assemble_vector_impl.h

// Copyright (C) 2018-2021 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 "DirichletBC.h"
#include "DofMap.h"
#include "Form.h"
#include "FunctionSpace.h"
#include "utils.h"
#include <dolfinx/common/IndexMap.h>
#include <dolfinx/common/utils.h>
#include <dolfinx/graph/AdjacencyList.h>
#include <dolfinx/mesh/Geometry.h>
#include <dolfinx/mesh/Mesh.h>
#include <dolfinx/mesh/Topology.h>
#include <functional>
#include <memory>
#include <vector>
#include <xtensor/xbuilder.hpp>
#include <xtensor/xtensor.hpp>
 
namespace dolfinx::fem::impl
{
 
 
template <typename T, int _bs0 = -1, int _bs1 = -1>
void _lift_bc_cells(
    xtl::span<T> b, const mesh::Geometry& geometry,
    const std::function<void(T*, const T*, const T*, const double*, const int*,
                             const std::uint8_t*)>& kernel,
    const xtl::span<const std::int32_t>& cells,
    const std::function<void(const xtl::span<T>&,
                             const xtl::span<const std::uint32_t>&,
                             std::int32_t, int)>& dof_transform,
    const graph::AdjacencyList<std::int32_t>& dofmap0, int bs0,
    const std::function<void(const xtl::span<T>&,
                             const xtl::span<const std::uint32_t>&,
                             std::int32_t, int)>& dof_transform_to_transpose,
    const graph::AdjacencyList<std::int32_t>& dofmap1, int bs1,
    const xtl::span<const T>& constants, const xtl::span<const T>& coeffs,
    int cstride, const xtl::span<const std::uint32_t>& cell_info,
    const xtl::span<const T>& bc_values1,
    const xtl::span<const std::int8_t>& bc_markers1,
    const xtl::span<const T>& x0, double scale)
{
  assert(_bs0 < 0 or _bs0 == bs0);
  assert(_bs1 < 0 or _bs1 == bs1);
 
  if (cells.empty())
    return;
 
  // Prepare cell geometry
  const graph::AdjacencyList<std::int32_t>& x_dofmap = geometry.dofmap();
  const std::size_t num_dofs_g = geometry.cmap().dim();
  xtl::span<const double> x_g = geometry.x();
 
  // Data structures used in bc application
  std::vector<double> coordinate_dofs(3 * num_dofs_g);
  std::vector<T> Ae, be;
  for (std::size_t index = 0; index < cells.size(); ++index)
  {
    std::int32_t c = cells[index];
 
    // Get dof maps for cell
    auto dmap1 = dofmap1.links(c);
 
    // Check if bc is applied to cell
    bool has_bc = false;
    for (std::size_t j = 0; j < dmap1.size(); ++j)
    {
      if constexpr (_bs1 > 0)
      {
        for (int k = 0; k < _bs1; ++k)
        {
          assert(_bs1 * dmap1[j] + k < (int)bc_markers1.size());
          if (bc_markers1[_bs1 * dmap1[j] + k])
          {
            has_bc = true;
            break;
          }
        }
      }
      else
      {
        for (int k = 0; k < bs1; ++k)
        {
          assert(bs1 * dmap1[j] + k < (int)bc_markers1.size());
          if (bc_markers1[bs1 * dmap1[j] + k])
          {
            has_bc = true;
            break;
          }
        }
      }
    }
 
    if (!has_bc)
      continue;
 
    // Get cell coordinates/geometry
    auto x_dofs = x_dofmap.links(c);
    for (std::size_t i = 0; i < x_dofs.size(); ++i)
    {
      common::impl::copy_N<3>(std::next(x_g.begin(), 3 * x_dofs[i]),
                              std::next(coordinate_dofs.begin(), 3 * i));
    }
 
    // Size data structure for assembly
    auto dmap0 = dofmap0.links(c);
 
    const int num_rows = bs0 * dmap0.size();
    const int num_cols = bs1 * dmap1.size();
 
    const T* coeff_array = coeffs.data() + index * cstride;
    Ae.resize(num_rows * num_cols);
    std::fill(Ae.begin(), Ae.end(), 0);
    kernel(Ae.data(), coeff_array, constants.data(), coordinate_dofs.data(),
           nullptr, nullptr);
    dof_transform(Ae, cell_info, c, num_cols);
    dof_transform_to_transpose(Ae, cell_info, c, num_rows);
 
