Model of a finite element. More...

#include <FiniteElement.h>

Public Types
using	geometry_type = T
	Geometry type of the Mesh that the FunctionSpace is defined on.

Public Member Functions
	FiniteElement (const ufcx_finite_element &e)
	Create finite element from UFC finite element.

	FiniteElement (const basix::FiniteElement< geometry_type > &element, const std::size_t block_size, const bool symmetric=false)
	Create finite element from a Basix finite element.

	FiniteElement (const FiniteElement &element)=delete
	Copy constructor.

	FiniteElement (FiniteElement &&element)=default
	Move constructor.

	~FiniteElement ()=default
	Destructor.

FiniteElement &	operator= (const FiniteElement &element)=delete
	Copy assignment.

FiniteElement &	operator= (FiniteElement &&element)=default
	Move assignment.

bool	operator== (const FiniteElement &e) const

bool	operator!= (const FiniteElement &e) const

std::string	signature () const noexcept

mesh::CellType	cell_shape () const noexcept

int	space_dimension () const noexcept

int	block_size () const noexcept

int	reference_value_size () const

std::span< const std::size_t >	reference_value_shape () const
	The reference value shape.

bool	symmetric () const
	Does the element represent a symmetric 2-tensor?

void	tabulate (std::span< geometry_type > values, std::span< const geometry_type > X, std::array< std::size_t, 2 > shape, int order) const
	Evaluate derivatives of the basis functions up to given order at points in the reference cell.

std::pair< std::vector< geometry_type >, std::array< std::size_t, 4 > >	tabulate (std::span< const geometry_type > X, std::array< std::size_t, 2 > shape, int order) const

int	num_sub_elements () const noexcept
	Number of sub elements (for a mixed or blocked element).

bool	is_mixed () const noexcept
	Check if element is a mixed element.

const std::vector< std::shared_ptr< const FiniteElement< geometry_type > > > &	sub_elements () const noexcept
	Get subelements (if any)

std::shared_ptr< const FiniteElement< geometry_type > >	extract_sub_element (const std::vector< int > &component) const
	Extract sub finite element for component.

const basix::FiniteElement< geometry_type > &	basix_element () const
	Return underlying Basix element (if it exists).

basix::maps::type	map_type () const
	Get the map type used by the element.

bool	interpolation_ident () const noexcept

bool	map_ident () const noexcept

std::pair< std::vector< geometry_type >, std::array< std::size_t, 2 > >	interpolation_points () const
	Points on the reference cell at which an expression needs to be evaluated in order to interpolate the expression in the finite element space.

std::pair< std::vector< geometry_type >, std::array< std::size_t, 2 > >	interpolation_operator () const

std::pair< std::vector< geometry_type >, std::array< std::size_t, 2 > >	create_interpolation_operator (const FiniteElement &from) const
	Create a matrix that maps degrees of freedom from one element to this element (interpolation).

bool	needs_dof_transformations () const noexcept
	Check if DOF transformations are needed for this element.

bool	needs_dof_permutations () const noexcept
	Check if DOF permutations are needed for this element.

template<typename U >
std::function< void(std::span< U >, std::span< const std::uint32_t >, std::int32_t, int)>	dof_transformation_fn (doftransform ttype, bool scalar_element=false) const
	Return a function that applies a DOF transformation operator to some data (see T_apply()).

template<typename U >
std::function< void(std::span< U >, std::span< const std::uint32_t >, std::int32_t, int)>	dof_transformation_right_fn (doftransform ttype, bool scalar_element=false) const
	Return a function that applies DOF transformation to some transposed data (see T_apply_right()).

template<typename U >
void	T_apply (std::span< U > data, std::uint32_t cell_permutation, int n) const
	Transform basis functions from the reference element ordering and orientation to the globally consistent physical element ordering and orientation.

template<typename U >
void	Tt_inv_apply (std::span< U > data, std::uint32_t cell_permutation, int n) const
	Apply the inverse transpose of the operator applied by T_apply().

template<typename U >
void	Tt_apply (std::span< U > data, std::uint32_t cell_permutation, int n) const
	Apply the transpose of the operator applied by T_apply().

template<typename U >
void	Tinv_apply (std::span< U > data, std::uint32_t cell_permutation, int n) const
	Apply the inverse of the operator applied by T_apply().

