Finite Element, containing the dof layout on a reference element, and various methods for evaluating and transforming the basis. More...

#include <FiniteElement.h>

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

	FiniteElement (const basix::FiniteElement &element, int bs)
	Create finite element from a Basix finite element. More...

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

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

virtual	~FiniteElement ()=default
	Destructor.

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

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

bool	operator== (const FiniteElement &e) const
	Check if two elements are equivalent. More...

bool	operator!= (const FiniteElement &e) const
	Check if two elements are not equivalent. More...

std::string	signature () const noexcept
	String identifying the finite element. More...

mesh::CellType	cell_shape () const noexcept
	Cell shape. More...

int	space_dimension () const noexcept
	Dimension of the finite element function space (the number of degrees-of-freedom for the element) More...

int	block_size () const noexcept
	Block size of the finite element function space. For VectorElements and TensorElements, this is the number of DOFs colocated at each DOF point. For other elements, this is always 1. More...

int	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. More...

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. More...

xtl::span< const int >	value_shape () const noexcept
	Shape of the value space. The rank is the size of the `value_shape`.

std::string	family () const noexcept
	The finite element family. More...

void	tabulate (xt::xtensor< double, 4 > &values, const xt::xtensor< double, 2 > &X, int order) const
	Evaluate all derivatives of the basis functions up to given order at given points in reference cell. More...

template<typename O , typename P , typename Q , typename R >
std::function< void(O &, const P &, const Q &, double, const R &)>	map_fn () const
	Return a function that performs the appropriate push-forward (pull-back) for the element type. More...

int	num_sub_elements () const noexcept
	Get the number of sub elements (for a mixed or blocked element) More...

bool	is_mixed () const noexcept
	Check if element is a mixed element, i.e. 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. More...

const std::vector< std::shared_ptr< const FiniteElement > > &	sub_elements () const noexcept
	Subelements (if any)

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

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

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. More...

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. More...

const xt::xtensor< double, 2 > &	interpolation_points () const
	Points on the reference cell at which an expression need 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. More...

const xt::xtensor< double, 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. More...

xt::xtensor< double, 2 >	create_interpolation_operator (const FiniteElement &from) const
	Create a matrix that maps degrees of freedom from one element to this element (interpolation). More...

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

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

template<typename T >
std::function< void(const xtl::span< T > &, const xtl::span< const std::uint32_t > &, std::int32_t, int)>	get_dof_transformation_function (bool inverse=false, bool transpose=false, bool scalar_element=false) const
	Return a function that applies DOF transformation to some data. More...

template<typename T >
std::function< void(const xtl::span< T > &, const xtl::span< const std::uint32_t > &, std::int32_t, int)>	get_dof_transformation_to_transpose_function (bool inverse=false, bool transpose=false, bool scalar_element=false) const
	Return a function that applies DOF transformation to some transposed data. More...

template<typename T >
void	apply_dof_transformation (const xtl::span< T > &data, std::uint32_t cell_permutation, int block_size) const
	Apply DOF transformation to some data. More...

template<typename T >
void	apply_inverse_transpose_dof_transformation (const xtl::span< T > &data, std::uint32_t cell_permutation, int block_size) const
	Apply inverse transpose transformation to some data. For VectorElements, this applies the transformations for the scalar subelement. More...

template<typename T >
void	apply_transpose_dof_transformation (const xtl::span< T > &data, std::uint32_t cell_permutation, int block_size) const
	Apply transpose transformation to some data. For VectorElements, this applies the transformations for the scalar subelement. More...

template<typename T >
void	apply_inverse_dof_transformation (const xtl::span< T > &data, std::uint32_t cell_permutation, int block_size) const
	Apply inverse transformation to some data. For VectorElements, this applies the transformations for the scalar subelement. More...

template<typename T >
void	apply_dof_transformation_to_transpose (const xtl::span< T > &data, std::uint32_t cell_permutation, int block_size) const
	Apply DOF transformation to some transposed data. More...

