symengine_tensor_operations.h

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// ---------------------------------------------------------------------
//
// Copyright (C) 2019 - 2022 by the deal.II authors
//
// This file is part of the deal.II library.
//
// The deal.II library is free software; you can use it, redistribute
// it, and/or modify it under the terms of the GNU Lesser General
// Public License as published by the Free Software Foundation; either
// version 2.1 of the License, or (at your option) any later version.
// The full text of the license can be found in the file LICENSE at
// the top level of the deal.II distribution.
//
// ---------------------------------------------------------------------
 
#ifndef dealii_differentiation_sd_symengine_tensor_operations_h
#define dealii_differentiation_sd_symengine_tensor_operations_h
 
#include <deal.II/base/config.h>
 
#ifdef DEAL_II_WITH_SYMENGINE
 
#  include <deal.II/base/symmetric_tensor.h>
#  include <deal.II/base/tensor.h>
 
#  include <deal.II/differentiation/sd/symengine_number_types.h>
#  include <deal.II/differentiation/sd/symengine_scalar_operations.h>
#  include <deal.II/differentiation/sd/symengine_types.h>
 
#  include <utility>
#  include <vector>
 
DEAL_II_NAMESPACE_OPEN
 
namespace Differentiation
{
  namespace SD
  {
    template <int dim>
    Tensor<1, dim, Expression>
    make_vector_of_symbols(const std::string &symbol);
 
    template <int rank, int dim>
    Tensor<rank, dim, Expression>
    make_tensor_of_symbols(const std::string &symbol);
 
    template <int rank, int dim>
    SymmetricTensor<rank, dim, Expression>
    make_symmetric_tensor_of_symbols(const std::string &symbol);
 
    template <int dim>
    Tensor<1, dim, Expression>
    make_vector_of_symbolic_functions(const std::string &            symbol,
                                      const types::substitution_map &arguments);
 
    template <int rank, int dim>
    Tensor<rank, dim, Expression>
    make_tensor_of_symbolic_functions(const std::string &            symbol,
                                      const types::substitution_map &arguments);
 
    template <int rank, int dim>
    SymmetricTensor<rank, dim, Expression>
    make_symmetric_tensor_of_symbolic_functions(
      const std::string &            symbol,
      const types::substitution_map &arguments);
 
    template <int rank, int dim>
    Tensor<rank, dim, Expression>
    differentiate(const Expression &f, const Tensor<rank, dim, Expression> &T);
 
    template <int rank, int dim>
    SymmetricTensor<rank, dim, Expression>
    differentiate(const Expression &                            f,
                  const SymmetricTensor<rank, dim, Expression> &S);
 
    template <int rank, int dim>
    Tensor<rank, dim, Expression>
    differentiate(const Tensor<0, dim, Expression> &   f,
                  const Tensor<rank, dim, Expression> &T);
 
    template <int rank, int dim>
    SymmetricTensor<rank, dim, Expression>
    differentiate(const Tensor<0, dim, Expression> &            f,
                  const SymmetricTensor<rank, dim, Expression> &S);
 
    template <int rank, int dim>
    Tensor<rank, dim, Expression>
    differentiate(const Tensor<rank, dim, Expression> &T, const Expression &x);
 
    template <int rank, int dim>
    SymmetricTensor<rank, dim, Expression>
    differentiate(const SymmetricTensor<rank, dim, Expression> &S,
                  const Expression &                            x);
 
    template <int rank, int dim>
    Tensor<rank, dim, Expression>
    differentiate(const Tensor<rank, dim, Expression> &T,
                  const Tensor<0, dim, Expression> &   x);
 
    template <int rank, int dim>
    SymmetricTensor<rank, dim, Expression>
    differentiate(const SymmetricTensor<rank, dim, Expression> &S,
                  const Tensor<0, dim, Expression> &            x);
 
    template <int rank_1, int rank_2, int dim>
    Tensor<rank_1 + rank_2, dim, Expression>
    differentiate(const Tensor<rank_1, dim, Expression> &T1,
                  const Tensor<rank_2, dim, Expression> &T2);
 
    template <int rank_1, int rank_2, int dim>
    SymmetricTensor<rank_1 + rank_2, dim, Expression>
    differentiate(const SymmetricTensor<rank_1, dim, Expression> &S1,
                  const SymmetricTensor<rank_2, dim, Expression> &S2);
 