template<typename U >
void	T_apply_right (std::span< U > data, std::uint32_t cell_permutation, int n) const
	Right(post)-apply the operator applied by T_apply().

template<typename U >
void	Tinv_apply_right (std::span< U > data, std::uint32_t cell_permutation, int n) const
	Right(post)-apply the inverse of the operator applied by T_apply().

template<typename U >
void	Tt_apply_right (std::span< U > data, std::uint32_t cell_permutation, int n) const
	Right(post)-apply the transpose of the operator applied by T_apply().

template<typename U >
void	Tt_inv_apply_right (std::span< U > data, std::uint32_t cell_permutation, int n) const
	Right(post)-apply the transpose inverse of the operator applied by T_apply().

void	permute (std::span< std::int32_t > doflist, std::uint32_t cell_permutation) const
	Permute indices associated with degree-of-freedoms on the reference element ordering to the globally consistent physical element degree-of-freedom ordering.

void	permute_inv (std::span< std::int32_t > doflist, std::uint32_t cell_permutation) const
	Perform the inverse of the operation applied by permute().

std::function< void(std::span< std::int32_t >, std::uint32_t)>	dof_permutation_fn (bool inverse=false, bool scalar_element=false) const
	Return a function that applies a degree-of-freedom permutation to some data.

Detailed Description

template<std::floating_point T>
class dolfinx::fem::FiniteElement< T >

Model of a finite element.

Provides the dof layout on a reference element, and various methods for evaluating and transforming the basis.

Constructor & Destructor Documentation

◆ FiniteElement() [1/2]

template<std::floating_point T>

FiniteElement ( const ufcx_finite_element & e )

explicit

Create finite element from UFC finite element.

Parameters

[in] e UFC finite element.

◆ FiniteElement() [2/2]

template<std::floating_point T>

FiniteElement	(	const basix::FiniteElement< geometry_type > &	element,
		const std::size_t	block_size,
		const bool	symmetric = false )

Create finite element from a Basix finite element.

Parameters

[in]	element	Basix finite element
[in]	block_size	The block size for the element
[in]	symmetric	Is the element a symmetric tensor?

Member Function Documentation

◆ basix_element()

template<std::floating_point T>

const basix::FiniteElement< T > & basix_element ( ) const

Return underlying Basix element (if it exists).

Exceptions

Throws a std::runtime_error is there no Basix element.

◆ block_size()

template<std::floating_point T>

int block_size ( ) const

noexcept

Block size of the finite element function space. For BlockedElements, this is the number of DOFs colocated at each DOF point. For other elements, this is always 1.

Returns: Block size of the finite element space

◆ cell_shape()

template<std::floating_point T>

mesh::CellType cell_shape ( ) const

noexcept

Cell shape

Returns: Element cell shape

◆ create_interpolation_operator()

template<std::floating_point T>

std::pair< std::vector< T >, std::array< std::size_t, 2 > > create_interpolation_operator ( const FiniteElement< T > & from ) const

Create a matrix that maps degrees of freedom from one element to this element (interpolation).

Parameters

[in] from The element to interpolate from

Returns: Matrix operator that maps the from degrees-of-freedom to the degrees-of-freedom of this element. The (0) matrix data (row-major storage) and (1) the shape (num_dofs of this element, num_dofs of from) are returned.

Precondition: The two elements must use the same mapping between the reference and physical cells

Note: Does not support mixed elements

◆ dof_permutation_fn()

template<std::floating_point T>

std::function< void(std::span< std::int32_t >, std::uint32_t)> dof_permutation_fn	(	bool	inverse = false,
		bool	scalar_element = false ) const

Return a function that applies a degree-of-freedom permutation to some data.

The returned function can apply permute() to mixed-elements.

The signature of the returned function has three arguments:

[in,out] doflist The numbers of the DOFs, a span of length num_dofs
[in] cell_permutation Permutation data for the cell
[in] block_size The block_size of the input data

Parameters

[in]	inverse	Indicates if the inverse transformation should be returned.
[in]	scalar_element	Indicates is the scalar transformations should be returned for a vector element.

