utils.h

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// Copyright (C) 2019-2024 Garth N. Wells
//
// This file is part of DOLFINx (https://www.fenicsproject.org)
//
// SPDX-License-Identifier:    LGPL-3.0-or-later
 
#pragma once
 
#include "EntityMap.h"
#include "Mesh.h"
#include "Topology.h"
#include "graphbuild.h"
#include <algorithm>
#include <basix/mdspan.hpp>
#include <concepts>
#include <cstdint>
#include <dolfinx/graph/AdjacencyList.h>
#include <dolfinx/graph/ordering.h>
#include <dolfinx/graph/partition.h>
#include <functional>
#include <mpi.h>
#include <numeric>
#include <optional>
#include <span>
#include <vector>

 
namespace dolfinx::fem
{
class ElementDofLayout;
}
 
namespace dolfinx::mesh
{
enum class CellType : std::int8_t;

enum class GhostMode : std::uint8_t
{
  none,
  shared_facet
};
 
namespace impl
{
template <typename T>
void reorder_list(std::span<T> list, std::span<const std::int32_t> nodemap)
{
  if (nodemap.empty())
    return;
 
  assert(list.size() % nodemap.size() == 0);
  std::size_t degree = list.size() / nodemap.size();
  const std::vector<T> orig(list.begin(), list.end());
  for (std::size_t n = 0; n < nodemap.size(); ++n)
  {
    std::span links_old(orig.data() + n * degree, degree);
    auto links_new = list.subspan(nodemap[n] * degree, degree);
    std::ranges::copy(links_old, links_new.begin());
  }
}

template <std::floating_point T>
std::tuple<std::vector<std::int32_t>, std::vector<T>, std::vector<std::int32_t>>
compute_vertex_coords_boundary(const mesh::Mesh<T>& mesh, int dim,
                               std::span<const std::int32_t> facets)
{
  auto topology = mesh.topology();
  assert(topology);
  const int tdim = topology->dim();
  if (dim == tdim)
  {
    throw std::runtime_error(
        "Cannot use mesh::locate_entities_boundary (boundary) for cells.");
  }
 
  // Build set of vertices on boundary and set of boundary entities
  mesh.topology_mutable()->create_connectivity(tdim - 1, 0);
  mesh.topology_mutable()->create_connectivity(tdim - 1, dim);
  std::vector<std::int32_t> vertices, entities;
  {
    auto f_to_v = topology->connectivity(tdim - 1, 0);
    assert(f_to_v);
    auto f_to_e = topology->connectivity(tdim - 1, dim);
    assert(f_to_e);
    for (auto f : facets)
    {
      auto v = f_to_v->links(f);
      vertices.insert(vertices.end(), v.begin(), v.end());
      auto e = f_to_e->links(f);
      entities.insert(entities.end(), e.begin(), e.end());
    }
 
    // Build vector of boundary vertices
    {
      std::ranges::sort(vertices);
      auto [unique_end, range_end] = std::ranges::unique(vertices);
      vertices.erase(unique_end, range_end);
    }
 
    {
      std::ranges::sort(entities);
      auto [unique_end, range_end] = std::ranges::unique(entities);
      entities.erase(unique_end, range_end);
    }
  }
 
  // Get geometry data
  auto x_dofmap = mesh.geometry().dofmap();
  std::span<const T> x_nodes = mesh.geometry().x();
 
  // Get all vertex 'node' indices
  mesh.topology_mutable()->create_connectivity(0, tdim);
  mesh.topology_mutable()->create_connectivity(tdim, 0);
  auto v_to_c = topology->connectivity(0, tdim);
  assert(v_to_c);
  auto c_to_v = topology->connectivity(tdim, 0);
  assert(c_to_v);
  std::vector<T> x_vertices(3 * vertices.size(), -1.0);
  std::vector<std::int32_t> vertex_to_pos(v_to_c->num_nodes(), -1);
  for (std::size_t i = 0; i < vertices.size(); ++i)
  {
    const std::int32_t v = vertices[i];
 
    // Get first cell and find position
    const std::int32_t c = v_to_c->links(v).front();
    auto cell_vertices = c_to_v->links(c);
    auto it = std::find(cell_vertices.begin(), cell_vertices.end(), v);
    assert(it != cell_vertices.end());
    const std::size_t local_pos = std::distance(cell_vertices.begin(), it);
 
    auto dofs = md::submdspan(x_dofmap, c, md::full_extent);
    for (std::size_t j = 0; j < 3; ++j)
      x_vertices[j * vertices.size() + i] = x_nodes[3 * dofs[local_pos] + j];
    vertex_to_pos[v] = i;
  }
 
  return {std::move(entities), std::move(x_vertices), std::move(vertex_to_pos)};
}
 
} // namespace impl

std::vector<std::int32_t> exterior_facet_indices(const Topology& topology,
                                                 int facet_type_idx);

std::vector<std::int32_t> exterior_facet_indices(const Topology& topology);

using CellPartitionFunction = std::function<graph::AdjacencyList<std::int32_t>(
    MPI_Comm comm, int nparts, const std::vector<CellType>& cell_types,
    const std::vector<std::span<const std::int64_t>>& cells)>;

using CellReorderFunction = std::function<std::vector<std::int32_t>(
    const graph::AdjacencyList<std::int32_t>&)>;

inline auto
create_boundary_vertices_fn(const CellReorderFunction& reorder_fn,
                            std::optional<std::int32_t> max_facet_to_cell_links
                            = 2)
{
  return [&, max_facet_to_cell_links](
             const std::vector<CellType>& celltypes,
             const std::vector<fem::ElementDofLayout>& doflayouts,
             const std::vector<std::vector<int>>& ghost_owners,
             std::vector<std::vector<std::int64_t>>& cells,
             std::vector<std::vector<std::int64_t>>& cells_v,
             std::vector<std::vector<std::int64_t>>& original_idx)
             -> std::vector<std::int64_t>
  {
    // Build local dual graph for owned cells to (i) get list of vertices
    // on the process boundary and (ii) apply re-ordering to cells for
    // locality
 
    spdlog::info("Build local dual graphs, re-order cells, and compute process "
                 "boundary vertices.");
 
    std::vector<std::pair<std::vector<std::int64_t>, int>> facets;
 