    template <int rank_1, int rank_2, int dim>
    Tensor<rank_1 + rank_2, dim, Expression>
    differentiate(const Tensor<rank_1, dim, Expression> &         T,
                  const SymmetricTensor<rank_2, dim, Expression> &S);
 
    template <int rank_1, int rank_2, int dim>
    Tensor<rank_1 + rank_2, dim, Expression>
    differentiate(const SymmetricTensor<rank_1, dim, Expression> &S,
                  const Tensor<rank_2, dim, Expression> &         T);
 
    template <bool ignore_invalid_symbols = false,
              typename ValueType          = double,
              int rank,
              int dim,
              typename SymbolicType>
    void
    add_to_symbol_map(types::substitution_map &              symbol_map,
                      const Tensor<rank, dim, SymbolicType> &symbol_tensor);
 
    template <bool ignore_invalid_symbols = false,
              typename ValueType          = double,
              int rank,
              int dim,
              typename SymbolicType>
    void
    add_to_symbol_map(
      types::substitution_map &                       symbol_map,
      const SymmetricTensor<rank, dim, SymbolicType> &symbol_tensor);
 
    template <int rank, int dim, typename SymbolicType, typename ValueType>
    void
    set_value_in_symbol_map(
      types::substitution_map &              substitution_map,
      const Tensor<rank, dim, SymbolicType> &symbol_tensor,
      const Tensor<rank, dim, ValueType> &   value_tensor);
 
    template <int rank, int dim, typename SymbolicType, typename ValueType>
    void
    set_value_in_symbol_map(
      types::substitution_map &                       substitution_map,
      const SymmetricTensor<rank, dim, SymbolicType> &symbol_tensor,
      const SymmetricTensor<rank, dim, ValueType> &   value_tensor);
 
    template <int rank, int dim, typename ExpressionType, typename ValueType>
    types::substitution_map
    make_substitution_map(
      const Tensor<rank, dim, ExpressionType> &symbol_tensor,
      const Tensor<rank, dim, ValueType> &     value_tensor);
 
    template <int rank, int dim, typename ExpressionType, typename ValueType>
    types::substitution_map
    make_substitution_map(
      const SymmetricTensor<rank, dim, ExpressionType> &symbol_tensor,
      const SymmetricTensor<rank, dim, ValueType> &     value_tensor);
 
    template <bool ignore_invalid_symbols = false,
              int  rank,
              int  dim,
              typename ExpressionType,
              typename ValueType>
    void
    add_to_substitution_map(
      types::substitution_map &                substitution_map,
      const Tensor<rank, dim, ExpressionType> &symbol_tensor,
      const Tensor<rank, dim, ValueType> &     value_tensor);
 
    template <bool ignore_invalid_symbols = false,
              int  rank,
              int  dim,
              typename ExpressionType,
              typename ValueType>
    void
    add_to_substitution_map(
      types::substitution_map &                         substitution_map,
      const SymmetricTensor<rank, dim, ExpressionType> &symbol_tensor,
      const SymmetricTensor<rank, dim, ValueType> &     value_tensor);
 
    template <int rank, int dim>
    Tensor<rank, dim, Expression>
    substitute(const Tensor<rank, dim, Expression> &expression_tensor,
               const types::substitution_map &      substitution_map);
 
    template <int rank, int dim>
    SymmetricTensor<rank, dim, Expression>
    substitute(const SymmetricTensor<rank, dim, Expression> &expression_tensor,
               const types::substitution_map &               substitution_map);
 
    template <typename ValueType, int rank, int dim>
    Tensor<rank, dim, ValueType>
    substitute_and_evaluate(
      const Tensor<rank, dim, Expression> &expression_tensor,
      const types::substitution_map &      substitution_map);
 
    template <typename ValueType, int rank, int dim>
    SymmetricTensor<rank, dim, ValueType>
    substitute_and_evaluate(
      const SymmetricTensor<rank, dim, Expression> &expression_tensor,
      const types::substitution_map &               substitution_map);
 
