#include <deal.II/fe/fe_poly.h>

Inheritance diagram for FE_Poly< PolynomialType, dim, spacedim >:

Classes
class	InternalData

Public Member Functions
	FE_Poly (const PolynomialType &poly_space, const FiniteElementData< dim > &fe_data, const std::vector< bool > &restriction_is_additive_flags, const std::vector< ComponentMask > &nonzero_components)

unsigned int	get_degree () const

virtual UpdateFlags	requires_update_flags (const UpdateFlags update_flags) const override

std::vector< unsigned int >	get_poly_space_numbering () const

std::vector< unsigned int >	get_poly_space_numbering_inverse () const

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

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

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

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

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

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

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

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

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

virtual Tensor< 4, dim >	shape_4th_derivative_component (const unsigned int i, const Point< dim > &p, const unsigned int component) const override

Public Member Functions inherited from FiniteElement< dim, spacedim >
	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

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

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

virtual std::size_t	memory_consumption () 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

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

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

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

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

Public Member Functions inherited from Subscriptor
	Subscriptor ()

	Subscriptor (const Subscriptor &)

	Subscriptor (Subscriptor &&) noexcept

virtual	~Subscriptor ()

Subscriptor &	operator= (const Subscriptor &)

Subscriptor &	operator= (Subscriptor &&) noexcept

void	subscribe (std::atomic< bool > *const validity, const std::string &identifier="") const

void	unsubscribe (std::atomic< bool > *const validity, const std::string &identifier="") const

unsigned int	n_subscriptions () const

template<typename StreamType >
void	list_subscribers (StreamType &stream) const

void	list_subscribers () const

template<class Archive >
void	serialize (Archive &ar, const unsigned int version)

Public Member Functions inherited from FiniteElementData< dim >
	FiniteElementData (const std::vector< unsigned int > &dofs_per_object, const unsigned int n_components, const unsigned int degree, const Conformity conformity=unknown, const BlockIndices &block_indices=BlockIndices())

unsigned int	n_dofs_per_vertex () const

unsigned int	n_dofs_per_line () const

unsigned int	n_dofs_per_quad () const

unsigned int	n_dofs_per_hex () const

unsigned int	n_dofs_per_face () const

unsigned int	n_dofs_per_cell () const

template<int structdim>
unsigned int	n_dofs_per_object () const

unsigned int	n_components () const

unsigned int	n_blocks () const

const BlockIndices &	block_indices () const

unsigned int	tensor_degree () const

bool	conforms (const Conformity) const

bool	operator== (const FiniteElementData &) const

Protected Member Functions
virtual std::unique_ptr< typename FiniteElement< dim, spacedim >::InternalDataBase >	get_data (const UpdateFlags update_flags, const Mapping< dim, spacedim > &, const Quadrature< dim > &quadrature, ::internal::FEValuesImplementation::FiniteElementRelatedData< dim, spacedim > &output_data) const override

void	correct_third_derivatives (internal::FEValuesImplementation::FiniteElementRelatedData< dim, spacedim > &output_data, const internal::FEValuesImplementation::MappingRelatedData< dim, spacedim > &mapping_data, const unsigned int n_q_points, const unsigned int dof) const

Protected Member Functions inherited from FiniteElement< dim, spacedim >
void	reinit_restriction_and_prolongation_matrices (const bool isotropic_restriction_only=false, const bool isotropic_prolongation_only=false)

TableIndices< 2 >	interface_constraints_size () 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 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

Protected Attributes
PolynomialType	poly_space

Protected Attributes inherited from FiniteElement< dim, spacedim >
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

Additional Inherited Members
Public Types inherited from FiniteElementData< dim >
enum	Conformity { unknown = 0x00, L2 = 0x01, Hcurl = 0x02, Hdiv = 0x04, H1 = Hcurl \| Hdiv, H2 = 0x0e }

Static Public Member Functions inherited from FiniteElement< dim, spacedim >
static ::ExceptionBase &	ExcShapeFunctionNotPrimitive (int arg1)

static ::ExceptionBase &	ExcFENotPrimitive ()

static ::ExceptionBase &	ExcUnitShapeValuesDoNotExist ()

static ::ExceptionBase &	ExcFEHasNoSupportPoints ()

static ::ExceptionBase &	ExcEmbeddingVoid ()

static ::ExceptionBase &	ExcProjectionVoid ()

static ::ExceptionBase &	ExcWrongInterfaceMatrixSize (int arg1, int arg2)

static ::ExceptionBase &	ExcInterpolationNotImplemented ()

