fe.h

// ---------------------------------------------------------------------
//
// Copyright (C) 1998 - 2019 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_fe_h
#define dealii_fe_h

#include <deal.II/base/config.h>

#include <deal.II/fe/block_mask.h>
#include <deal.II/fe/component_mask.h>
#include <deal.II/fe/fe_base.h>
#include <deal.II/fe/fe_update_flags.h>
#include <deal.II/fe/fe_values_extractors.h>
#include <deal.II/fe/mapping.h>

#include <deal.II/lac/full_matrix.h>

#include <memory>


DEAL_II_NAMESPACE_OPEN

template <int dim, int spacedim>
class FEValuesBase;
template <int dim, int spacedim>
class FEValues;
template <int dim, int spacedim>
class FEFaceValues;
template <int dim, int spacedim>
class FESubfaceValues;
template <int dim, int spacedim>
class FESystem;

template <int dim, int spacedim = dim>
class FiniteElement : public Subscriptor, public FiniteElementData<dim>
{
public:
  static const unsigned int space_dimension = spacedim;

  class InternalDataBase
  {
  public:
    InternalDataBase();

    virtual ~InternalDataBase() = default;

    InternalDataBase(const InternalDataBase &) = delete;

    UpdateFlags update_each;

    virtual std::size_t
    memory_consumption() const;
  };

public:
  FiniteElement(const FiniteElementData<dim> &    fe_data,
                const std::vector<bool> &         restriction_is_additive_flags,
                const std::vector<ComponentMask> &nonzero_components);

  FiniteElement(FiniteElement<dim, spacedim> &&) = default; // NOLINT

  FiniteElement(const FiniteElement<dim, spacedim> &) = default;

  virtual ~FiniteElement() override = default;

  std::pair<std::unique_ptr<FiniteElement<dim, spacedim>>, unsigned int>
  operator^(const unsigned int multiplicity) const;

  virtual std::unique_ptr<FiniteElement<dim, spacedim>>
  clone() const = 0;

  virtual std::string
  get_name() const = 0;

  const FiniteElement<dim, spacedim> &
  operator[](const unsigned int fe_index) const;

  virtual double
  shape_value(const unsigned int i, const Point<dim> &p) const;

  virtual double
  shape_value_component(const unsigned int i,
                        const Point<dim> & p,
                        const unsigned int component) const;

  virtual Tensor<1, dim>
  shape_grad(const unsigned int i, const Point<dim> &p) const;

  virtual Tensor<1, dim>
  shape_grad_component(const unsigned int i,
                       const Point<dim> & p,
                       const unsigned int component) const;

  virtual Tensor<2, dim>
  shape_grad_grad(const unsigned int i, const Point<dim> &p) const;

  virtual Tensor<2, dim>
  shape_grad_grad_component(const unsigned int i,
                            const Point<dim> & p,
                            const unsigned int component) const;

  virtual Tensor<3, dim>
  shape_3rd_derivative(const unsigned int i, const Point<dim> &p) const;

  virtual Tensor<3, dim>
  shape_3rd_derivative_component(const unsigned int i,
                                 const Point<dim> & p,
                                 const unsigned int component) const;

  virtual Tensor<4, dim>
  shape_4th_derivative(const unsigned int i, const Point<dim> &p) const;

  virtual Tensor<4, dim>
  shape_4th_derivative_component(const unsigned int i,
                                 const Point<dim> & p,
                                 const unsigned int component) const;
  virtual bool
  has_support_on_face(const unsigned int shape_index,
                      const unsigned int face_index) const;


  virtual const FullMatrix<double> &
  get_restriction_matrix(const unsigned int         child,
                         const RefinementCase<dim> &refinement_case =
                           RefinementCase<dim>::isotropic_refinement) const;

  virtual const FullMatrix<double> &
  get_prolongation_matrix(const unsigned int         child,
                          const RefinementCase<dim> &refinement_case =
                            RefinementCase<dim>::isotropic_refinement) const;

  bool
  prolongation_is_implemented() const;

  bool
  isotropic_prolongation_is_implemented() const;

  bool
  restriction_is_implemented() const;

  bool
  isotropic_restriction_is_implemented() const;


  bool
  restriction_is_additive(const unsigned int index) const;

