utils.h

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// Copyright (C) 2013-2020 Johan Hake, Jan Blechta and Garth N. Wells
//
// This file is part of DOLFINx (https://www.fenicsproject.org)
//
// SPDX-License-Identifier:    LGPL-3.0-or-later
 
#pragma once
 
#include "Constant.h"
#include "CoordinateElement.h"
#include "DofMap.h"
#include "ElementDofLayout.h"
#include "Expression.h"
#include "Form.h"
#include "Function.h"
#include "FunctionSpace.h"
#include "sparsitybuild.h"
#include <algorithm>
#include <array>
#include <concepts>
#include <dolfinx/common/types.h>
#include <dolfinx/la/SparsityPattern.h>
#include <dolfinx/mesh/Topology.h>
#include <dolfinx/mesh/cell_types.h>
#include <dolfinx/mesh/utils.h>
#include <functional>
#include <memory>
#include <set>
#include <span>
#include <stdexcept>
#include <string>
#include <type_traits>
#include <ufcx.h>
#include <utility>
#include <vector>
 
namespace basix
{
template <std::floating_point T>
class FiniteElement;
}
 
namespace dolfinx::common
{
class IndexMap;
}
 
namespace dolfinx::fem
{
 
namespace impl
{
template <int num_cells>
std::array<std::int32_t, 2 * num_cells>
get_cell_facet_pairs(std::int32_t f, std::span<const std::int32_t> cells,
                     const graph::AdjacencyList<std::int32_t>& c_to_f)
{
  // Loop over cells sharing facet
  assert(cells.size() == num_cells);
  std::array<std::int32_t, 2 * num_cells> cell_local_facet_pairs;
  for (int c = 0; c < num_cells; ++c)
  {
    // Get local index of facet with respect to the cell
    std::int32_t cell = cells[c];
    auto cell_facets = c_to_f.links(cell);
    auto facet_it = std::find(cell_facets.begin(), cell_facets.end(), f);
    assert(facet_it != cell_facets.end());
    int local_f = std::distance(cell_facets.begin(), facet_it);
    cell_local_facet_pairs[2 * c] = cell;
    cell_local_facet_pairs[2 * c + 1] = local_f;
  }
 
  return cell_local_facet_pairs;
}
 
} // namespace impl
 
std::vector<std::int32_t>
compute_integration_domains(IntegralType integral_type,
                            const mesh::Topology& topology,
                            std::span<const std::int32_t> entities, int dim);
 
template <dolfinx::scalar T, std::floating_point U>
std::vector<std::vector<std::array<std::shared_ptr<const FunctionSpace<U>>, 2>>>
extract_function_spaces(const std::vector<std::vector<const Form<T, U>*>>& a)
{
  std::vector<
      std::vector<std::array<std::shared_ptr<const FunctionSpace<U>>, 2>>>
      spaces(
          a.size(),
          std::vector<std::array<std::shared_ptr<const FunctionSpace<U>>, 2>>(
              a.front().size()));
  for (std::size_t i = 0; i < a.size(); ++i)
  {
    for (std::size_t j = 0; j < a[i].size(); ++j)
    {
      if (const Form<T, U>* form = a[i][j]; form)
        spaces[i][j] = {form->function_spaces()[0], form->function_spaces()[1]};
    }
  }
  return spaces;
}
 
template <dolfinx::scalar T, std::floating_point U>
la::SparsityPattern create_sparsity_pattern(const Form<T, U>& a)
{
  if (a.rank() != 2)
  {
    throw std::runtime_error(
        "Cannot create sparsity pattern. Form is not a bilinear.");
  }
 
  // Get dof maps and mesh
  std::array<std::reference_wrapper<const DofMap>, 2> dofmaps{
      *a.function_spaces().at(0)->dofmap(),
      *a.function_spaces().at(1)->dofmap()};
  std::shared_ptr mesh = a.mesh();
  assert(mesh);
 
  std::shared_ptr mesh0 = a.function_spaces().at(0)->mesh();
  assert(mesh0);
  std::shared_ptr mesh1 = a.function_spaces().at(1)->mesh();
  assert(mesh1);
 
  const std::set<IntegralType> types = a.integral_types();
  if (types.find(IntegralType::interior_facet) != types.end()
      or types.find(IntegralType::exterior_facet) != types.end())
  {
    // FIXME: cleanup these calls? Some of the happen internally again.
    int tdim = mesh->topology()->dim();
    mesh->topology_mutable()->create_entities(tdim - 1);
    mesh->topology_mutable()->create_connectivity(tdim - 1, tdim);
  }
 
  common::Timer t0("Build sparsity");
 
  // Get common::IndexMaps for each dimension
  const std::array index_maps{dofmaps[0].get().index_map,
                              dofmaps[1].get().index_map};
  const std::array bs
      = {dofmaps[0].get().index_map_bs(), dofmaps[1].get().index_map_bs()};
 
  auto extract_cells = [](std::span<const std::int32_t> facets)
  {
    assert(facets.size() % 2 == 0);
    std::vector<std::int32_t> cells;
    cells.reserve(facets.size() / 2);
    for (std::size_t i = 0; i < facets.size(); i += 2)
      cells.push_back(facets[i]);
    return cells;
  };
 
