interpolate.h

// Copyright (C) 2020-2021 Garth N. Wells
//
// This file is part of DOLFINx (https://www.fenicsproject.org)
//
// SPDX-License-Identifier:    LGPL-3.0-or-later
 
#pragma once
 
#include "FunctionSpace.h"
#include <dolfinx/fem/DofMap.h>
#include <dolfinx/fem/FiniteElement.h>
#include <dolfinx/mesh/Mesh.h>
#include <functional>
#include <variant>
#include <xtensor/xadapt.hpp>
#include <xtensor/xarray.hpp>
#include <xtensor/xtensor.hpp>
#include <xtensor/xview.hpp>
#include <xtl/xspan.hpp>
 
namespace dolfinx::fem
{
 
template <typename T>
class Function;
 
xt::xtensor<double, 2>
interpolation_coords(const fem::FiniteElement& element, const mesh::Mesh& mesh,
                     const xtl::span<const std::int32_t>& cells);
 
template <typename T>
void interpolate(Function<T>& u, const Function<T>& v);
 
template <typename T>
void interpolate(
    Function<T>& u,
    const std::function<xt::xarray<T>(const xt::xtensor<double, 2>&)>& f,
    const xt::xtensor<double, 2>& x,
    const xtl::span<const std::int32_t>& cells);
 
template <typename T>
void interpolate_c(
    Function<T>& u,
    const std::function<void(xt::xarray<T>&, const xt::xtensor<double, 2>&)>& f,
    const xt::xtensor<double, 2>& x,
    const xtl::span<const std::int32_t>& cells);
 
namespace detail
{
 
template <typename T>
void interpolate_from_any(Function<T>& u, const Function<T>& v)
{
  assert(v.function_space());
  const auto element = u.function_space()->element();
  assert(element);
  if (v.function_space()->element()->hash() != element->hash())
  {
    throw std::runtime_error("Restricting finite elements function in "
                             "different elements not supported.");
  }
 
  const auto mesh = u.function_space()->mesh();
  assert(mesh);
  assert(v.function_space()->mesh());
  if (mesh->id() != v.function_space()->mesh()->id())
  {
    throw std::runtime_error(
        "Interpolation on different meshes not supported (yet).");
  }
  const int tdim = mesh->topology().dim();
 
  // Get dofmaps
  assert(v.function_space());
  std::shared_ptr<const fem::DofMap> dofmap_v = v.function_space()->dofmap();
  assert(dofmap_v);
  auto map = mesh->topology().index_map(tdim);
  assert(map);
 
  std::vector<T>& coeffs = u.x()->mutable_array();
 
  // Iterate over mesh and interpolate on each cell
  const auto dofmap_u = u.function_space()->dofmap();
  const std::vector<T>& v_array = v.x()->array();
  const int num_cells = map->size_local() + map->num_ghosts();
  const int bs = dofmap_v->bs();
  assert(bs == dofmap_u->bs());
  for (int c = 0; c < num_cells; ++c)
  {
    xtl::span<const std::int32_t> dofs_v = dofmap_v->cell_dofs(c);
    xtl::span<const std::int32_t> cell_dofs = dofmap_u->cell_dofs(c);
    assert(dofs_v.size() == cell_dofs.size());
    for (std::size_t i = 0; i < dofs_v.size(); ++i)
    {
      for (int k = 0; k < bs; ++k)
        coeffs[bs * cell_dofs[i] + k] = v_array[bs * dofs_v[i] + k];
    }
  }
}
 
} // namespace detail
 
//----------------------------------------------------------------------------
template <typename T>
void interpolate(Function<T>& u, const Function<T>& v)
{
  assert(u.function_space());
  const std::shared_ptr<const FiniteElement> element
      = u.function_space()->element();
  assert(element);
 
  // Check that function ranks match
  if (int rank_v = v.function_space()->element()->value_rank();
      element->value_rank() != rank_v)
  {
    throw std::runtime_error("Cannot interpolate function into function space. "
                             "Rank of function ("
                             + std::to_string(rank_v)
                             + ") does not match rank of function space ("
                             + std::to_string(element->value_rank()) + ")");
  }
 
