests and reply with the ghost indices
      std::vector<std::vector<typename CellId::binary_type>> cell_data_to_send(
        ghost_owners.size());
      std::vector<std::vector<types::global_dof_index>>
                                     send_dof_numbers_and_indices(ghost_owners.size());
      std::vector<MPI_Request>       reply_requests(ghost_owners.size());
      std::vector<std::vector<char>> sendbuffers(ghost_owners.size());
 
      for (unsigned int idx = 0; idx < ghost_owners.size(); ++idx)
        {
          MPI_Status status;
          int        ierr = MPI_Probe(MPI_ANY_SOURCE,
                               mpi_tag,
                               tria->get_communicator(),
                               &status);
          AssertThrowMPI(ierr);
 
          int len;
          ierr = MPI_Get_count(&status, MPI_BYTE, &len);
          AssertThrowMPI(ierr);
          Assert(len % sizeof(cell_data_to_send[idx][0]) == 0,
                 ExcInternalError());
 
          const unsigned int n_cells =
            len / sizeof(typename CellId::binary_type);
          cell_data_to_send[idx].resize(n_cells);
 
          ierr = MPI_Recv(cell_data_to_send[idx].data(),
                          len,
                          MPI_BYTE,
                          status.MPI_SOURCE,
                          status.MPI_TAG,
                          tria->get_communicator(),
                          &status);
          AssertThrowMPI(ierr);
 
          // store data for each cell
          for (unsigned int c = 0; c < static_cast<unsigned int>(n_cells); ++c)
            {
              const auto cell =
                tria->create_cell_iterator(CellId(cell_data_to_send[idx][c]));
 
              MeshCellIteratorType                mesh_it(tria,
                                           cell->level(),
                                           cell->index(),
                                           &mesh);
              const std_cxx17::optional<DataType> data = pack(mesh_it);
 
              if (data)
                {
                  typename DestinationToBufferMap::iterator p =
                    destination_to_data_buffer_map
                      .insert(std::make_pair(
                        idx,
                        GridTools::CellDataTransferBuffer<dim, DataType>()))
                      .first;
 
                  p->second.cell_ids.emplace_back(cell->id());
                  p->second.data.emplace_back(*data);
                }
            }
 
          // send reply
          GridTools::CellDataTransferBuffer<dim, DataType> &data =
            destination_to_data_buffer_map[idx];
 
          sendbuffers[idx] =
            Utilities::pack(data, /*enable_compression*/ false);
          ierr = MPI_Isend(sendbuffers[idx].data(),
                           sendbuffers[idx].size(),
                           MPI_BYTE,
                           status.MPI_SOURCE,
                           mpi_tag_reply,
                           tria->get_communicator(),
                           &reply_requests[idx]);
          AssertThrowMPI(ierr);
        }
 
      // finally receive the replies
      std::vector<char> receive;
      for (unsigned int idx = 0; idx < ghost_owners.size(); ++idx)
        {
          MPI_Status status;
          int        ierr = MPI_Probe(MPI_ANY_SOURCE,
                               mpi_tag_reply,
                               tria->get_communicator(),
                               &status);
          AssertThrowMPI(ierr);
 
          int len;
          ierr = MPI_Get_count(&status, MPI_BYTE, &len);
          AssertThrowMPI(ierr);
 
          receive.resize(len);
 
          char *ptr = receive.data();
          ierr      = MPI_Recv(ptr,
                          len,
                          MPI_BYTE,
                          status.MPI_SOURCE,
                          status.MPI_TAG,
                          tria->get_communicator(),
                          &status);
          AssertThrowMPI(ierr);
 
          auto cellinfo =
            Utilities::unpack<CellDataTransferBuffer<dim, DataType>>(
              receive, /*enable_compression*/ false);
 
          DataType *data = cellinfo.data.data();
          for (unsigned int c = 0; c < cellinfo.cell_ids.size(); ++c, ++data)
            {
              const typename Triangulation<dim, spacedim>::cell_iterator
                tria_cell = tria->create_cell_iterator(cellinfo.cell_ids[c]);
 
              MeshCellIteratorType cell(tria,
                                        tria_cell->level(),
                                        tria_cell->index(),
                                        &mesh);
 
              unpack(cell, *data);
            }
        }
 
      // make sure that all communication is finished
      // when we leave this function.
      if (requests.size() > 0)
        {
          const int ierr =
            MPI_Waitall(requests.size(), requests.data(), MPI_STATUSES_IGNORE);
          AssertThrowMPI(ierr);
        }
      if (reply_requests.size() > 0)
        {
          const int ierr = MPI_Waitall(reply_requests.size(),
                                       reply_requests.data(),
                                       MPI_STATUSES_IGNORE);
          AssertThrowMPI(ierr);
        }
 
 
#    endif // DEAL_II_WITH_MPI
    }
 
  } // namespace internal
 
  template <typename DataType, typename MeshType>
  inline void
  exchange_cell_data_to_ghosts(
    const MeshType &                                     mesh,
    const std::function<std_cxx17::optional<DataType>(
      const typename MeshType::active_cell_iterator &)> &pack,
    const std::function<void(const typename MeshType::active_cell_iterator &,
                             const DataType &)> &        unpack,
    const std::function<bool(const typename MeshType::active_cell_iterator &)>
      &cell_filter)
  {
#    ifndef DEAL_II_WITH_MPI
    (void)mesh;
    (void)pack;
    (void)unpack;
    (void)cell_filter;
    Assert(false, ExcNeedsMPI());
#    else
    internal::exchange_cell_data<DataType,
                                 MeshType,
                                 typename MeshType::active_cell_iterator>(
      mesh,
      pack,
      unpack,
      cell_filter,
      [&](const auto &process) {
        for (const auto &cell : mesh.active_cell_iterators())
          if (cell->is_ghost())
            process(cell, cell->subdomain_id());
      },
      [](const auto &tria) { return tria.ghost_owners(); });
#    endif
  }
 
 
 
  template <typename DataType, typename MeshType>
  inline void
  exchange_cell_data_to_level_ghosts(
    const MeshType &                                    mesh,
    const std::function<std_cxx17::optional<DataType>(
      const typename MeshType::level_cell_iterator &)> &pack,
    const std::function<void(const typename MeshType::level_cell_iterator &,
                             const DataType &)> &       unpack,
    const std::function<bool(const typename MeshType::level_cell_iterator &)>
      &cell_filter)
  {
#    ifndef DEAL_II_WITH_MPI
    (void)mesh;
    (void)pack;
    (void)unpack;
    (void)cell_filter;
    Assert(false, ExcNeedsMPI());
#    else
    internal::exchange_cell_data<DataType,
                                 MeshType,
                                 typename MeshType::level_cell_iterator>(
      mesh,
      pack,
      unpack,
      cell_filter,
      [&](const auto &process) {
        for (const auto &cell : mesh.cell_iterators())
          if (cell->level_subdomain_id() !=
                ::numbers::artificial_subdomain_id &&
              !cell->is_locally_owned_on_level())
            process(cell, cell->level_subdomain_id());
      },
      [](const auto &tria) { return tria.level_ghost_owners(); });
#    endif
  }
} // namespace GridTools
 
#  endif
 
DEAL_II_NAMESPACE_CLOSE
 
/*----------------------------   grid_tools.h     ---------------------------*/
/* end of #ifndef dealii_grid_tools_h */
#endif
/*----------------------------   grid_tools.h     ---------------------------*/