  // Create and build sparsity pattern
  la::SparsityPattern pattern(mesh->comm(), index_maps, bs);
  for (auto type : types)
  {
    std::vector<int> ids = a.integral_ids(type);
    switch (type)
    {
    case IntegralType::cell:
      for (int id : ids)
      {
        sparsitybuild::cells(
            pattern, {a.domain(type, id, *mesh0), a.domain(type, id, *mesh1)},
            {{dofmaps[0], dofmaps[1]}});
      }
      break;
    case IntegralType::interior_facet:
      for (int id : ids)
      {
        sparsitybuild::interior_facets(
            pattern,
            {extract_cells(a.domain(type, id, *mesh0)),
             extract_cells(a.domain(type, id, *mesh1))},
            {{dofmaps[0], dofmaps[1]}});
      }
      break;
    case IntegralType::exterior_facet:
      for (int id : ids)
      {
        sparsitybuild::cells(pattern,
                             {extract_cells(a.domain(type, id, *mesh0)),
                              extract_cells(a.domain(type, id, *mesh1))},
                             {{dofmaps[0], dofmaps[1]}});
      }
      break;
    default:
      throw std::runtime_error("Unsupported integral type");
    }
  }
 
  t0.stop();
 
  return pattern;
}
 
template <std::floating_point T>
ElementDofLayout create_element_dof_layout(const fem::FiniteElement<T>& element,
                                           const std::vector<int>& parent_map
                                           = {})
{
  // Create subdofmaps and compute offset
  std::vector<int> offsets(1, 0);
  std::vector<dolfinx::fem::ElementDofLayout> sub_doflayout;
  int bs = element.block_size();
  for (int i = 0; i < element.num_sub_elements(); ++i)
  {
    // The ith sub-element. For mixed elements this is subelements()[i]. For
    // blocked elements, the sub-element will always be the same, so we'll use
    // sub_elements()[0]
    std::shared_ptr<const fem::FiniteElement<T>> sub_e
        = element.sub_elements()[bs > 1 ? 0 : i];
 
    // In a mixed element DOFs are ordered element by element, so the offset to
    // the next sub-element is sub_e->space_dimension(). Blocked elements use
    // xxyyzz ordering, so the offset to the next sub-element is 1
 
    std::vector<int> parent_map_sub(sub_e->space_dimension(), offsets.back());
    for (std::size_t j = 0; j < parent_map_sub.size(); ++j)
      parent_map_sub[j] += bs * j;
    offsets.push_back(offsets.back() + (bs > 1 ? 1 : sub_e->space_dimension()));
    sub_doflayout.push_back(
        dolfinx::fem::create_element_dof_layout(*sub_e, parent_map_sub));
  }
 
  return ElementDofLayout(bs, element.entity_dofs(),
                          element.entity_closure_dofs(), parent_map,
                          sub_doflayout);
}
 
DofMap create_dofmap(
    MPI_Comm comm, const ElementDofLayout& layout, mesh::Topology& topology,
    std::function<void(std::span<std::int32_t>, std::uint32_t)> permute_inv,
    std::function<std::vector<int>(const graph::AdjacencyList<std::int32_t>&)>
        reorder_fn);
 
std::vector<DofMap> create_dofmaps(
    MPI_Comm comm, const std::vector<ElementDofLayout>& layouts,
    mesh::Topology& topology,
    std::function<void(std::span<std::int32_t>, std::uint32_t)> permute_inv,
    std::function<std::vector<int>(const graph::AdjacencyList<std::int32_t>&)>
        reorder_fn);
 
std::vector<std::string> get_coefficient_names(const ufcx_form& ufcx_form);
 
std::vector<std::string> get_constant_names(const ufcx_form& ufcx_form);
 
template <dolfinx::scalar T, std::floating_point U = scalar_value_type_t<T>>
Form<T, U> create_form_factory(
    const ufcx_form& ufcx_form,
    const std::vector<std::shared_ptr<const FunctionSpace<U>>>& spaces,
    const std::vector<std::shared_ptr<const Function<T, U>>>& coefficients,
    const std::vector<std::shared_ptr<const Constant<T>>>& constants,
    const std::map<
        IntegralType,
        std::vector<std::pair<std::int32_t, std::span<const std::int32_t>>>>&
        subdomains,
    const std::map<std::shared_ptr<const mesh::Mesh<U>>,
                   std::span<const std::int32_t>>& entity_maps,
    std::shared_ptr<const mesh::Mesh<U>> mesh = nullptr)
{
  if (ufcx_form.rank != (int)spaces.size())
    throw std::runtime_error("Wrong number of argument spaces for Form.");
  if (ufcx_form.num_coefficients != (int)coefficients.size())
  {
    throw std::runtime_error(
        "Mismatch between number of expected and provided Form coefficients.");
  }
  if (ufcx_form.num_constants != (int)constants.size())
  {
    throw std::runtime_error(
        "Mismatch between number of expected and provided Form constants.");
  }
 
  // Check argument function spaces
  for (std::size_t i = 0; i < spaces.size(); ++i)
  {
    assert(spaces[i]->element());
    if (auto element_hash = ufcx_form.finite_element_hashes[i];
        element_hash != 0
        and element_hash != spaces[i]->element()->basix_element().hash())
    {
      throw std::runtime_error("Cannot create form. Elements are different to "
                               "those used to compile the form.");
    }
  }
 
  // Extract mesh from FunctionSpace, and check they are the same
  if (!mesh and !spaces.empty())
    mesh = spaces[0]->mesh();
  for (auto& V : spaces)
  {
    if (mesh != V->mesh() and entity_maps.find(V->mesh()) == entity_maps.end())
      throw std::runtime_error(
          "Incompatible mesh. entity_maps must be provided.");
  }
  if (!mesh)
    throw std::runtime_error("No mesh could be associated with the Form.");
 
  auto topology = mesh->topology();
  assert(topology);
  const int tdim = topology->dim();
 
  const int* integral_offsets = ufcx_form.form_integral_offsets;
  std::vector<int> num_integrals_type(3);
  for (int i = 0; i < 3; ++i)
    num_integrals_type[i] = integral_offsets[i + 1] - integral_offsets[i];
 
