tensor.h

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// ---------------------------------------------------------------------
//
// Copyright (C) 1998 - 2020 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.md at
// the top level directory of deal.II.
//
// ---------------------------------------------------------------------
 
#ifndef dealii_tensor_h
#define dealii_tensor_h
 
#include <deal.II/base/config.h>
 
#include <deal.II/base/exceptions.h>
#include <deal.II/base/numbers.h>
#include <deal.II/base/std_cxx14/utility.h>
#include <deal.II/base/table_indices.h>
#include <deal.II/base/template_constraints.h>
#include <deal.II/base/tensor_accessors.h>
#include <deal.II/base/utilities.h>
 
#include <deal.II/lac/lapack_full_matrix.h>
 
#ifdef DEAL_II_WITH_ADOLC
#  include <adolc/adouble.h> // Taped double
#endif
 
#include <cmath>
#include <ostream>
#include <vector>
 
 
DEAL_II_NAMESPACE_OPEN
 
// Forward declarations:
#ifndef DOXYGEN
template <int dim, typename Number>
class Point;
template <int rank_, int dim, typename Number = double>
class Tensor;
template <typename Number>
class Vector;
template <typename number>
class FullMatrix;
namespace Differentiation
{
  namespace SD
  {
    class Expression;
  }
} // namespace Differentiation
#endif
 
 
template <int dim, typename Number>
class Tensor<0, dim, Number>
{
public:
  static_assert(dim >= 0,
                "Tensors must have a dimension greater than or equal to one.");
 
  static constexpr unsigned int dimension = dim;
 
  static constexpr unsigned int rank = 0;
 
  static constexpr unsigned int n_independent_components = 1;
 
  using real_type = typename numbers::NumberTraits<Number>::real_type;
 
  using value_type = Number;
 
  using array_type = Number;
 
  constexpr DEAL_II_CUDA_HOST_DEV
  Tensor();
 
  template <typename OtherNumber>
  constexpr DEAL_II_CUDA_HOST_DEV
  Tensor(const Tensor<0, dim, OtherNumber> &initializer);
 
  template <typename OtherNumber>
  constexpr DEAL_II_CUDA_HOST_DEV
  Tensor(const OtherNumber &initializer);
 
  Number *
  begin_raw();
 
  const Number *
  begin_raw() const;
 
  Number *
  end_raw();
 
  const Number *
  end_raw() const;
 
  DEAL_II_CONSTEXPR DEAL_II_CUDA_HOST_DEV operator Number &();
 
  DEAL_II_CONSTEXPR DEAL_II_CUDA_HOST_DEV operator const Number &() const;
 
  template <typename OtherNumber>
  DEAL_II_CONSTEXPR DEAL_II_CUDA_HOST_DEV Tensor &
                                          operator=(const Tensor<0, dim, OtherNumber> &rhs);
 
#ifdef __INTEL_COMPILER
 
  DEAL_II_CONSTEXPR DEAL_II_CUDA_HOST_DEV Tensor &
                                          operator=(const Tensor<0, dim, Number> &rhs);
#endif
 
  template <typename OtherNumber>
  DEAL_II_CONSTEXPR DEAL_II_CUDA_HOST_DEV Tensor &
                                          operator=(const OtherNumber &d);
 
  template <typename OtherNumber>
  DEAL_II_CONSTEXPR bool
  operator==(const Tensor<0, dim, OtherNumber> &rhs) const;
 
  template <typename OtherNumber>
  constexpr bool
  operator!=(const Tensor<0, dim, OtherNumber> &rhs) const;
 
  template <typename OtherNumber>
  DEAL_II_CONSTEXPR DEAL_II_CUDA_HOST_DEV Tensor &
                                          operator+=(const Tensor<0, dim, OtherNumber> &rhs);
 
  template <typename OtherNumber>
  DEAL_II_CONSTEXPR DEAL_II_CUDA_HOST_DEV Tensor &
                                          operator-=(const Tensor<0, dim, OtherNumber> &rhs);
 
  template <typename OtherNumber>
  DEAL_II_CONSTEXPR DEAL_II_CUDA_HOST_DEV Tensor &
                                          operator*=(const OtherNumber &factor);
 
  template <typename OtherNumber>
  DEAL_II_CONSTEXPR DEAL_II_CUDA_HOST_DEV Tensor &
                                          operator/=(const OtherNumber &factor);
 
  constexpr DEAL_II_CUDA_HOST_DEV Tensor
                                  operator-() const;
 
  DEAL_II_CONSTEXPR void
  clear();
 
  real_type
  norm() const;
 
  DEAL_II_CONSTEXPR DEAL_II_CUDA_HOST_DEV real_type
                                          norm_square() const;
 
  template <class Archive>
  void
  serialize(Archive &ar, const unsigned int version);
 
  using tensor_type = Number;
 
private:
  Number value;
 
  template <typename OtherNumber>
  void
  unroll_recursion(Vector<OtherNumber> &result,
                   unsigned int &       start_index) const;
 
  // Allow an arbitrary Tensor to access the underlying values.
  template <int, int, typename>
  friend class Tensor;
};
 
 
 
template <int rank_, int dim, typename Number>
class Tensor
{
public:
  static_assert(rank_ >= 0,
                "Tensors must have a rank greater than or equal to one.");
  static_assert(dim >= 0,
                "Tensors must have a dimension greater than or equal to one.");
  static constexpr unsigned int dimension = dim;
 
  static constexpr unsigned int rank = rank_;
 
  static constexpr unsigned int n_independent_components =
    Tensor<rank_ - 1, dim>::n_independent_components * dim;
 
  using value_type = typename Tensor<rank_ - 1, dim, Number>::tensor_type;
 
  using array_type =
    typename Tensor<rank_ - 1, dim, Number>::array_type[(dim != 0) ? dim : 1];
 
  constexpr DEAL_II_ALWAYS_INLINE DEAL_II_CUDA_HOST_DEV
                                  Tensor()
#ifdef DEAL_II_MSVC
    : values{}
  {}
#else
    = default;
#endif
 
  constexpr DEAL_II_CUDA_HOST_DEV explicit Tensor(
    const array_type &initializer);
 
  template <typename OtherNumber>
  constexpr DEAL_II_CUDA_HOST_DEV
  Tensor(const Tensor<rank_, dim, OtherNumber> &initializer);
 
  template <typename OtherNumber>
  constexpr Tensor(
    const Tensor<1, dim, Tensor<rank_ - 1, dim, OtherNumber>> &initializer);
 
  template <typename OtherNumber>
  constexpr
  operator Tensor<1, dim, Tensor<rank_ - 1, dim, OtherNumber>>() const;
 
  DEAL_II_CONSTEXPR DEAL_II_CUDA_HOST_DEV value_type &
                                          operator[](const unsigned int i);
 
  constexpr DEAL_II_CUDA_HOST_DEV const value_type &
                                        operator[](const unsigned int i) const;
 
  DEAL_II_CONSTEXPR const Number &
                          operator[](const TableIndices<rank_> &indices) const;
 
  DEAL_II_CONSTEXPR Number &operator[](const TableIndices<rank_> &indices);
 
  Number *
  begin_raw();
 
  const Number *
  begin_raw() const;
 
  Number *
  end_raw();
 
  const Number *
  end_raw() const;
 
  template <typename OtherNumber>
  DEAL_II_CONSTEXPR DEAL_II_CUDA_HOST_DEV Tensor &
                                          operator=(const Tensor<rank_, dim, OtherNumber> &rhs);
 
  DEAL_II_CONSTEXPR Tensor &
                    operator=(const Number &d);
 
  template <typename OtherNumber>
  DEAL_II_CONSTEXPR bool
  operator==(const Tensor<rank_, dim, OtherNumber> &) const;
 
  template <typename OtherNumber>
  constexpr bool
  operator!=(const Tensor<rank_, dim, OtherNumber> &) const;
 
  template <typename OtherNumber>
  DEAL_II_CONSTEXPR DEAL_II_CUDA_HOST_DEV Tensor &
                                          operator+=(const Tensor<rank_, dim, OtherNumber> &);
 
  template <typename OtherNumber>
  DEAL_II_CONSTEXPR DEAL_II_CUDA_HOST_DEV Tensor &
                                          operator-=(const Tensor<rank_, dim, OtherNumber> &);
 
  template <typename OtherNumber>
  DEAL_II_CONSTEXPR DEAL_II_CUDA_HOST_DEV Tensor &
                                          operator*=(const OtherNumber &factor);
 
  template <typename OtherNumber>
  DEAL_II_CONSTEXPR DEAL_II_CUDA_HOST_DEV Tensor &
                                          operator/=(const OtherNumber &factor);
 