    // Size data structure for assembly
    be.resize(num_rows);
    std::fill(be.begin(), be.end(), 0);
    for (std::size_t j = 0; j < dmap1.size(); ++j)
    {
      if constexpr (_bs1 > 0)
      {
        for (int k = 0; k < _bs1; ++k)
        {
          const std::int32_t jj = _bs1 * dmap1[j] + k;
          assert(jj < (int)bc_markers1.size());
          if (bc_markers1[jj])
          {
            const T bc = bc_values1[jj];
            const T _x0 = x0.empty() ? 0.0 : x0[jj];
            // const T _x0 = 0.0;
            // be -= Ae.col(bs1 * j + k) * scale * (bc - _x0);
            for (int m = 0; m < num_rows; ++m)
              be[m] -= Ae[m * num_cols + _bs1 * j + k] * scale * (bc - _x0);
          }
        }
      }
      else
      {
        for (int k = 0; k < bs1; ++k)
        {
          const std::int32_t jj = bs1 * dmap1[j] + k;
          assert(jj < (int)bc_markers1.size());
          if (bc_markers1[jj])
          {
            const T bc = bc_values1[jj];
            const T _x0 = x0.empty() ? 0.0 : x0[jj];
            // be -= Ae.col(bs1 * j + k) * scale * (bc - _x0);
            for (int m = 0; m < num_rows; ++m)
              be[m] -= Ae[m * num_cols + bs1 * j + k] * scale * (bc - _x0);
          }
        }
      }
    }
 
    for (std::size_t i = 0; i < dmap0.size(); ++i)
    {
      if constexpr (_bs0 > 0)
      {
        for (int k = 0; k < _bs0; ++k)
          b[_bs0 * dmap0[i] + k] += be[_bs0 * i + k];
      }
      else
      {
        for (int k = 0; k < bs0; ++k)
          b[bs0 * dmap0[i] + k] += be[bs0 * i + k];
      }
    }
  }
}
 
template <typename T, int _bs = -1>
void _lift_bc_exterior_facets(
    xtl::span<T> b, const mesh::Mesh& mesh,
    const std::function<void(T*, const T*, const T*, const double*, const int*,
                             const std::uint8_t*)>& kernel,
    const xtl::span<const std::pair<std::int32_t, int>>& facets,
    const std::function<void(const xtl::span<T>&,
                             const xtl::span<const std::uint32_t>&,
                             std::int32_t, int)>& dof_transform,
    const graph::AdjacencyList<std::int32_t>& dofmap0, int bs0,
    const std::function<void(const xtl::span<T>&,
                             const xtl::span<const std::uint32_t>&,
                             std::int32_t, int)>& dof_transform_to_transpose,
    const graph::AdjacencyList<std::int32_t>& dofmap1, int bs1,
    const xtl::span<const T>& constants, const xtl::span<const T>& coeffs,
    int cstride, const xtl::span<const std::uint32_t>& cell_info,
    const xtl::span<const T>& bc_values1,
    const xtl::span<const std::int8_t>& bc_markers1,
    const xtl::span<const T>& x0, double scale)
{
  if (facets.empty())
    return;
 
  // Prepare cell geometry
  const graph::AdjacencyList<std::int32_t>& x_dofmap = mesh.geometry().dofmap();
 
  const std::size_t num_dofs_g = mesh.geometry().cmap().dim();
 
  xtl::span<const double> x_g = mesh.geometry().x();
 
  // Data structures used in bc application
  std::vector<double> coordinate_dofs(3 * num_dofs_g);
  std::vector<T> Ae, be;
 
  for (std::size_t index = 0; index < facets.size(); ++index)
  {
    std::int32_t cell = facets[index].first;
    int local_facet = facets[index].second;
 
    // Get dof maps for cell
    auto dmap1 = dofmap1.links(cell);
 
    // Check if bc is applied to cell
    bool has_bc = false;
    for (std::size_t j = 0; j < dmap1.size(); ++j)
    {
      for (int k = 0; k < bs1; ++k)
      {
        if (bc_markers1[bs1 * dmap1[j] + k])
        {
          has_bc = true;
          break;
        }
      }
    }
 
    if (!has_bc)
      continue;
 
    // Get cell coordinates/geometry
    auto x_dofs = x_dofmap.links(cell);
    for (std::size_t i = 0; i < x_dofs.size(); ++i)
    {
      common::impl::copy_N<3>(std::next(x_g.begin(), 3 * x_dofs[i]),
                              std::next(coordinate_dofs.begin(), 3 * i));
    }
 
    // Size data structure for assembly
    auto dmap0 = dofmap0.links(cell);
 
    const int num_rows = bs0 * dmap0.size();
    const int num_cols = bs1 * dmap1.size();
 
    const T* coeff_array = coeffs.data() + index * cstride;
    Ae.resize(num_rows * num_cols);
    std::fill(Ae.begin(), Ae.end(), 0);
    kernel(Ae.data(), coeff_array, constants.data(), coordinate_dofs.data(),
           &local_facet, nullptr);
    dof_transform(Ae, cell_info, cell, num_cols);
    dof_transform_to_transpose(Ae, cell_info, cell, num_rows);
 