  const FullMatrix<double> &
  constraints(const ::internal::SubfaceCase<dim> &subface_case =
                ::internal::SubfaceCase<dim>::case_isotropic) const;

  bool
  constraints_are_implemented(
    const ::internal::SubfaceCase<dim> &subface_case =
      ::internal::SubfaceCase<dim>::case_isotropic) const;


  virtual bool
  hp_constraints_are_implemented() const;


  virtual void
  get_interpolation_matrix(const FiniteElement<dim, spacedim> &source,
                           FullMatrix<double> &                matrix) const;

  virtual void
  get_face_interpolation_matrix(const FiniteElement<dim, spacedim> &source,
                                FullMatrix<double> &matrix) const;


  virtual void
  get_subface_interpolation_matrix(const FiniteElement<dim, spacedim> &source,
                                   const unsigned int                  subface,
                                   FullMatrix<double> &matrix) const;


  virtual std::vector<std::pair<unsigned int, unsigned int>>
  hp_vertex_dof_identities(const FiniteElement<dim, spacedim> &fe_other) const;

  virtual std::vector<std::pair<unsigned int, unsigned int>>
  hp_line_dof_identities(const FiniteElement<dim, spacedim> &fe_other) const;

  virtual std::vector<std::pair<unsigned int, unsigned int>>
  hp_quad_dof_identities(const FiniteElement<dim, spacedim> &fe_other) const;

  DEAL_II_DEPRECATED virtual FiniteElementDomination::Domination
  compare_for_face_domination(
    const FiniteElement<dim, spacedim> &fe_other) const final;

  virtual FiniteElementDomination::Domination
  compare_for_domination(const FiniteElement<dim, spacedim> &fe_other,
                         const unsigned int                  codim = 0) const;


  virtual bool
  operator==(const FiniteElement<dim, spacedim> &fe) const;

  bool
  operator!=(const FiniteElement<dim, spacedim> &) const;

  std::pair<unsigned int, unsigned int>
  system_to_component_index(const unsigned int index) const;

  unsigned int
  component_to_system_index(const unsigned int component,
                            const unsigned int index) const;

  std::pair<unsigned int, unsigned int>
  face_system_to_component_index(const unsigned int index) const;

  unsigned int
  adjust_quad_dof_index_for_face_orientation(const unsigned int index,
                                             const bool face_orientation,
                                             const bool face_flip,
                                             const bool face_rotation) const;

  virtual unsigned int
  face_to_cell_index(const unsigned int face_dof_index,
                     const unsigned int face,
                     const bool         face_orientation = true,
                     const bool         face_flip        = false,
                     const bool         face_rotation    = false) const;

  unsigned int
  adjust_line_dof_index_for_line_orientation(const unsigned int index,
                                             const bool line_orientation) const;

  const ComponentMask &
  get_nonzero_components(const unsigned int i) const;

  unsigned int
  n_nonzero_components(const unsigned int i) const;

  bool
  is_primitive() const;

  bool
  is_primitive(const unsigned int i) const;

  unsigned int
  n_base_elements() const;

  virtual const FiniteElement<dim, spacedim> &
  base_element(const unsigned int index) const;

  unsigned int
  element_multiplicity(const unsigned int index) const;

  const FiniteElement<dim, spacedim> &
  get_sub_fe(const ComponentMask &mask) const;

  virtual const FiniteElement<dim, spacedim> &
  get_sub_fe(const unsigned int first_component,
             const unsigned int n_selected_components) const;

  std::pair<std::pair<unsigned int, unsigned int>, unsigned int>
  system_to_base_index(const unsigned int index) const;

  std::pair<std::pair<unsigned int, unsigned int>, unsigned int>
  face_system_to_base_index(const unsigned int index) const;

  types::global_dof_index
  first_block_of_base(const unsigned int b) const;

  std::pair<unsigned int, unsigned int>
  component_to_base_index(const unsigned int component) const;


  std::pair<unsigned int, unsigned int>
  block_to_base_index(const unsigned int block) const;

  std::pair<unsigned int, types::global_dof_index>
  system_to_block_index(const unsigned int component) const;

  unsigned int
  component_to_block_index(const unsigned int component) const;


  ComponentMask
  component_mask(const FEValuesExtractors::Scalar &scalar) const;