  } // namespace SD
} // namespace Differentiation
 
 
/* -------------------- inline and template functions ------------------ */
 
 
#  ifndef DOXYGEN
 
namespace Differentiation
{
  namespace SD
  {
    /* ---------------- Symbolic differentiation --------------*/
 
 
    namespace internal
    {
      template <int dim>
      TableIndices<4>
      make_rank_4_tensor_indices(const unsigned int idx_i,
                                 const unsigned int idx_j)
      {
        const TableIndices<2> indices_i(
          SymmetricTensor<2, dim>::unrolled_to_component_indices(idx_i));
        const TableIndices<2> indices_j(
          SymmetricTensor<2, dim>::unrolled_to_component_indices(idx_j));
        return TableIndices<4>(indices_i[0],
                               indices_i[1],
                               indices_j[0],
                               indices_j[1]);
      }
 
 
      template <int rank_1, int rank_2>
      TableIndices<rank_1 + rank_2>
      concatenate_indices(const TableIndices<rank_1> &indices_1,
                          const TableIndices<rank_2> &indices_2)
      {
        TableIndices<rank_1 + rank_2> indices_out;
        for (unsigned int i = 0; i < rank_1; ++i)
          indices_out[i] = indices_1[i];
        for (unsigned int j = 0; j < rank_2; ++j)
          indices_out[rank_1 + j] = indices_2[j];
        return indices_out;
      }
 
 
      template <int rank>
      TableIndices<rank>
      transpose_indices(const TableIndices<rank> &indices)
      {
        return indices;
      }
 
 
      template <>
      inline TableIndices<2>
      transpose_indices(const TableIndices<2> &indices)
      {
        return TableIndices<2>(indices[1], indices[0]);
      }
 
 
      template <int rank, int dim, typename ValueType>
      bool
      is_symmetric_component(const TableIndices<rank> &,
                             const Tensor<rank, dim, ValueType> &)
      {
        return false;
      }
 
 
      template <int rank, int dim, typename ValueType>
      bool
      is_symmetric_component(const TableIndices<rank> &,
                             const SymmetricTensor<rank, dim, ValueType> &)
      {
        static_assert(
          rank == 0 || rank == 2,
          "Querying symmetric component for non rank-2 symmetric tensor index is not allowed.");
        return false;
      }
 
 
      template <int dim, typename ValueType>
      bool
      is_symmetric_component(const TableIndices<2> &table_indices,
                             const SymmetricTensor<2, dim, ValueType> &)
      {
        return table_indices[0] != table_indices[1];
      }
 
 
      template <int dim,
                typename ValueType = Expression,
                template <int, int, typename>
                class TensorType>
      TensorType<0, dim, ValueType>
      scalar_diff_tensor(const ValueType &                    func,
                         const TensorType<0, dim, ValueType> &op)
      {
        return differentiate(func, op);
      }
 
 
      template <int rank,
                int dim,
                typename ValueType = Expression,
                template <int, int, typename>
                class TensorType>
      TensorType<rank, dim, ValueType>
      scalar_diff_tensor(const ValueType &                       func,
                         const TensorType<rank, dim, ValueType> &op)
      {
        TensorType<rank, dim, ValueType> out;
        for (unsigned int i = 0; i < out.n_independent_components; ++i)
          {
            const TableIndices<rank> indices(
              out.unrolled_to_component_indices(i));
            out[indices] = differentiate(func, op[indices]);
 
            if (is_symmetric_component(indices, op))
              out[indices] *= 0.5;
          }
        return out;
      }
 
 
      // Specialization for rank-0 tensor
      template <int rank,
                int dim,
                typename ValueType = Expression,
                template <int, int, typename>
                class TensorType>
      TensorType<rank, dim, ValueType>
      tensor_diff_tensor(const TensorType<0, dim, ValueType> &   func,
                         const TensorType<rank, dim, ValueType> &op)
      {
        TensorType<rank, dim, ValueType> out;
        for (unsigned int i = 0; i < out.n_independent_components; ++i)
          {
            const TableIndices<rank> indices(
              out.unrolled_to_component_indices(i));
            out[indices] = differentiate(func, op[indices]);
 
            if (is_symmetric_component(indices, op))
              out[indices] *= 0.5;
          }
        return out;
      }
 
 
      template <int rank,
                int dim,
                typename ValueType = Expression,
                template <int, int, typename>
                class TensorType>
      TensorType<rank, dim, ValueType>
      tensor_diff_scalar(const TensorType<rank, dim, ValueType> &funcs,
                         const ValueType &                       op)
      {
        TensorType<rank, dim, ValueType> out;
        for (unsigned int i = 0; i < out.n_independent_components; ++i)
          {
            const TableIndices<rank> indices(
              out.unrolled_to_component_indices(i));
            out[indices] = differentiate(funcs[indices], op);
          }
        return out;
      }
 