  // Check that function dimension match
  for (int i = 0; i < element->value_rank(); ++i)
  {
    if (int v_dim = v.function_space()->element()->value_dimension(i);
        element->value_dimension(i) != v_dim)
    {
      throw std::runtime_error(
          "Cannot interpolate function into function space. "
          "Dimension "
          + std::to_string(i) + " of function (" + std::to_string(v_dim)
          + ") does not match dimension " + std::to_string(i)
          + " of function space(" + std::to_string(element->value_dimension(i))
          + ")");
    }
  }
 
  detail::interpolate_from_any(u, v);
}
//----------------------------------------------------------------------------
template <typename T>
void interpolate(
    Function<T>& u,
    const std::function<xt::xarray<T>(const xt::xtensor<double, 2>&)>& f,
    const xt::xtensor<double, 2>& x, const xtl::span<const std::int32_t>& cells)
{
  const std::shared_ptr<const FiniteElement> element
      = u.function_space()->element();
  assert(element);
  const int element_bs = element->block_size();
  if (int num_sub = element->num_sub_elements();
      num_sub > 0 and num_sub != element_bs)
  {
    throw std::runtime_error("Cannot directly interpolate a mixed space. "
                             "Interpolate into subspaces.");
  }
 
  // Get mesh
  assert(u.function_space());
  auto mesh = u.function_space()->mesh();
  assert(mesh);
 
  const int gdim = mesh->geometry().dim();
  const int tdim = mesh->topology().dim();
 
  // Get the interpolation points on the reference cells
  const xt::xtensor<double, 2>& X = element->interpolation_points();
 
  if (X.shape(0) == 0)
  {
    throw std::runtime_error(
        "Interpolation into this space is not yet supported.");
  }
 
  xtl::span<const std::uint32_t> cell_info;
  if (element->needs_dof_transformations())
  {
    mesh->topology_mutable().create_entity_permutations();
    cell_info = xtl::span(mesh->topology().get_cell_permutation_info());
  }
 
  // Evaluate function at physical points. The returned array has a
  // number of rows equal to the number of components of the function,
  // and the number of columns is equal to the number of evaluation
  // points.
  xt::xarray<T> values = f(x);
 
  if (values.dimension() == 1)
  {
    if (element->value_size() != 1)
      throw std::runtime_error("Interpolation data has the wrong shape.");
    values.reshape(
        {static_cast<std::size_t>(element->value_size()), x.shape(1)});
  }
 
  if (values.shape(0) != element->value_size())
    throw std::runtime_error("Interpolation data has the wrong shape.");
 
  if (values.shape(1) != cells.size() * X.shape(0))
    throw std::runtime_error("Interpolation data has the wrong shape.");
 
  // Get dofmap
  const auto dofmap = u.function_space()->dofmap();
  assert(dofmap);
  const int dofmap_bs = dofmap->bs();
 
  // Loop over cells and compute interpolation dofs
  const int num_scalar_dofs = element->space_dimension() / element_bs;
  const int value_size = element->value_size() / element_bs;
 
  std::vector<T>& coeffs = u.x()->mutable_array();
  std::vector<T> _coeffs(num_scalar_dofs);
 
  const std::function<void(const xtl::span<T>&,
                           const xtl::span<const std::uint32_t>&, std::int32_t,
                           int)>
      apply_inverse_transpose_dof_transformation
      = element->get_dof_transformation_function<T>(true, true, true);
 
  // This assumes that any element with an identity interpolation matrix is a
  // point evaluation
  if (element->interpolation_ident())
  {
    for (std::int32_t c : cells)
    {
      xtl::span<const std::int32_t> dofs = dofmap->cell_dofs(c);
      for (int k = 0; k < element_bs; ++k)
      {
        for (int i = 0; i < num_scalar_dofs; ++i)
          _coeffs[i] = values(k, c * num_scalar_dofs + i);
        apply_inverse_transpose_dof_transformation(_coeffs, cell_info, c, 1);
        for (int i = 0; i < num_scalar_dofs; ++i)
        {
          const int dof = i * element_bs + k;
          std::div_t pos = std::div(dof, dofmap_bs);
          coeffs[dofmap_bs * dofs[pos.quot] + pos.rem] = _coeffs[i];
        }
      }
    }
  }
  else
  {
    // Get coordinate map
    const fem::CoordinateElement& cmap = mesh->geometry().cmap();
 