    // Build lists of cells (by cell type) that excludes ghosts
    std::vector<std::span<const std::int64_t>> cells1_v_local;
    for (std::size_t i = 0; i < celltypes.size(); ++i)
    {
      int num_cell_vertices = mesh::num_cell_vertices(celltypes[i]);
      std::size_t num_owned_cells
          = cells_v[i].size() / num_cell_vertices - ghost_owners[i].size();
      cells1_v_local.emplace_back(cells_v[i].data(),
                                  num_owned_cells * num_cell_vertices);
 
      // Build local dual graph for cell type
      auto [graph, unmatched_facets, max_v, _facet_attached_cells]
          = build_local_dual_graph(std::vector{celltypes[i]},
                                   std::vector{cells1_v_local.back()},
                                   max_facet_to_cell_links);
 
      // Store unmatched_facets for current cell type
      facets.emplace_back(std::move(unmatched_facets), max_v);
 
      // Compute re-ordering of graph
      const std::vector<std::int32_t> remap = reorder_fn(graph);
 
      // Update 'original' indices
      const std::vector<std::int64_t>& orig_idx = original_idx[i];
      std::vector<std::int64_t> _original_idx(orig_idx.size());
      std::copy_n(orig_idx.rbegin(), ghost_owners[i].size(),
                  _original_idx.rbegin());
      {
        for (std::size_t j = 0; j < remap.size(); ++j)
          _original_idx[remap[j]] = orig_idx[j];
      }
      original_idx[i] = _original_idx;
 
      // Reorder cells
      impl::reorder_list(
          std::span(cells_v[i].data(), remap.size() * num_cell_vertices),
          remap);
      impl::reorder_list(
          std::span(cells[i].data(), remap.size() * doflayouts[i].num_dofs()),
          remap);
    }
 
    if (facets.size() == 1) // Optimisation for single cell type
    {
      std::vector<std::int64_t>& vertices = facets.front().first;
 
      // Remove duplicated vertex indices
      std::ranges::sort(vertices);
      auto [unique_end, range_end] = std::ranges::unique(vertices);
      vertices.erase(unique_end, range_end);
 
      // Remove -1 if it appears as first entity. This can happen in
      // mixed topology meshes where '-1' is used to pad facet data when
      // cells facets have differing numbers of vertices.
      if (!vertices.empty() and vertices.front() == -1)
        vertices.erase(vertices.begin());
 
      return vertices;
    }
    else
    {
      // Pack 'unmatched' facets for all cell types into single array
      // (facets0)
      std::vector<std::int64_t> facets0;
      facets0.reserve(std::accumulate(facets.begin(), facets.end(),
                                      std::size_t(0), [](std::size_t x, auto& y)
                                      { return x + y.first.size(); }));
      int max_v = std::ranges::max_element(facets, [](auto& a, auto& b)
                                           { return a.second < b.second; })
                      ->second;
      for (const auto& [v_data, num_v] : facets)
      {
        for (auto it = v_data.begin(); it != v_data.end(); it += num_v)
        {
          facets0.insert(facets0.end(), it, std::next(it, num_v));
          facets0.insert(facets0.end(), max_v - num_v, -1);
        }
      }
 
      // Compute row permutation
      const std::vector<std::int32_t> perm = dolfinx::sort_by_perm(
          std::span<const std::int64_t>(facets0), max_v);
 
      // For facets in facets0 that appear only once, store the facet
      // vertices
      std::vector<std::int64_t> vertices;
      // TODO: allocate memory for vertices
      auto it = perm.begin();
      while (it != perm.end())
      {
        // Find iterator to next facet different from f and trim any  -1
        // padding
        std::span _f(facets0.data() + (*it) * max_v, max_v);
        auto end = std::find_if(_f.rbegin(), _f.rend(),
                                [](auto a) { return a >= 0; });
        auto f = _f.first(std::distance(end, _f.rend()));
 
        auto it1 = std::find_if_not(
            it, perm.end(),
            [f, max_v, it0 = facets0.begin()](auto p) -> bool
            {
              return std::equal(f.begin(), f.end(), std::next(it0, p * max_v));
            });
 
        // If no repeated facet found, insert f vertices
        if (std::distance(it, it1) == 1)
          vertices.insert(vertices.end(), f.begin(), f.end());
        else if (std::distance(it, it1) > 2)
          throw std::runtime_error("More than two matching facets found.");
 