    // Size data structure for assembly
    be.resize(num_rows);
    std::fill(be.begin(), be.end(), 0);
    for (std::size_t j = 0; j < dmap1.size(); ++j)
    {
      for (int k = 0; k < bs1; ++k)
      {
        const std::int32_t jj = bs1 * dmap1[j] + k;
        if (bc_markers1[jj])
        {
          const T bc = bc_values1[jj];
          const T _x0 = x0.empty() ? 0.0 : x0[jj];
          // be -= Ae.col(bs1 * j + k) * scale * (bc - _x0);
          for (int m = 0; m < num_rows; ++m)
            be[m] -= Ae[m * num_cols + bs1 * j + k] * scale * (bc - _x0);
        }
      }
    }
 
    for (std::size_t i = 0; i < dmap0.size(); ++i)
      for (int k = 0; k < bs0; ++k)
        b[bs0 * dmap0[i] + k] += be[bs0 * i + k];
  }
}
 
template <typename T, int _bs = -1>
void _lift_bc_interior_facets(
    xtl::span<T> b, const mesh::Mesh& mesh,
    const std::function<void(T*, const T*, const T*, const double*, const int*,
                             const std::uint8_t*)>& kernel,
    const xtl::span<const std::tuple<std::int32_t, int, std::int32_t, int>>&
        facets,
    const std::function<void(const xtl::span<T>&,
                             const xtl::span<const std::uint32_t>&,
                             std::int32_t, int)>& dof_transform,
    const graph::AdjacencyList<std::int32_t>& dofmap0, int bs0,
    const std::function<void(const xtl::span<T>&,
                             const xtl::span<const std::uint32_t>&,
                             std::int32_t, int)>& dof_transform_to_transpose,
    const graph::AdjacencyList<std::int32_t>& dofmap1, int bs1,
    const xtl::span<const T>& constants, const xtl::span<const T>& coeffs,
    int cstride, const xtl::span<const std::uint32_t>& cell_info,
    const std::function<std::uint8_t(std::size_t)>& get_perm,
    const xtl::span<const T>& bc_values1,
    const xtl::span<const std::int8_t>& bc_markers1,
    const xtl::span<const T>& x0, double scale)
{
  if (facets.empty())
    return;
 
  const int tdim = mesh.topology().dim();
 
  // Prepare cell geometry
  const graph::AdjacencyList<std::int32_t>& x_dofmap = mesh.geometry().dofmap();
  const std::size_t num_dofs_g = mesh.geometry().cmap().dim();
  xtl::span<const double> x_g = mesh.geometry().x();
 
  const int num_cell_facets
      = mesh::cell_num_entities(mesh.topology().cell_type(), tdim - 1);
 
  // Data structures used in assembly
  xt::xtensor<double, 3> coordinate_dofs({2, num_dofs_g, 3});
  std::vector<T> Ae, be;
 
  // Temporaries for joint dofmaps
  std::vector<std::int32_t> dmapjoint0, dmapjoint1;
 
  for (std::size_t index = 0; index < facets.size(); ++index)
  {
    const std::array<std::int32_t, 2> cells
        = {std::get<0>(facets[index]), std::get<2>(facets[index])};
    const std::array<int, 2> local_facet
        = {std::get<1>(facets[index]), std::get<3>(facets[index])};
 
    // Get cell geometry
    auto x_dofs0 = x_dofmap.links(cells[0]);
    for (std::size_t i = 0; i < x_dofs0.size(); ++i)
    {
      common::impl::copy_N<3>(
          std::next(x_g.begin(), 3 * x_dofs0[i]),
          xt::view(coordinate_dofs, 0, i, xt::all()).begin());
    }
    auto x_dofs1 = x_dofmap.links(cells[1]);
    for (std::size_t i = 0; i < x_dofs1.size(); ++i)
    {
      common::impl::copy_N<3>(
          std::next(x_g.begin(), 3 * x_dofs1[i]),
          xt::view(coordinate_dofs, 1, i, xt::all()).begin());
    }
 
    // Get dof maps for cells and pack
    const xtl::span<const std::int32_t> dmap0_cell0 = dofmap0.links(cells[0]);
    const xtl::span<const std::int32_t> dmap0_cell1 = dofmap0.links(cells[1]);
    dmapjoint0.resize(dmap0_cell0.size() + dmap0_cell1.size());
    std::copy(dmap0_cell0.begin(), dmap0_cell0.end(), dmapjoint0.begin());
    std::copy(dmap0_cell1.begin(), dmap0_cell1.end(),
              std::next(dmapjoint0.begin(), dmap0_cell0.size()));
 
    const xtl::span<const std::int32_t> dmap1_cell0 = dofmap1.links(cells[0]);
    const xtl::span<const std::int32_t> dmap1_cell1 = dofmap1.links(cells[1]);
    dmapjoint1.resize(dmap1_cell0.size() + dmap1_cell1.size());
    std::copy(dmap1_cell0.begin(), dmap1_cell0.end(), dmapjoint1.begin());
    std::copy(dmap1_cell1.begin(), dmap1_cell1.end(),
              std::next(dmapjoint1.begin(), dmap1_cell0.size()));
 
    // Check if bc is applied to cell0
    bool has_bc = false;
    for (std::size_t j = 0; j < dmap1_cell0.size(); ++j)
    {
      for (int k = 0; k < bs1; ++k)
      {
        if (bc_markers1[bs1 * dmap1_cell0[j] + k])
        {
          has_bc = true;
          break;
        }
      }
    }
 
    // Check if bc is applied to cell1
    for (std::size_t j = 0; j < dmap1_cell1.size(); ++j)
    {
      for (int k = 0; k < bs1; ++k)
      {
        if (bc_markers1[bs1 * dmap1_cell1[j] + k])
        {
          has_bc = true;
          break;
        }
      }
    }
 
    if (!has_bc)
      continue;
 
    const int num_rows = bs0 * dmapjoint0.size();
    const int num_cols = bs1 * dmapjoint1.size();
 