template<typename T >
void	apply_inverse_dof_transformation_to_transpose (const xtl::span< T > &data, std::uint32_t cell_permutation, int block_size) const
	Apply inverse of DOF transformation to some transposed data. More...

template<typename T >
void	apply_transpose_dof_transformation_to_transpose (const xtl::span< T > &data, std::uint32_t cell_permutation, int block_size) const
	Apply transpose of transformation to some transposed data. More...

template<typename T >
void	apply_inverse_transpose_dof_transformation_to_transpose (const xtl::span< T > &data, std::uint32_t cell_permutation, int block_size) const
	Apply inverse transpose transformation to some transposed data. More...

void	permute_dofs (const xtl::span< std::int32_t > &doflist, std::uint32_t cell_permutation) const
	Permute the DOFs of the element. More...

void	unpermute_dofs (const xtl::span< std::int32_t > &doflist, std::uint32_t cell_permutation) const
	Unpermute the DOFs of the element. More...

std::function< void(const xtl::span< std::int32_t > &, std::uint32_t)>	get_dof_permutation_function (bool inverse=false, bool scalar_element=false) const
	Return a function that applies DOF permutation to some data. More...

Detailed Description

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

Constructor & Destructor Documentation

◆ FiniteElement() [1/2]

FiniteElement ( const ufcx_finite_element & e )

explicit

Create finite element from UFC finite element.

Parameters

[in] e UFC finite element

◆ FiniteElement() [2/2]

FiniteElement	(	const basix::FiniteElement &	element,
		int	bs
	)

Create finite element from a Basix finite element.

Parameters

[in]	element	Basix finite element
[in]	bs	The block size

Member Function Documentation

◆ apply_dof_transformation()

void apply_dof_transformation	(	const xtl::span< T > &	data,
		std::uint32_t	cell_permutation,
		int	block_size
	)		const

inline

Apply DOF transformation to some data.

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]	block_size	The block_size of the input data

◆ apply_dof_transformation_to_transpose()

void apply_dof_transformation_to_transpose	(	const xtl::span< T > &	data,
		std::uint32_t	cell_permutation,
		int	block_size
	)		const

inline

Apply DOF transformation to some transposed data.

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]	block_size	The block_size of the input data

◆ apply_inverse_dof_transformation()

void apply_inverse_dof_transformation	(	const xtl::span< T > &	data,
		std::uint32_t	cell_permutation,
		int	block_size
	)		const

inline

Apply inverse transformation to some data. For VectorElements, this applies the transformations for the scalar subelement.

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]	block_size	The block_size of the input data

◆ apply_inverse_dof_transformation_to_transpose()

void apply_inverse_dof_transformation_to_transpose	(	const xtl::span< T > &	data,
		std::uint32_t	cell_permutation,
		int	block_size
	)		const

inline

Apply inverse of DOF transformation to some transposed data.

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]	block_size	The block_size of the input data

◆ apply_inverse_transpose_dof_transformation()

void apply_inverse_transpose_dof_transformation	(	const xtl::span< T > &	data,
		std::uint32_t	cell_permutation,
		int	block_size
	)		const

inline

Apply inverse transpose transformation to some data. For VectorElements, this applies the transformations for the scalar subelement.

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]	block_size	The block_size of the input data

◆ apply_inverse_transpose_dof_transformation_to_transpose()

void apply_inverse_transpose_dof_transformation_to_transpose	(	const xtl::span< T > &	data,
		std::uint32_t	cell_permutation,
		int	block_size
	)		const

inline

Apply inverse transpose transformation to some transposed data.

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]	block_size	The block_size of the input data

◆ apply_transpose_dof_transformation()

void apply_transpose_dof_transformation	(	const xtl::span< T > &	data,
		std::uint32_t	cell_permutation,
		int	block_size
	)		const

inline

Apply transpose transformation to some data. For VectorElements, this applies the transformations for the scalar subelement.

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]	block_size	The block_size of the input data

◆ apply_transpose_dof_transformation_to_transpose()

void apply_transpose_dof_transformation_to_transpose	(	const xtl::span< T > &	data,
		std::uint32_t	cell_permutation,
		int	block_size
	)		const

inline

Apply transpose of transformation to some transposed data.