Static Public Member Functions inherited from Subscriptor
static ::ExceptionBase &	ExcInUse (int arg1, std::string arg2, std::string arg3)

static ::ExceptionBase &	ExcNoSubscriber (std::string arg1, std::string arg2)

Public Attributes inherited from FiniteElementData< dim >
const unsigned int	dofs_per_vertex

const unsigned int	dofs_per_line

const unsigned int	dofs_per_quad

const unsigned int	dofs_per_hex

const unsigned int	first_line_index

const unsigned int	first_quad_index

const unsigned int	first_hex_index

const unsigned int	first_face_line_index

const unsigned int	first_face_quad_index

const unsigned int	dofs_per_face

const unsigned int	dofs_per_cell

const unsigned int	components

const unsigned int	degree

const Conformity	conforming_space

const BlockIndices	block_indices_data

Static Public Attributes inherited from FiniteElement< dim, spacedim >
static const unsigned int	space_dimension = spacedim

Static Public Attributes inherited from FiniteElementData< dim >
static const unsigned int	dimension = dim

Static Protected Member Functions inherited from FiniteElement< dim, spacedim >
static std::vector< unsigned int >	compute_n_nonzero_components (const std::vector< ComponentMask > &nonzero_components)

Detailed Description

template<class PolynomialType, int dim = PolynomialType::dimension, int spacedim = dim>
class FE_Poly< PolynomialType, dim, spacedim >

This class gives a unified framework for the implementation of FiniteElement classes based on polynomial spaces like the TensorProductPolynomials or PolynomialSpace classes.

Every class that has the following public member variables and functions can be used as template parameter PolynomialType.

static const unsigned int dimension;
void compute (const Point<dim>            &unit_point,
              std::vector<double>         &values,
              std::vector<Tensor<1,dim> > &grads,
              std::vector<Tensor<2,dim> > &grad_grads,
              std::vector<Tensor<3,dim> > &third_derivatives,
              std::vector<Tensor<4,dim> > &fourth_derivatives) const;
double compute_value (const unsigned int i,
                      const Point<dim> &p) const;
template <int order>
Tensor<order,dim> compute_derivative (const unsigned int i,
                                      const Point<dim> &p) const;

Example classes are TensorProductPolynomials, PolynomialSpace or PolynomialsP.

This class is not a fully implemented FiniteElement class. Instead there are several pure virtual functions declared in the FiniteElement and FiniteElement classes which cannot be implemented by this class but are left for implementation in derived classes.

Todo:: Since nearly all functions for spacedim != dim are specialized, this class needs cleaning up.

Author: Ralf Hartmann 2004, Guido Kanschat, 2009

Definition at line 72 of file fe_poly.h.

Constructor & Destructor Documentation

◆ FE_Poly()

template<class PolynomialType, int dim = PolynomialType::dimension, int spacedim = dim>

FE_Poly< PolynomialType, dim, spacedim >::FE_Poly	(	const PolynomialType &	poly_space,
		const FiniteElementData< dim > &	fe_data,
		const std::vector< bool > &	restriction_is_additive_flags,
		const std::vector< ComponentMask > &	nonzero_components
	)

Constructor.

Member Function Documentation

◆ get_degree()

template<class PolynomialType, int dim = PolynomialType::dimension, int spacedim = dim>

unsigned int FE_Poly< PolynomialType, dim, spacedim >::get_degree ( ) const

Return the polynomial degree of this finite element, i.e. the value passed to the constructor.

◆ requires_update_flags()

template<class PolynomialType, int dim = PolynomialType::dimension, int spacedim = dim>

virtual UpdateFlags FE_Poly< PolynomialType, dim, spacedim >::requires_update_flags ( const UpdateFlags update_flags ) const

overridevirtual

Given a set of update flags, compute which other quantities also need to be computed in order to satisfy the request by the given flags. Then return the combination of the original set of flags and those just computed.

As an example, if update_flags contains update_gradients a finite element class will typically require the computation of the inverse of the Jacobian matrix in order to rotate the gradient of shape functions on the reference cell to the real cell. It would then return not just update_gradients, but also update_covariant_transformation, the flag that makes the mapping class produce the inverse of the Jacobian matrix.

An extensive discussion of the interaction between this function and FEValues can be found in the How Mapping, FiniteElement, and FEValues work together documentation module.

See also: UpdateFlags

Implements FiniteElement< dim, spacedim >.