    // Tabulate tensor
    Ae.resize(num_rows * num_cols);
    std::fill(Ae.begin(), Ae.end(), 0);
    const std::array perm{
        get_perm(cells[0] * num_cell_facets + local_facet[0]),
        get_perm(cells[1] * num_cell_facets + local_facet[1])};
    kernel(Ae.data(), coeffs.data() + index * 2 * cstride, constants.data(),
           coordinate_dofs.data(), local_facet.data(), perm.data());
 
    const xtl::span<T> _Ae(Ae);
 
    const xtl::span<T> sub_Ae0
        = _Ae.subspan(bs0 * dmap0_cell0.size() * num_cols,
                      bs0 * dmap0_cell1.size() * num_cols);
    const xtl::span<T> sub_Ae1
        = _Ae.subspan(bs1 * dmap1_cell0.size(),
                      num_rows * num_cols - bs1 * dmap1_cell0.size());
 
    dof_transform(_Ae, cell_info, cells[0], num_cols);
    dof_transform(sub_Ae0, cell_info, cells[1], num_cols);
    dof_transform_to_transpose(_Ae, cell_info, cells[0], num_rows);
    dof_transform_to_transpose(sub_Ae1, cell_info, cells[1], num_rows);
 
    be.resize(num_rows);
    std::fill(be.begin(), be.end(), 0);
 
    // Compute b = b - A*b for cell0
    for (std::size_t j = 0; j < dmap1_cell0.size(); ++j)
    {
      for (int k = 0; k < bs1; ++k)
      {
        const std::int32_t jj = bs1 * dmap1_cell0[j] + k;
        if (bc_markers1[jj])
        {
          const T bc = bc_values1[jj];
          const T _x0 = x0.empty() ? 0.0 : x0[jj];
          // be -= Ae.col(bs1 * j + k) * scale * (bc - _x0);
          for (int m = 0; m < num_rows; ++m)
            be[m] -= Ae[m * num_cols + bs1 * j + k] * scale * (bc - _x0);
        }
      }
    }
 
    // Compute b = b - A*b for cell1
    const int offset = bs1 * dmap1_cell0.size();
    for (std::size_t j = 0; j < dmap1_cell1.size(); ++j)
    {
      for (int k = 0; k < bs1; ++k)
      {
        const std::int32_t jj = bs1 * dmap1_cell1[j] + k;
        if (bc_markers1[jj])
        {
          const T bc = bc_values1[jj];
          const T _x0 = x0.empty() ? 0.0 : x0[jj];
          // be -= Ae.col(offset + bs1 * j + k) * scale * (bc - x0[jj]);
          for (int m = 0; m < num_rows; ++m)
          {
            be[m]
                -= Ae[m * num_cols + offset + bs1 * j + k] * scale * (bc - _x0);
          }
        }
      }
    }
 
    for (std::size_t i = 0; i < dmap0_cell0.size(); ++i)
      for (int k = 0; k < bs0; ++k)
        b[bs0 * dmap0_cell0[i] + k] += be[bs0 * i + k];
 
    const int offset_be = bs0 * dmap0_cell0.size();
    for (std::size_t i = 0; i < dmap0_cell1.size(); ++i)
      for (int k = 0; k < bs0; ++k)
        b[bs0 * dmap0_cell1[i] + k] += be[offset_be + bs0 * i + k];
  }
}
template <typename T, int _bs = -1>
void assemble_cells(
    const std::function<void(const xtl::span<T>&,
                             const xtl::span<const std::uint32_t>&,
                             std::int32_t, int)>& dof_transform,
    xtl::span<T> b, const mesh::Geometry& geometry,
    const xtl::span<const std::int32_t>& cells,
    const graph::AdjacencyList<std::int32_t>& dofmap, int bs,
    const std::function<void(T*, const T*, const T*, const double*, const int*,
                             const std::uint8_t*)>& kernel,
    const xtl::span<const T>& constants, const xtl::span<const T>& coeffs,
    int cstride, const xtl::span<const std::uint32_t>& cell_info)
{
  assert(_bs < 0 or _bs == bs);
 
  if (cells.empty())
    return;
 
  // Prepare cell geometry
  const graph::AdjacencyList<std::int32_t>& x_dofmap = geometry.dofmap();
  const std::size_t num_dofs_g = geometry.cmap().dim();
  xtl::span<const double> x_g = geometry.x();
 
  // FIXME: Add proper interface for num_dofs
  // Create data structures used in assembly
  const int num_dofs = dofmap.links(0).size();
  std::vector<double> coordinate_dofs(3 * num_dofs_g);
  std::vector<T> be(bs * num_dofs);
  const xtl::span<T> _be(be);
 
  // Iterate over active cells
  for (std::size_t index = 0; index < cells.size(); ++index)
  {
    std::int32_t c = cells[index];
 