  DEAL_II_CONSTEXPR DEAL_II_CUDA_HOST_DEV Tensor
                                          operator-() const;
 
  DEAL_II_CONSTEXPR void
  clear();
 
  DEAL_II_CUDA_HOST_DEV
  typename numbers::NumberTraits<Number>::real_type
  norm() const;
 
  DEAL_II_CONSTEXPR DEAL_II_CUDA_HOST_DEV
    typename numbers::NumberTraits<Number>::real_type
    norm_square() const;
 
  template <typename OtherNumber>
  void
  unroll(Vector<OtherNumber> &result) const;
 
  static DEAL_II_CONSTEXPR unsigned int
  component_to_unrolled_index(const TableIndices<rank_> &indices);
 
  static DEAL_II_CONSTEXPR TableIndices<rank_>
                           unrolled_to_component_indices(const unsigned int i);
 
  static constexpr std::size_t
  memory_consumption();
 
  template <class Archive>
  void
  serialize(Archive &ar, const unsigned int version);
 
  using tensor_type = Tensor<rank_, dim, Number>;
 
private:
  Tensor<rank_ - 1, dim, Number> values[(dim != 0) ? dim : 1];
  // ... avoid a compiler warning in case of dim == 0 and ensure that the
  // array always has positive size.
 
  template <typename OtherNumber>
  void
  unroll_recursion(Vector<OtherNumber> &result,
                   unsigned int &       start_index) const;
 
  template <typename ArrayLike, std::size_t... Indices>
  constexpr DEAL_II_CUDA_HOST_DEV
  Tensor(const ArrayLike &initializer, std_cxx14::index_sequence<Indices...>);
 
  // Allow an arbitrary Tensor to access the underlying values.
  template <int, int, typename>
  friend class Tensor;
 
  // Point is allowed access to the coordinates. This is supposed to improve
  // speed.
  friend class Point<dim, Number>;
};
 
 
#ifndef DOXYGEN
namespace internal
{
  // Workaround: The following 4 overloads are necessary to be able to
  // compile the library with Apple Clang 8 and older. We should remove
  // these overloads again when we bump the minimal required version to
  // something later than clang-3.6 / Apple Clang 6.3.
  // - Jean-Paul Pelteret, Matthias Maier, Daniel Arndt 2020
  template <int rank, int dim, typename T, typename U>
  struct ProductTypeImpl<Tensor<rank, dim, T>, std::complex<U>>
  {
    using type =
      Tensor<rank, dim, std::complex<typename ProductType<T, U>::type>>;
  };
 
  template <int rank, int dim, typename T, typename U>
  struct ProductTypeImpl<Tensor<rank, dim, std::complex<T>>, std::complex<U>>
  {
    using type =
      Tensor<rank, dim, std::complex<typename ProductType<T, U>::type>>;
  };
 
  template <typename T, int rank, int dim, typename U>
  struct ProductTypeImpl<std::complex<T>, Tensor<rank, dim, U>>
  {
    using type =
      Tensor<rank, dim, std::complex<typename ProductType<T, U>::type>>;
  };
 
  template <int rank, int dim, typename T, typename U>
  struct ProductTypeImpl<std::complex<T>, Tensor<rank, dim, std::complex<U>>>
  {
    using type =
      Tensor<rank, dim, std::complex<typename ProductType<T, U>::type>>;
  };
  // end workaround
 
  template <int rank, int dim, typename T>
  struct NumberType<Tensor<rank, dim, T>>
  {
    static constexpr DEAL_II_ALWAYS_INLINE const Tensor<rank, dim, T> &
                                                 value(const Tensor<rank, dim, T> &t)
    {
      return t;
    }
 
    static DEAL_II_CONSTEXPR DEAL_II_ALWAYS_INLINE Tensor<rank, dim, T>
                                                   value(const T &t)
    {
      Tensor<rank, dim, T> tmp;
      tmp = t;
      return tmp;
    }
  };
} // namespace internal
 
 
/*---------------------- Inline functions: Tensor<0,dim> ---------------------*/
 
 
template <int dim, typename Number>
constexpr DEAL_II_ALWAYS_INLINE DEAL_II_CUDA_HOST_DEV
                                Tensor<0, dim, Number>::Tensor()
  // Some auto-differentiable numbers need explicit
  // zero initialization such as adtl::adouble.
  : Tensor{0.0}
{}
 
 
 
template <int dim, typename Number>
template <typename OtherNumber>
constexpr DEAL_II_ALWAYS_INLINE DEAL_II_CUDA_HOST_DEV
                                Tensor<0, dim, Number>::Tensor(const OtherNumber &initializer)
  : value(internal::NumberType<Number>::value(initializer))
{}
 
 
 
template <int dim, typename Number>
template <typename OtherNumber>
constexpr DEAL_II_ALWAYS_INLINE DEAL_II_CUDA_HOST_DEV
                                Tensor<0, dim, Number>::Tensor(const Tensor<0, dim, OtherNumber> &p)
  : Tensor{p.value}
{}
 
 
 
template <int dim, typename Number>
inline Number *
Tensor<0, dim, Number>::begin_raw()
{
  return std::addressof(value);
}
 
 
 
template <int dim, typename Number>
inline const Number *
Tensor<0, dim, Number>::begin_raw() const
{
  return std::addressof(value);
}
 
 
 
template <int dim, typename Number>
inline Number *
Tensor<0, dim, Number>::end_raw()
{
  return begin_raw() + n_independent_components;
}
 
 
 
template <int dim, typename Number>
const Number *
Tensor<0, dim, Number>::end_raw() const
{
  return begin_raw() + n_independent_components;
}
 
 
 
template <int dim, typename Number>
DEAL_II_CONSTEXPR inline DEAL_II_ALWAYS_INLINE
  DEAL_II_CUDA_HOST_DEV Tensor<0, dim, Number>::operator Number &()
{
  // We cannot use Assert inside a CUDA kernel
#  ifndef __CUDA_ARCH__
  Assert(dim != 0,
         ExcMessage("Cannot access an object of type Tensor<0,0,Number>"));
#  endif
  return value;
}
 
 
template <int dim, typename Number>
DEAL_II_CONSTEXPR inline DEAL_II_ALWAYS_INLINE
  DEAL_II_CUDA_HOST_DEV Tensor<0, dim, Number>::operator const Number &() const
{
  // We cannot use Assert inside a CUDA kernel
#  ifndef __CUDA_ARCH__
  Assert(dim != 0,
         ExcMessage("Cannot access an object of type Tensor<0,0,Number>"));
#  endif
  return value;
}
 
 
template <int dim, typename Number>
template <typename OtherNumber>
DEAL_II_CONSTEXPR inline DEAL_II_ALWAYS_INLINE
  DEAL_II_CUDA_HOST_DEV Tensor<0, dim, Number> &
  Tensor<0, dim, Number>::operator=(const Tensor<0, dim, OtherNumber> &p)
{
  value = internal::NumberType<Number>::value(p);
  return *this;
}
 