  ComponentMask
  component_mask(const FEValuesExtractors::Vector &vector) const;

  ComponentMask
  component_mask(
    const FEValuesExtractors::SymmetricTensor<2> &sym_tensor) const;

  ComponentMask
  component_mask(const BlockMask &block_mask) const;

  BlockMask
  block_mask(const FEValuesExtractors::Scalar &scalar) const;

  BlockMask
  block_mask(const FEValuesExtractors::Vector &vector) const;

  BlockMask
  block_mask(const FEValuesExtractors::SymmetricTensor<2> &sym_tensor) const;

  BlockMask
  block_mask(const ComponentMask &component_mask) const;

  virtual std::pair<Table<2, bool>, std::vector<unsigned int>>
  get_constant_modes() const;


  const std::vector<Point<dim>> &
  get_unit_support_points() const;

  bool
  has_support_points() const;

  virtual Point<dim>
  unit_support_point(const unsigned int index) const;

  const std::vector<Point<dim - 1>> &
  get_unit_face_support_points() const;

  bool
  has_face_support_points() const;

  virtual Point<dim - 1>
  unit_face_support_point(const unsigned int index) const;

  const std::vector<Point<dim>> &
  get_generalized_support_points() const;

  bool
  has_generalized_support_points() const;

  DEAL_II_DEPRECATED
  const std::vector<Point<dim - 1>> &
  get_generalized_face_support_points() const;

  DEAL_II_DEPRECATED
  bool
  has_generalized_face_support_points() const;

  GeometryPrimitive
  get_associated_geometry_primitive(const unsigned int cell_dof_index) const;


  virtual void
  convert_generalized_support_point_values_to_dof_values(
    const std::vector<Vector<double>> &support_point_values,
    std::vector<double> &              nodal_values) const;


  virtual std::size_t
  memory_consumption() const;

  DeclException1(ExcShapeFunctionNotPrimitive,
                 int,
                 << "The shape function with index " << arg1
                 << " is not primitive, i.e. it is vector-valued and "
                 << "has more than one non-zero vector component. This "
                 << "function cannot be called for these shape functions. "
                 << "Maybe you want to use the same function with the "
                 << "_component suffix?");
  DeclException0(ExcFENotPrimitive);
  DeclExceptionMsg(
    ExcUnitShapeValuesDoNotExist,
    "You are trying to access the values or derivatives of shape functions "
    "on the reference cell of an element that does not define its shape "
    "functions through mapping from the reference cell. Consequently, "
    "you cannot ask for shape function values or derivatives on the "
    "reference cell.");

  DeclExceptionMsg(ExcFEHasNoSupportPoints,
                   "You are trying to access the support points of a finite "
                   "element that either has no support points at all, or for "
                   "which the corresponding tables have not been implemented.");

  DeclExceptionMsg(ExcEmbeddingVoid,
                   "You are trying to access the matrices that describe how "
                   "to embed a finite element function on one cell into the "
                   "finite element space on one of its children (i.e., the "
                   "'embedding' or 'prolongation' matrices). However, the "
                   "current finite element can either not define this sort of "
                   "operation, or it has not yet been implemented.");

  DeclExceptionMsg(ExcProjectionVoid,
                   "You are trying to access the matrices that describe how "
                   "to restrict a finite element function from the children "
                   "of one cell to the finite element space defined on their "
                   "parent (i.e., the 'restriction' or 'projection' matrices). "
                   "However, the current finite element can either not define "
                   "this sort of operation, or it has not yet been "
                   "implemented.");

  DeclException2(ExcWrongInterfaceMatrixSize,
                 int,
                 int,
                 << "The interface matrix has a size of " << arg1 << "x" << arg2
                 << ", which is not reasonable for the current element "
                    "in the present dimension.");
  DeclException0(ExcInterpolationNotImplemented);

protected:
  void
  reinit_restriction_and_prolongation_matrices(
    const bool isotropic_restriction_only  = false,
    const bool isotropic_prolongation_only = false);

  std::vector<std::vector<FullMatrix<double>>> restriction;

  std::vector<std::vector<FullMatrix<double>>> prolongation;