  // Create facets, if required
  if (num_integrals_type[exterior_facet] > 0
      or num_integrals_type[interior_facet] > 0)
  {
    mesh->topology_mutable()->create_entities(tdim - 1);
    mesh->topology_mutable()->create_connectivity(tdim - 1, tdim);
    mesh->topology_mutable()->create_connectivity(tdim, tdim - 1);
  }
 
  // Get list of integral IDs, and load tabulate tensor into memory for
  // each
  using kern_t = std::function<void(T*, const T*, const T*, const U*,
                                    const int*, const std::uint8_t*)>;
  std::map<IntegralType, std::vector<integral_data<T, U>>> integrals;
 
  // Attach cell kernels
  bool needs_facet_permutations = false;
  std::vector<std::int32_t> default_cells;
  {
    std::span<const int> ids(ufcx_form.form_integral_ids
                                 + integral_offsets[cell],
                             num_integrals_type[cell]);
    auto itg = integrals.insert({IntegralType::cell, {}});
    auto sd = subdomains.find(IntegralType::cell);
    for (int i = 0; i < num_integrals_type[cell]; ++i)
    {
      const int id = ids[i];
      ufcx_integral* integral
          = ufcx_form.form_integrals[integral_offsets[cell] + i];
      assert(integral);
 
      // Build list of active coefficients
      std::vector<int> active_coeffs;
      for (int j = 0; j < ufcx_form.num_coefficients; ++j)
      {
        if (integral->enabled_coefficients[j])
          active_coeffs.push_back(j);
      }
 
      kern_t k = nullptr;
      if constexpr (std::is_same_v<T, float>)
        k = integral->tabulate_tensor_float32;
#ifndef DOLFINX_NO_STDC_COMPLEX_KERNELS
      else if constexpr (std::is_same_v<T, std::complex<float>>)
      {
        k = reinterpret_cast<void (*)(
            T*, const T*, const T*,
            const typename scalar_value_type<T>::value_type*, const int*,
            const unsigned char*)>(integral->tabulate_tensor_complex64);
      }
#endif // DOLFINX_NO_STDC_COMPLEX_KERNELS
      else if constexpr (std::is_same_v<T, double>)
        k = integral->tabulate_tensor_float64;
#ifndef DOLFINX_NO_STDC_COMPLEX_KERNELS
      else if constexpr (std::is_same_v<T, std::complex<double>>)
      {
        k = reinterpret_cast<void (*)(
            T*, const T*, const T*,
            const typename scalar_value_type<T>::value_type*, const int*,
            const unsigned char*)>(integral->tabulate_tensor_complex128);
      }
#endif // DOLFINX_NO_STDC_COMPLEX_KERNELS
 
      if (!k)
      {
        throw std::runtime_error(
            "UFCx kernel function is NULL. Check requested types.");
      }
 
      // Build list of entities to assemble over
      if (id == -1)
      {
        // Default kernel, operates on all (owned) cells
        assert(topology->index_map(tdim));
        default_cells.resize(topology->index_map(tdim)->size_local(), 0);
        std::iota(default_cells.begin(), default_cells.end(), 0);
        itg.first->second.emplace_back(id, k, default_cells, active_coeffs);
      }
      else if (sd != subdomains.end())
      {
        // NOTE: This requires that pairs are sorted
        auto it
            = std::ranges::lower_bound(sd->second, id, std::less<>{},
                                       [](const auto& a) { return a.first; });
        if (it != sd->second.end() and it->first == id)
          itg.first->second.emplace_back(id, k, it->second, active_coeffs);
      }
 
      if (integral->needs_facet_permutations)
        needs_facet_permutations = true;
    }
  }
 
  // Attach exterior facet kernels
  std::vector<std::int32_t> default_facets_ext;
  {
    std::span<const int> ids(ufcx_form.form_integral_ids
                                 + integral_offsets[exterior_facet],
                             num_integrals_type[exterior_facet]);
    auto itg = integrals.insert({IntegralType::exterior_facet, {}});
    auto sd = subdomains.find(IntegralType::exterior_facet);
    for (int i = 0; i < num_integrals_type[exterior_facet]; ++i)
    {
      const int id = ids[i];
      ufcx_integral* integral
          = ufcx_form.form_integrals[integral_offsets[exterior_facet] + i];
      assert(integral);
      std::vector<int> active_coeffs;
      for (int j = 0; j < ufcx_form.num_coefficients; ++j)
      {
        if (integral->enabled_coefficients[j])
          active_coeffs.push_back(j);
      }
 
      kern_t k = nullptr;
      if constexpr (std::is_same_v<T, float>)
        k = integral->tabulate_tensor_float32;
#ifndef DOLFINX_NO_STDC_COMPLEX_KERNELS
      else if constexpr (std::is_same_v<T, std::complex<float>>)
      {
        k = reinterpret_cast<void (*)(
            T*, const T*, const T*,
            const typename scalar_value_type<T>::value_type*, const int*,
            const unsigned char*)>(integral->tabulate_tensor_complex64);
      }
#endif // DOLFINX_NO_STDC_COMPLEX_KERNELS
      else if constexpr (std::is_same_v<T, double>)
        k = integral->tabulate_tensor_float64;
#ifndef DOLFINX_NO_STDC_COMPLEX_KERNELS
      else if constexpr (std::is_same_v<T, std::complex<double>>)
      {
        k = reinterpret_cast<void (*)(
            T*, const T*, const T*,
            const typename scalar_value_type<T>::value_type*, const int*,
            const unsigned char*)>(integral->tabulate_tensor_complex128);
      }
#endif // DOLFINX_NO_STDC_COMPLEX_KERNELS
      assert(k);
 