        // Advance iterator
        it = it1;
      }
 
      // Remove duplicate indices
      std::ranges::sort(vertices);
      auto [unique_end, range_end] = std::ranges::unique(vertices);
      vertices.erase(unique_end, range_end);
 
      return vertices;
    }
  };
}

std::vector<std::int64_t> extract_topology(CellType cell_type,
                                           const fem::ElementDofLayout& layout,
                                           std::span<const std::int64_t> cells);

template <std::floating_point T>
std::vector<T> h(const Mesh<T>& mesh, std::span<const std::int32_t> entities,
                 int dim)
{
  if (entities.empty())
    return std::vector<T>();
  if (dim == 0)
    return std::vector<T>(entities.size(), 0);
 
  // Get the geometry dofs for the vertices of each entity
  const auto [vertex_xdofs, xdof_shape]
      = entities_to_geometry(mesh, dim, entities, false);
 
  // Get the  geometry coordinate
  std::span<const T> x = mesh.geometry().x();
 
  // Function to compute the length of (p0 - p1)
  auto delta_norm = [](auto&& p0, auto&& p1)
  {
    T norm = 0;
    for (std::size_t i = 0; i < 3; ++i)
      norm += (p0[i] - p1[i]) * (p0[i] - p1[i]);
    return std::sqrt(norm);
  };
 
  // Compute greatest distance between any to vertices
  assert(dim > 0);
  std::vector<T> h(entities.size(), 0);
  for (std::size_t e = 0; e < entities.size(); ++e)
  {
    // Get geometry 'dof' for each vertex of entity e
    std::span<const std::int32_t> e_vertices(
        vertex_xdofs.data() + e * xdof_shape[1], xdof_shape[1]);
 
    // Compute maximum distance between any two vertices
    for (std::size_t i = 0; i < e_vertices.size(); ++i)
    {
      std::span<const T, 3> p0(x.data() + 3 * e_vertices[i], 3);
      for (std::size_t j = i + 1; j < e_vertices.size(); ++j)
      {
        std::span<const T, 3> p1(x.data() + 3 * e_vertices[j], 3);
        h[e] = std::max(h[e], delta_norm(p0, p1));
      }
    }
  }
 
  return h;
}

template <std::floating_point T>
std::vector<T> cell_normals(const Mesh<T>& mesh, int dim,
                            std::span<const std::int32_t> entities)
{
  if (entities.empty())
    return std::vector<T>();
 
  auto topology = mesh.topology();
  assert(topology);
  if (topology->cell_type() == CellType::prism and dim == 2)
  {
    throw std::runtime_error(
        "Cell normal computation for prism cells not yet supported.");
  }
 
  const int gdim = mesh.geometry().dim();
  const CellType type = cell_entity_type(topology->cell_type(), dim, 0);
 
  // Find geometry nodes for topology entities
  std::span<const T> x = mesh.geometry().x();
  const auto [geometry_entities, eshape]
      = entities_to_geometry(mesh, dim, entities, false);
 
  std::vector<T> n(entities.size() * 3);
  switch (type)
  {
  case CellType::interval:
  {
    if (gdim > 2)
      throw std::invalid_argument("Interval cell normal undefined in 3D.");
    for (std::size_t i = 0; i < entities.size(); ++i)
    {
      // Get the two vertices as points
      std::array vertices{geometry_entities[i * eshape[1]],
                          geometry_entities[i * eshape[1] + 1]};
      std::array p = {std::span<const T, 3>(x.data() + 3 * vertices[0], 3),
                      std::span<const T, 3>(x.data() + 3 * vertices[1], 3)};
 
      // Define normal by rotating tangent counter-clockwise
      std::array<T, 3> t;
      std::ranges::transform(p[1], p[0], t.begin(),
                             [](auto x, auto y) { return x - y; });
 
      T norm = std::sqrt(t[0] * t[0] + t[1] * t[1]);
      std::span<T, 3> ni(n.data() + 3 * i, 3);
      ni[0] = -t[1] / norm;
      ni[1] = t[0] / norm;
      ni[2] = 0.0;
    }
    return n;
  }
  case CellType::triangle:
  {
    for (std::size_t i = 0; i < entities.size(); ++i)
    {
      // Get the three vertices as points
      std::array vertices = {geometry_entities[i * eshape[1] + 0],
                             geometry_entities[i * eshape[1] + 1],
                             geometry_entities[i * eshape[1] + 2]};
      std::array p = {std::span<const T, 3>(x.data() + 3 * vertices[0], 3),
                      std::span<const T, 3>(x.data() + 3 * vertices[1], 3),
                      std::span<const T, 3>(x.data() + 3 * vertices[2], 3)};
 
      // Compute (p1 - p0) and (p2 - p0)
      std::array<T, 3> dp1, dp2;
      std::ranges::transform(p[1], p[0], dp1.begin(),
                             [](auto x, auto y) { return x - y; });
      std::ranges::transform(p[2], p[0], dp2.begin(),
                             [](auto x, auto y) { return x - y; });
 
      // Define cell normal via cross product of first two edges
      std::array<T, 3> ni = math::cross(dp1, dp2);
      T norm = std::sqrt(ni[0] * ni[0] + ni[1] * ni[1] + ni[2] * ni[2]);
      std::ranges::transform(ni, std::next(n.begin(), 3 * i),
                             [norm](auto x) { return x / norm; });
    }
 
    return n;
  }
  case CellType::quadrilateral:
  {
    // TODO: check
    for (std::size_t i = 0; i < entities.size(); ++i)
    {
      // Get the three vertices as points
      std::array vertices = {geometry_entities[i * eshape[1] + 0],
                             geometry_entities[i * eshape[1] + 1],
                             geometry_entities[i * eshape[1] + 2]};
      std::array p = {std::span<const T, 3>(x.data() + 3 * vertices[0], 3),
                      std::span<const T, 3>(x.data() + 3 * vertices[1], 3),
                      std::span<const T, 3>(x.data() + 3 * vertices[2], 3)};
 