 
#  ifdef __INTEL_COMPILER
template <int dim, typename Number>
DEAL_II_CONSTEXPR inline DEAL_II_ALWAYS_INLINE
  DEAL_II_CUDA_HOST_DEV Tensor<0, dim, Number> &
  Tensor<0, dim, Number>::operator=(const Tensor<0, dim, Number> &p)
{
  value = p.value;
  return *this;
}
#  endif
 
 
template <int dim, typename Number>
template <typename OtherNumber>
DEAL_II_CONSTEXPR inline DEAL_II_ALWAYS_INLINE
  DEAL_II_CUDA_HOST_DEV Tensor<0, dim, Number> &
  Tensor<0, dim, Number>::operator=(const OtherNumber &d)
{
  value = internal::NumberType<Number>::value(d);
  return *this;
}
 
 
template <int dim, typename Number>
template <typename OtherNumber>
DEAL_II_CONSTEXPR inline bool
Tensor<0, dim, Number>::operator==(const Tensor<0, dim, OtherNumber> &p) const
{
#  if defined(DEAL_II_ADOLC_WITH_ADVANCED_BRANCHING)
  Assert(!(std::is_same<Number, adouble>::value ||
           std::is_same<OtherNumber, adouble>::value),
         ExcMessage(
           "The Tensor equality operator for ADOL-C taped numbers has not yet "
           "been extended to support advanced branching."));
#  endif
 
  return numbers::values_are_equal(value, p.value);
}
 
 
template <int dim, typename Number>
template <typename OtherNumber>
constexpr bool
Tensor<0, dim, Number>::operator!=(const Tensor<0, dim, OtherNumber> &p) const
{
  return !((*this) == p);
}
 
 
template <int dim, typename Number>
template <typename OtherNumber>
DEAL_II_CONSTEXPR inline DEAL_II_ALWAYS_INLINE
  DEAL_II_CUDA_HOST_DEV Tensor<0, dim, Number> &
  Tensor<0, dim, Number>::operator+=(const Tensor<0, dim, OtherNumber> &p)
{
  value += p.value;
  return *this;
}
 
 
template <int dim, typename Number>
template <typename OtherNumber>
DEAL_II_CONSTEXPR inline DEAL_II_ALWAYS_INLINE
  DEAL_II_CUDA_HOST_DEV Tensor<0, dim, Number> &
  Tensor<0, dim, Number>::operator-=(const Tensor<0, dim, OtherNumber> &p)
{
  value -= p.value;
  return *this;
}
 
 
 
namespace internal
{
  namespace ComplexWorkaround
  {
    template <typename Number, typename OtherNumber>
    DEAL_II_CONSTEXPR inline DEAL_II_ALWAYS_INLINE DEAL_II_CUDA_HOST_DEV void
                                                   multiply_assign_scalar(Number &val, const OtherNumber &s)
    {
      val *= s;
    }
 
#  ifdef __CUDA_ARCH__
    template <typename Number, typename OtherNumber>
    DEAL_II_CONSTEXPR inline DEAL_II_ALWAYS_INLINE DEAL_II_CUDA_HOST_DEV void
                                                   multiply_assign_scalar(std::complex<Number> &, const OtherNumber &)
    {
      printf("This function is not implemented for std::complex<Number>!\n");
      assert(false);
    }
#  endif
  } // namespace ComplexWorkaround
} // namespace internal
 
 
template <int dim, typename Number>
template <typename OtherNumber>
DEAL_II_CONSTEXPR inline DEAL_II_ALWAYS_INLINE
  DEAL_II_CUDA_HOST_DEV Tensor<0, dim, Number> &
  Tensor<0, dim, Number>::operator*=(const OtherNumber &s)
{
  internal::ComplexWorkaround::multiply_assign_scalar(value, s);
  return *this;
}
 
 
 
template <int dim, typename Number>
template <typename OtherNumber>
DEAL_II_CONSTEXPR inline DEAL_II_CUDA_HOST_DEV Tensor<0, dim, Number> &
Tensor<0, dim, Number>::operator/=(const OtherNumber &s)
{
  value /= s;
  return *this;
}
 
 
template <int dim, typename Number>
constexpr DEAL_II_ALWAYS_INLINE DEAL_II_CUDA_HOST_DEV Tensor<0, dim, Number>
Tensor<0, dim, Number>::operator-() const
{
  return -value;
}
 
 
template <int dim, typename Number>
inline DEAL_II_ALWAYS_INLINE typename Tensor<0, dim, Number>::real_type
Tensor<0, dim, Number>::norm() const
{
  Assert(dim != 0,
         ExcMessage("Cannot access an object of type Tensor<0,0,Number>"));
  return numbers::NumberTraits<Number>::abs(value);
}
 
 
template <int dim, typename Number>
DEAL_II_CONSTEXPR DEAL_II_CUDA_HOST_DEV inline DEAL_II_ALWAYS_INLINE
  typename Tensor<0, dim, Number>::real_type
  Tensor<0, dim, Number>::norm_square() const
{
  // We cannot use Assert inside a CUDA kernel
#  ifndef __CUDA_ARCH__
  Assert(dim != 0,
         ExcMessage("Cannot access an object of type Tensor<0,0,Number>"));
#  endif
  return numbers::NumberTraits<Number>::abs_square(value);
}
 
 
template <int dim, typename Number>
template <typename OtherNumber>
inline void
Tensor<0, dim, Number>::unroll_recursion(Vector<OtherNumber> &result,
                                         unsigned int &       index) const
{
  Assert(dim != 0,
         ExcMessage("Cannot unroll an object of type Tensor<0,0,Number>"));
  result[index] = value;
  ++index;
}
 
 
template <int dim, typename Number>
DEAL_II_CONSTEXPR inline void
Tensor<0, dim, Number>::clear()
{
  // Some auto-differentiable numbers need explicit
  // zero initialization.
  value = internal::NumberType<Number>::value(0.0);
}
 
 
template <int dim, typename Number>
template <class Archive>
inline void
Tensor<0, dim, Number>::serialize(Archive &ar, const unsigned int)
{
  ar &value;
}
 
 
template <int dim, typename Number>
constexpr unsigned int Tensor<0, dim, Number>::n_independent_components;
 
 
/*-------------------- Inline functions: Tensor<rank,dim> --------------------*/
 
template <int rank_, int dim, typename Number>
template <typename ArrayLike, std::size_t... indices>
constexpr DEAL_II_ALWAYS_INLINE DEAL_II_CUDA_HOST_DEV
                                Tensor<rank_, dim, Number>::Tensor(const ArrayLike &initializer,
                                   std_cxx14::index_sequence<indices...>)
  : values{Tensor<rank_ - 1, dim, Number>(initializer[indices])...}
{
  static_assert(sizeof...(indices) == dim,
                "dim should match the number of indices");
}
 
 
template <int rank_, int dim, typename Number>
constexpr DEAL_II_ALWAYS_INLINE DEAL_II_CUDA_HOST_DEV
                                Tensor<rank_, dim, Number>::Tensor(const array_type &initializer)
  : Tensor(initializer, std_cxx14::make_index_sequence<dim>{})
{}
 
 
template <int rank_, int dim, typename Number>
template <typename OtherNumber>
constexpr DEAL_II_ALWAYS_INLINE DEAL_II_CUDA_HOST_DEV
                                Tensor<rank_, dim, Number>::Tensor(
  const Tensor<rank_, dim, OtherNumber> &initializer)
  : Tensor(initializer, std_cxx14::make_index_sequence<dim>{})
{}
 
 
template <int rank_, int dim, typename Number>
template <typename OtherNumber>
constexpr DEAL_II_ALWAYS_INLINE
Tensor<rank_, dim, Number>::Tensor(
  const Tensor<1, dim, Tensor<rank_ - 1, dim, OtherNumber>> &initializer)
  : Tensor(initializer, std_cxx14::make_index_sequence<dim>{})
{}
 
 
template <int rank_, int dim, typename Number>
template <typename OtherNumber>
constexpr DEAL_II_ALWAYS_INLINE Tensor<rank_, dim, Number>::
                                operator Tensor<1, dim, Tensor<rank_ - 1, dim, OtherNumber>>() const
{
  return Tensor<1, dim, Tensor<rank_ - 1, dim, Number>>(values);
}
 
 
 
namespace internal
{
  namespace TensorSubscriptor
  {
    template <typename ArrayElementType, int dim>
    DEAL_II_CONSTEXPR inline DEAL_II_ALWAYS_INLINE
      DEAL_II_CUDA_HOST_DEV ArrayElementType &
                            subscript(ArrayElementType * values,
                                      const unsigned int i,
                                      std::integral_constant<int, dim>)
    {
      // We cannot use Assert in a CUDA kernel
#  ifndef __CUDA_ARCH__
      AssertIndexRange(i, dim);
#  endif
      return values[i];
    }
 