◆ dof_transformation_fn()

template<std::floating_point T>

template<typename U >

std::function< void(std::span< U >, std::span< const std::uint32_t >, std::int32_t, int)> dof_transformation_fn	(	doftransform	ttype,
		bool	scalar_element = false ) const

inline

Return a function that applies a DOF transformation operator to some data (see T_apply()).

The transformation is applied from the left-hand side, i.e.

\[ u \leftarrow T u. \]

If the transformation for the (sub)element is a permutation only, the returned function will do change the ordering for the (sub)element as it is assumed that permutations are incorporated into the degree-of-freedom map.

See the documentation for T_apply() for a description of the transformation for a single element type. This function generates a function that can apply the transformation to a mixed element.

The signature of the returned function has four arguments:

[in,out] data The data to be transformed. This data is flattened with row-major layout, shape=(num_dofs, block_size)
[in] cell_info Permutation data for the cell. The size of this is num_cells. For elements where no transformations are required, an empty span can be passed in.
[in] cell The cell number.
[in] n The block_size of the input data.

Parameters

[in]	ttype	The transformation type. Typical usage is: doftransform::standard Transforms basis function data from the reference element to the conforming 'physical' element, e.g. \(\phi = T \tilde{\phi}\). doftransform::transpose Transforms degree-of-freedom data from the conforming (physical) ordering to the reference ordering, e.g. \(\tilde{u} = T^{T} u\). doftransform::inverse: Transforms basis function data from the the conforming (physical) ordering to the reference ordering, e.g. \(\tilde{\phi} = T^{-1} \phi\). doftransform::inverse_transpose: Transforms degree-of-freedom data from the reference element to the conforming (physical) ordering, e.g. \(u = T^{-t} \tilde{u}\).
[in]	scalar_element	Indicates whether the scalar transformations should be returned for a vector element

◆ dof_transformation_right_fn()

template<std::floating_point T>

template<typename U >

std::function< void(std::span< U >, std::span< const std::uint32_t >, std::int32_t, int)> dof_transformation_right_fn	(	doftransform	ttype,
		bool	scalar_element = false ) const

inline

Return a function that applies DOF transformation to some transposed data (see T_apply_right()).

The transformation is applied from the right-hand side, i.e.

\[ u^{t} \leftarrow u^{t} T. \]

If the transformation for the (sub)element is a permutation only, the returned function will do change the ordering for the (sub)element as it is assumed that permutations are incorporated into the degree-of-freedom map.

The signature of the returned function has four arguments:

[in,out] data The data to be transformed. This data is flattened with row-major layout, shape=(num_dofs, block_size)
[in] cell_info Permutation data for the cell. The size of this is num_cells. For elements where no transformations are required, an empty span can be passed in.
[in] cell The cell number
[in] block_size The block_size of the input data

Parameters

[in]	ttype	Transformation type. See dof_transformation_fn().
[in]	scalar_element	Indicate if the scalar transformations should be returned for a vector element.

◆ interpolation_ident()

template<std::floating_point T>

bool interpolation_ident ( ) const

noexcept

Check if interpolation into the finite element space is an identity operation given the evaluation on an expression at specific points, i.e. the degree-of-freedom are equal to point evaluations. The function will return true for Lagrange elements.

Returns: True if interpolation is an identity operation

◆ interpolation_operator()

template<std::floating_point T>

std::pair< std::vector< T >, std::array< std::size_t, 2 > > interpolation_operator ( ) const

Interpolation operator (matrix) Pi that maps a function evaluated at the points provided by FiniteElement::interpolation_points to the element degrees of freedom, i.e. dofs = Pi f_x. See the Basix documentation for basix::FiniteElement::interpolation_matrix for how the data in f_x should be ordered.

Returns: The interpolation operator Pi, returning the data for Pi (row-major storage) and the shape (num_dofs, num_points * value_size)

◆ interpolation_points()

template<std::floating_point T>

std::pair< std::vector< T >, std::array< std::size_t, 2 > > interpolation_points ( ) const

Points on the reference cell at which an expression needs to be evaluated in order to interpolate the expression in the finite element space.

For Lagrange elements the points will just be the nodal positions. For other elements the points will typically be the quadrature points used to evaluate moment degrees of freedom.

Returns: Interpolation point coordinates on the reference cell, returning the (0) coordinates data (row-major) storage and (1) the shape (num_points, tdim).