      // Build list of entities to assembler over
      const std::vector bfacets = mesh::exterior_facet_indices(*topology);
      auto f_to_c = topology->connectivity(tdim - 1, tdim);
      assert(f_to_c);
      auto c_to_f = topology->connectivity(tdim, tdim - 1);
      assert(c_to_f);
      if (id == -1)
      {
        // Default kernel, operates on all (owned) exterior facets
        default_facets_ext.reserve(2 * bfacets.size());
        for (std::int32_t f : bfacets)
        {
          // There will only be one pair for an exterior facet integral
          auto pair
              = impl::get_cell_facet_pairs<1>(f, f_to_c->links(f), *c_to_f);
          default_facets_ext.insert(default_facets_ext.end(), pair.begin(),
                                    pair.end());
        }
        itg.first->second.emplace_back(id, k, default_facets_ext,
                                       active_coeffs);
      }
      else if (sd != subdomains.end())
      {
        // NOTE: This requires that pairs are sorted
        auto it
            = std::ranges::lower_bound(sd->second, id, std::less<>{},
                                       [](const auto& a) { return a.first; });
        if (it != sd->second.end() and it->first == id)
          itg.first->second.emplace_back(id, k, it->second, active_coeffs);
      }
 
      if (integral->needs_facet_permutations)
        needs_facet_permutations = true;
    }
  }
 
  // Attach interior facet kernels
  std::vector<std::int32_t> default_facets_int;
  {
    std::span<const int> ids(ufcx_form.form_integral_ids
                                 + integral_offsets[interior_facet],
                             num_integrals_type[interior_facet]);
    auto itg = integrals.insert({IntegralType::interior_facet, {}});
    auto sd = subdomains.find(IntegralType::interior_facet);
 
    // Create indicator for interprocess facets
    std::vector<std::int8_t> interprocess_marker;
    if (num_integrals_type[interior_facet] > 0)
    {
      assert(topology->index_map(tdim - 1));
      const std::vector<std::int32_t>& interprocess_facets
          = topology->interprocess_facets();
      std::int32_t num_facets = topology->index_map(tdim - 1)->size_local()
                                + topology->index_map(tdim - 1)->num_ghosts();
      interprocess_marker.resize(num_facets, 0);
      std::ranges::for_each(interprocess_facets, [&interprocess_marker](auto f)
                            { interprocess_marker[f] = 1; });
    }
 
    for (int i = 0; i < num_integrals_type[interior_facet]; ++i)
    {
      const int id = ids[i];
      ufcx_integral* integral
          = ufcx_form.form_integrals[integral_offsets[interior_facet] + i];
      assert(integral);
      std::vector<int> active_coeffs;
      for (int j = 0; j < ufcx_form.num_coefficients; ++j)
      {
        if (integral->enabled_coefficients[j])
          active_coeffs.push_back(j);
      }
 
      kern_t k = nullptr;
      if constexpr (std::is_same_v<T, float>)
        k = integral->tabulate_tensor_float32;
#ifndef DOLFINX_NO_STDC_COMPLEX_KERNELS
      else if constexpr (std::is_same_v<T, std::complex<float>>)
      {
        k = reinterpret_cast<void (*)(
            T*, const T*, const T*,
            const typename scalar_value_type<T>::value_type*, const int*,
            const unsigned char*)>(integral->tabulate_tensor_complex64);
      }
#endif // DOLFINX_NO_STDC_COMPLEX_KERNELS
      else if constexpr (std::is_same_v<T, double>)
        k = integral->tabulate_tensor_float64;
#ifndef DOLFINX_NO_STDC_COMPLEX_KERNELS
      else if constexpr (std::is_same_v<T, std::complex<double>>)
      {
        k = reinterpret_cast<void (*)(
            T*, const T*, const T*,
            const typename scalar_value_type<T>::value_type*, const int*,
            const unsigned char*)>(integral->tabulate_tensor_complex128);
      }
#endif // DOLFINX_NO_STDC_COMPLEX_KERNELS
      assert(k);
 
      // Build list of entities to assembler over
      auto f_to_c = topology->connectivity(tdim - 1, tdim);
      assert(f_to_c);
      auto c_to_f = topology->connectivity(tdim, tdim - 1);
      assert(c_to_f);
      if (id == -1)
      {
        // Default kernel, operates on all (owned) interior facets
        assert(topology->index_map(tdim - 1));
        std::int32_t num_facets = topology->index_map(tdim - 1)->size_local();
        default_facets_int.reserve(4 * num_facets);
        for (std::int32_t f = 0; f < num_facets; ++f)
        {
          if (f_to_c->num_links(f) == 2)
          {
            auto pairs
                = impl::get_cell_facet_pairs<2>(f, f_to_c->links(f), *c_to_f);
            default_facets_int.insert(default_facets_int.end(), pairs.begin(),
                                      pairs.end());
          }
          else if (interprocess_marker[f])
          {
            throw std::runtime_error(
                "Cannot compute interior facet integral over interprocess "
                "facet. Please use ghost mode shared facet when creating the "
                "mesh");
          }
        }
        itg.first->second.emplace_back(id, k, default_facets_int,
                                       active_coeffs);
      }
      else if (sd != subdomains.end())
      {
        auto it
            = std::ranges::lower_bound(sd->second, id, std::less<>{},
                                       [](const auto& a) { return a.first; });
        if (it != sd->second.end() and it->first == id)
          itg.first->second.emplace_back(id, k, it->second, active_coeffs);
      }
 
      if (integral->needs_facet_permutations)
        needs_facet_permutations = true;
    }
  }
 