 
      // Specialization for rank-0 tensor
      template <int rank,
                int dim,
                typename ValueType = Expression,
                template <int, int, typename>
                class TensorType>
      TensorType<rank, dim, ValueType>
      tensor_diff_tensor(const TensorType<rank, dim, ValueType> &funcs,
                         const TensorType<0, dim, ValueType> &   op)
      {
        TensorType<rank, dim, ValueType> out;
        for (unsigned int i = 0; i < out.n_independent_components; ++i)
          {
            const TableIndices<rank> indices(
              out.unrolled_to_component_indices(i));
            out[indices] = differentiate(funcs[indices], op);
          }
        return out;
      }
 
 
      // For either symmetric or normal tensors
      template <int rank_1,
                int rank_2,
                int dim,
                typename ValueType = Expression,
                template <int, int, typename>
                class TensorType>
      TensorType<rank_1 + rank_2, dim, ValueType>
      tensor_diff_tensor(const TensorType<rank_1, dim, ValueType> &funcs,
                         const TensorType<rank_2, dim, ValueType> &op)
      {
        TensorType<rank_1 + rank_2, dim, ValueType> out;
        for (unsigned int i = 0; i < funcs.n_independent_components; ++i)
          {
            const TableIndices<rank_1> indices_i(
              funcs.unrolled_to_component_indices(i));
            for (unsigned int j = 0; j < op.n_independent_components; ++j)
              {
                const TableIndices<rank_2> indices_j(
                  op.unrolled_to_component_indices(j));
                const TableIndices<rank_1 + rank_2> indices_out =
                  concatenate_indices(indices_i, indices_j);
 
                out[indices_out] =
                  differentiate(funcs[indices_i], op[indices_j]);
 
                if (is_symmetric_component(indices_j, op))
                  out[indices_out] *= 0.5;
              }
          }
        return out;
      }
 
 
      // For mixed symmetric/standard tensors
      // The return type is always a standard tensor, since we cannot be sure
      // that any symmetries exist in either the function tensor or the
      // differential operator.
      template <int rank_1,
                int rank_2,
                int dim,
                typename ValueType = Expression,
                template <int, int, typename>
                class TensorType_1,
                template <int, int, typename>
                class TensorType_2>
      Tensor<rank_1 + rank_2, dim, ValueType>
      tensor_diff_tensor(const TensorType_1<rank_1, dim, ValueType> &funcs,
                         const TensorType_2<rank_2, dim, ValueType> &op)
      {
        Tensor<rank_1 + rank_2, dim, ValueType> out;
        for (unsigned int i = 0; i < funcs.n_independent_components; ++i)
          {
            const TableIndices<rank_1> indices_i(
              funcs.unrolled_to_component_indices(i));
            for (unsigned int j = 0; j < op.n_independent_components; ++j)
              {
                const TableIndices<rank_2> indices_j(
                  op.unrolled_to_component_indices(j));
                const TableIndices<rank_1 + rank_2> indices_out =
                  concatenate_indices(indices_i, indices_j);
 
                out[indices_out] =
                  differentiate(funcs[indices_i], op[indices_j]);
 
                if (is_symmetric_component(indices_j, op))
                  out[indices_out] *= 0.5;
 
                // TODO: Implement for SymmetricTensor<4,dim,...>
                if (std::is_same<TensorType_1<rank_1, dim, ValueType>,
                                 SymmetricTensor<2, dim, ValueType>>::
                      value) // Symmetric function
                  {
                    const TableIndices<rank_1 + rank_2> indices_out_t =
                      concatenate_indices(transpose_indices(indices_i),
                                          indices_j);
                    out[indices_out_t] = out[indices_out];
                  }
                else if (std::is_same<TensorType_2<rank_2, dim, ValueType>,
                                      SymmetricTensor<2, dim, ValueType>>::
                           value) // Symmetric operator
                  {
                    const TableIndices<rank_1 + rank_2> indices_out_t =
                      concatenate_indices(indices_i,
                                          transpose_indices(indices_j));
                    out[indices_out_t] = out[indices_out];
                  }
                else
                  {
                    Assert(
                      false,
                      ExcMessage(
                        "Expect mixed tensor differentiation to have at least "
                        "one component stemming from a symmetric tensor."));
                  }
              }
          }
        return out;
      }
 