◆ is_mixed()

template<std::floating_point T>

bool is_mixed ( ) const

noexcept

Check if element is a mixed element.

A mixed element i composed of two or more elements of different types (a block element, e.g. a Lagrange element with block size >= 1 is not considered mixed).

Returns: True if element is mixed.

◆ map_ident()

template<std::floating_point T>

bool map_ident ( ) const

noexcept

Check if the push forward/pull back map from the values on reference to the values on a physical cell for this element is the identity map.

Returns: True if the map is the identity

◆ needs_dof_permutations()

template<std::floating_point T>

bool needs_dof_permutations ( ) const

noexcept

Check if DOF permutations are needed for this element.

DOF permutations will be needed for elements which might not be continuous when two neighbouring cells disagree on the orientation of a shared subentity, and when this can be corrected for by permuting the DOF numbering in the dofmap.

For example, higher order Lagrange elements will need DOF permutations, as the arrangement of DOFs on a shared sub-entity may be different from the point of view of neighbouring cells, and this can be corrected for by permuting the DOF numbers on each cell.

Returns: True if DOF transformations are required

◆ needs_dof_transformations()

template<std::floating_point T>

bool needs_dof_transformations ( ) const

noexcept

Check if DOF transformations are needed for this element.

DOF transformations will be needed for elements which might not be continuous when two neighbouring cells disagree on the orientation of a shared sub-entity, and when this cannot be corrected for by permuting the DOF numbering in the dofmap.

For example, Raviart-Thomas elements will need DOF transformations, as the neighbouring cells may disagree on the orientation of a basis function, and this orientation cannot be corrected for by permuting the DOF numbers on each cell.

Returns: True if DOF transformations are required

◆ num_sub_elements()

template<std::floating_point T>

int num_sub_elements ( ) const

noexcept

Number of sub elements (for a mixed or blocked element).

Returns: The number of sub elements

◆ operator!=()

template<std::floating_point T>

bool operator!= ( const FiniteElement< T > & e ) const

Check if two elements are not equivalent

Returns: True is the two elements are not the same

Note: Equality can be checked only for non-mixed elements. For a mixed element, this function will raise an exception.

◆ operator==()

template<std::floating_point T>

bool operator== ( const FiniteElement< T > & e ) const

Check if two elements are equivalent

Returns: True is the two elements are the same

Note: Equality can be checked only for non-mixed elements. For a mixed element, this function will raise an exception.

◆ permute()

template<std::floating_point T>

void permute	(	std::span< std::int32_t >	doflist,
		std::uint32_t	cell_permutation ) const

Permute indices associated with degree-of-freedoms on the reference element ordering to the globally consistent physical element degree-of-freedom ordering.

Given an array \(\tilde{d}\) that holds an integer associated with each degree-of-freedom and following the reference element degree-of-freedom ordering, this function computes

\[ d = P \tilde{d},\]

where \(P\) is a permutation matrix and \(d\) holds the integers in \(\tilde{d}\) but permuted to follow the globally consistent physical element degree-of-freedom ordering. The permutation is computed in-place.

Parameters

[in,out]	doflist	Indicies associated with the degrees-of-freedom. Size=`num_dofs`.
[in]	cell_permutation	Permutation data for the cell.

◆ permute_inv()

template<std::floating_point T>

void permute_inv	(	std::span< std::int32_t >	doflist,
		std::uint32_t	cell_permutation ) const

Perform the inverse of the operation applied by permute().

Given an array \(d\) that holds an integer associated with each degree-of-freedom and following the globally consistent physical element degree-of-freedom ordering, this function computes

\[ \tilde{d} = P^{T} d, \]

where \(P^{T}\) is a permutation matrix and \(\tilde{d}\) holds the integers in \(d\) but permuted to follow the reference element degree-of-freedom ordering. The permutation is computed in-place.

Parameters

[in,out]	doflist	Indicies associated with the degrees-of-freedom. Size=`num_dofs`.
[in]	cell_permutation	Permutation data for the cell.