    // Get cell coordinates/geometry
    auto x_dofs = x_dofmap.links(c);
    for (std::size_t i = 0; i < x_dofs.size(); ++i)
    {
      common::impl::copy_N<3>(std::next(x_g.begin(), 3 * x_dofs[i]),
                              std::next(coordinate_dofs.begin(), 3 * i));
    }
 
    // Tabulate vector for cell
    std::fill(be.begin(), be.end(), 0);
    kernel(be.data(), coeffs.data() + index * cstride, constants.data(),
           coordinate_dofs.data(), nullptr, nullptr);
    dof_transform(_be, cell_info, c, 1);
 
    // Scatter cell vector to 'global' vector array
    auto dofs = dofmap.links(c);
    if constexpr (_bs > 0)
    {
      for (int i = 0; i < num_dofs; ++i)
        for (int k = 0; k < _bs; ++k)
          b[_bs * dofs[i] + k] += be[_bs * i + k];
    }
    else
    {
      for (int i = 0; i < num_dofs; ++i)
        for (int k = 0; k < bs; ++k)
          b[bs * dofs[i] + k] += be[bs * i + k];
    }
  }
}
 
template <typename T, int _bs = -1>
void assemble_exterior_facets(
    const std::function<void(const xtl::span<T>&,
                             const xtl::span<const std::uint32_t>&,
                             std::int32_t, int)>& dof_transform,
    xtl::span<T> b, const mesh::Mesh& mesh,
    const xtl::span<const std::pair<std::int32_t, int>>& facets,
    const graph::AdjacencyList<std::int32_t>& dofmap, int bs,
    const std::function<void(T*, const T*, const T*, const double*, const int*,
                             const std::uint8_t*)>& fn,
    const xtl::span<const T>& constants, const xtl::span<const T>& coeffs,
    int cstride, const xtl::span<const std::uint32_t>& cell_info)
{
  assert(_bs < 0 or _bs == bs);
 
  if (facets.empty())
    return;
 
  // Prepare cell geometry
  const graph::AdjacencyList<std::int32_t>& x_dofmap = mesh.geometry().dofmap();
  const std::size_t num_dofs_g = mesh.geometry().cmap().dim();
  xtl::span<const double> x_g = mesh.geometry().x();
 
  // FIXME: Add proper interface for num_dofs
  // Create data structures used in assembly
  const int num_dofs = dofmap.links(0).size();
  std::vector<double> coordinate_dofs(3 * num_dofs_g);
  std::vector<T> be(bs * num_dofs);
  const xtl::span<T> _be(be);
 
  for (std::size_t index = 0; index < facets.size(); ++index)
  {
    std::int32_t cell = facets[index].first;
    int local_facet = facets[index].second;
 
    // Get cell coordinates/geometry
    auto x_dofs = x_dofmap.links(cell);
    for (std::size_t i = 0; i < x_dofs.size(); ++i)
    {
      common::impl::copy_N<3>(std::next(x_g.begin(), 3 * x_dofs[i]),
                              std::next(coordinate_dofs.begin(), 3 * i));
    }
 
    // Tabulate element vector
    std::fill(be.begin(), be.end(), 0);
    fn(be.data(), coeffs.data() + index * cstride, constants.data(),
       coordinate_dofs.data(), &local_facet, nullptr);
 
    dof_transform(_be, cell_info, cell, 1);
 
    // Add element vector to global vector
    auto dofs = dofmap.links(cell);
    if constexpr (_bs > 0)
    {
      for (int i = 0; i < num_dofs; ++i)
        for (int k = 0; k < _bs; ++k)
          b[_bs * dofs[i] + k] += be[_bs * i + k];
    }
    else
    {
      for (int i = 0; i < num_dofs; ++i)
        for (int k = 0; k < bs; ++k)
          b[bs * dofs[i] + k] += be[bs * i + k];
    }
  }
}
 
template <typename T, int _bs = -1>
void assemble_interior_facets(
    const std::function<void(const xtl::span<T>&,
                             const xtl::span<const std::uint32_t>&,
                             std::int32_t, int)>& dof_transform,
    xtl::span<T> b, const mesh::Mesh& mesh,
    const xtl::span<const std::tuple<std::int32_t, int, std::int32_t, int>>&
        facets,
    const fem::DofMap& dofmap,
    const std::function<void(T*, const T*, const T*, const double*, const int*,
                             const std::uint8_t*)>& fn,
    const xtl::span<const T>& constants, const xtl::span<const T>& coeffs,
    int cstride, const xtl::span<const std::uint32_t>& cell_info,
    const std::function<std::uint8_t(std::size_t)>& get_perm)
{
  const int tdim = mesh.topology().dim();
 
  // Prepare cell geometry
  const graph::AdjacencyList<std::int32_t>& x_dofmap = mesh.geometry().dofmap();
  const std::size_t num_dofs_g = mesh.geometry().cmap().dim();
  xtl::span<const double> x_g = mesh.geometry().x();
 