◆ get_poly_space_numbering()

template<class PolynomialType, int dim = PolynomialType::dimension, int spacedim = dim>

std::vector<unsigned int> FE_Poly< PolynomialType, dim, spacedim >::get_poly_space_numbering ( ) const

Return the numbering of the underlying polynomial space compared to lexicographic ordering of the basis functions. Returns PolynomialType::get_numbering().

◆ get_poly_space_numbering_inverse()

template<class PolynomialType, int dim = PolynomialType::dimension, int spacedim = dim>

std::vector<unsigned int> FE_Poly< PolynomialType, dim, spacedim >::get_poly_space_numbering_inverse ( ) const

Return the inverse numbering of the underlying polynomial space. Returns PolynomialType::get_numbering_inverse().

◆ shape_value()

template<class PolynomialType, int dim = PolynomialType::dimension, int spacedim = dim>

virtual double FE_Poly< PolynomialType, dim, spacedim >::shape_value	(	const unsigned int	i,
		const Point< dim > &	p
	)		const

overridevirtual

Return the value of the ith shape function at the point p. See the FiniteElement base class for more information about the semantics of this function.

Reimplemented from FiniteElement< dim, spacedim >.

◆ shape_value_component()

template<class PolynomialType, int dim = PolynomialType::dimension, int spacedim = dim>

virtual double FE_Poly< PolynomialType, dim, spacedim >::shape_value_component	(	const unsigned int	i,
		const Point< dim > &	p,
		const unsigned int	component
	)		const

overridevirtual

Return the value of the componentth vector component of the ith shape function at the point p. See the FiniteElement base class for more information about the semantics of this function.

Since this element is scalar, the returned value is the same as if the function without the _component suffix were called, provided that the specified component is zero.

Reimplemented from FiniteElement< dim, spacedim >.

◆ shape_grad()

template<class PolynomialType, int dim = PolynomialType::dimension, int spacedim = dim>

virtual Tensor<1, dim> FE_Poly< PolynomialType, dim, spacedim >::shape_grad	(	const unsigned int	i,
		const Point< dim > &	p
	)		const

overridevirtual

Return the gradient of the ith shape function at the point p. See the FiniteElement base class for more information about the semantics of this function.

Reimplemented from FiniteElement< dim, spacedim >.

◆ shape_grad_component()

template<class PolynomialType, int dim = PolynomialType::dimension, int spacedim = dim>

virtual Tensor<1, dim> FE_Poly< PolynomialType, dim, spacedim >::shape_grad_component	(	const unsigned int	i,
		const Point< dim > &	p,
		const unsigned int	component
	)		const

overridevirtual

Return the gradient of the componentth vector component of the ith shape function at the point p. See the FiniteElement base class for more information about the semantics of this function.

Since this element is scalar, the returned value is the same as if the function without the _component suffix were called, provided that the specified component is zero.

Reimplemented from FiniteElement< dim, spacedim >.

◆ shape_grad_grad()

template<class PolynomialType, int dim = PolynomialType::dimension, int spacedim = dim>

virtual Tensor<2, dim> FE_Poly< PolynomialType, dim, spacedim >::shape_grad_grad	(	const unsigned int	i,
		const Point< dim > &	p
	)		const

overridevirtual

Return the tensor of second derivatives of the ith shape function at point p on the unit cell. See the FiniteElement base class for more information about the semantics of this function.

Reimplemented from FiniteElement< dim, spacedim >.

◆ shape_grad_grad_component()

template<class PolynomialType, int dim = PolynomialType::dimension, int spacedim = dim>

virtual Tensor<2, dim> FE_Poly< PolynomialType, dim, spacedim >::shape_grad_grad_component	(	const unsigned int	i,
		const Point< dim > &	p,
		const unsigned int	component
	)		const

overridevirtual

Return the second derivative of the componentth vector component of the ith shape function at the point p. See the FiniteElement base class for more information about the semantics of this function.

Since this element is scalar, the returned value is the same as if the function without the _component suffix were called, provided that the specified component is zero.

Reimplemented from FiniteElement< dim, spacedim >.

◆ shape_3rd_derivative()

template<class PolynomialType, int dim = PolynomialType::dimension, int spacedim = dim>

virtual Tensor<3, dim> FE_Poly< PolynomialType, dim, spacedim >::shape_3rd_derivative	(	const unsigned int	i,
		const Point< dim > &	p
	)		const

overridevirtual

Return the tensor of third derivatives of the ith shape function at point p on the unit cell. See the FiniteElement base class for more information about the semantics of this function.

