fe.h

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// ------------------------------------------------------------------------
//
// SPDX-License-Identifier: LGPL-2.1-or-later
// Copyright (C) 1998 - 2024 by the deal.II authors
//
// This file is part of the deal.II library.
//
// Part of the source code is dual licensed under Apache-2.0 WITH
// LLVM-exception OR LGPL-2.1-or-later. Detailed license information
// governing the source code and code contributions can be found in
// LICENSE.md and CONTRIBUTING.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_data.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/fe/mapping_related_data.h>
 
#include <deal.II/lac/full_matrix.h>
#include <deal.II/lac/vector.h>
 
#include <memory>
 
 
DEAL_II_NAMESPACE_OPEN
 
// Forward declarations:
#ifndef DOXYGEN
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;
namespace NonMatching
{
  template <int dim>
  class FEImmersedSurfaceValues;
}
template <int dim, int spacedim>
class FESystem;
#endif
 
template <int dim, int spacedim = dim>
class FiniteElement : public EnableObserverPointer,
                      public FiniteElementData<dim>
{
public:
  static constexpr 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;
 
  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 unsigned int face_no = 0) const;
 
 
  virtual void
  get_subface_interpolation_matrix(const FiniteElement<dim, spacedim> &source,
                                   const unsigned int                  subface,
                                   FullMatrix<double>                 &matrix,
                                   const unsigned int face_no = 0) 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 unsigned int                  face_no = 0) const;
 
  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 face_no = 0) const;
 
  unsigned int
  adjust_quad_dof_index_for_face_orientation(
    const unsigned int                 index,
    const unsigned int                 face_no,
    const types::geometric_orientation combined_orientation) const;
 
  virtual unsigned int
  face_to_cell_index(
    const unsigned int                 face_dof_index,
    const unsigned int                 face,
    const types::geometric_orientation combined_orientation =
      ReferenceCell::default_combined_face_orientation()) const;
 
  unsigned int
  adjust_line_dof_index_for_line_orientation(
    const unsigned int                 index,
    const types::geometric_orientation combined_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;
 
  virtual const Table<2, bool> &
  get_local_dof_sparsity_pattern() 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 unsigned int face_no = 0) 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;
 
  bool
  shape_function_belongs_to(const unsigned int                shape_function,
                            const FEValuesExtractors::Scalar &component) const;
 
  bool
  shape_function_belongs_to(const unsigned int                shape_function,
                            const FEValuesExtractors::Vector &components) const;
 
 
  bool
  shape_function_belongs_to(
    const unsigned int                            shape_function,
    const FEValuesExtractors::SymmetricTensor<2> &components) const;
 
 
  bool
  shape_function_belongs_to(
    const unsigned int                   shape_function,
    const FEValuesExtractors::Tensor<2> &components) 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 unsigned int face_no = 0) const;
 
  bool
  has_face_support_points(const unsigned int face_no = 0) const;
 
  virtual Point<dim - 1>
  unit_face_support_point(const unsigned int index,
                          const unsigned int face_no = 0) const;
 
  const std::vector<Point<dim>> &
  get_generalized_support_points() const;
 
  bool
  has_generalized_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<std::vector<Point<dim - 1>>> unit_face_support_points;
 
  std::vector<Point<dim>> generalized_support_points;
 
  std::vector<std::vector<Point<dim - 1>>> generalized_face_support_points;
 
  std::vector<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::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::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;
 
  Table<2, bool> local_dof_sparsity_pattern;
 
  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 hp::QCollection<dim - 1> &quadrature,
                ::internal::FEValuesImplementation::
                  FiniteElementRelatedData<dim, spacedim> &output_data) const;
 
  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 hp::QCollection<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;
 
  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;
 
  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 NonMatching::FEImmersedSurfaceValues<dim>;
  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
};
 
 
//----------------------------------------------------------------------//
#ifndef DOXYGEN
 
template <int dim, int spacedim>
inline std::pair<unsigned int, unsigned int>
FiniteElement<dim, spacedim>::system_to_component_index(
  const unsigned int index) const
{
  AssertIndexRange(index, 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 unsigned int face_no) const
{
  AssertIndexRange(
    index,
    face_system_to_component_table[this->n_unique_faces() == 1 ? 0 : face_no]
      .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, face_no)),
         (typename FiniteElement<dim, spacedim>::ExcShapeFunctionNotPrimitive(
           index)));
 
  return face_system_to_component_table[this->n_unique_faces() == 1 ?
                                          0 :
                                          face_no][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
{
  AssertIndexRange(index, 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 unsigned int face_no) const
{
  AssertIndexRange(
    index,
    face_system_to_base_table[this->n_unique_faces() == 1 ? 0 : face_no]
      .size());
  return face_system_to_base_table[this->n_unique_faces() == 1 ? 0 : face_no]
                                  [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
{
  AssertIndexRange(index, 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
{
  AssertIndexRange(index, this->n_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>::shape_function_belongs_to(
  const unsigned int                shape_function,
  const FEValuesExtractors::Scalar &component) const
{
  AssertIndexRange(shape_function, this->n_dofs_per_cell());
  AssertIndexRange(component.component, this->n_components());
 
  if (is_primitive(shape_function))
    return (system_to_component_table[shape_function].first ==
            component.component);
  else
    return nonzero_components[shape_function][component.component];
}
 