      // Compute (p1 - p0) and (p2 - p0)
      std::array<T, 3> dp1, dp2;
      std::ranges::transform(p[1], p[0], dp1.begin(),
                             [](auto x, auto y) { return x - y; });
      std::ranges::transform(p[2], p[0], dp2.begin(),
                             [](auto x, auto y) { return x - y; });
 
      // Define cell normal via cross product of first two edges
      std::array<T, 3> ni = math::cross(dp1, dp2);
      T norm = std::sqrt(ni[0] * ni[0] + ni[1] * ni[1] + ni[2] * ni[2]);
      std::ranges::transform(ni, std::next(n.begin(), 3 * i),
                             [norm](auto x) { return x / norm; });
    }
 
    return n;
  }
  default:
    throw std::invalid_argument(
        "cell_normal not supported for this cell type.");
  }
}

template <std::floating_point T>
std::vector<T> compute_midpoints(const Mesh<T>& mesh, int dim,
                                 std::span<const std::int32_t> entities)
{
  if (entities.empty())
    return std::vector<T>();
 
  std::span<const T> x = mesh.geometry().x();
 
  // Build map from entity -> geometry dof
  const auto [e_to_g, eshape]
      = entities_to_geometry(mesh, dim, entities, false);
 
  std::vector<T> x_mid(entities.size() * 3, 0);
  for (std::size_t e = 0; e < entities.size(); ++e)
  {
    std::span<T, 3> p(x_mid.data() + 3 * e, 3);
    std::span<const std::int32_t> rows(e_to_g.data() + e * eshape[1],
                                       eshape[1]);
    for (auto row : rows)
    {
      std::span<const T, 3> xg(x.data() + 3 * row, 3);
      std::ranges::transform(p, xg, p.begin(),
                             [size = rows.size()](auto x, auto y)
                             { return x + y / size; });
    }
  }
 
  return x_mid;
}
 
namespace impl
{
template <std::floating_point T>
std::pair<std::vector<T>, std::array<std::size_t, 2>>
compute_vertex_coords(const mesh::Mesh<T>& mesh)
{
  auto topology = mesh.topology();
  assert(topology);
  const int tdim = topology->dim();
 
  // Create entities and connectivities
 
  // Get all vertex 'node' indices
  const std::int32_t num_vertices = topology->index_map(0)->size_local()
                                    + topology->index_map(0)->num_ghosts();
 
  std::vector<std::int32_t> vertex_to_node(num_vertices);
  for (int cell_type_idx = 0,
           num_cell_types = topology->entity_types(tdim).size();
       cell_type_idx < num_cell_types; ++cell_type_idx)
  {
    auto x_dofmap = mesh.geometry().dofmap(cell_type_idx);
    auto c_to_v = topology->connectivity({tdim, cell_type_idx}, {0, 0});
    assert(c_to_v);
    for (int c = 0; c < c_to_v->num_nodes(); ++c)
    {
      auto x_dofs = md::submdspan(x_dofmap, c, md::full_extent);
      auto vertices = c_to_v->links(c);
      for (std::size_t i = 0; i < vertices.size(); ++i)
        vertex_to_node[vertices[i]] = x_dofs[i];
    }
  }
 
  // Pack coordinates of vertices
  std::span<const T> x_nodes = mesh.geometry().x();
  std::vector<T> x_vertices(3 * vertex_to_node.size(), 0.0);
  for (std::size_t i = 0; i < vertex_to_node.size(); ++i)
  {
    std::int32_t pos = 3 * vertex_to_node[i];
    for (std::size_t j = 0; j < 3; ++j)
      x_vertices[j * vertex_to_node.size() + i] = x_nodes[pos + j];
  }
 
  return {std::move(x_vertices), {3, vertex_to_node.size()}};
}
 
} // namespace impl

template <typename Fn, typename T>
concept MarkerFn = std::is_invocable_r<
    std::vector<std::int8_t>, Fn,
    md::mdspan<const T,
               md::extents<std::size_t, 3, md::dynamic_extent>>>::value;

template <std::floating_point T, MarkerFn<T> U>
std::vector<std::int32_t> locate_entities(const Mesh<T>& mesh, int dim,
                                          U marker, int entity_type_idx)
{
 
  using cmdspan3x_t
      = md::mdspan<const T, md::extents<std::size_t, 3, md::dynamic_extent>>;
 
  // Run marker function on vertex coordinates
  const auto [xdata, xshape] = impl::compute_vertex_coords(mesh);
 
  cmdspan3x_t x(xdata.data(), xshape);
  const std::vector<std::int8_t> marked = marker(x);
  if (marked.size() != x.extent(1))
    throw std::runtime_error("Length of array of markers is wrong.");
 
  auto topology = mesh.topology();
  assert(topology);
  const int tdim = topology->dim();
 
  mesh.topology_mutable()->create_entities(dim);
  if (dim < tdim)
    mesh.topology_mutable()->create_connectivity(dim, 0);
 