  FullMatrix<double> interface_constraints;

  std::vector<Point<dim>> unit_support_points;

  std::vector<Point<dim - 1>> unit_face_support_points;

  std::vector<Point<dim>> generalized_support_points;

  std::vector<Point<dim - 1>> generalized_face_support_points;

  Table<2, int> adjust_quad_dof_index_for_face_orientation_table;

  std::vector<int> adjust_line_dof_index_for_line_orientation_table;

  std::vector<std::pair<unsigned int, unsigned int>> system_to_component_table;

  std::vector<std::pair<unsigned int, unsigned int>>
    face_system_to_component_table;

  std::vector<std::pair<std::pair<unsigned int, unsigned int>, unsigned int>>
    system_to_base_table;

  std::vector<std::pair<std::pair<unsigned int, unsigned int>, unsigned int>>
    face_system_to_base_table;

  BlockIndices base_to_block_indices;

  std::vector<std::pair<std::pair<unsigned int, unsigned int>, unsigned int>>
    component_to_base_table;

  const std::vector<bool> restriction_is_additive_flags;

  const std::vector<ComponentMask> nonzero_components;

  const std::vector<unsigned int> n_nonzero_components_table;

  const bool cached_primitivity;

  TableIndices<2>
  interface_constraints_size() const;

  static std::vector<unsigned int>
  compute_n_nonzero_components(
    const std::vector<ComponentMask> &nonzero_components);

  virtual UpdateFlags
  requires_update_flags(const UpdateFlags update_flags) const = 0;

  virtual std::unique_ptr<InternalDataBase>
  get_data(const UpdateFlags             update_flags,
           const Mapping<dim, spacedim> &mapping,
           const Quadrature<dim> &       quadrature,
           ::internal::FEValuesImplementation::
             FiniteElementRelatedData<dim, spacedim> &output_data) const = 0;

  virtual std::unique_ptr<InternalDataBase>
  get_face_data(const UpdateFlags             update_flags,
                const Mapping<dim, spacedim> &mapping,
                const Quadrature<dim - 1> &   quadrature,
                ::internal::FEValuesImplementation::
                  FiniteElementRelatedData<dim, spacedim> &output_data) const;

  virtual std::unique_ptr<InternalDataBase>
  get_subface_data(
    const UpdateFlags             update_flags,
    const Mapping<dim, spacedim> &mapping,
    const Quadrature<dim - 1> &   quadrature,
    ::internal::FEValuesImplementation::FiniteElementRelatedData<dim,
                                                                       spacedim>
      &output_data) const;

  virtual void
  fill_fe_values(
    const typename Triangulation<dim, spacedim>::cell_iterator &cell,
    const CellSimilarity::Similarity                            cell_similarity,
    const Quadrature<dim> &                                     quadrature,
    const Mapping<dim, spacedim> &                              mapping,
    const typename Mapping<dim, spacedim>::InternalDataBase &mapping_internal,
    const ::internal::FEValuesImplementation::MappingRelatedData<dim,
                                                                       spacedim>
      &                     mapping_data,
    const InternalDataBase &fe_internal,
    ::internal::FEValuesImplementation::FiniteElementRelatedData<dim,
                                                                       spacedim>
      &output_data) const = 0;

  virtual void
  fill_fe_face_values(
    const typename Triangulation<dim, spacedim>::cell_iterator &cell,
    const unsigned int                                          face_no,
    const Quadrature<dim - 1> &                                 quadrature,
    const Mapping<dim, spacedim> &                              mapping,
    const typename Mapping<dim, spacedim>::InternalDataBase &mapping_internal,
    const ::internal::FEValuesImplementation::MappingRelatedData<dim,
                                                                       spacedim>
      &                     mapping_data,
    const InternalDataBase &fe_internal,
    ::internal::FEValuesImplementation::FiniteElementRelatedData<dim,
                                                                       spacedim>
      &output_data) const = 0;

  virtual void
  fill_fe_subface_values(
    const typename Triangulation<dim, spacedim>::cell_iterator &cell,
    const unsigned int                                          face_no,
    const unsigned int                                          sub_no,
    const Quadrature<dim - 1> &                                 quadrature,
    const Mapping<dim, spacedim> &                              mapping,
    const typename Mapping<dim, spacedim>::InternalDataBase &mapping_internal,
    const ::internal::FEValuesImplementation::MappingRelatedData<dim,
                                                                       spacedim>
      &                     mapping_data,
    const InternalDataBase &fe_internal,
    ::internal::FEValuesImplementation::FiniteElementRelatedData<dim,
                                                                       spacedim>
      &output_data) const = 0;

  friend class InternalDataBase;
  friend class FEValuesBase<dim, spacedim>;
  friend class FEValues<dim, spacedim>;
  friend class FEFaceValues<dim, spacedim>;
  friend class FESubfaceValues<dim, spacedim>;
  friend class FESystem<dim, spacedim>;