◆ reference_value_size()

template<std::floating_point T>

int reference_value_size ( ) const

The value size, e.g. 1 for a scalar function, 2 for a 2D vector, 9 for a second-order tensor in 3D, for the reference element

Returns: The value size for the reference element

◆ signature()

template<std::floating_point T>

std::string signature ( ) const

noexcept

String identifying the finite element

Returns: Element signature

Warning: The function is provided for convenience, but it should not be relied upon for determining the element type. Use other functions, commonly returning enums, to determine element properties.

◆ space_dimension()

template<std::floating_point T>

int space_dimension ( ) const

noexcept

Dimension of the finite element function space (the number of degrees-of-freedom for the element)

Returns: Dimension of the finite element space

◆ T_apply()

template<std::floating_point T>

template<typename U >

void T_apply	(	std::span< U >	data,
		std::uint32_t	cell_permutation,
		int	n ) const

inline

Transform basis functions from the reference element ordering and orientation to the globally consistent physical element ordering and orientation.

Consider that the value of a finite element function \(f_{h}\) at a point is given by

\[ f_{h} = \phi^{T} c, \]

where \(f_{h}\) has shape \(r \times 1\), \(\phi\) has shape \(d \times r\) and holds the finite element basis functions, and \(c\) has shape \(d \times 1\) and holds the degrees-of-freedom. The basis functions and degree-of-freedom are with respect to the physical element orientation. If the degrees-of-freedom on the physical element orientation are given by

\[ \phi = T \tilde{\phi}, \]

where \(T\) is a \(d \times d\) matrix, it follows from \(f_{h} = \phi^{T} c = \tilde{\phi}^{T} T^{T} c\) that

\[ \tilde{c} = T^{T} c. \]

This function applies \(T\) to data. The transformation is performed in-place. The operator \(T\) is orthogonal for many elements, but not all.

This function calls the corresponding Basix function.

Parameters

[in,out]	data	Data to transform. The shape is `(m, n)`, where `m` is the number of dgerees-of-freedom and the storage is row-major.
[in]	cell_permutation	Permutation data for the cell
[in]	n	Number of columns in `data`.

◆ T_apply_right()

template<std::floating_point T>

template<typename U >

void T_apply_right	(	std::span< U >	data,
		std::uint32_t	cell_permutation,
		int	n ) const

inline

Right(post)-apply the operator applied by T_apply().

Computes

\[ v^{T} = u^{T} T \]

in-place.

Parameters

[in,out]	data	The data to be transformed. This data is flattened with row-major layout, `shape=(num_dofs, block_size)`.
[in]	cell_permutation	Permutation data for the cell
[in]	n	Block size of the input data

◆ tabulate() [1/2]

template<std::floating_point T>

std::pair< std::vector< T >, std::array< std::size_t, 4 > > tabulate	(	std::span< const geometry_type >	X,
		std::array< std::size_t, 2 >	shape,
		int	order ) const

Evaluate all derivatives of the basis functions up to given order at given points in reference cell

Parameters

[in]	X	The reference coordinates at which to evaluate the basis functions. Shape is `(num_points, topological dimension)` (row-major storage)
[in]	shape	The shape of `X`
[in]	order	The number of derivatives (up to and including this order) to tabulate for

Returns: Basis function values and array shape (row-major storage)

◆ tabulate() [2/2]

template<std::floating_point T>

void tabulate	(	std::span< geometry_type >	values,
		std::span< const geometry_type >	X,
		std::array< std::size_t, 2 >	shape,
		int	order ) const

Evaluate derivatives of the basis functions up to given order at points in the reference cell.

Parameters

[in,out]	values	Array that will be filled with the tabulated basis values. Must have shape `(num_derivatives, num_points, num_dofs, reference_value_size)` (row-major storage)
[in]	X	The reference coordinates at which to evaluate the basis functions. Shape is `(num_points, topological dimension)` (row-major storage)
[in]	shape	The shape of `X`
[in]	order	The number of derivatives (up to and including this order) to tabulate for

◆ Tinv_apply()

template<std::floating_point T>

template<typename U >

void Tinv_apply	(	std::span< U >	data,
		std::uint32_t	cell_permutation,
		int	n ) const

inline

Apply the inverse of the operator applied by T_apply().

The transformation

\[ v = T^{-1} u \]

is performed in-place.

Parameters

[in,out]	data	The data to be transformed. This data is flattened with row-major layout, `shape=(num_dofs, block_size)`.
[in]	cell_permutation	Permutation data for the cell.
[in]	n	Block size of the input data.