  // Iterate over entities of dimension 'dim' to build vector of marked
  // entities
  auto e_to_v = topology->connectivity({dim, entity_type_idx}, {0, 0});
  assert(e_to_v);
  std::vector<std::int32_t> entities;
  for (int e = 0; e < e_to_v->num_nodes(); ++e)
  {
    // Iterate over entity vertices
    bool all_vertices_marked = true;
    for (std::int32_t v : e_to_v->links(e))
    {
      if (!marked[v])
      {
        all_vertices_marked = false;
        break;
      }
    }
 
    if (all_vertices_marked)
      entities.push_back(e);
  }
 
  return entities;
}

template <std::floating_point T, MarkerFn<T> U>
std::vector<std::int32_t> locate_entities(const Mesh<T>& mesh, int dim,
                                          U marker)
{
  const int num_entity_types = mesh.topology()->entity_types(dim).size();
  if (num_entity_types > 1)
  {
    throw std::runtime_error(
        "Multiple entity types of this dimension. Specify entity type index");
  }
  return locate_entities(mesh, dim, marker, 0);
}

template <std::floating_point T, MarkerFn<T> U>
std::vector<std::int32_t> locate_entities_boundary(const Mesh<T>& mesh, int dim,
                                                   U marker)
{
  // TODO Rewrite this function, it should be possible to simplify considerably
  auto topology = mesh.topology();
  assert(topology);
  int tdim = topology->dim();
  if (dim == tdim)
  {
    throw std::runtime_error(
        "Cannot use mesh::locate_entities_boundary (boundary) for cells.");
  }
 
  // Compute list of boundary facets
  mesh.topology_mutable()->create_entities(tdim - 1);
  mesh.topology_mutable()->create_connectivity(tdim - 1, tdim);
  std::vector<std::int32_t> boundary_facets = exterior_facet_indices(*topology);
 
  using cmdspan3x_t
      = md::mdspan<const T, md::extents<std::size_t, 3, md::dynamic_extent>>;
 
  // Run marker function on the vertex coordinates
  auto [facet_entities, xdata, vertex_to_pos]
      = impl::compute_vertex_coords_boundary(mesh, dim, boundary_facets);
  cmdspan3x_t x(xdata.data(), 3, xdata.size() / 3);
  std::vector<std::int8_t> marked = marker(x);
  if (marked.size() != x.extent(1))
    throw std::runtime_error("Length of array of markers is wrong.");
 
  // Loop over entities and check vertex markers
  mesh.topology_mutable()->create_entities(dim);
  auto e_to_v = topology->connectivity(dim, 0);
  assert(e_to_v);
  std::vector<std::int32_t> entities;
  for (auto e : facet_entities)
  {
    // Iterate over entity vertices
    bool all_vertices_marked = true;
    for (auto v : e_to_v->links(e))
    {
      const std::int32_t pos = vertex_to_pos[v];
      if (!marked[pos])
      {
        all_vertices_marked = false;
        break;
      }
    }
 
    // Mark facet with all vertices marked
    if (all_vertices_marked)
      entities.push_back(e);
  }
 
  return entities;
}

template <std::floating_point T>
std::pair<std::vector<std::int32_t>, std::array<std::size_t, 2>>
entities_to_geometry(const Mesh<T>& mesh, int dim,
                     std::span<const std::int32_t> entities,
                     bool permute = false)
{
  auto topology = mesh.topology();
  assert(topology);
  CellType cell_type = topology->cell_type();
  if (cell_type == CellType::prism and dim == 2)
  {
    throw std::runtime_error(
        "mesh::entities_to_geometry for prism cells not yet supported.");
  }
 
  const int tdim = topology->dim();
  const Geometry<T>& geometry = mesh.geometry();
  auto xdofs = geometry.dofmap();
 
  // Get the DOF layout and the number of DOFs per entity
  const fem::CoordinateElement<T>& coord_ele = geometry.cmap();
  const fem::ElementDofLayout layout = coord_ele.create_dof_layout();
  const std::size_t num_entity_dofs = layout.num_entity_closure_dofs(dim);
  std::vector<std::int32_t> entity_xdofs;
  entity_xdofs.reserve(entities.size() * num_entity_dofs);
  std::array<std::size_t, 2> eshape{entities.size(), num_entity_dofs};
 
  // Get the element's closure DOFs
  const std::vector<std::vector<std::vector<int>>>& closure_dofs_all
      = layout.entity_closure_dofs_all();
 
  // Special case when dim == tdim (cells)
  if (dim == tdim)
  {
    for (std::int32_t c : entities)
    {
      // Extract degrees of freedom
      auto x_c = md::submdspan(xdofs, c, md::full_extent);
      for (std::int32_t entity_dof : closure_dofs_all[tdim][0])
        entity_xdofs.push_back(x_c[entity_dof]);
    }
 
    return {std::move(entity_xdofs), eshape};
  }
 
  assert(dim != tdim);
 
  auto e_to_c = topology->connectivity(dim, tdim);
  if (!e_to_c)
  {
    throw std::runtime_error(
        "Entity-to-cell connectivity has not been computed. Missing dims "
        + std::to_string(dim) + "->" + std::to_string(tdim));
  }
 
  auto c_to_e = topology->connectivity(tdim, dim);
  if (!c_to_e)
  {
    throw std::runtime_error(
        "Cell-to-entity connectivity has not been computed. Missing dims "
        + std::to_string(tdim) + "->" + std::to_string(dim));
  }
 
  // Get the cell info, which is needed to permute the closure dofs
  std::span<const std::uint32_t> cell_info;
  if (permute)
    cell_info = std::span(mesh.topology()->get_cell_permutation_info());
 
  for (std::int32_t e : entities)
  {
    // Get a cell connected to the entity
    assert(!e_to_c->links(e).empty());
    std::int32_t c = e_to_c->links(e).front();
 