  // Create data structures used in assembly
  xt::xtensor<double, 3> coordinate_dofs({2, num_dofs_g, 3});
  std::vector<T> be;
 
  const int num_cell_facets
      = mesh::cell_num_entities(mesh.topology().cell_type(), tdim - 1);
 
  const int bs = dofmap.bs();
  assert(_bs < 0 or _bs == bs);
  for (std::size_t index = 0; index < facets.size(); ++index)
  {
    const std::array<std::int32_t, 2> cells
        = {std::get<0>(facets[index]), std::get<2>(facets[index])};
    const std::array<int, 2> local_facet
        = {std::get<1>(facets[index]), std::get<3>(facets[index])};
 
    // Get cell geometry
    auto x_dofs0 = x_dofmap.links(cells[0]);
    for (std::size_t i = 0; i < x_dofs0.size(); ++i)
    {
      common::impl::copy_N<3>(
          std::next(x_g.begin(), 3 * x_dofs0[i]),
          xt::view(coordinate_dofs, 0, i, xt::all()).begin());
    }
    auto x_dofs1 = x_dofmap.links(cells[1]);
    for (std::size_t i = 0; i < x_dofs1.size(); ++i)
    {
      common::impl::copy_N<3>(
          std::next(x_g.begin(), 3 * x_dofs1[i]),
          xt::view(coordinate_dofs, 1, i, xt::all()).begin());
    }
 
    // Get dofmaps for cells
    xtl::span<const std::int32_t> dmap0 = dofmap.cell_dofs(cells[0]);
    xtl::span<const std::int32_t> dmap1 = dofmap.cell_dofs(cells[1]);
 
    // Tabulate element vector
    be.resize(bs * (dmap0.size() + dmap1.size()));
    std::fill(be.begin(), be.end(), 0);
    const std::array perm{
        get_perm(cells[0] * num_cell_facets + local_facet[0]),
        get_perm(cells[1] * num_cell_facets + local_facet[1])};
    fn(be.data(), coeffs.data() + index * 2 * cstride, constants.data(),
       coordinate_dofs.data(), local_facet.data(), perm.data());
 
    const xtl::span<T> _be(be);
    const xtl::span<T> sub_be
        = _be.subspan(bs * dmap0.size(), bs * dmap1.size());
 
    dof_transform(be, cell_info, cells[0], 1);
    dof_transform(sub_be, cell_info, cells[1], 1);
 
    // Add element vector to global vector
    if constexpr (_bs > 0)
    {
      for (std::size_t i = 0; i < dmap0.size(); ++i)
        for (int k = 0; k < _bs; ++k)
          b[_bs * dmap0[i] + k] += be[_bs * i + k];
      for (std::size_t i = 0; i < dmap1.size(); ++i)
        for (int k = 0; k < _bs; ++k)
          b[_bs * dmap1[i] + k] += be[_bs * (i + dmap0.size()) + k];
    }
    else
    {
      for (std::size_t i = 0; i < dmap0.size(); ++i)
        for (int k = 0; k < bs; ++k)
          b[bs * dmap0[i] + k] += be[bs * i + k];
      for (std::size_t i = 0; i < dmap1.size(); ++i)
        for (int k = 0; k < bs; ++k)
          b[bs * dmap1[i] + k] += be[bs * (i + dmap0.size()) + k];
    }
  }
}
 
template <typename T>
void lift_bc(xtl::span<T> b, const Form<T>& a,
             const xtl::span<const T>& constants,
             const std::map<std::pair<IntegralType, int>,
                            std::pair<xtl::span<const T>, int>>& coefficients,
             const xtl::span<const T>& bc_values1,
             const xtl::span<const std::int8_t>& bc_markers1,
             const xtl::span<const T>& x0, double scale)
{
  std::shared_ptr<const mesh::Mesh> mesh = a.mesh();
  assert(mesh);
 
  // Get dofmap for columns and rows of a
  assert(a.function_spaces().at(0));
  assert(a.function_spaces().at(1));
  const graph::AdjacencyList<std::int32_t>& dofmap0
      = a.function_spaces()[0]->dofmap()->list();
  const int bs0 = a.function_spaces()[0]->dofmap()->bs();
  std::shared_ptr<const fem::FiniteElement> element0
      = a.function_spaces()[0]->element();
  const graph::AdjacencyList<std::int32_t>& dofmap1
      = a.function_spaces()[1]->dofmap()->list();
  const int bs1 = a.function_spaces()[1]->dofmap()->bs();
  std::shared_ptr<const fem::FiniteElement> element1
      = a.function_spaces()[1]->element();
 
  const bool needs_transformation_data
      = element0->needs_dof_transformations()
        or element1->needs_dof_transformations()
        or a.needs_facet_permutations();
 
  xtl::span<const std::uint32_t> cell_info;
  if (needs_transformation_data)
  {
    mesh->topology_mutable().create_entity_permutations();
    cell_info = xtl::span(mesh->topology().get_cell_permutation_info());
  }
 
  const std::function<void(const xtl::span<T>&,
                           const xtl::span<const std::uint32_t>&, std::int32_t,
                           int)>
      dof_transform = element0->get_dof_transformation_function<T>();
  const std::function<void(const xtl::span<T>&,
                           const xtl::span<const std::uint32_t>&, std::int32_t,
                           int)>
      dof_transform_to_transpose
      = element1->get_dof_transformation_to_transpose_function<T>();
 