  std::map<IntegralType,
           std::vector<std::pair<std::int32_t, std::vector<std::int32_t>>>>
      sd;
  for (auto& [itg, data] : subdomains)
  {
    std::vector<std::pair<std::int32_t, std::vector<std::int32_t>>> x;
    for (auto& [id, idx] : data)
      x.emplace_back(id, std::vector(idx.data(), idx.data() + idx.size()));
    sd.insert({itg, std::move(x)});
  }
 
  return Form<T, U>(spaces, integrals, coefficients, constants,
                    needs_facet_permutations, entity_maps, mesh);
}
 
template <dolfinx::scalar T, std::floating_point U = scalar_value_type_t<T>>
Form<T, U> create_form(
    const ufcx_form& ufcx_form,
    const std::vector<std::shared_ptr<const FunctionSpace<U>>>& spaces,
    const std::map<std::string, std::shared_ptr<const Function<T, U>>>&
        coefficients,
    const std::map<std::string, std::shared_ptr<const Constant<T>>>& constants,
    const std::map<
        IntegralType,
        std::vector<std::pair<std::int32_t, std::span<const std::int32_t>>>>&
        subdomains,
    const std::map<std::shared_ptr<const mesh::Mesh<U>>,
                   std::span<const std::int32_t>>& entity_maps,
    std::shared_ptr<const mesh::Mesh<U>> mesh = nullptr)
{
  // Place coefficients in appropriate order
  std::vector<std::shared_ptr<const Function<T, U>>> coeff_map;
  for (const std::string& name : get_coefficient_names(ufcx_form))
  {
    if (auto it = coefficients.find(name); it != coefficients.end())
      coeff_map.push_back(it->second);
    else
    {
      throw std::runtime_error("Form coefficient \"" + name
                               + "\" not provided.");
    }
  }
 
  // Place constants in appropriate order
 
  std::vector<std::shared_ptr<const Constant<T>>> const_map;
  for (const std::string& name : get_constant_names(ufcx_form))
  {
    if (auto it = constants.find(name); it != constants.end())
      const_map.push_back(it->second);
    else
      throw std::runtime_error("Form constant \"" + name + "\" not provided.");
  }
 
  return create_form_factory(ufcx_form, spaces, coeff_map, const_map,
                             subdomains, entity_maps, mesh);
}
 
template <dolfinx::scalar T, std::floating_point U = scalar_value_type_t<T>>
Form<T, U> create_form(
    ufcx_form* (*fptr)(),
    const std::vector<std::shared_ptr<const FunctionSpace<U>>>& spaces,
    const std::map<std::string, std::shared_ptr<const Function<T, U>>>&
        coefficients,
    const std::map<std::string, std::shared_ptr<const Constant<T>>>& constants,
    const std::map<
        IntegralType,
        std::vector<std::pair<std::int32_t, std::span<const std::int32_t>>>>&
        subdomains,
    const std::map<std::shared_ptr<const mesh::Mesh<U>>,
                   std::span<const std::int32_t>>& entity_maps,
    std::shared_ptr<const mesh::Mesh<U>> mesh = nullptr)
{
  ufcx_form* form = fptr();
  Form<T, U> L = create_form<T, U>(*form, spaces, coefficients, constants,
                                   subdomains, entity_maps, mesh);
  std::free(form);
  return L;
}
 
template <std::floating_point T>
FunctionSpace<T> create_functionspace(
    std::shared_ptr<mesh::Mesh<T>> mesh, const basix::FiniteElement<T>& e,
    const std::vector<std::size_t>& value_shape = {},
    std::function<std::vector<int>(const graph::AdjacencyList<std::int32_t>&)>
        reorder_fn
    = nullptr)
{
  if (!e.value_shape().empty() and !value_shape.empty())
  {
    throw std::runtime_error(
        "Cannot specify value shape for non-scalar base element.");
  }
 
  if (mesh::cell_type_from_basix_type(e.cell_type())
      != mesh->topology()->cell_type())
    throw std::runtime_error("Cell type of element and mesh must match.");
 
  std::size_t bs = value_shape.empty()
                       ? 1
                       : std::accumulate(value_shape.begin(), value_shape.end(),
                                         1, std::multiplies{});
 
  // Create a DOLFINx element
  auto _e = std::make_shared<const FiniteElement<T>>(e, bs);
  assert(_e);
 
  const std::vector<std::size_t> _value_shape
      = (value_shape.empty() and !e.value_shape().empty())
            ? fem::compute_value_shape(_e, mesh->topology()->dim(),
                                       mesh->geometry().dim())
            : value_shape;
 
  // Create UFC subdofmaps and compute offset
  const int num_sub_elements = _e->num_sub_elements();
  std::vector<ElementDofLayout> sub_doflayout;
  sub_doflayout.reserve(num_sub_elements);
  for (int i = 0; i < num_sub_elements; ++i)
  {
    auto sub_element = _e->extract_sub_element({i});
    std::vector<int> parent_map_sub(sub_element->space_dimension());
    for (std::size_t j = 0; j < parent_map_sub.size(); ++j)
      parent_map_sub[j] = i + _e->block_size() * j;
    sub_doflayout.emplace_back(1, e.entity_dofs(), e.entity_closure_dofs(),
                               parent_map_sub, std::vector<ElementDofLayout>());
  }
 