 
 
template <int dim, int spacedim>
inline bool
FiniteElement<dim, spacedim>::shape_function_belongs_to(
  const unsigned int                shape_function,
  const FEValuesExtractors::Vector &components) const
{
  AssertIndexRange(shape_function, this->n_dofs_per_cell());
  AssertIndexRange(components.first_vector_component, this->n_components());
  AssertIndexRange(components.first_vector_component + dim,
                   this->n_components() + 1);
 
  if (is_primitive(shape_function))
    return ((system_to_component_table[shape_function].first >=
             components.first_vector_component) &&
            (system_to_component_table[shape_function].first <
             components.first_vector_component + dim));
  else
    // Return whether there is overlap between the nonzero components
    // of the current shape function and the selected components of
    // the extractor:
    {
      for (unsigned int i = components.first_vector_component;
           i < components.first_vector_component + dim;
           ++i)
        if (nonzero_components[shape_function][i])
          return true;
      return false;
    }
}
 
 
 
template <int dim, int spacedim>
inline bool
FiniteElement<dim, spacedim>::shape_function_belongs_to(
  const unsigned int                            shape_function,
  const FEValuesExtractors::SymmetricTensor<2> &components) const
{
  AssertIndexRange(shape_function, this->n_dofs_per_cell());
  AssertIndexRange(components.first_tensor_component, this->n_components());
  AssertIndexRange(components.first_tensor_component + dim,
                   this->n_components() + 1);
 
  if (is_primitive(shape_function))
    return ((system_to_component_table[shape_function].first >=
             components.first_tensor_component) &&
            (system_to_component_table[shape_function].first <
             components.first_tensor_component +
               SymmetricTensor<2, dim>::n_independent_components));
  else
    // Return whether there is overlap between the nonzero components
    // of the current shape function and the selected components of
    // the extractor:
    {
      for (unsigned int i = components.first_tensor_component;
           i < components.first_tensor_component +
                 SymmetricTensor<2, dim>::n_independent_components;
           ++i)
        if (nonzero_components[shape_function][i])
          return true;
      return false;
    }
}
 
 
 
template <int dim, int spacedim>
inline bool
FiniteElement<dim, spacedim>::shape_function_belongs_to(
  const unsigned int                   shape_function,
  const FEValuesExtractors::Tensor<2> &components) const
{
  AssertIndexRange(shape_function, this->n_dofs_per_cell());
  AssertIndexRange(components.first_tensor_component, this->n_components());
  AssertIndexRange(components.first_tensor_component + dim,
                   this->n_components() + 1);
 
  if (is_primitive(shape_function))
    return ((system_to_component_table[shape_function].first >=
             components.first_tensor_component) &&
            (system_to_component_table[shape_function].first <
             components.first_tensor_component +
               Tensor<2, dim>::n_independent_components));
  else
    // Return whether there is overlap between the nonzero components
    // of the current shape function and the selected components of
    // the extractor:
    {
      for (unsigned int i = components.first_tensor_component;
           i < components.first_tensor_component +
                 Tensor<2, dim>::n_independent_components;
           ++i)
        if (nonzero_components[shape_function][i])
          return true;
      return false;
    }
}
 
 
 
template <int dim, int spacedim>
inline bool
FiniteElement<dim, spacedim>::restriction_is_additive(
  const unsigned int index) const
{
  AssertIndexRange(index, this->n_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
{
  AssertIndexRange(i, this->n_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
{
  AssertIndexRange(i, this->n_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
{
  AssertIndexRange(i, this->n_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 const Table<2, bool> &
FiniteElement<dim, spacedim>::get_local_dof_sparsity_pattern() const
{
  return local_dof_sparsity_pattern;
}
 
 
 
template <int dim, int spacedim>
inline GeometryPrimitive
FiniteElement<dim, spacedim>::get_associated_geometry_primitive(
  const unsigned int cell_dof_index) const
{
  AssertIndexRange(cell_dof_index, this->n_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->get_first_line_index())
    return GeometryPrimitive::vertex;
  else if (cell_dof_index < this->get_first_quad_index(0))
    return GeometryPrimitive::line;
  else if (cell_dof_index < this->get_first_hex_index())
    return GeometryPrimitive::quad;
  else
    return GeometryPrimitive::hex;
}
 
#endif
 
DEAL_II_NAMESPACE_CLOSE
 
#endif