    // Get the local index of the entity
    std::span<const std::int32_t> cell_entities = c_to_e->links(c);
    auto it = std::find(cell_entities.begin(), cell_entities.end(), e);
    assert(it != cell_entities.end());
    std::size_t local_entity = std::distance(cell_entities.begin(), it);
 
    // Cell sub-entities must be permuted so that their local
    // orientation agrees with their global orientation
    std::vector<std::int32_t> closure_dofs(closure_dofs_all[dim][local_entity]);
    if (permute)
    {
      mesh::CellType entity_type
          = mesh::cell_entity_type(cell_type, dim, local_entity);
      coord_ele.permute_subentity_closure(closure_dofs, cell_info[c],
                                          entity_type, local_entity);
    }
 
    // Extract degrees of freedom
    auto x_c = md::submdspan(xdofs, c, md::full_extent);
    for (std::int32_t entity_dof : closure_dofs)
      entity_xdofs.push_back(x_c[entity_dof]);
  }
 
  return {std::move(entity_xdofs), eshape};
}

CellPartitionFunction create_cell_partitioner(
    mesh::GhostMode ghost_mode = mesh::GhostMode::none,
    const graph::partition_fn& partfn = &graph::partition_graph,
    std::optional<std::int32_t> max_facet_to_cell_links = 2);

std::vector<std::int32_t>
compute_incident_entities(const Topology& topology,
                          std::span<const std::int32_t> entities, int d0,
                          int d1);

template <typename U>
Mesh<typename std::remove_reference_t<typename U::value_type>> create_mesh(
    MPI_Comm comm, MPI_Comm commt,
    std::vector<std::span<const std::int64_t>> cells,
    const std::vector<fem::CoordinateElement<
        typename std::remove_reference_t<typename U::value_type>>>& elements,
    MPI_Comm commg, const U& x, std::array<std::size_t, 2> xshape,
    const CellPartitionFunction& partitioner,
    std::optional<std::int32_t> max_facet_to_cell_links,
    const CellReorderFunction& reorder_fn = graph::reorder_gps)
{
  assert(cells.size() == elements.size());
  std::vector<CellType> celltypes;
  std::ranges::transform(elements, std::back_inserter(celltypes),
                         [](auto& e) { return e.cell_shape(); });
  std::vector<fem::ElementDofLayout> doflayouts;
  std::ranges::transform(elements, std::back_inserter(doflayouts),
                         [](auto& e) { return e.create_dof_layout(); });
 
  // Note: `extract_topology` extracts topology data, i.e. just the
  // vertices. For P1 geometry this should just be the identity
  // operator. For other elements the filtered lists may have 'gaps',
  // i.e. the indices might not be contiguous.
  //
  // `extract_topology` could be skipped for 'P1 geometry' elements
 
  std::int32_t num_cell_types = cells.size();
 
  // -- Partition topology across ranks of comm
  std::vector<std::vector<std::int64_t>> cells1(num_cell_types);
  std::vector<std::vector<std::int64_t>> original_idx1(num_cell_types);
  std::vector<std::vector<int>> ghost_owners(num_cell_types);
  if (partitioner)
  {
    spdlog::info("Using partitioner with cell data ({} cell types)",
                 num_cell_types);
    graph::AdjacencyList<std::int32_t> dest(0);
    if (commt != MPI_COMM_NULL)
    {
      int size = dolfinx::MPI::size(comm);
      std::vector<std::vector<std::int64_t>> t(num_cell_types);
      std::vector<std::span<const std::int64_t>> tspan(num_cell_types);
      for (std::int32_t i = 0; i < num_cell_types; ++i)
      {
        t[i] = extract_topology(celltypes[i], doflayouts[i], cells[i]);
        tspan[i] = std::span(t[i]);
      }
      dest = partitioner(commt, size, celltypes, tspan);
    }
 
    std::int32_t cell_offset = 0;
    for (std::int32_t i = 0; i < num_cell_types; ++i)
    {
      std::size_t num_cell_nodes = doflayouts[i].num_dofs();
      assert(cells[i].size() % num_cell_nodes == 0);
      std::size_t num_cells = cells[i].size() / num_cell_nodes;
 
      // Extract destination AdjacencyList for this cell type
      std::vector<std::int32_t> offsets_i(
          std::next(dest.offsets().begin(), cell_offset),
          std::next(dest.offsets().begin(), cell_offset + num_cells + 1));
      std::vector<std::int32_t> data_i(
          std::next(dest.array().begin(), offsets_i.front()),
          std::next(dest.array().begin(), offsets_i.back()));
      std::int32_t offset_0 = offsets_i.front();
      std::ranges::for_each(offsets_i,
                            [&offset_0](std::int32_t& j) { j -= offset_0; });
      graph::AdjacencyList<std::int32_t> dest_i(data_i, offsets_i);
      cell_offset += num_cells;
 