  // Create a dofmap
  ElementDofLayout layout(_e->block_size(), e.entity_dofs(),
                          e.entity_closure_dofs(), {}, sub_doflayout);
  std::function<void(std::span<std::int32_t>, std::uint32_t)> permute_inv
      = nullptr;
  if (_e->needs_dof_permutations())
    permute_inv = _e->dof_permutation_fn(true, true);
  assert(mesh);
  assert(mesh->topology());
  auto dofmap = std::make_shared<const DofMap>(create_dofmap(
      mesh->comm(), layout, *mesh->topology(), permute_inv, reorder_fn));
  return FunctionSpace(mesh, _e, dofmap, _value_shape);
}
 
namespace impl
{
template <dolfinx::scalar T, std::floating_point U>
std::span<const std::uint32_t>
get_cell_orientation_info(const Function<T, U>& coefficient)
{
  std::span<const std::uint32_t> cell_info;
  auto element = coefficient.function_space()->element();
  if (element->needs_dof_transformations())
  {
    auto mesh = coefficient.function_space()->mesh();
    mesh->topology_mutable()->create_entity_permutations();
    cell_info = std::span(mesh->topology()->get_cell_permutation_info());
  }
 
  return cell_info;
}
 
template <int _bs, dolfinx::scalar T>
void pack(std::span<T> coeffs, std::int32_t cell, int bs, std::span<const T> v,
          std::span<const std::uint32_t> cell_info, const DofMap& dofmap,
          auto transform)
{
  auto dofs = dofmap.cell_dofs(cell);
  for (std::size_t i = 0; i < dofs.size(); ++i)
  {
    if constexpr (_bs < 0)
    {
      const int pos_c = bs * i;
      const int pos_v = bs * dofs[i];
      for (int k = 0; k < bs; ++k)
        coeffs[pos_c + k] = v[pos_v + k];
    }
    else
    {
      assert(_bs == bs);
      const int pos_c = _bs * i;
      const int pos_v = _bs * dofs[i];
      for (int k = 0; k < _bs; ++k)
        coeffs[pos_c + k] = v[pos_v + k];
    }
  }
 
  transform(coeffs, cell_info, cell, 1);
}
 
template <typename F>
concept FetchCells = requires(F&& f, std::span<const std::int32_t> v) {
  requires std::invocable<F, std::span<const std::int32_t>>;
  { f(v) } -> std::convertible_to<std::int32_t>;
};
 
template <dolfinx::scalar T, std::floating_point U>
void pack_coefficient_entity(std::span<T> c, int cstride,
                             const Function<T, U>& u,
                             std::span<const std::uint32_t> cell_info,
                             std::span<const std::int32_t> entities,
                             std::size_t estride, FetchCells auto&& fetch_cells,
                             std::int32_t offset)
{
  // Read data from coefficient Function u
  std::span<const T> v = u.x()->array();
  const DofMap& dofmap = *u.function_space()->dofmap();
  auto element = u.function_space()->element();
  assert(element);
  int space_dim = element->space_dimension();
 
  // Transformation from conforming degrees-of-freedom to reference
  // degrees-of-freedom
  auto transformation
      = element->template dof_transformation_fn<T>(doftransform::transpose);
  const int bs = dofmap.bs();
  switch (bs)
  {
  case 1:
    for (std::size_t e = 0; e < entities.size(); e += estride)
    {
      auto entity = entities.subspan(e, estride);
      if (std::int32_t cell = fetch_cells(entity); cell >= 0)
      {
        auto cell_coeff
            = c.subspan((e / estride) * cstride + offset, space_dim);
        pack<1>(cell_coeff, cell, bs, v, cell_info, dofmap, transformation);
      }
    }
    break;
  case 2:
    for (std::size_t e = 0; e < entities.size(); e += estride)
    {
      auto entity = entities.subspan(e, estride);
      if (std::int32_t cell = fetch_cells(entity); cell >= 0)
      {
        auto cell_coeff
            = c.subspan((e / estride) * cstride + offset, space_dim);
        pack<2>(cell_coeff, cell, bs, v, cell_info, dofmap, transformation);
      }
    }
    break;
  case 3:
    for (std::size_t e = 0; e < entities.size(); e += estride)
    {
      auto entity = entities.subspan(e, estride);
      if (std::int32_t cell = fetch_cells(entity); cell >= 0)
      {
        auto cell_coeff = c.subspan(e / estride * cstride + offset, space_dim);
        pack<3>(cell_coeff, cell, bs, v, cell_info, dofmap, transformation);
      }
    }
    break;
  default:
    for (std::size_t e = 0; e < entities.size(); e += estride)
    {
      auto entity = entities.subspan(e, estride);
      if (std::int32_t cell = fetch_cells(entity); cell >= 0)
      {
        auto cell_coeff
            = c.subspan((e / estride) * cstride + offset, space_dim);
        pack<-1>(cell_coeff, cell, bs, v, cell_info, dofmap, transformation);
      }
    }
    break;
  }
}
 
} // namespace impl
 
template <dolfinx::scalar T, std::floating_point U>
std::pair<std::vector<T>, int>
allocate_coefficient_storage(const Form<T, U>& form, IntegralType integral_type,
                             int id)
{
  // Get form coefficient offsets and dofmaps
  const std::vector<std::shared_ptr<const Function<T, U>>>& coefficients
      = form.coefficients();
  const std::vector<int> offsets = form.coefficient_offsets();
 
  std::size_t num_entities = 0;
  int cstride = 0;
  if (!coefficients.empty())
  {
    cstride = offsets.back();
    num_entities = form.domain(integral_type, id).size();
    if (integral_type == IntegralType::exterior_facet
        or integral_type == IntegralType::interior_facet)
    {
      num_entities /= 2;
    }
  }
 
  return {std::vector<T>(num_entities * cstride), cstride};
}
 
template <dolfinx::scalar T, std::floating_point U>
std::map<std::pair<IntegralType, int>, std::pair<std::vector<T>, int>>
allocate_coefficient_storage(const Form<T, U>& form)
{
  std::map<std::pair<IntegralType, int>, std::pair<std::vector<T>, int>> coeffs;
  for (auto integral_type : form.integral_types())
  {
    for (int id : form.integral_ids(integral_type))
    {
      coeffs.emplace_hint(
          coeffs.end(), std::pair(integral_type, id),
          allocate_coefficient_storage(form, integral_type, id));
    }
  }
 
  return coeffs;
}
 
template <dolfinx::scalar T, std::floating_point U>
void pack_coefficients(const Form<T, U>& form, IntegralType integral_type,
                       int id, std::span<T> c, int cstride)
{
  // Get form coefficient offsets and dofmaps
  const std::vector<std::shared_ptr<const Function<T, U>>>& coefficients
      = form.coefficients();
  const std::vector<int> offsets = form.coefficient_offsets();
 