    // The variables within this struct will be referenced in the next function.
    // It is a workaround that allows returning a reference to a static variable
    // while allowing constexpr evaluation of the function.
    // It has to be defined outside the function because constexpr functions
    // cannot define static variables
    template <typename ArrayElementType>
    struct Uninitialized
    {
      static ArrayElementType value;
    };
 
    template <typename Type>
    Type Uninitialized<Type>::value;
 
    template <typename ArrayElementType>
    DEAL_II_CONSTEXPR inline DEAL_II_ALWAYS_INLINE
      DEAL_II_CUDA_HOST_DEV ArrayElementType &
                            subscript(ArrayElementType *,
                                      const unsigned int,
                                      std::integral_constant<int, 0>)
    {
      // We cannot use Assert in a CUDA kernel
#  ifndef __CUDA_ARCH__
      Assert(
        false,
        ExcMessage(
          "Cannot access elements of an object of type Tensor<rank,0,Number>."));
#  endif
      return Uninitialized<ArrayElementType>::value;
    }
  } // namespace TensorSubscriptor
} // namespace internal
 
 
template <int rank_, int dim, typename Number>
DEAL_II_CONSTEXPR inline DEAL_II_ALWAYS_INLINE     DEAL_II_CUDA_HOST_DEV
  typename Tensor<rank_, dim, Number>::value_type &Tensor<rank_, dim, Number>::
                                                   operator[](const unsigned int i)
{
  return ::internal::TensorSubscriptor::subscript(
    values, i, std::integral_constant<int, dim>());
}
 
 
template <int rank_, int dim, typename Number>
constexpr DEAL_II_ALWAYS_INLINE
    DEAL_II_CUDA_HOST_DEV const typename Tensor<rank_, dim, Number>::value_type &
    Tensor<rank_, dim, Number>::operator[](const unsigned int i) const
{
  return values[i];
}
 
 
template <int rank_, int dim, typename Number>
DEAL_II_CONSTEXPR inline DEAL_II_ALWAYS_INLINE const Number &
                                                     Tensor<rank_, dim, Number>::
                                                     operator[](const TableIndices<rank_> &indices) const
{
  Assert(dim != 0,
         ExcMessage("Cannot access an object of type Tensor<rank_,0,Number>"));
 
  return TensorAccessors::extract<rank_>(*this, indices);
}
 
 
 
template <int rank_, int dim, typename Number>
DEAL_II_CONSTEXPR inline DEAL_II_ALWAYS_INLINE Number &
  Tensor<rank_, dim, Number>::operator[](const TableIndices<rank_> &indices)
{
  Assert(dim != 0,
         ExcMessage("Cannot access an object of type Tensor<rank_,0,Number>"));
 
  return TensorAccessors::extract<rank_>(*this, indices);
}
 
 
 
template <int rank_, int dim, typename Number>
inline Number *
Tensor<rank_, dim, Number>::begin_raw()
{
  return std::addressof(
    this->operator[](this->unrolled_to_component_indices(0)));
}
 
 
 
template <int rank_, int dim, typename Number>
inline const Number *
Tensor<rank_, dim, Number>::begin_raw() const
{
  return std::addressof(
    this->operator[](this->unrolled_to_component_indices(0)));
}
 
 
 
template <int rank_, int dim, typename Number>
inline Number *
Tensor<rank_, dim, Number>::end_raw()
{
  return begin_raw() + n_independent_components;
}
 
 
 
template <int rank_, int dim, typename Number>
inline const Number *
Tensor<rank_, dim, Number>::end_raw() const
{
  return begin_raw() + n_independent_components;
}
 
 
 
template <int rank_, int dim, typename Number>
template <typename OtherNumber>
DEAL_II_CONSTEXPR inline DEAL_II_ALWAYS_INLINE Tensor<rank_, dim, Number> &
Tensor<rank_, dim, Number>::operator=(const Tensor<rank_, dim, OtherNumber> &t)
{
  // The following loop could be written more concisely using std::copy, but
  // that function is only constexpr from C++20 on.
  for (unsigned int i = 0; i < dim; ++i)
    values[i] = t.values[i];
  return *this;
}
 
 
template <int rank_, int dim, typename Number>
DEAL_II_CONSTEXPR inline DEAL_II_ALWAYS_INLINE Tensor<rank_, dim, Number> &
Tensor<rank_, dim, Number>::operator=(const Number &d)
{
  Assert(numbers::value_is_zero(d),
         ExcMessage("Only assignment with zero is allowed"));
  (void)d;
 
  for (unsigned int i = 0; i < dim; ++i)
    values[i] = internal::NumberType<Number>::value(0.0);
  return *this;
}
 
 
template <int rank_, int dim, typename Number>
template <typename OtherNumber>
DEAL_II_CONSTEXPR inline bool
Tensor<rank_, dim, Number>::
operator==(const Tensor<rank_, dim, OtherNumber> &p) const
{
  for (unsigned int i = 0; i < dim; ++i)
    if (values[i] != p.values[i])
      return false;
  return true;
}
 
 
// At some places in the library, we have Point<0> for formal reasons
// (e.g., we sometimes have Quadrature<dim-1> for faces, so we have
// Quadrature<0> for dim=1, and then we have Point<0>). To avoid warnings
// in the above function that the loop end check always fails, we
// implement this function here
template <>
template <>
DEAL_II_CONSTEXPR inline bool
Tensor<1, 0, double>::operator==(const Tensor<1, 0, double> &) const
{
  return true;
}
 
 
template <int rank_, int dim, typename Number>
template <typename OtherNumber>
constexpr bool
Tensor<rank_, dim, Number>::
operator!=(const Tensor<rank_, dim, OtherNumber> &p) const
{
  return !((*this) == p);
}
 
 
template <int rank_, int dim, typename Number>
template <typename OtherNumber>
DEAL_II_CONSTEXPR inline DEAL_II_ALWAYS_INLINE
  DEAL_II_CUDA_HOST_DEV Tensor<rank_, dim, Number> &
                        Tensor<rank_, dim, Number>::
                        operator+=(const Tensor<rank_, dim, OtherNumber> &p)
{
  for (unsigned int i = 0; i < dim; ++i)
    values[i] += p.values[i];
  return *this;
}
 
 
template <int rank_, int dim, typename Number>
template <typename OtherNumber>
DEAL_II_CONSTEXPR inline DEAL_II_ALWAYS_INLINE
  DEAL_II_CUDA_HOST_DEV Tensor<rank_, dim, Number> &
                        Tensor<rank_, dim, Number>::
                        operator-=(const Tensor<rank_, dim, OtherNumber> &p)
{
  for (unsigned int i = 0; i < dim; ++i)
    values[i] -= p.values[i];
  return *this;
}
 
 
template <int rank_, int dim, typename Number>
template <typename OtherNumber>
DEAL_II_CONSTEXPR inline DEAL_II_ALWAYS_INLINE
  DEAL_II_CUDA_HOST_DEV Tensor<rank_, dim, Number> &
  Tensor<rank_, dim, Number>::operator*=(const OtherNumber &s)
{
  for (unsigned int i = 0; i < dim; ++i)
    values[i] *= s;
  return *this;
}
 
 
namespace internal
{
  namespace TensorImplementation
  {
    template <int rank,
              int dim,
              typename Number,
              typename OtherNumber,
              typename std::enable_if<
                !std::is_integral<
                  typename ProductType<Number, OtherNumber>::type>::value &&
                  !std::is_same<Number, Differentiation::SD::Expression>::value,
                int>::type = 0>
    DEAL_II_CONSTEXPR DEAL_II_CUDA_HOST_DEV inline DEAL_II_ALWAYS_INLINE void
                      division_operator(Tensor<rank, dim, Number> (&t)[dim],
                                        const OtherNumber &factor)
    {
      const Number inverse_factor = Number(1.) / factor;
      // recurse over the base objects
      for (unsigned int d = 0; d < dim; ++d)
        t[d] *= inverse_factor;
    }
 
 
    template <int rank,
              int dim,
              typename Number,
              typename OtherNumber,
              typename std::enable_if<
                std::is_integral<
                  typename ProductType<Number, OtherNumber>::type>::value ||
                  std::is_same<Number, Differentiation::SD::Expression>::value,
                int>::type = 0>
    DEAL_II_CONSTEXPR DEAL_II_CUDA_HOST_DEV inline DEAL_II_ALWAYS_INLINE void
                      division_operator(Tensor<rank, dim, Number> (&t)[dim],
                                        const OtherNumber &factor)
    {
      // recurse over the base objects
      for (unsigned int d = 0; d < dim; ++d)
        t[d] /= factor;
    }
  } // namespace TensorImplementation
} // namespace internal
 