◆ Tinv_apply_right()

template<std::floating_point T>

template<typename U >

void Tinv_apply_right	(	std::span< U >	data,
		std::uint32_t	cell_permutation,
		int	n ) const

inline

Right(post)-apply the inverse of the operator applied by T_apply().

Computes

\[ v^{T} = u^{T} T^{-1} \]

in-place.

Parameters

[in,out]	data	Data to be transformed. This data is flattened with row-major layout, `shape=(num_dofs, block_size)`.
[in]	cell_permutation	Permutation data for the cell
[in]	n	Block size of the input data

◆ Tt_apply()

template<std::floating_point T>

template<typename U >

void Tt_apply	(	std::span< U >	data,
		std::uint32_t	cell_permutation,
		int	n ) const

inline

Apply the transpose of the operator applied by T_apply().

The transformation

\[ u \leftarrow T^{T} u \]

is performed in-place.

Parameters

[in,out]	data	The data to be transformed. This data is flattened with row-major layout, `shape=(num_dofs, block_size)`.
[in]	cell_permutation	Permutation data for the cell.
[in]	n	The block size of the input data.

◆ Tt_apply_right()

template<std::floating_point T>

template<typename U >

void Tt_apply_right	(	std::span< U >	data,
		std::uint32_t	cell_permutation,
		int	n ) const

inline

Right(post)-apply the transpose of the operator applied by T_apply().

Computes

\[ v^{T} = u^{T} T^{T} \]

in-place.

Parameters

[in,out]	data	Data to be transformed. The data is flattened with row-major layout, `shape=(num_dofs, block_size)`.
[in]	cell_permutation	Permutation data for the cell
[in]	n	Block size of the input data.

◆ Tt_inv_apply()

template<std::floating_point T>

template<typename U >

void Tt_inv_apply	(	std::span< U >	data,
		std::uint32_t	cell_permutation,
		int	n ) const

inline

Apply the inverse transpose of the operator applied by T_apply().

The transformation

\[ v = T^{-T} u \]

is performed in-place.

Parameters

[in,out]	data	The data to be transformed. This data is flattened with row-major layout, `shape=(num_dofs, block_size)`.
[in]	cell_permutation	Permutation data for the cell.
[in]	n	Block_size of the input data.

◆ Tt_inv_apply_right()

template<std::floating_point T>

template<typename U >

void Tt_inv_apply_right	(	std::span< U >	data,
		std::uint32_t	cell_permutation,
		int	n ) const

inline

Right(post)-apply the transpose inverse of the operator applied by T_apply().

Computes

\[ v^{T} = u^{T} T^{-T} \]

in-place.

Parameters

[in,out]	data	Data to be transformed. This data is flattened with row-major layout, `shape=(num_dofs, block_size)`.
[in]	cell_permutation	Permutation data for the cell.
[in]	n	Block size of the input data.

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

/__w/dolfinx/dolfinx/cpp/dolfinx/fem/FiniteElement.h
/__w/dolfinx/dolfinx/cpp/dolfinx/fem/FiniteElement.cpp

Public Types

Public Member Functions

Detailed Description

Constructor & Destructor Documentation

◆ FiniteElement() [1/2]

◆ FiniteElement() [2/2]

Member Function Documentation

◆ basix_element()

◆ block_size()

◆ cell_shape()

◆ create_interpolation_operator()

◆ dof_permutation_fn()

◆ dof_transformation_fn()

◆ dof_transformation_right_fn()

◆ interpolation_ident()

◆ interpolation_operator()

◆ interpolation_points()

◆ is_mixed()

◆ map_ident()

◆ needs_dof_permutations()

◆ needs_dof_transformations()

◆ num_sub_elements()

◆ operator!=()

◆ operator==()

◆ permute()

◆ permute_inv()

◆ reference_value_size()

◆ signature()

◆ space_dimension()

◆ T_apply()

◆ T_apply_right()

◆ tabulate() [1/2]

◆ tabulate() [2/2]

◆ Tinv_apply()

◆ Tinv_apply_right()

◆ Tt_apply()

◆ Tt_apply_right()

◆ Tt_inv_apply()

◆ Tt_inv_apply_right()