      // Distribute cells (topology, includes higher-order 'nodes') to
      // destination rank
      std::vector<int> src_ranks;
      std::tie(cells1[i], src_ranks, original_idx1[i], ghost_owners[i])
          = graph::build::distribute(comm, cells[i],
                                     {num_cells, num_cell_nodes}, dest_i);
      spdlog::debug("Got {} cells from distribution", cells1[i].size());
    }
  }
  else
  {
    // No partitioning, construct a global index
    std::int64_t num_owned = 0;
    for (std::int32_t i = 0; i < num_cell_types; ++i)
    {
      cells1[i] = std::vector<std::int64_t>(cells[i].begin(), cells[i].end());
      std::int32_t num_cell_nodes = doflayouts[i].num_dofs();
      assert(cells1[i].size() % num_cell_nodes == 0);
      original_idx1[i].resize(cells1[i].size() / num_cell_nodes);
      num_owned += original_idx1[i].size();
    }
 
    // Add on global offset
    std::int64_t global_offset = 0;
    MPI_Exscan(&num_owned, &global_offset, 1, MPI_INT64_T, MPI_SUM, comm);
    for (std::int32_t i = 0; i < num_cell_types; ++i)
    {
      std::iota(original_idx1[i].begin(), original_idx1[i].end(),
                global_offset);
      global_offset += original_idx1[i].size();
    }
  }
 
  // Extract cell 'topology', i.e. extract the vertices for each cell
  // and discard any 'higher-order' nodes
  std::vector<std::vector<std::int64_t>> cells1_v(num_cell_types);
  for (std::int32_t i = 0; i < num_cell_types; ++i)
  {
    cells1_v[i] = extract_topology(celltypes[i], doflayouts[i], cells1[i]);
    spdlog::info("Extract basic topology: {}->{}", cells1[i].size(),
                 cells1_v[i].size());
  }
 
  auto boundary_v_fn
      = create_boundary_vertices_fn(reorder_fn, max_facet_to_cell_links);
  const std::vector<std::int64_t> boundary_v = boundary_v_fn(
      celltypes, doflayouts, ghost_owners, cells1, cells1_v, original_idx1);
 
  spdlog::debug("Got {} boundary vertices", boundary_v.size());
 
  // Create Topology
  std::vector<std::span<const std::int64_t>> cells1_v_span;
  std::ranges::transform(cells1_v, std::back_inserter(cells1_v_span),
                         [](auto& c) { return std::span(c); });
  std::vector<std::span<const std::int64_t>> original_idx1_span;
  std::ranges::transform(original_idx1, std::back_inserter(original_idx1_span),
                         [](auto& c) { return std::span(c); });
  std::vector<std::span<const int>> ghost_owners_span;
  std::ranges::transform(ghost_owners, std::back_inserter(ghost_owners_span),
                         [](auto& c) { return std::span(c); });
  Topology topology
      = create_topology(comm, celltypes, cells1_v_span, original_idx1_span,
                        ghost_owners_span, boundary_v);
 
  // Create connectivities required higher-order geometries for creating
  // a Geometry object
  for (int i = 0; i < num_cell_types; ++i)
  {
    for (int e = 1; e < topology.dim(); ++e)
    {
      if (doflayouts[i].num_entity_dofs(e) > 0)
        topology.create_entities(e);
    }
 
    if (elements[i].needs_dof_permutations())
      topology.create_entity_permutations();
  }
 
  // Build list of unique (global) node indices from cells1 and
  // distribute coordinate data
  std::vector<std::int64_t> nodes1, nodes2;
  for (std::vector<std::int64_t>& c : cells1)
    nodes1.insert(nodes1.end(), c.begin(), c.end());
  for (std::vector<std::int64_t>& c : cells1)
    nodes2.insert(nodes2.end(), c.begin(), c.end());
 
  dolfinx::radix_sort(nodes1);
  auto [unique_end, range_end] = std::ranges::unique(nodes1);
  nodes1.erase(unique_end, range_end);
 
  std::vector coords
      = dolfinx::MPI::distribute_data(comm, nodes1, commg, x, xshape[1]);
 
  // Create geometry object
  Geometry geometry
      = create_geometry(topology, elements, nodes1, nodes2, coords, xshape[1]);
 
  return Mesh(comm, std::make_shared<Topology>(std::move(topology)),
              std::move(geometry));
}

template <typename U>
Mesh<typename std::remove_reference_t<typename U::value_type>> create_mesh(
    MPI_Comm comm, MPI_Comm commt, std::span<const std::int64_t> cells,
    const fem::CoordinateElement<
        typename std::remove_reference_t<typename U::value_type>>& element,
    MPI_Comm commg, const U& x, std::array<std::size_t, 2> xshape,
    const CellPartitionFunction& partitioner,
    std::optional<std::int32_t> max_facet_to_cell_links = 2,
    const CellReorderFunction& reorder_fn = graph::reorder_gps)
{
  return create_mesh(comm, commt, std::vector{cells}, std::vector{element},
                     commg, x, xshape, partitioner, max_facet_to_cell_links,
                     reorder_fn);
}

template <typename U>
Mesh<typename std::remove_reference_t<typename U::value_type>>
create_mesh(MPI_Comm comm, std::span<const std::int64_t> cells,
            const fem::CoordinateElement<
                std::remove_reference_t<typename U::value_type>>& elements,
            const U& x, std::array<std::size_t, 2> xshape, GhostMode ghost_mode,
            std::optional<std::int32_t> max_facet_to_cell_links = 2)
{
  if (dolfinx::MPI::size(comm) == 1)
  {
    return create_mesh(comm, comm, std::vector{cells}, std::vector{elements},
                       comm, x, xshape, nullptr, max_facet_to_cell_links);
  }
  else
  {
    return create_mesh(comm, comm, std::vector{cells}, std::vector{elements},
                       comm, x, xshape, create_cell_partitioner(ghost_mode),
                       max_facet_to_cell_links);
  }
}

template <std::floating_point T>
std::pair<Geometry<T>, std::vector<int32_t>>
create_subgeometry(const Mesh<T>& mesh, int dim,
                   std::span<const std::int32_t> subentity_to_entity)
{
  const Geometry<T>& geometry = mesh.geometry();
 