 
template <int rank_, int dim, typename Number>
template <typename OtherNumber>
DEAL_II_CONSTEXPR inline DEAL_II_ALWAYS_INLINE
  DEAL_II_CUDA_HOST_DEV Tensor<rank_, dim, Number> &
  Tensor<rank_, dim, Number>::operator/=(const OtherNumber &s)
{
  internal::TensorImplementation::division_operator(values, s);
  return *this;
}
 
 
template <int rank_, int dim, typename Number>
DEAL_II_CONSTEXPR inline DEAL_II_ALWAYS_INLINE
  DEAL_II_CUDA_HOST_DEV Tensor<rank_, dim, Number>
  Tensor<rank_, dim, Number>::operator-() const
{
  Tensor<rank_, dim, Number> tmp;
 
  for (unsigned int i = 0; i < dim; ++i)
    tmp.values[i] = -values[i];
 
  return tmp;
}
 
 
template <int rank_, int dim, typename Number>
inline typename numbers::NumberTraits<Number>::real_type
Tensor<rank_, dim, Number>::norm() const
{
  return std::sqrt(norm_square());
}
 
 
template <int rank_, int dim, typename Number>
DEAL_II_CONSTEXPR inline DEAL_II_ALWAYS_INLINE DEAL_II_CUDA_HOST_DEV
  typename numbers::NumberTraits<Number>::real_type
  Tensor<rank_, dim, Number>::norm_square() const
{
  typename numbers::NumberTraits<Number>::real_type s = internal::NumberType<
    typename numbers::NumberTraits<Number>::real_type>::value(0.0);
  for (unsigned int i = 0; i < dim; ++i)
    s += values[i].norm_square();
 
  return s;
}
 
 
template <int rank_, int dim, typename Number>
template <typename OtherNumber>
inline void
Tensor<rank_, dim, Number>::unroll(Vector<OtherNumber> &result) const
{
  AssertDimension(result.size(),
                  (Utilities::fixed_power<rank_, unsigned int>(dim)));
 
  unsigned int index = 0;
  unroll_recursion(result, index);
}
 
 
template <int rank_, int dim, typename Number>
template <typename OtherNumber>
inline void
Tensor<rank_, dim, Number>::unroll_recursion(Vector<OtherNumber> &result,
                                             unsigned int &       index) const
{
  for (unsigned int i = 0; i < dim; ++i)
    values[i].unroll_recursion(result, index);
}
 
 
template <int rank_, int dim, typename Number>
DEAL_II_CONSTEXPR inline unsigned int
Tensor<rank_, dim, Number>::component_to_unrolled_index(
  const TableIndices<rank_> &indices)
{
  unsigned int index = 0;
  for (int r = 0; r < rank_; ++r)
    index = index * dim + indices[r];
 
  return index;
}
 
 
 
namespace internal
{
  // unrolled_to_component_indices is instantiated from DataOut for dim==0
  // and rank=2. Make sure we don't have compiler warnings.
 
  template <int dim>
  inline DEAL_II_CONSTEXPR unsigned int
  mod(const unsigned int x)
  {
    return x % dim;
  }
 
  template <>
  inline unsigned int
  mod<0>(const unsigned int x)
  {
    Assert(false, ExcInternalError());
    return x;
  }
 
  template <int dim>
  inline DEAL_II_CONSTEXPR unsigned int
  div(const unsigned int x)
  {
    return x / dim;
  }
 
  template <>
  inline unsigned int
  div<0>(const unsigned int x)
  {
    Assert(false, ExcInternalError());
    return x;
  }
 
} // namespace internal
 
 
 
template <int rank_, int dim, typename Number>
DEAL_II_CONSTEXPR inline TableIndices<rank_>
Tensor<rank_, dim, Number>::unrolled_to_component_indices(const unsigned int i)
{
  AssertIndexRange(i, n_independent_components);
 
  TableIndices<rank_> indices;
 
  unsigned int remainder = i;
  for (int r = rank_ - 1; r >= 0; --r)
    {
      indices[r] = internal::mod<dim>(remainder);
      remainder  = internal::div<dim>(remainder);
    }
  Assert(remainder == 0, ExcInternalError());
 
  return indices;
}
 
 
template <int rank_, int dim, typename Number>
DEAL_II_CONSTEXPR inline void
Tensor<rank_, dim, Number>::clear()
{
  for (unsigned int i = 0; i < dim; ++i)
    values[i] = internal::NumberType<Number>::value(0.0);
}
 
 
template <int rank_, int dim, typename Number>
constexpr std::size_t
Tensor<rank_, dim, Number>::memory_consumption()
{
  return sizeof(Tensor<rank_, dim, Number>);
}
 
 
template <int rank_, int dim, typename Number>
template <class Archive>
inline void
Tensor<rank_, dim, Number>::serialize(Archive &ar, const unsigned int)
{
  ar &values;
}
 
 
template <int rank_, int dim, typename Number>
constexpr unsigned int Tensor<rank_, dim, Number>::n_independent_components;
 
#endif // DOXYGEN
 
/* ----------------- Non-member functions operating on tensors. ------------ */
 
 
template <int rank_, int dim, typename Number>
inline std::ostream &
operator<<(std::ostream &out, const Tensor<rank_, dim, Number> &p)
{
  for (unsigned int i = 0; i < dim; ++i)
    {
      out << p[i];
      if (i != dim - 1)
        out << ' ';
    }
 
  return out;
}
 
 
template <int dim, typename Number>
inline std::ostream &
operator<<(std::ostream &out, const Tensor<0, dim, Number> &p)
{
  out << static_cast<const Number &>(p);
  return out;
}
 
 
 
 
 
template <int dim, typename Number, typename Other>
DEAL_II_CONSTEXPR DEAL_II_CUDA_HOST_DEV inline DEAL_II_ALWAYS_INLINE
  typename ProductType<Other, Number>::type
  operator*(const Other &object, const Tensor<0, dim, Number> &t)
{
  return object * static_cast<const Number &>(t);
}
 
 
 
template <int dim, typename Number, typename Other>
DEAL_II_CONSTEXPR DEAL_II_CUDA_HOST_DEV inline DEAL_II_ALWAYS_INLINE
  typename ProductType<Number, Other>::type
  operator*(const Tensor<0, dim, Number> &t, const Other &object)
{
  return static_cast<const Number &>(t) * object;
}
 
 
template <int dim, typename Number, typename OtherNumber>
DEAL_II_CUDA_HOST_DEV constexpr DEAL_II_ALWAYS_INLINE
  typename ProductType<Number, OtherNumber>::type
  operator*(const Tensor<0, dim, Number> &     src1,
            const Tensor<0, dim, OtherNumber> &src2)
{
  return static_cast<const Number &>(src1) *
         static_cast<const OtherNumber &>(src2);
}
 
 
template <int dim, typename Number, typename OtherNumber>
DEAL_II_CUDA_HOST_DEV constexpr DEAL_II_ALWAYS_INLINE
  Tensor<0,
         dim,
         typename ProductType<Number,
                              typename EnableIfScalar<OtherNumber>::type>::type>
  operator/(const Tensor<0, dim, Number> &t, const OtherNumber &factor)
{
  return static_cast<const Number &>(t) / factor;
}
 
 
template <int dim, typename Number, typename OtherNumber>
constexpr DEAL_II_ALWAYS_INLINE DEAL_II_CUDA_HOST_DEV
                                Tensor<0, dim, typename ProductType<Number, OtherNumber>::type>
                                operator+(const Tensor<0, dim, Number> &     p,
            const Tensor<0, dim, OtherNumber> &q)
{
  return static_cast<const Number &>(p) + static_cast<const OtherNumber &>(q);
}
 
 
template <int dim, typename Number, typename OtherNumber>
constexpr DEAL_II_ALWAYS_INLINE DEAL_II_CUDA_HOST_DEV
                                Tensor<0, dim, typename ProductType<Number, OtherNumber>::type>
                                operator-(const Tensor<0, dim, Number> &     p,
            const Tensor<0, dim, OtherNumber> &q)
{
  return static_cast<const Number &>(p) - static_cast<const OtherNumber &>(q);
}
 