  // Indicator for packing coefficients
  std::vector<int> active_coefficient(coefficients.size(), 0);
  if (!coefficients.empty())
  {
    switch (integral_type)
    {
    case IntegralType::cell:
    {
      // Get indicator for all coefficients that are active in cell
      // integrals
      for (std::size_t i = 0; i < form.num_integrals(IntegralType::cell); ++i)
      {
        for (auto idx : form.active_coeffs(IntegralType::cell, i))
          active_coefficient[idx] = 1;
      }
 
      // Iterate over coefficients
      for (std::size_t coeff = 0; coeff < coefficients.size(); ++coeff)
      {
        if (!active_coefficient[coeff])
          continue;
 
        // Get coefficient mesh
        auto mesh = coefficients[coeff]->function_space()->mesh();
        assert(mesh);
 
        // Other integrals in the form might have coefficients defined over
        // entities of codim > 0, which don't make sense for cell integrals, so
        // don't pack them.
        if (const int codim
            = form.mesh()->topology()->dim() - mesh->topology()->dim();
            codim > 0)
        {
          throw std::runtime_error("Should not be packing coefficients with "
                                   "codim>0 in a cell integral");
        }
 
        std::vector<std::int32_t> cells
            = form.domain(IntegralType::cell, id, *mesh);
        std::span<const std::uint32_t> cell_info
            = impl::get_cell_orientation_info(*coefficients[coeff]);
        impl::pack_coefficient_entity(
            c, cstride, *coefficients[coeff], cell_info, cells, 1,
            [](auto entity) { return entity.front(); }, offsets[coeff]);
      }
      break;
    }
    case IntegralType::exterior_facet:
    {
      // Get indicator for all coefficients that are active in exterior
      // facet integrals
      for (std::size_t i = 0;
           i < form.num_integrals(IntegralType::exterior_facet); ++i)
      {
        for (auto idx : form.active_coeffs(IntegralType::exterior_facet, i))
          active_coefficient[idx] = 1;
      }
 
      // Iterate over coefficients
      for (std::size_t coeff = 0; coeff < coefficients.size(); ++coeff)
      {
        if (!active_coefficient[coeff])
          continue;
 
        auto mesh = coefficients[coeff]->function_space()->mesh();
        std::vector<std::int32_t> facets
            = form.domain(IntegralType::exterior_facet, id, *mesh);
        std::span<const std::uint32_t> cell_info
            = impl::get_cell_orientation_info(*coefficients[coeff]);
        impl::pack_coefficient_entity(
            c, cstride, *coefficients[coeff], cell_info, facets, 2,
            [](auto entity) { return entity.front(); }, offsets[coeff]);
      }
      break;
    }
    case IntegralType::interior_facet:
    {
      // Get indicator for all coefficients that are active in interior
      // facet integrals
      for (std::size_t i = 0;
           i < form.num_integrals(IntegralType::interior_facet); ++i)
      {
        for (auto idx : form.active_coeffs(IntegralType::interior_facet, i))
          active_coefficient[idx] = 1;
      }
 
      // Iterate over coefficients
      for (std::size_t coeff = 0; coeff < coefficients.size(); ++coeff)
      {
        if (!active_coefficient[coeff])
          continue;
 
        auto mesh = coefficients[coeff]->function_space()->mesh();
        std::vector<std::int32_t> facets
            = form.domain(IntegralType::interior_facet, id, *mesh);
 
        std::span<const std::uint32_t> cell_info
            = impl::get_cell_orientation_info(*coefficients[coeff]);
 
        // Pack coefficient ['+']
        impl::pack_coefficient_entity(
            c, 2 * cstride, *coefficients[coeff], cell_info, facets, 4,
            [](auto entity) { return entity[0]; }, 2 * offsets[coeff]);
 
        // Pack coefficient ['-']
        impl::pack_coefficient_entity(
            c, 2 * cstride, *coefficients[coeff], cell_info, facets, 4,
            [](auto entity) { return entity[2]; },
            offsets[coeff] + offsets[coeff + 1]);
      }
      break;
    }
    default:
      throw std::runtime_error(
          "Could not pack coefficient. Integral type not supported.");
    }
  }
}
 
template <dolfinx::scalar T, std::floating_point U = scalar_value_type_t<T>>
Expression<T, U> create_expression(
    const ufcx_expression& e,
    const std::vector<std::shared_ptr<const Function<T, U>>>& coefficients,
    const std::vector<std::shared_ptr<const Constant<T>>>& constants,
    std::shared_ptr<const FunctionSpace<U>> argument_function_space = nullptr)
{
  if (e.rank > 0 and !argument_function_space)
  {
    throw std::runtime_error("Expression has Argument but no Argument "
                             "function space was provided.");
  }
 