Reimplemented from FiniteElement< dim, spacedim >.

◆ shape_3rd_derivative_component()

template<class PolynomialType, int dim = PolynomialType::dimension, int spacedim = dim>

virtual Tensor<3, dim> FE_Poly< PolynomialType, dim, spacedim >::shape_3rd_derivative_component	(	const unsigned int	i,
		const Point< dim > &	p,
		const unsigned int	component
	)		const

overridevirtual

Return the third derivative of the componentth vector component of the ith shape function at the point p. See the FiniteElement base class for more information about the semantics of this function.

Since this element is scalar, the returned value is the same as if the function without the _component suffix were called, provided that the specified component is zero.

Reimplemented from FiniteElement< dim, spacedim >.

◆ shape_4th_derivative()

template<class PolynomialType, int dim = PolynomialType::dimension, int spacedim = dim>

virtual Tensor<4, dim> FE_Poly< PolynomialType, dim, spacedim >::shape_4th_derivative	(	const unsigned int	i,
		const Point< dim > &	p
	)		const

overridevirtual

Return the tensor of fourth derivatives of the ith shape function at point p on the unit cell. See the FiniteElement base class for more information about the semantics of this function.

Reimplemented from FiniteElement< dim, spacedim >.

◆ shape_4th_derivative_component()

template<class PolynomialType, int dim = PolynomialType::dimension, int spacedim = dim>

virtual Tensor<4, dim> FE_Poly< PolynomialType, dim, spacedim >::shape_4th_derivative_component	(	const unsigned int	i,
		const Point< dim > &	p,
		const unsigned int	component
	)		const

overridevirtual

Return the fourth derivative of the componentth vector component of the ith shape function at the point p. See the FiniteElement base class for more information about the semantics of this function.

Since this element is scalar, the returned value is the same as if the function without the _component suffix were called, provided that the specified component is zero.

Reimplemented from FiniteElement< dim, spacedim >.

◆ get_data()

template<class PolynomialType, int dim = PolynomialType::dimension, int spacedim = dim>

virtual std::unique_ptr< typename FiniteElement<dim, spacedim>::InternalDataBase> FE_Poly< PolynomialType, dim, spacedim >::get_data	(	const UpdateFlags	update_flags,
		const Mapping< dim, spacedim > &	mapping,
		const Quadrature< dim > &	quadrature,
		::internal::FEValuesImplementation::FiniteElementRelatedData< dim, spacedim > &	output_data
	)		const

inlineoverrideprotectedvirtual

Create an internal data object and return a pointer to it of which the caller of this function then assumes ownership. This object will then be passed to the FiniteElement::fill_fe_values() every time the finite element shape functions and their derivatives are evaluated on a concrete cell. The object created here is therefore used by derived classes as a place for scratch objects that are used in evaluating shape functions, as well as to store information that can be pre-computed once and re-used on every cell (e.g., for evaluating the values and gradients of shape functions on the reference cell, for later re-use when transforming these values to a concrete cell).

This function is the first one called in the process of initializing a FEValues object for a given mapping and finite element object. The returned object will later be passed to FiniteElement::fill_fe_values() for a concrete cell, which will itself place its output into an object of type internal::FEValuesImplementation::FiniteElementRelatedData. Since there may be data that can already be computed in its final form on the reference cell, this function also receives a reference to the internal::FEValuesImplementation::FiniteElementRelatedData object as its last argument. This output argument is guaranteed to always be the same one when used with the InternalDataBase object returned by this function. In other words, the subdivision of scratch data and final data in the returned object and the output_data object is as follows: If data can be pre- computed on the reference cell in the exact form in which it will later be needed on a concrete cell, then this function should already emplace it in the output_data object. An example are the values of shape functions at quadrature points for the usual Lagrange elements which on a concrete cell are identical to the ones on the reference cell. On the other hand, if some data can be pre-computed to make computations on a concrete cell cheaper, then it should be put into the returned object for later re-use in a derive class's implementation of FiniteElement::fill_fe_values(). An example are the gradients of shape functions on the reference cell for Lagrange elements: to compute the gradients of the shape functions on a concrete cell, one has to multiply the gradients on the reference cell by the inverse of the Jacobian of the mapping; consequently, we cannot already compute the gradients on a concrete cell at the time the current function is called, but we can at least pre-compute the gradients on the reference cell, and store it in the object returned.

An extensive discussion of the interaction between this function and FEValues can be found in the How Mapping, FiniteElement, and FEValues work together documentation module. See also the documentation of the InternalDataBase class.