 
template <int rank, int dim, typename Number, typename OtherNumber>
DEAL_II_CONSTEXPR DEAL_II_CUDA_HOST_DEV inline DEAL_II_ALWAYS_INLINE
                  Tensor<rank,
         dim,
         typename ProductType<Number,
                              typename EnableIfScalar<OtherNumber>::type>::type>
                  operator*(const Tensor<rank, dim, Number> &t, const OtherNumber &factor)
{
  // recurse over the base objects
  Tensor<rank, dim, typename ProductType<Number, OtherNumber>::type> tt;
  for (unsigned int d = 0; d < dim; ++d)
    tt[d] = t[d] * factor;
  return tt;
}
 
 
template <int rank, int dim, typename Number, typename OtherNumber>
DEAL_II_CUDA_HOST_DEV DEAL_II_CONSTEXPR inline DEAL_II_ALWAYS_INLINE
                      Tensor<rank,
         dim,
         typename ProductType<typename EnableIfScalar<Number>::type,
                              OtherNumber>::type>
                      operator*(const Number &factor, const Tensor<rank, dim, OtherNumber> &t)
{
  // simply forward to the operator above
  return t * factor;
}
 
 
namespace internal
{
  namespace TensorImplementation
  {
    template <int rank,
              int dim,
              typename Number,
              typename OtherNumber,
              typename std::enable_if<
                !std::is_integral<
                  typename ProductType<Number, OtherNumber>::type>::value,
                int>::type = 0>
    DEAL_II_CONSTEXPR DEAL_II_CUDA_HOST_DEV inline DEAL_II_ALWAYS_INLINE
                      Tensor<rank, dim, typename ProductType<Number, OtherNumber>::type>
                      division_operator(const Tensor<rank, dim, Number> &t,
                                        const OtherNumber &              factor)
    {
      Tensor<rank, dim, typename ProductType<Number, OtherNumber>::type> tt;
      const Number inverse_factor = Number(1.) / factor;
      // recurse over the base objects
      for (unsigned int d = 0; d < dim; ++d)
        tt[d] = t[d] * inverse_factor;
      return tt;
    }
 
 
    template <int rank,
              int dim,
              typename Number,
              typename OtherNumber,
              typename std::enable_if<
                std::is_integral<
                  typename ProductType<Number, OtherNumber>::type>::value,
                int>::type = 0>
    DEAL_II_CONSTEXPR DEAL_II_CUDA_HOST_DEV inline DEAL_II_ALWAYS_INLINE
                      Tensor<rank, dim, typename ProductType<Number, OtherNumber>::type>
                      division_operator(const Tensor<rank, dim, Number> &t,
                                        const OtherNumber &              factor)
    {
      Tensor<rank, dim, typename ProductType<Number, OtherNumber>::type> tt;
      // recurse over the base objects
      for (unsigned int d = 0; d < dim; ++d)
        tt[d] = t[d] / factor;
      return tt;
    }
  } // namespace TensorImplementation
} // namespace internal
 
 
template <int rank, int dim, typename Number, typename OtherNumber>
DEAL_II_CONSTEXPR DEAL_II_CUDA_HOST_DEV inline DEAL_II_ALWAYS_INLINE
                  Tensor<rank,
         dim,
         typename ProductType<Number,
                              typename EnableIfScalar<OtherNumber>::type>::type>
                  operator/(const Tensor<rank, dim, Number> &t, const OtherNumber &factor)
{
  return internal::TensorImplementation::division_operator(t, factor);
}
 
 
template <int rank, int dim, typename Number, typename OtherNumber>
DEAL_II_CONSTEXPR DEAL_II_CUDA_HOST_DEV inline DEAL_II_ALWAYS_INLINE
                  Tensor<rank, dim, typename ProductType<Number, OtherNumber>::type>
                  operator+(const Tensor<rank, dim, Number> &     p,
            const Tensor<rank, dim, OtherNumber> &q)
{
  Tensor<rank, dim, typename ProductType<Number, OtherNumber>::type> tmp(p);
 
  for (unsigned int i = 0; i < dim; ++i)
    tmp[i] += q[i];
 
  return tmp;
}
 
 
template <int rank, int dim, typename Number, typename OtherNumber>
DEAL_II_CONSTEXPR DEAL_II_CUDA_HOST_DEV inline DEAL_II_ALWAYS_INLINE
                  Tensor<rank, dim, typename ProductType<Number, OtherNumber>::type>
                  operator-(const Tensor<rank, dim, Number> &     p,
            const Tensor<rank, dim, OtherNumber> &q)
{
  Tensor<rank, dim, typename ProductType<Number, OtherNumber>::type> tmp(p);
 
  for (unsigned int i = 0; i < dim; ++i)
    tmp[i] -= q[i];
 
  return tmp;
}
 
template <int dim, typename Number, typename OtherNumber>
inline DEAL_II_CONSTEXPR DEAL_II_ALWAYS_INLINE
                         Tensor<0, dim, typename ProductType<Number, OtherNumber>::type>
                         schur_product(const Tensor<0, dim, Number> &     src1,
                                       const Tensor<0, dim, OtherNumber> &src2)
{
  Tensor<0, dim, typename ProductType<Number, OtherNumber>::type> tmp(src1);
 
  tmp *= src2;
 
  return tmp;
}
 
template <int rank, int dim, typename Number, typename OtherNumber>
inline DEAL_II_CONSTEXPR DEAL_II_ALWAYS_INLINE
                         Tensor<rank, dim, typename ProductType<Number, OtherNumber>::type>
                         schur_product(const Tensor<rank, dim, Number> &     src1,
                                       const Tensor<rank, dim, OtherNumber> &src2)
{
  Tensor<rank, dim, typename ProductType<Number, OtherNumber>::type> tmp;
 
  for (unsigned int i = 0; i < dim; ++i)
    tmp[i] = schur_product(Tensor<rank - 1, dim, Number>(src1[i]),
                           Tensor<rank - 1, dim, OtherNumber>(src2[i]));
 
  return tmp;
}
 
 
 
 
template <int rank_1,
          int rank_2,
          int dim,
          typename Number,
          typename OtherNumber,
          typename = typename std::enable_if<rank_1 >= 1 && rank_2 >= 1>::type>
DEAL_II_CONSTEXPR inline DEAL_II_ALWAYS_INLINE
  typename Tensor<rank_1 + rank_2 - 2,
                  dim,
                  typename ProductType<Number, OtherNumber>::type>::tensor_type
  operator*(const Tensor<rank_1, dim, Number> &     src1,
            const Tensor<rank_2, dim, OtherNumber> &src2)
{
  typename Tensor<rank_1 + rank_2 - 2,
                  dim,
                  typename ProductType<Number, OtherNumber>::type>::tensor_type
    result{};
 
  TensorAccessors::internal::
    ReorderedIndexView<0, rank_2, const Tensor<rank_2, dim, OtherNumber>>
      reordered = TensorAccessors::reordered_index_view<0, rank_2>(src2);
  TensorAccessors::contract<1, rank_1, rank_2, dim>(result, src1, reordered);
 
  return result;
}
 
 
template <int index_1,
          int index_2,
          int rank_1,
          int rank_2,
          int dim,
          typename Number,
          typename OtherNumber>
DEAL_II_CONSTEXPR inline DEAL_II_ALWAYS_INLINE
  typename Tensor<rank_1 + rank_2 - 2,
                  dim,
                  typename ProductType<Number, OtherNumber>::type>::tensor_type
  contract(const Tensor<rank_1, dim, Number> &     src1,
           const Tensor<rank_2, dim, OtherNumber> &src2)
{
  Assert(0 <= index_1 && index_1 < rank_1,
         ExcMessage(
           "The specified index_1 must lie within the range [0,rank_1)"));
  Assert(0 <= index_2 && index_2 < rank_2,
         ExcMessage(
           "The specified index_2 must lie within the range [0,rank_2)"));
 
  using namespace TensorAccessors;
  using namespace TensorAccessors::internal;
 
  // Reorder index_1 to the end of src1:
  ReorderedIndexView<index_1, rank_1, const Tensor<rank_1, dim, Number>>
    reord_01 = reordered_index_view<index_1, rank_1>(src1);
 