  // explicitly check for sensible template arguments, but not on windows
  // because MSVC creates bogus warnings during normal compilation
#ifndef DEAL_II_MSVC
  static_assert(dim <= spacedim,
                "The dimension <dim> of a FiniteElement must be less than or "
                "equal to the space dimension <spacedim> in which it lives.");
#endif
};


//----------------------------------------------------------------------//


template <int dim, int spacedim>
inline const FiniteElement<dim, spacedim> &FiniteElement<dim, spacedim>::
                                           operator[](const unsigned int fe_index) const
{
  (void)fe_index;
  Assert(fe_index == 0,
         ExcMessage("A fe_index of zero is the only index allowed here"));
  return *this;
}



template <int dim, int spacedim>
inline std::pair<unsigned int, unsigned int>
FiniteElement<dim, spacedim>::system_to_component_index(
  const unsigned int index) const
{
  Assert(index < system_to_component_table.size(),
         ExcIndexRange(index, 0, system_to_component_table.size()));
  Assert(is_primitive(index),
         (typename FiniteElement<dim, spacedim>::ExcShapeFunctionNotPrimitive(
           index)));
  return system_to_component_table[index];
}



template <int dim, int spacedim>
inline unsigned int
FiniteElement<dim, spacedim>::n_base_elements() const
{
  return base_to_block_indices.size();
}



template <int dim, int spacedim>
inline unsigned int
FiniteElement<dim, spacedim>::element_multiplicity(
  const unsigned int index) const
{
  return static_cast<unsigned int>(base_to_block_indices.block_size(index));
}



template <int dim, int spacedim>
inline unsigned int
FiniteElement<dim, spacedim>::component_to_system_index(
  const unsigned int component,
  const unsigned int index) const
{
  AssertIndexRange(component, this->n_components());
  const std::vector<std::pair<unsigned int, unsigned int>>::const_iterator it =
    std::find(system_to_component_table.begin(),
              system_to_component_table.end(),
              std::pair<unsigned int, unsigned int>(component, index));

  Assert(it != system_to_component_table.end(),
         ExcMessage("You are asking for the number of the shape function "
                    "within a system element that corresponds to vector "
                    "component " +
                    Utilities::int_to_string(component) +
                    " and within this to "
                    "index " +
                    Utilities::int_to_string(index) +
                    ". But no such "
                    "shape function exists."));
  return std::distance(system_to_component_table.begin(), it);
}



template <int dim, int spacedim>
inline std::pair<unsigned int, unsigned int>
FiniteElement<dim, spacedim>::face_system_to_component_index(
  const unsigned int index) const
{
  Assert(index < face_system_to_component_table.size(),
         ExcIndexRange(index, 0, face_system_to_component_table.size()));

  // in debug mode, check whether the
  // function is primitive, since
  // otherwise the result may have no
  // meaning
  //
  // since the primitivity tables are
  // all geared towards cell dof
  // indices, rather than face dof
  // indices, we have to work a
  // little bit...
  //
  // in 1d, the face index is equal
  // to the cell index
  Assert(is_primitive(this->face_to_cell_index(index, 0)),
         (typename FiniteElement<dim, spacedim>::ExcShapeFunctionNotPrimitive(
           index)));

  return face_system_to_component_table[index];
}



template <int dim, int spacedim>
inline std::pair<std::pair<unsigned int, unsigned int>, unsigned int>
FiniteElement<dim, spacedim>::system_to_base_index(
  const unsigned int index) const
{
  Assert(index < system_to_base_table.size(),
         ExcIndexRange(index, 0, system_to_base_table.size()));
  return system_to_base_table[index];
}