    } // namespace internal
 
 
    template <int rank, int dim>
    Tensor<rank, dim, Expression>
    differentiate(const Expression &                   func,
                  const Tensor<rank, dim, Expression> &op)
    {
      return internal::scalar_diff_tensor(func, op);
    }
 
 
    template <int rank, int dim>
    SymmetricTensor<rank, dim, Expression>
    differentiate(const Expression &                            func,
                  const SymmetricTensor<rank, dim, Expression> &op)
    {
      return internal::scalar_diff_tensor(func, op);
    }
 
 
    template <int rank, int dim>
    Tensor<rank, dim, Expression>
    differentiate(const Tensor<0, dim, Expression> &   func,
                  const Tensor<rank, dim, Expression> &op)
    {
      return internal::scalar_diff_tensor(func, op);
    }
 
 
    template <int rank, int dim>
    SymmetricTensor<rank, dim, Expression>
    differentiate(const Tensor<0, dim, Expression> &            func,
                  const SymmetricTensor<rank, dim, Expression> &op)
    {
      // Ensure that this returns a symmetric tensor by
      // invoking the scalar value function
      const Expression tmp = func;
      return internal::scalar_diff_tensor(tmp, op);
    }
 
 
    template <int rank, int dim>
    Tensor<rank, dim, Expression>
    differentiate(const Tensor<rank, dim, Expression> &symbol_tensor,
                  const Expression &                   op)
    {
      return internal::tensor_diff_scalar(symbol_tensor, op);
    }
 
 
    template <int rank, int dim>
    Tensor<rank, dim, Expression>
    differentiate(const Tensor<rank, dim, Expression> &symbol_tensor,
                  const Tensor<0, dim, Expression> &   op)
    {
      return internal::tensor_diff_scalar(symbol_tensor, op);
    }
 
 
    template <int rank, int dim>
    SymmetricTensor<rank, dim, Expression>
    differentiate(const SymmetricTensor<rank, dim, Expression> &symbol_tensor,
                  const Expression &                            op)
    {
      return internal::tensor_diff_scalar(symbol_tensor, op);
    }
 
 
    template <int rank, int dim>
    SymmetricTensor<rank, dim, Expression>
    differentiate(const SymmetricTensor<rank, dim, Expression> &symbol_tensor,
                  const Tensor<0, dim, Expression> &            op)
    {
      return internal::tensor_diff_scalar(symbol_tensor, op);
    }
 
 
    template <int rank_1, int rank_2, int dim>
    Tensor<rank_1 + rank_2, dim, Expression>
    differentiate(const Tensor<rank_1, dim, Expression> &symbol_tensor,
                  const Tensor<rank_2, dim, Expression> &op)
    {
      return internal::tensor_diff_tensor(symbol_tensor, op);
    }
 
 
    template <int rank_1, int rank_2, int dim>
    SymmetricTensor<rank_1 + rank_2, dim, Expression>
    differentiate(const SymmetricTensor<rank_1, dim, Expression> &symbol_tensor,
                  const SymmetricTensor<rank_2, dim, Expression> &op)
    {
      return internal::tensor_diff_tensor(symbol_tensor, op);
    }
 
 
    template <int rank_1, int rank_2, int dim>
    Tensor<rank_1 + rank_2, dim, Expression>
    differentiate(const Tensor<rank_1, dim, Expression> &         symbol_tensor,
                  const SymmetricTensor<rank_2, dim, Expression> &op)
    {
      return internal::tensor_diff_tensor(symbol_tensor, op);
    }
 
 
    template <int rank_1, int rank_2, int dim>
    Tensor<rank_1 + rank_2, dim, Expression>
    differentiate(const SymmetricTensor<rank_1, dim, Expression> &symbol_tensor,
                  const Tensor<rank_2, dim, Expression> &         op)
    {
      return internal::tensor_diff_tensor(symbol_tensor, op);
    }
 