  // Reorder index_2 to the end of src2:
  ReorderedIndexView<index_2, rank_2, const Tensor<rank_2, dim, OtherNumber>>
    reord_02 = reordered_index_view<index_2, rank_2>(src2);
 
  typename Tensor<rank_1 + rank_2 - 2,
                  dim,
                  typename ProductType<Number, OtherNumber>::type>::tensor_type
    result{};
  TensorAccessors::contract<1, rank_1, rank_2, dim>(result, reord_01, reord_02);
  return result;
}
 
 
template <int index_1,
          int index_2,
          int index_3,
          int index_4,
          int rank_1,
          int rank_2,
          int dim,
          typename Number,
          typename OtherNumber>
DEAL_II_CONSTEXPR inline
  typename Tensor<rank_1 + rank_2 - 4,
                  dim,
                  typename ProductType<Number, OtherNumber>::type>::tensor_type
  double_contract(const Tensor<rank_1, dim, Number> &     src1,
                  const Tensor<rank_2, dim, OtherNumber> &src2)
{
  Assert(0 <= index_1 && index_1 < rank_1,
         ExcMessage(
           "The specified index_1 must lie within the range [0,rank_1)"));
  Assert(0 <= index_3 && index_3 < rank_1,
         ExcMessage(
           "The specified index_3 must lie within the range [0,rank_1)"));
  Assert(index_1 != index_3,
         ExcMessage("index_1 and index_3 must not be the same"));
  Assert(0 <= index_2 && index_2 < rank_2,
         ExcMessage(
           "The specified index_2 must lie within the range [0,rank_2)"));
  Assert(0 <= index_4 && index_4 < rank_2,
         ExcMessage(
           "The specified index_4 must lie within the range [0,rank_2)"));
  Assert(index_2 != index_4,
         ExcMessage("index_2 and index_4 must not be the same"));
 
  using namespace TensorAccessors;
  using namespace TensorAccessors::internal;
 
  // Reorder index_1 to the end of src1:
  ReorderedIndexView<index_1, rank_1, const Tensor<rank_1, dim, Number>>
    reord_1 = TensorAccessors::reordered_index_view<index_1, rank_1>(src1);
 
  // Reorder index_2 to the end of src2:
  ReorderedIndexView<index_2, rank_2, const Tensor<rank_2, dim, OtherNumber>>
    reord_2 = TensorAccessors::reordered_index_view<index_2, rank_2>(src2);
 
  // Now, reorder index_3 to the end of src1. We have to make sure to
  // preserve the original ordering: index_1 has been removed. If
  // index_3 > index_1, we have to use (index_3 - 1) instead:
  ReorderedIndexView<
    (index_3 < index_1 ? index_3 : index_3 - 1),
    rank_1,
    ReorderedIndexView<index_1, rank_1, const Tensor<rank_1, dim, Number>>>
    reord_3 =
      TensorAccessors::reordered_index_view < index_3 < index_1 ? index_3 :
                                                                  index_3 - 1,
    rank_1 > (reord_1);
 
  // Now, reorder index_4 to the end of src2. We have to make sure to
  // preserve the original ordering: index_2 has been removed. If
  // index_4 > index_2, we have to use (index_4 - 1) instead:
  ReorderedIndexView<
    (index_4 < index_2 ? index_4 : index_4 - 1),
    rank_2,
    ReorderedIndexView<index_2, rank_2, const Tensor<rank_2, dim, OtherNumber>>>
    reord_4 =
      TensorAccessors::reordered_index_view < index_4 < index_2 ? index_4 :
                                                                  index_4 - 1,
    rank_2 > (reord_2);
 
  typename Tensor<rank_1 + rank_2 - 4,
                  dim,
                  typename ProductType<Number, OtherNumber>::type>::tensor_type
    result{};
  TensorAccessors::contract<2, rank_1, rank_2, dim>(result, reord_3, reord_4);
  return result;
}
 
 
template <int rank, int dim, typename Number, typename OtherNumber>
DEAL_II_CONSTEXPR inline DEAL_II_ALWAYS_INLINE
  typename ProductType<Number, OtherNumber>::type
  scalar_product(const Tensor<rank, dim, Number> &     left,
                 const Tensor<rank, dim, OtherNumber> &right)
{
  typename ProductType<Number, OtherNumber>::type result{};
  TensorAccessors::contract<rank, rank, rank, dim>(result, left, right);
  return result;
}
 
 
template <template <int, int, typename> class TensorT1,
          template <int, int, typename> class TensorT2,
          template <int, int, typename> class TensorT3,
          int rank_1,
          int rank_2,
          int dim,
          typename T1,
          typename T2,
          typename T3>
DEAL_II_CONSTEXPR inline DEAL_II_ALWAYS_INLINE
  typename ProductType<T1, typename ProductType<T2, T3>::type>::type
  contract3(const TensorT1<rank_1, dim, T1> &         left,
            const TensorT2<rank_1 + rank_2, dim, T2> &middle,
            const TensorT3<rank_2, dim, T3> &         right)
{
  using return_type =
    typename ProductType<T1, typename ProductType<T2, T3>::type>::type;
  return TensorAccessors::contract3<rank_1, rank_2, dim, return_type>(left,
                                                                      middle,
                                                                      right);
}
 
 
template <int rank_1,
          int rank_2,
          int dim,
          typename Number,
          typename OtherNumber>
DEAL_II_CONSTEXPR inline DEAL_II_ALWAYS_INLINE
  Tensor<rank_1 + rank_2, dim, typename ProductType<Number, OtherNumber>::type>
  outer_product(const Tensor<rank_1, dim, Number> &     src1,
                const Tensor<rank_2, dim, OtherNumber> &src2)
{
  typename Tensor<rank_1 + rank_2,
                  dim,
                  typename ProductType<Number, OtherNumber>::type>::tensor_type
    result{};
  TensorAccessors::contract<0, rank_1, rank_2, dim>(result, src1, src2);
  return result;
}
 
 
 
 
 
template <int dim, typename Number>
DEAL_II_CONSTEXPR inline DEAL_II_ALWAYS_INLINE Tensor<1, dim, Number>
                                               cross_product_2d(const Tensor<1, dim, Number> &src)
{
  Assert(dim == 2, ExcInternalError());
 
  Tensor<1, dim, Number> result;
 
  result[0] = src[1];
  result[1] = -src[0];
 
  return result;
}
 
 
template <int dim, typename Number1, typename Number2>
DEAL_II_CONSTEXPR inline DEAL_II_ALWAYS_INLINE
  Tensor<1, dim, typename ProductType<Number1, Number2>::type>
  cross_product_3d(const Tensor<1, dim, Number1> &src1,
                   const Tensor<1, dim, Number2> &src2)
{
  Assert(dim == 3, ExcInternalError());
 
  Tensor<1, dim, typename ProductType<Number1, Number2>::type> result;
 
  // avoid compiler warnings
  constexpr int s0 = 0 % dim;
  constexpr int s1 = 1 % dim;
  constexpr int s2 = 2 % dim;
 
  result[s0] = src1[s1] * src2[s2] - src1[s2] * src2[s1];
  result[s1] = src1[s2] * src2[s0] - src1[s0] * src2[s2];
  result[s2] = src1[s0] * src2[s1] - src1[s1] * src2[s0];
 
  return result;
}
 
 
 
 
 
template <int dim, typename Number>
DEAL_II_CONSTEXPR inline DEAL_II_ALWAYS_INLINE Number
                                               determinant(const Tensor<2, dim, Number> &t)
{
  // Compute the determinant using the Laplace expansion of the
  // determinant. We expand along the last row.
  Number det = internal::NumberType<Number>::value(0.0);
 
  for (unsigned int k = 0; k < dim; ++k)
    {
      Tensor<2, dim - 1, Number> minor;
      for (unsigned int i = 0; i < dim - 1; ++i)
        for (unsigned int j = 0; j < dim - 1; ++j)
          minor[i][j] = t[i][j < k ? j : j + 1];
 
      const Number cofactor = ((k % 2 == 0) ? -1. : 1.) * determinant(minor);
 
      det += t[dim - 1][k] * cofactor;
    }
 
  return ((dim % 2 == 0) ? 1. : -1.) * det;
}
 
template <typename Number>
constexpr DEAL_II_ALWAYS_INLINE Number
                                determinant(const Tensor<2, 1, Number> &t)
{
  return t[0][0];
}
 
 
template <int dim, typename Number>
DEAL_II_CONSTEXPR inline DEAL_II_ALWAYS_INLINE Number
                                               trace(const Tensor<2, dim, Number> &d)
{
  Number t = d[0][0];
  for (unsigned int i = 1; i < dim; ++i)
    t += d[i][i];
  return t;
}
 
 
template <int dim, typename Number>
DEAL_II_CONSTEXPR inline Tensor<2, dim, Number>
invert(const Tensor<2, dim, Number> &)
{
  Number return_tensor[dim][dim];
 