  for (int i : a.integral_ids(IntegralType::cell))
  {
    const auto& kernel = a.kernel(IntegralType::cell, i);
    const auto& [coeffs, cstride] = coefficients.at({IntegralType::cell, i});
    const std::vector<std::int32_t>& cells = a.cell_domains(i);
    if (bs0 == 1 and bs1 == 1)
    {
      _lift_bc_cells<T, 1, 1>(b, mesh->geometry(), kernel, cells, dof_transform,
                              dofmap0, bs0, dof_transform_to_transpose, dofmap1,
                              bs1, constants, coeffs, cstride, cell_info,
                              bc_values1, bc_markers1, x0, scale);
    }
    else if (bs0 == 3 and bs1 == 3)
    {
      _lift_bc_cells<T, 3, 3>(b, mesh->geometry(), kernel, cells, dof_transform,
                              dofmap0, bs0, dof_transform_to_transpose, dofmap1,
                              bs1, constants, coeffs, cstride, cell_info,
                              bc_values1, bc_markers1, x0, scale);
    }
    else
    {
      _lift_bc_cells(b, mesh->geometry(), kernel, cells, dof_transform, dofmap0,
                     bs0, dof_transform_to_transpose, dofmap1, bs1, constants,
                     coeffs, cstride, cell_info, bc_values1, bc_markers1, x0,
                     scale);
    }
  }
 
  for (int i : a.integral_ids(IntegralType::exterior_facet))
  {
    const auto& kernel = a.kernel(IntegralType::exterior_facet, i);
    const auto& [coeffs, cstride]
        = coefficients.at({IntegralType::exterior_facet, i});
    const std::vector<std::pair<std::int32_t, int>>& facets
        = a.exterior_facet_domains(i);
    _lift_bc_exterior_facets(b, *mesh, kernel, facets, dof_transform, dofmap0,
                             bs0, dof_transform_to_transpose, dofmap1, bs1,
                             constants, coeffs, cstride, cell_info, bc_values1,
                             bc_markers1, x0, scale);
  }
 
  if (a.num_integrals(IntegralType::interior_facet) > 0)
  {
    std::function<std::uint8_t(std::size_t)> get_perm;
    if (a.needs_facet_permutations())
    {
      mesh->topology_mutable().create_entity_permutations();
      const std::vector<std::uint8_t>& perms
          = mesh->topology().get_facet_permutations();
      get_perm = [&perms](std::size_t i) { return perms[i]; };
    }
    else
      get_perm = [](std::size_t) { return 0; };
 
    for (int i : a.integral_ids(IntegralType::interior_facet))
    {
      const auto& kernel = a.kernel(IntegralType::interior_facet, i);
      const auto& [coeffs, cstride]
          = coefficients.at({IntegralType::interior_facet, i});
      const std::vector<std::tuple<std::int32_t, int, std::int32_t, int>>&
          facets
          = a.interior_facet_domains(i);
      _lift_bc_interior_facets(b, *mesh, kernel, facets, dof_transform, dofmap0,
                               bs0, dof_transform_to_transpose, dofmap1, bs1,
                               constants, coeffs, cstride, cell_info, get_perm,
                               bc_values1, bc_markers1, x0, scale);
    }
  }
}
 
template <typename T>
void apply_lifting(
    xtl::span<T> b, const std::vector<std::shared_ptr<const Form<T>>> a,
    const std::vector<xtl::span<const T>>& constants,
    const std::vector<std::map<std::pair<IntegralType, int>,
                               std::pair<xtl::span<const T>, int>>>& coeffs,
    const std::vector<std::vector<std::shared_ptr<const DirichletBC<T>>>>& bcs1,
    const std::vector<xtl::span<const T>>& x0, double scale)
{
  // FIXME: make changes to reactivate this check
  if (!x0.empty() and x0.size() != a.size())
  {
    throw std::runtime_error(
        "Mismatch in size between x0 and bilinear form in assembler.");
  }
 
  if (a.size() != bcs1.size())
  {
    throw std::runtime_error(
        "Mismatch in size between a and bcs in assembler.");
  }
 
  for (std::size_t j = 0; j < a.size(); ++j)
  {
    std::vector<std::int8_t> bc_markers1;
    std::vector<T> bc_values1;
    if (a[j] and !bcs1[j].empty())
    {
      assert(a[j]->function_spaces().at(0));
 
      auto V1 = a[j]->function_spaces()[1];
      assert(V1);
      auto map1 = V1->dofmap()->index_map;
      const int bs1 = V1->dofmap()->index_map_bs();
      assert(map1);
      const int crange = bs1 * (map1->size_local() + map1->num_ghosts());
      bc_markers1.assign(crange, false);
      bc_values1.assign(crange, 0.0);
      for (const std::shared_ptr<const DirichletBC<T>>& bc : bcs1[j])
      {
        bc->mark_dofs(bc_markers1);
        bc->dof_values(bc_values1);
      }
 
      if (!x0.empty())
      {
        lift_bc<T>(b, *a[j], constants[j], coeffs[j], bc_values1, bc_markers1,
                   x0[j], scale);
      }
      else
      {
        lift_bc<T>(b, *a[j], constants[j], coeffs[j], bc_values1, bc_markers1,
                   xtl::span<const T>(), scale);
      }
    }
  }
}
 
template <typename T>
void assemble_vector(
    xtl::span<T> b, const Form<T>& L, const xtl::span<const T>& constants,
    const std::map<std::pair<IntegralType, int>,
                   std::pair<xtl::span<const T>, int>>& coefficients)
{
  std::shared_ptr<const mesh::Mesh> mesh = L.mesh();
  assert(mesh);
 