 
    /* ---------------- Symbol map creation and manipulation --------------*/
 
 
    namespace internal
    {
      template <typename SymbolicType,
                typename ValueType,
                int rank,
                int dim,
                template <int, int, typename>
                class TensorType>
      void
      set_tensor_value_in_symbol_map(
        types::substitution_map &                  substitution_map,
        const TensorType<rank, dim, SymbolicType> &symbol_tensor,
        const TensorType<rank, dim, ValueType> &   value_tensor)
      {
        TensorType<rank, dim, Expression> out;
        for (unsigned int i = 0; i < out.n_independent_components; ++i)
          {
            const TableIndices<rank> indices(
              out.unrolled_to_component_indices(i));
            set_value_in_symbol_map(substitution_map,
                                    symbol_tensor[indices],
                                    value_tensor[indices]);
          }
      }
 
 
      template <typename SymbolicType, typename ValueType, int dim>
      void
      set_tensor_value_in_symbol_map(
        types::substitution_map &                    substitution_map,
        const SymmetricTensor<4, dim, SymbolicType> &symbol_tensor,
        const SymmetricTensor<4, dim, ValueType> &   value_tensor)
      {
        SymmetricTensor<4, dim, Expression> out;
        for (unsigned int i = 0;
             i < SymmetricTensor<2, dim>::n_independent_components;
             ++i)
          for (unsigned int j = 0;
               j < SymmetricTensor<2, dim>::n_independent_components;
               ++j)
            {
              const TableIndices<4> indices =
                make_rank_4_tensor_indices<dim>(i, j);
              set_value_in_symbol_map(substitution_map,
                                      symbol_tensor[indices],
                                      value_tensor[indices]);
            }
      }
    } // namespace internal
 
 
    template <bool ignore_invalid_symbols,
              typename ValueType,
              int rank,
              int dim,
              typename SymbolicType>
    void
    add_to_symbol_map(types::substitution_map &              symbol_map,
                      const Tensor<rank, dim, SymbolicType> &symbol_tensor)
    {
      // Call the above function
      add_to_substitution_map<ignore_invalid_symbols>(
        symbol_map, symbol_tensor, Tensor<rank, dim, ValueType>());
    }
 
 
    template <bool ignore_invalid_symbols,
              typename ValueType,
              int rank,
              int dim,
              typename SymbolicType>
    void
    add_to_symbol_map(
      types::substitution_map &                       symbol_map,
      const SymmetricTensor<rank, dim, SymbolicType> &symbol_tensor)
    {
      // Call the above function
      add_to_substitution_map<ignore_invalid_symbols>(
        symbol_map, symbol_tensor, SymmetricTensor<rank, dim, ValueType>());
    }
 
 
    template <int rank, int dim, typename SymbolicType, typename ValueType>
    void
    set_value_in_symbol_map(
      types::substitution_map &              substitution_map,
      const Tensor<rank, dim, SymbolicType> &symbol_tensor,
      const Tensor<rank, dim, ValueType> &   value_tensor)
    {
      internal::set_tensor_value_in_symbol_map(substitution_map,
                                               symbol_tensor,
                                               value_tensor);
    }
 
 
    template <int rank, int dim, typename SymbolicType, typename ValueType>
    void
    set_value_in_symbol_map(
      types::substitution_map &                       substitution_map,
      const SymmetricTensor<rank, dim, SymbolicType> &symbol_tensor,
      const SymmetricTensor<rank, dim, ValueType> &   value_tensor)
    {
      internal::set_tensor_value_in_symbol_map(substitution_map,
                                               symbol_tensor,
                                               value_tensor);
    }
 
 
    /* ------------------ Symbol substitution map creation ----------------*/
 
 
    template <int rank, int dim, typename ExpressionType, typename ValueType>
    types::substitution_map
    make_substitution_map(
      const Tensor<rank, dim, ExpressionType> &symbol_tensor,
      const Tensor<rank, dim, ValueType> &     value_tensor)
    {
      types::substitution_map substitution_map;
      add_to_substitution_map(substitution_map, symbol_tensor, value_tensor);
      return substitution_map;
    }
 
 
    template <int rank, int dim, typename ExpressionType, typename ValueType>
    types::substitution_map
    make_substitution_map(
      const SymmetricTensor<rank, dim, ExpressionType> &symbol_tensor,
      const SymmetricTensor<rank, dim, ValueType> &     value_tensor)
    {
      types::substitution_map substitution_map;
      add_to_substitution_map(substitution_map, symbol_tensor, value_tensor);
      return substitution_map;
    }
 