  // if desired, take over the
  // inversion of a 4x4 tensor
  // from the FullMatrix
  AssertThrow(false, ExcNotImplemented());
 
  return Tensor<2, dim, Number>(return_tensor);
}
 
 
#ifndef DOXYGEN
 
template <typename Number>
DEAL_II_CONSTEXPR inline DEAL_II_ALWAYS_INLINE Tensor<2, 1, Number>
                                               invert(const Tensor<2, 1, Number> &t)
{
  Tensor<2, 1, Number> return_tensor;
 
  return_tensor[0][0] = internal::NumberType<Number>::value(1.0 / t[0][0]);
 
  return return_tensor;
}
 
 
template <typename Number>
DEAL_II_CONSTEXPR inline DEAL_II_ALWAYS_INLINE Tensor<2, 2, Number>
                                               invert(const Tensor<2, 2, Number> &t)
{
  Tensor<2, 2, Number> return_tensor;
 
  // this is Maple output,
  // thus a bit unstructured
  const Number inv_det_t = internal::NumberType<Number>::value(
    1.0 / (t[0][0] * t[1][1] - t[1][0] * t[0][1]));
  return_tensor[0][0] = t[1][1];
  return_tensor[0][1] = -t[0][1];
  return_tensor[1][0] = -t[1][0];
  return_tensor[1][1] = t[0][0];
  return_tensor *= inv_det_t;
 
  return return_tensor;
}
 
 
template <typename Number>
DEAL_II_CONSTEXPR inline DEAL_II_ALWAYS_INLINE Tensor<2, 3, Number>
                                               invert(const Tensor<2, 3, Number> &t)
{
  Tensor<2, 3, Number> return_tensor;
 
  const Number t4  = internal::NumberType<Number>::value(t[0][0] * t[1][1]),
               t6  = internal::NumberType<Number>::value(t[0][0] * t[1][2]),
               t8  = internal::NumberType<Number>::value(t[0][1] * t[1][0]),
               t00 = internal::NumberType<Number>::value(t[0][2] * t[1][0]),
               t01 = internal::NumberType<Number>::value(t[0][1] * t[2][0]),
               t04 = internal::NumberType<Number>::value(t[0][2] * t[2][0]),
               inv_det_t = internal::NumberType<Number>::value(
                 1.0 / (t4 * t[2][2] - t6 * t[2][1] - t8 * t[2][2] +
                        t00 * t[2][1] + t01 * t[1][2] - t04 * t[1][1]));
  return_tensor[0][0] = internal::NumberType<Number>::value(t[1][1] * t[2][2]) -
                        internal::NumberType<Number>::value(t[1][2] * t[2][1]);
  return_tensor[0][1] = internal::NumberType<Number>::value(t[0][2] * t[2][1]) -
                        internal::NumberType<Number>::value(t[0][1] * t[2][2]);
  return_tensor[0][2] = internal::NumberType<Number>::value(t[0][1] * t[1][2]) -
                        internal::NumberType<Number>::value(t[0][2] * t[1][1]);
  return_tensor[1][0] = internal::NumberType<Number>::value(t[1][2] * t[2][0]) -
                        internal::NumberType<Number>::value(t[1][0] * t[2][2]);
  return_tensor[1][1] =
    internal::NumberType<Number>::value(t[0][0] * t[2][2]) - t04;
  return_tensor[1][2] = t00 - t6;
  return_tensor[2][0] = internal::NumberType<Number>::value(t[1][0] * t[2][1]) -
                        internal::NumberType<Number>::value(t[1][1] * t[2][0]);
  return_tensor[2][1] =
    t01 - internal::NumberType<Number>::value(t[0][0] * t[2][1]);
  return_tensor[2][2] = internal::NumberType<Number>::value(t4 - t8);
  return_tensor *= inv_det_t;
 
  return return_tensor;
}
 
#endif /* DOXYGEN */
 
 
template <int dim, typename Number>
DEAL_II_CONSTEXPR inline DEAL_II_ALWAYS_INLINE Tensor<2, dim, Number>
                                               transpose(const Tensor<2, dim, Number> &t)
{
  Tensor<2, dim, Number> tt;
  for (unsigned int i = 0; i < dim; ++i)
    {
      tt[i][i] = t[i][i];
      for (unsigned int j = i + 1; j < dim; ++j)
        {
          tt[i][j] = t[j][i];
          tt[j][i] = t[i][j];
        };
    }
  return tt;
}
 
 
template <int dim, typename Number>
constexpr Tensor<2, dim, Number>
adjugate(const Tensor<2, dim, Number> &t)
{
  return determinant(t) * invert(t);
}
 
 
template <int dim, typename Number>
constexpr Tensor<2, dim, Number>
cofactor(const Tensor<2, dim, Number> &t)
{
  return transpose(adjugate(t));
}
 
 
template <int dim, typename Number>
Tensor<2, dim, Number>
project_onto_orthogonal_tensors(const Tensor<2, dim, Number> &tensor)
{
  Tensor<2, dim, Number>   output_tensor;
  FullMatrix<Number>       matrix(dim);
  LAPACKFullMatrix<Number> lapack_matrix(dim);
  LAPACKFullMatrix<Number> result(dim);
 
  // todo: find or add dealii functionality to copy in one step.
  matrix.copy_from(tensor);
  lapack_matrix.copy_from(matrix);
 
  // now compute the svd of the matrices
  lapack_matrix.compute_svd();
 
  // Use the SVD results to orthogonalize: @f$U V^T@f$
  lapack_matrix.get_svd_u().mmult(result, lapack_matrix.get_svd_vt());
 
  // todo: find or add dealii functionality to copy in one step.
  matrix = result;
  matrix.copy_to(output_tensor);
  return output_tensor;
}
 
 
template <int dim, typename Number>
inline Number
l1_norm(const Tensor<2, dim, Number> &t)
{
  Number max = internal::NumberType<Number>::value(0.0);
  for (unsigned int j = 0; j < dim; ++j)
    {
      Number sum = internal::NumberType<Number>::value(0.0);
      for (unsigned int i = 0; i < dim; ++i)
        sum += std::fabs(t[i][j]);
 
      if (sum > max)
        max = sum;
    }
 
  return max;
}
 
 
template <int dim, typename Number>
inline Number
linfty_norm(const Tensor<2, dim, Number> &t)
{
  Number max = internal::NumberType<Number>::value(0.0);
  for (unsigned int i = 0; i < dim; ++i)
    {
      Number sum = internal::NumberType<Number>::value(0.0);
      for (unsigned int j = 0; j < dim; ++j)
        sum += std::fabs(t[i][j]);
 
      if (sum > max)
        max = sum;
    }
 
  return max;
}
 
 
 
#ifndef DOXYGEN
 
 
#  ifdef DEAL_II_ADOLC_WITH_ADVANCED_BRANCHING
 
// Specialization of functions for ADOL-C number types when
// the advanced branching feature is used
template <int dim>
inline adouble
l1_norm(const Tensor<2, dim, adouble> &t)
{
  adouble max = internal::NumberType<adouble>::value(0.0);
  for (unsigned int j = 0; j < dim; ++j)
    {
      adouble sum = internal::NumberType<adouble>::value(0.0);
      for (unsigned int i = 0; i < dim; ++i)
        sum += std::fabs(t[i][j]);
 
      condassign(max, (sum > max), sum, max);
    }
 
  return max;
}
 
 
template <int dim>
inline adouble
linfty_norm(const Tensor<2, dim, adouble> &t)
{
  adouble max = internal::NumberType<adouble>::value(0.0);
  for (unsigned int i = 0; i < dim; ++i)
    {
      adouble sum = internal::NumberType<adouble>::value(0.0);
      for (unsigned int j = 0; j < dim; ++j)
        sum += std::fabs(t[i][j]);
 
      condassign(max, (sum > max), sum, max);
    }
 
  return max;
}
 
#  endif // DEAL_II_ADOLC_WITH_ADVANCED_BRANCHING
 
 
#endif // DOXYGEN
 
DEAL_II_NAMESPACE_CLOSE
 
#endif