  std::vector<U> X(e.points, e.points + e.num_points * e.entity_dimension);
  std::array<std::size_t, 2> Xshape
      = {static_cast<std::size_t>(e.num_points),
         static_cast<std::size_t>(e.entity_dimension)};
  std::vector<int> value_shape(e.value_shape, e.value_shape + e.num_components);
  std::function<void(T*, const T*, const T*,
                     const typename scalar_value_type<T>::value_type*,
                     const int*, const std::uint8_t*)>
      tabulate_tensor = nullptr;
  if constexpr (std::is_same_v<T, float>)
    tabulate_tensor = e.tabulate_tensor_float32;
#ifndef DOLFINX_NO_STDC_COMPLEX_KERNELS
  else if constexpr (std::is_same_v<T, std::complex<float>>)
  {
    tabulate_tensor = reinterpret_cast<void (*)(
        T*, const T*, const T*,
        const typename scalar_value_type<T>::value_type*, const int*,
        const unsigned char*)>(e.tabulate_tensor_complex64);
  }
#endif // DOLFINX_NO_STDC_COMPLEX_KERNELS
  else if constexpr (std::is_same_v<T, double>)
    tabulate_tensor = e.tabulate_tensor_float64;
#ifndef DOLFINX_NO_STDC_COMPLEX_KERNELS
  else if constexpr (std::is_same_v<T, std::complex<double>>)
  {
    tabulate_tensor = reinterpret_cast<void (*)(
        T*, const T*, const T*,
        const typename scalar_value_type<T>::value_type*, const int*,
        const unsigned char*)>(e.tabulate_tensor_complex128);
  }
#endif // DOLFINX_NO_STDC_COMPLEX_KERNELS
  else
    throw std::runtime_error("Type not supported.");
 
  assert(tabulate_tensor);
  return Expression(coefficients, constants, std::span<const U>(X), Xshape,
                    tabulate_tensor, value_shape, argument_function_space);
}
 
template <dolfinx::scalar T, std::floating_point U = scalar_value_type_t<T>>
Expression<T, U> create_expression(
    const ufcx_expression& e,
    const std::map<std::string, std::shared_ptr<const Function<T, U>>>&
        coefficients,
    const std::map<std::string, std::shared_ptr<const Constant<T>>>& constants,
    std::shared_ptr<const FunctionSpace<U>> argument_function_space = nullptr)
{
  // Place coefficients in appropriate order
  std::vector<std::shared_ptr<const Function<T, U>>> coeff_map;
  std::vector<std::string> coefficient_names;
  for (int i = 0; i < e.num_coefficients; ++i)
    coefficient_names.push_back(e.coefficient_names[i]);
 
  for (const std::string& name : coefficient_names)
  {
    if (auto it = coefficients.find(name); it != coefficients.end())
      coeff_map.push_back(it->second);
    else
    {
      throw std::runtime_error("Expression coefficient \"" + name
                               + "\" not provided.");
    }
  }
 
  // Place constants in appropriate order
  std::vector<std::shared_ptr<const Constant<T>>> const_map;
  std::vector<std::string> constant_names;
  for (int i = 0; i < e.num_constants; ++i)
    constant_names.push_back(e.constant_names[i]);
 
  for (const std::string& name : constant_names)
  {
    if (auto it = constants.find(name); it != constants.end())
      const_map.push_back(it->second);
    else
    {
      throw std::runtime_error("Expression constant \"" + name
                               + "\" not provided.");
    }
  }
 
  return create_expression(e, coeff_map, const_map, argument_function_space);
}
 
template <dolfinx::scalar T, std::floating_point U>
void pack_coefficients(const Form<T, U>& form,
                       std::map<std::pair<IntegralType, int>,
                                std::pair<std::vector<T>, int>>& coeffs)
{
  for (auto& [key, val] : coeffs)
    pack_coefficients<T>(form, key.first, key.second, val.first, val.second);
}
 
template <dolfinx::scalar T, std::floating_point U>
std::pair<std::vector<T>, int>
pack_coefficients(const Expression<T, U>& e,
                  std::span<const std::int32_t> entities, std::size_t estride)
{
  // Get form coefficient offsets and dofmaps
  const std::vector<std::shared_ptr<const Function<T, U>>>& coeffs
      = e.coefficients();
  const std::vector<int> offsets = e.coefficient_offsets();
 
  // Copy data into coefficient array
  const int cstride = offsets.back();
  std::vector<T> c(entities.size() / estride * offsets.back());
  if (!coeffs.empty())
  {
    // Iterate over coefficients
    for (std::size_t coeff = 0; coeff < coeffs.size(); ++coeff)
    {
      std::span<const std::uint32_t> cell_info
          = impl::get_cell_orientation_info(*coeffs[coeff]);
 
      impl::pack_coefficient_entity(
          std::span(c), cstride, *coeffs[coeff], cell_info, entities, estride,
          [](auto entity) { return entity[0]; }, offsets[coeff]);
    }
  }
  return {std::move(c), cstride};
}
 
template <typename U>
std::vector<typename U::scalar_type> pack_constants(const U& u)
{
  using T = typename U::scalar_type;
  const std::vector<std::shared_ptr<const Constant<T>>>& constants
      = u.constants();
 
  // Calculate size of array needed to store packed constants
  std::int32_t size = std::accumulate(constants.cbegin(), constants.cend(), 0,
                                      [](std::int32_t sum, auto& constant)
                                      { return sum + constant->value.size(); });
 
  // Pack constants
  std::vector<T> constant_values(size);
  std::int32_t offset = 0;
  for (auto& constant : constants)
  {
    const std::vector<T>& value = constant->value;
    std::ranges::copy(value, std::next(constant_values.begin(), offset));
    offset += value.size();
  }
 
  return constant_values;
}
 
} // namespace dolfinx::fem