    // Get geometry data
    const graph::AdjacencyList<std::int32_t>& x_dofmap
        = mesh->geometry().dofmap();
    // FIXME: Add proper interface for num coordinate dofs
    const int num_dofs_g = x_dofmap.num_links(0);
    const xt::xtensor<double, 2>& x_g = mesh->geometry().x();
 
    // Create data structures for Jacobian info
    xt::xtensor<double, 3> J = xt::empty<double>({int(X.shape(0)), gdim, tdim});
    xt::xtensor<double, 3> K = xt::empty<double>({int(X.shape(0)), tdim, gdim});
    xt::xtensor<double, 1> detJ = xt::empty<double>({X.shape(0)});
 
    xt::xtensor<double, 2> coordinate_dofs
        = xt::empty<double>({num_dofs_g, gdim});
 
    xt::xtensor<T, 3> reference_data({X.shape(0), 1, value_size});
    xt::xtensor<T, 3> _vals({X.shape(0), 1, value_size});
 
    // Tabulate 1st order derivatives of shape functions at interpolation coords
    xt::xtensor<double, 4> dphi
        = xt::view(cmap.tabulate(1, X), xt::range(1, tdim + 1), xt::all(),
                   xt::all(), xt::all());
 
    const std::function<void(const xtl::span<T>&,
                             const xtl::span<const std::uint32_t>&,
                             std::int32_t, int)>
        apply_inverse_transpose_dof_transformation
        = element->get_dof_transformation_function<T>(true, true);
 
    for (std::int32_t c : cells)
    {
      auto x_dofs = x_dofmap.links(c);
      for (int i = 0; i < num_dofs_g; ++i)
        for (int j = 0; j < gdim; ++j)
          coordinate_dofs(i, j) = x_g(x_dofs[i], j);
 
      // Compute J, detJ and K
      cmap.compute_jacobian(dphi, coordinate_dofs, J);
      cmap.compute_jacobian_inverse(J, K);
      cmap.compute_jacobian_determinant(J, detJ);
 
      xtl::span<const std::int32_t> dofs = dofmap->cell_dofs(c);
      for (int k = 0; k < element_bs; ++k)
      {
        // Extract computed expression values for element block k
        for (int m = 0; m < value_size; ++m)
        {
          std::copy_n(&values(k * value_size + m, c * X.shape(0)), X.shape(0),
                      xt::view(_vals, xt::all(), 0, m).begin());
        }
 
        // Get element degrees of freedom for block
        element->map_pull_back(_vals, J, detJ, K, reference_data);
 
        xt::xtensor<T, 2> ref_data
            = xt::transpose(xt::view(reference_data, xt::all(), 0, xt::all()));
        element->interpolate(ref_data, tcb::make_span(_coeffs));
        apply_inverse_transpose_dof_transformation(_coeffs, cell_info, c, 1);
 
        assert(_coeffs.size() == num_scalar_dofs);
 
        // Copy interpolation dofs into coefficient vector
        for (int i = 0; i < num_scalar_dofs; ++i)
        {
          const int dof = i * element_bs + k;
          std::div_t pos = std::div(dof, dofmap_bs);
          coeffs[dofmap_bs * dofs[pos.quot] + pos.rem] = _coeffs[i];
        }
      }
    }
  }
}
//----------------------------------------------------------------------------
template <typename T>
void interpolate_c(
    Function<T>& u,
    const std::function<void(xt::xarray<T>&, const xt::xtensor<double, 2>&)>& f,
    const xt::xtensor<double, 2>& x, const xtl::span<const std::int32_t>& cells)
{
  const std::shared_ptr<const FiniteElement> element
      = u.function_space()->element();
  assert(element);
  std::vector<int> vshape(element->value_rank(), 1);
  for (std::size_t i = 0; i < vshape.size(); ++i)
    vshape[i] = element->value_dimension(i);
  const std::size_t value_size = std::reduce(
      std::begin(vshape), std::end(vshape), 1, std::multiplies<>());
 
  auto fn = [value_size, &f](const xt::xtensor<double, 2>& x)
  {
    xt::xarray<T> values = xt::empty<T>({value_size, x.shape(1)});
    f(values, x);
    return values;
  };
 
  interpolate<T>(u, fn, x, cells);
}
//----------------------------------------------------------------------------
 
} // namespace dolfinx::fem