Parameters

[in]	update_flags	A set of UpdateFlags values that describe what kind of information the FEValues object requests the finite element to compute. This set of flags may also include information that the finite element can not compute, e.g., flags that pertain to data produced by the mapping. An implementation of this function needs to set up all data fields in the returned object that are necessary to produce the finite- element related data specified by these flags, and may already pre- compute part of this information as discussed above. Elements may want to store these update flags (or a subset of these flags) in InternalDataBase::update_each so they know at the time when FiniteElement::fill_fe_values() is called what they are supposed to compute
[in]	mapping	A reference to the mapping used for computing values and derivatives of shape functions.
[in]	quadrature	A reference to the object that describes where the shape functions should be evaluated.
[out]	output_data	A reference to the object that FEValues will use in conjunction with the object returned here and where an implementation of FiniteElement::fill_fe_values() will place the requested information. This allows the current function to already pre-compute pieces of information that can be computed on the reference cell, as discussed above. FEValues guarantees that this output object and the object returned by the current function will always be used together.

Returns: A pointer to an object of a type derived from InternalDataBase and that derived classes can use to store scratch data that can be pre- computed, or for scratch arrays that then only need to be allocated once. The calling site assumes ownership of this object and will delete it when it is no longer necessary.

Implements FiniteElement< dim, spacedim >.

Definition at line 236 of file fe_poly.h.

◆ correct_third_derivatives()

template<class PolynomialType, int dim = PolynomialType::dimension, int spacedim = dim>

void FE_Poly< PolynomialType, dim, spacedim >::correct_third_derivatives	(	internal::FEValuesImplementation::FiniteElementRelatedData< dim, spacedim > &	output_data,
		const internal::FEValuesImplementation::MappingRelatedData< dim, spacedim > &	mapping_data,
		const unsigned int	n_q_points,
		const unsigned int	dof
	)		const

protected

Correct the shape third derivatives by subtracting the terms corresponding to the Jacobian pushed forward gradient and second derivative.

Before the correction, the third derivatives would be given by

\[ D_{ijkl} = \frac{d^3\phi_i}{d \hat x_J d \hat x_K d \hat x_L} (J_{jJ})^{-1} (J_{kK})^{-1} (J_{lL})^{-1}, \]

where \(J_{iI}=\frac{d x_i}{d \hat x_I}\). After the correction, the correct third derivative would be given by

\[ \frac{d^3\phi_i}{d x_j d x_k d x_l} = D_{ijkl} - H_{mjl} \frac{d^2 \phi_i}{d x_k d x_m} - H_{mkl} \frac{d^2 \phi_i}{d x_j d x_m} - H_{mjk} \frac{d^2 \phi_i}{d x_l d x_m} - K_{mjkl} \frac{d \phi_i}{d x_m}, \]

where \(H_{ijk}\) is the Jacobian pushed-forward derivative and \(K_{ijkl}\) is the Jacobian pushed-forward second derivative.

Member Data Documentation

◆ poly_space

template<class PolynomialType, int dim = PolynomialType::dimension, int spacedim = dim>

PolynomialType FE_Poly< PolynomialType, dim, spacedim >::poly_space

protected

The polynomial space. Its type is given by the template parameter PolynomialType.

Definition at line 482 of file fe_poly.h.

The documentation for this class was generated from the following file:

deal.II/fe/fe_poly.h

Classes

Public Member Functions

Protected Member Functions

Protected Attributes

Additional Inherited Members

Detailed Description

template<class PolynomialType, int dim = PolynomialType::dimension, int spacedim = dim> class FE_Poly< PolynomialType, dim, spacedim >

Constructor & Destructor Documentation

◆ FE_Poly()

Member Function Documentation

◆ get_degree()

◆ requires_update_flags()

◆ get_poly_space_numbering()

◆ get_poly_space_numbering_inverse()

◆ shape_value()

◆ shape_value_component()

◆ shape_grad()

◆ shape_grad_component()

◆ shape_grad_grad()

◆ shape_grad_grad_component()

◆ shape_3rd_derivative()

◆ shape_3rd_derivative_component()

◆ shape_4th_derivative()

◆ shape_4th_derivative_component()

◆ get_data()

◆ correct_third_derivatives()

Member Data Documentation

◆ poly_space

template<class PolynomialType, int dim = PolynomialType::dimension, int spacedim = dim>
class FE_Poly< PolynomialType, dim, spacedim >