  // Get the geometry dofs in the sub-geometry based on the entities in
  // sub-geometry
  const fem::ElementDofLayout layout = geometry.cmap().create_dof_layout();
 
  const std::vector<std::int32_t> x_indices
      = entities_to_geometry(mesh, dim, subentity_to_entity, true).first;
 
  std::vector<std::int32_t> sub_x_dofs = x_indices;
  std::ranges::sort(sub_x_dofs);
  auto [unique_end, range_end] = std::ranges::unique(sub_x_dofs);
  sub_x_dofs.erase(unique_end, range_end);
 
  // Get the sub-geometry dofs owned by this process
  auto x_index_map = geometry.index_map();
  assert(x_index_map);
 
  std::shared_ptr<common::IndexMap> sub_x_dof_index_map;
  std::vector<std::int32_t> subx_to_x_dofmap;
  {
    auto [map, new_to_old] = common::create_sub_index_map(
        *x_index_map, sub_x_dofs, common::IndexMapOrder::any, true);
    sub_x_dof_index_map = std::make_shared<common::IndexMap>(std::move(map));
    subx_to_x_dofmap = std::move(new_to_old);
  }
 
  // Create sub-geometry coordinates
  std::span<const T> x = geometry.x();
  std::int32_t sub_num_x_dofs = subx_to_x_dofmap.size();
  std::vector<T> sub_x(3 * sub_num_x_dofs);
  for (std::int32_t i = 0; i < sub_num_x_dofs; ++i)
  {
    std::copy_n(std::next(x.begin(), 3 * subx_to_x_dofmap[i]), 3,
                std::next(sub_x.begin(), 3 * i));
  }
 
  // Create geometry to sub-geometry  map
  std::vector<std::int32_t> x_to_subx_dof_map(
      x_index_map->size_local() + x_index_map->num_ghosts(), -1);
  for (std::size_t i = 0; i < subx_to_x_dofmap.size(); ++i)
    x_to_subx_dof_map[subx_to_x_dofmap[i]] = i;
 
  // Create sub-geometry dofmap
  std::vector<std::int32_t> sub_x_dofmap;
  sub_x_dofmap.reserve(x_indices.size());
  std::ranges::transform(x_indices, std::back_inserter(sub_x_dofmap),
                         [&x_to_subx_dof_map](auto x_dof)
                         {
                           assert(x_to_subx_dof_map[x_dof] != -1);
                           return x_to_subx_dof_map[x_dof];
                         });
 
  // Sub-geometry coordinate element
  CellType sub_xcell = cell_entity_type(geometry.cmap().cell_shape(), dim, 0);
 
  // Special handling of point meshes, as they only support constant
  // basis functions
  int degree = (sub_xcell == CellType::point) ? 0 : geometry.cmap().degree();
  fem::CoordinateElement<T> sub_cmap(sub_xcell, degree,
                                     geometry.cmap().variant());
 
  // Sub-geometry input_global_indices
  const std::vector<std::int64_t>& igi = geometry.input_global_indices();
  std::vector<std::int64_t> sub_igi;
  sub_igi.reserve(subx_to_x_dofmap.size());
  std::ranges::transform(subx_to_x_dofmap, std::back_inserter(sub_igi),
                         [&igi](auto sub_x_dof) { return igi[sub_x_dof]; });
 
  // Create geometry
  return {Geometry(
              sub_x_dof_index_map,
              std::vector<std::vector<std::int32_t>>{std::move(sub_x_dofmap)},
              {sub_cmap}, std::move(sub_x), geometry.dim(), std::move(sub_igi)),
          std::move(subx_to_x_dofmap)};
}

template <std::floating_point T>
std::tuple<Mesh<T>, EntityMap, EntityMap, std::vector<std::int32_t>>
create_submesh(const Mesh<T>& mesh, int dim,
               std::span<const std::int32_t> entities)
{
  // Create sub-topology
  mesh.topology_mutable()->create_connectivity(dim, 0);
  auto [topology, subentity_to_entity, subvertex_to_vertex]
      = mesh::create_subtopology(*mesh.topology(), dim, entities);
 
  // Create sub-geometry
  const int tdim = mesh.topology()->dim();
  mesh.topology_mutable()->create_entities(dim);
  mesh.topology_mutable()->create_connectivity(dim, tdim);
  mesh.topology_mutable()->create_connectivity(tdim, dim);
  mesh.topology_mutable()->create_entity_permutations();
  auto [geometry, subx_to_x_dofmap]
      = mesh::create_subgeometry(mesh, dim, subentity_to_entity);
 
  Mesh<T> submesh
      = Mesh(mesh.comm(), std::make_shared<Topology>(std::move(topology)),
             std::move(geometry));
  EntityMap entity_map(mesh.topology(), submesh.topology(), dim,
                       subentity_to_entity);
  EntityMap vertex_map(mesh.topology(), submesh.topology(), 0,
                       subvertex_to_vertex);
  return {std::move(submesh), std::move(entity_map), std::move(vertex_map),
          std::move(subx_to_x_dofmap)};
}
 
} // namespace dolfinx::mesh