  // Get dofmap data
  assert(L.function_spaces().at(0));
  std::shared_ptr<const fem::FiniteElement> element
      = L.function_spaces().at(0)->element();
  std::shared_ptr<const fem::DofMap> dofmap
      = L.function_spaces().at(0)->dofmap();
  assert(dofmap);
  const graph::AdjacencyList<std::int32_t>& dofs = dofmap->list();
  const int bs = dofmap->bs();
 
  const std::function<void(const xtl::span<T>&,
                           const xtl::span<const std::uint32_t>&, std::int32_t,
                           int)>
      dof_transform = element->get_dof_transformation_function<T>();
 
  const bool needs_transformation_data
      = element->needs_dof_transformations() or L.needs_facet_permutations();
  xtl::span<const std::uint32_t> cell_info;
  if (needs_transformation_data)
  {
    mesh->topology_mutable().create_entity_permutations();
    cell_info = xtl::span(mesh->topology().get_cell_permutation_info());
  }
 
  for (int i : L.integral_ids(IntegralType::cell))
  {
    const auto& fn = L.kernel(IntegralType::cell, i);
    const auto& [coeffs, cstride] = coefficients.at({IntegralType::cell, i});
    const std::vector<std::int32_t>& cells = L.cell_domains(i);
    if (bs == 1)
    {
      impl::assemble_cells<T, 1>(dof_transform, b, mesh->geometry(), cells,
                                 dofs, bs, fn, constants, coeffs, cstride,
                                 cell_info);
    }
    else if (bs == 3)
    {
      impl::assemble_cells<T, 3>(dof_transform, b, mesh->geometry(), cells,
                                 dofs, bs, fn, constants, coeffs, cstride,
                                 cell_info);
    }
    else
    {
      impl::assemble_cells(dof_transform, b, mesh->geometry(), cells, dofs, bs,
                           fn, constants, coeffs, cstride, cell_info);
    }
  }
 
  for (int i : L.integral_ids(IntegralType::exterior_facet))
  {
    const auto& fn = L.kernel(IntegralType::exterior_facet, i);
    const auto& [coeffs, cstride]
        = coefficients.at({IntegralType::exterior_facet, i});
    const std::vector<std::pair<std::int32_t, int>>& facets
        = L.exterior_facet_domains(i);
    if (bs == 1)
    {
      impl::assemble_exterior_facets<T, 1>(dof_transform, b, *mesh, facets,
                                           dofs, bs, fn, constants, coeffs,
                                           cstride, cell_info);
    }
    else if (bs == 3)
    {
      impl::assemble_exterior_facets<T, 3>(dof_transform, b, *mesh, facets,
                                           dofs, bs, fn, constants, coeffs,
                                           cstride, cell_info);
    }
    else
    {
      impl::assemble_exterior_facets(dof_transform, b, *mesh, facets, dofs, bs,
                                     fn, constants, coeffs, cstride, cell_info);
    }
  }
 
  if (L.num_integrals(IntegralType::interior_facet) > 0)
  {
    std::function<std::uint8_t(std::size_t)> get_perm;
    if (L.needs_facet_permutations())
    {
      mesh->topology_mutable().create_entity_permutations();
      const std::vector<std::uint8_t>& perms
          = mesh->topology().get_facet_permutations();
      get_perm = [&perms](std::size_t i) { return perms[i]; };
    }
    else
      get_perm = [](std::size_t) { return 0; };
 
    for (int i : L.integral_ids(IntegralType::interior_facet))
    {
      const auto& fn = L.kernel(IntegralType::interior_facet, i);
      const auto& [coeffs, cstride]
          = coefficients.at({IntegralType::interior_facet, i});
      const std::vector<std::tuple<std::int32_t, int, std::int32_t, int>>&
          facets
          = L.interior_facet_domains(i);
      if (bs == 1)
      {
        impl::assemble_interior_facets<T, 1>(dof_transform, b, *mesh, facets,
                                             *dofmap, fn, constants, coeffs,
                                             cstride, cell_info, get_perm);
      }
      else if (bs == 3)
      {
        impl::assemble_interior_facets<T, 3>(dof_transform, b, *mesh, facets,
                                             *dofmap, fn, constants, coeffs,
                                             cstride, cell_info, get_perm);
      }
      else
      {
        impl::assemble_interior_facets(dof_transform, b, *mesh, facets, *dofmap,
                                       fn, constants, coeffs, cstride,
                                       cell_info, get_perm);
      }
    }
  }
}
} // namespace dolfinx::fem::impl