 
    /* ---------------- Symbolic substitution map enlargement --------------*/
 
 
    namespace internal
    {
      template <int rank,
                int dim,
                typename ExpressionType,
                typename ValueType,
                template <int, int, typename>
                class TensorType>
      std::vector<std::pair<ExpressionType, ValueType>>
      make_tensor_entries_for_substitution_map(
        const TensorType<rank, dim, ExpressionType> &symbol_tensor,
        const TensorType<rank, dim, ValueType> &     value_tensor)
      {
        std::vector<std::pair<ExpressionType, ValueType>> symbol_values;
        for (unsigned int i = 0; i < symbol_tensor.n_independent_components;
             ++i)
          {
            const TableIndices<rank> indices(
              symbol_tensor.unrolled_to_component_indices(i));
            symbol_values.push_back(
              std::make_pair(symbol_tensor[indices], value_tensor[indices]));
          }
        return symbol_values;
      }
 
 
      template <int dim, typename ExpressionType, typename ValueType>
      std::vector<std::pair<ExpressionType, ValueType>>
      make_tensor_entries_for_substitution_map(
        const Tensor<0, dim, ExpressionType> &symbol_tensor,
        const Tensor<0, dim, ValueType> &     value_tensor)
      {
        const ExpressionType &expression = symbol_tensor;
        const ValueType &     value      = value_tensor;
        return {std::make_pair(expression, value)};
      }
 
 
      template <int dim, typename ExpressionType, typename ValueType>
      std::vector<std::pair<ExpressionType, ValueType>>
      make_tensor_entries_for_substitution_map(
        const SymmetricTensor<4, dim, ExpressionType> &symbol_tensor,
        const SymmetricTensor<4, dim, ValueType> &     value_tensor)
      {
        std::vector<std::pair<ExpressionType, ValueType>> symbol_values;
        for (unsigned int i = 0;
             i < SymmetricTensor<2, dim>::n_independent_components;
             ++i)
          for (unsigned int j = 0;
               j < SymmetricTensor<2, dim>::n_independent_components;
               ++j)
            {
              const TableIndices<4> indices =
                make_rank_4_tensor_indices<dim>(i, j);
              symbol_values.push_back(
                std::make_pair(symbol_tensor[indices], value_tensor[indices]));
            }
        return symbol_values;
      }
    } // namespace internal
 
 
    template <bool ignore_invalid_symbols,
              int  rank,
              int  dim,
              typename ExpressionType,
              typename ValueType>
    void
    add_to_substitution_map(
      types::substitution_map &                substitution_map,
      const Tensor<rank, dim, ExpressionType> &symbol_tensor,
      const Tensor<rank, dim, ValueType> &     value_tensor)
    {
      add_to_substitution_map<ignore_invalid_symbols>(
        substitution_map,
        internal::make_tensor_entries_for_substitution_map(symbol_tensor,
                                                           value_tensor));
    }
 
 
    template <bool ignore_invalid_symbols,
              int  rank,
              int  dim,
              typename ExpressionType,
              typename ValueType>
    void
    add_to_substitution_map(
      types::substitution_map &                         substitution_map,
      const SymmetricTensor<rank, dim, ExpressionType> &symbol_tensor,
      const SymmetricTensor<rank, dim, ValueType> &     value_tensor)
    {
      add_to_substitution_map<ignore_invalid_symbols>(
        substitution_map,
        internal::make_tensor_entries_for_substitution_map(symbol_tensor,
                                                           value_tensor));
    }
 
 
    /* ---------------- Symbol substitution and evaluation --------------*/
 
 
    namespace internal
    {
      template <int rank,
                int dim,
                template <int, int, typename>
                class TensorType>
      TensorType<rank, dim, Expression>
      substitute_tensor(
        const TensorType<rank, dim, Expression> &expression_tensor,
        const types::substitution_map &          substitution_map)
      {
        TensorType<rank, dim, Expression> out;
        for (unsigned int i = 0; i < out.n_independent_components; ++i)
          {
            const TableIndices<rank> indices(
              out.unrolled_to_component_indices(i));
            out[indices] =
              substitute(expression_tensor[indices], substitution_map);
          }
        return out;
      }
 
 
      template <int dim>
      Tensor<0, dim, Expression>
      substitute_tensor(const Tensor<0, dim, Expression> &expression_tensor,
                        const types::substitution_map &   substitution_map)
      {
        const Expression &expression = expression_tensor;
        return substitute(expression, substitution_map);
      }
 