template <int dim, int spacedim>
inline std::pair<std::pair<unsigned int, unsigned int>, unsigned int>
FiniteElement<dim, spacedim>::face_system_to_base_index(
  const unsigned int index) const
{
  Assert(index < face_system_to_base_table.size(),
         ExcIndexRange(index, 0, face_system_to_base_table.size()));
  return face_system_to_base_table[index];
}



template <int dim, int spacedim>
inline types::global_dof_index
FiniteElement<dim, spacedim>::first_block_of_base(
  const unsigned int index) const
{
  return base_to_block_indices.block_start(index);
}



template <int dim, int spacedim>
inline std::pair<unsigned int, unsigned int>
FiniteElement<dim, spacedim>::component_to_base_index(
  const unsigned int index) const
{
  Assert(index < component_to_base_table.size(),
         ExcIndexRange(index, 0, component_to_base_table.size()));

  return component_to_base_table[index].first;
}



template <int dim, int spacedim>
inline std::pair<unsigned int, unsigned int>
FiniteElement<dim, spacedim>::block_to_base_index(
  const unsigned int index) const
{
  return base_to_block_indices.global_to_local(index);
}



template <int dim, int spacedim>
inline std::pair<unsigned int, types::global_dof_index>
FiniteElement<dim, spacedim>::system_to_block_index(
  const unsigned int index) const
{
  Assert(index < this->dofs_per_cell,
         ExcIndexRange(index, 0, this->dofs_per_cell));
  // The block is computed simply as
  // first block of this base plus
  // the index within the base blocks
  return std::pair<unsigned int, types::global_dof_index>(
    first_block_of_base(system_to_base_table[index].first.first) +
      system_to_base_table[index].first.second,
    system_to_base_table[index].second);
}



template <int dim, int spacedim>
inline bool
FiniteElement<dim, spacedim>::restriction_is_additive(
  const unsigned int index) const
{
  Assert(index < this->dofs_per_cell,
         ExcIndexRange(index, 0, this->dofs_per_cell));
  return restriction_is_additive_flags[index];
}



template <int dim, int spacedim>
inline const ComponentMask &
FiniteElement<dim, spacedim>::get_nonzero_components(const unsigned int i) const
{
  Assert(i < this->dofs_per_cell, ExcIndexRange(i, 0, this->dofs_per_cell));
  return nonzero_components[i];
}



template <int dim, int spacedim>
inline unsigned int
FiniteElement<dim, spacedim>::n_nonzero_components(const unsigned int i) const
{
  Assert(i < this->dofs_per_cell, ExcIndexRange(i, 0, this->dofs_per_cell));
  return n_nonzero_components_table[i];
}



template <int dim, int spacedim>
inline bool
FiniteElement<dim, spacedim>::is_primitive() const
{
  return cached_primitivity;
}



template <int dim, int spacedim>
inline bool
FiniteElement<dim, spacedim>::is_primitive(const unsigned int i) const
{
  Assert(i < this->dofs_per_cell, ExcIndexRange(i, 0, this->dofs_per_cell));

  // return primitivity of a shape
  // function by checking whether it
  // has more than one non-zero
  // component or not. we could cache
  // this value in an array of bools,
  // but accessing a bit-vector (as
  // std::vector<bool> is) is
  // probably more expensive than
  // just comparing against 1
  //
  // for good measure, short circuit the test
  // if the entire FE is primitive
  return (is_primitive() || (n_nonzero_components_table[i] == 1));
}



template <int dim, int spacedim>
inline GeometryPrimitive
FiniteElement<dim, spacedim>::get_associated_geometry_primitive(
  const unsigned int cell_dof_index) const
{
  Assert(cell_dof_index < this->dofs_per_cell,
         ExcIndexRange(cell_dof_index, 0, this->dofs_per_cell));

  // just go through the usual cases, taking into account how DoFs
  // are enumerated on the reference cell
  if (cell_dof_index < this->first_line_index)
    return GeometryPrimitive::vertex;
  else if (cell_dof_index < this->first_quad_index)
    return GeometryPrimitive::line;
  else if (cell_dof_index < this->first_hex_index)
    return GeometryPrimitive::quad;
  else
    return GeometryPrimitive::hex;
}



DEAL_II_NAMESPACE_CLOSE

#endif