 
      template <int dim>
      SymmetricTensor<4, dim, Expression>
      substitute_tensor(
        const SymmetricTensor<4, dim, Expression> &expression_tensor,
        const types::substitution_map &            substitution_map)
      {
        SymmetricTensor<4, dim, Expression> out;
        for (unsigned int i = 0;
             i < SymmetricTensor<2, dim>::n_independent_components;
             ++i)
          for (unsigned int j = 0;
               j < SymmetricTensor<2, dim>::n_independent_components;
               ++j)
            {
              const TableIndices<4> indices =
                make_rank_4_tensor_indices<dim>(i, j);
              out[indices] =
                substitute(expression_tensor[indices], substitution_map);
            }
        return out;
      }
 
 
      template <typename ValueType,
                int rank,
                int dim,
                template <int, int, typename>
                class TensorType>
      TensorType<rank, dim, ValueType>
      substitute_and_evaluate_tensor(
        const TensorType<rank, dim, Expression> &expression_tensor,
        const types::substitution_map &          substitution_map)
      {
        TensorType<rank, dim, ValueType> out;
        for (unsigned int i = 0; i < out.n_independent_components; ++i)
          {
            const TableIndices<rank> indices(
              out.unrolled_to_component_indices(i));
            out[indices] =
              substitute_and_evaluate<ValueType>(expression_tensor[indices],
                                                 substitution_map);
          }
        return out;
      }
 
 
      template <typename ValueType, int dim>
      Tensor<0, dim, ValueType>
      substitute_and_evaluate_tensor(
        const Tensor<0, dim, Expression> &expression_tensor,
        const types::substitution_map &   substitution_map)
      {
        const Expression &expression = expression_tensor;
        return substitute_and_evaluate<ValueType>(expression, substitution_map);
      }
 
 
      template <typename ValueType, int dim>
      SymmetricTensor<4, dim, ValueType>
      substitute_and_evaluate_tensor(
        const SymmetricTensor<4, dim, Expression> &expression_tensor,
        const types::substitution_map &            substitution_map)
      {
        SymmetricTensor<4, dim, ValueType> out;
        for (unsigned int i = 0;
             i < SymmetricTensor<2, dim>::n_independent_components;
             ++i)
          for (unsigned int j = 0;
               j < SymmetricTensor<2, dim>::n_independent_components;
               ++j)
            {
              const TableIndices<4> indices =
                make_rank_4_tensor_indices<dim>(i, j);
              out[indices] =
                substitute_and_evaluate<ValueType>(expression_tensor[indices],
                                                   substitution_map);
            }
        return out;
      }
    } // namespace internal
 
 
    template <int rank, int dim>
    Tensor<rank, dim, Expression>
    substitute(const Tensor<rank, dim, Expression> &expression_tensor,
               const types::substitution_map &      substitution_map)
    {
      return internal::substitute_tensor(expression_tensor, substitution_map);
    }
 
 
    template <int rank, int dim>
    SymmetricTensor<rank, dim, Expression>
    substitute(const SymmetricTensor<rank, dim, Expression> &expression_tensor,
               const types::substitution_map &               substitution_map)
    {
      return internal::substitute_tensor(expression_tensor, substitution_map);
    }
 
 
    template <typename ValueType, int rank, int dim>
    Tensor<rank, dim, ValueType>
    substitute_and_evaluate(
      const Tensor<rank, dim, Expression> &expression_tensor,
      const types::substitution_map &      substitution_map)
    {
      return internal::substitute_and_evaluate_tensor<ValueType>(
        expression_tensor, substitution_map);
    }
 
 
    template <typename ValueType, int rank, int dim>
    SymmetricTensor<rank, dim, ValueType>
    substitute_and_evaluate(
      const SymmetricTensor<rank, dim, Expression> &expression_tensor,
      const types::substitution_map &               substitution_map)
    {
      return internal::substitute_and_evaluate_tensor<ValueType>(
        expression_tensor, substitution_map);
    }
 
 
 
  } // namespace SD
} // namespace Differentiation
 
#  endif // DOXYGEN
 
DEAL_II_NAMESPACE_CLOSE
 
#endif // DEAL_II_WITH_SYMENGINE
 
#endif