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]	block_size	The block_size of the input data

◆ block_size()

int block_size ( ) const

noexcept

Block size of the finite element function space. For VectorElements and TensorElements, 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()

mesh::CellType cell_shape ( ) const

noexcept

Cell shape.

Returns: Element cell shape

◆ create_interpolation_operator()

xt::xtensor< double, 2 > create_interpolation_operator ( const FiniteElement & 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. Shape is (num_dofs of this element, num_dofs of from).

Note: The two elements must use the same mapping between the reference and physical cells; Does not support mixed elements

◆ family()

std::string family ( ) const

noexcept

The finite element family.

Returns: The string of the finite element family

◆ get_dof_permutation_function()

std::function< void(const xtl::span< std::int32_t > &, std::uint32_t)> get_dof_permutation_function	(	bool	inverse = `false`,
		bool	scalar_element = `false`
	)		const

Return a function that applies DOF permutation to some data.

The returned function will take three inputs:

[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 whether the inverse transformations should be returned
[in]	scalar_element	Indicated whether the scalar transformations should be returned for a vector element

◆ get_dof_transformation_function()

std::function<void(const xtl::span<T>&, const xtl::span<const std::uint32_t>&, std::int32_t, int)> get_dof_transformation_function	(	bool	inverse = `false`,
		bool	transpose = `false`,
		bool	scalar_element = `false`
	)		const

inline

Return a function that applies DOF transformation to some data.

The returned function will take four inputs:

[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]	inverse	Indicates whether the inverse transformations should be returned
[in]	transpose	Indicates whether the transpose transformations should be returned
[in]	scalar_element	Indicates whether the scalar transformations should be returned for a vector element

◆ get_dof_transformation_to_transpose_function()

std::function<void(const xtl::span<T>&, const xtl::span<const std::uint32_t>&, std::int32_t, int)> get_dof_transformation_to_transpose_function	(	bool	inverse = `false`,
		bool	transpose = `false`,
		bool	scalar_element = `false`
	)		const

inline

Return a function that applies DOF transformation to some transposed data.

The returned function will take three inputs:

[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]	inverse	Indicates whether the inverse transformations should be returned
[in]	transpose	Indicates whether the transpose transformations should be returned
[in]	scalar_element	Indicated whether the scalar transformations should be returned for a vector element

◆ interpolation_ident()

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()

const xt::xtensor< double, 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. Shape is (num_dofs, num_points*value_size)

◆ interpolation_points()

const xt::xtensor< double, 2 > & interpolation_points ( ) const

Points on the reference cell at which an expression need 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: Points on the reference cell. Shape is (num_points, tdim).

◆ is_mixed()

bool is_mixed ( ) const

noexcept

Check if element is a mixed element, i.e. 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 is element is mixed.

◆ map_fn()

std::function<void(O&, const P&, const Q&, double, const R&)> map_fn ( ) const

inline

Return a function that performs the appropriate push-forward (pull-back) for the element type.

Template Parameters

O	The type that hold the computed pushed-forward (pulled-back) data (ndim==1)
P	The type that hold the data to be pulled back (pushed forwarded) (ndim==1)
Q	The type that holds the Jacobian (inverse Jacobian) matrix (ndim==2)
R	The type that holds the inverse Jacobian (Jacobian) matrix (ndim==2)

Returns

A function that for a push-forward takes arguments

u [out] The data on the physical cell after the push-forward flattened with row-major layout, shape=(num_points, value_size)
U [in] The data on the reference cell physical field to push forward, flattened with row-major layout, shape=(num_points, ref_value_size)
J [in] The Jacobian matrix of the map ,shape=(gdim, tdim)
detJ [in] det(J)
K [in] The inverse of the Jacobian matrix, shape=(tdim, gdim)

For a pull-back the passed arguments should be:

U [out] The data on the reference cell after the pull-back, flattened with row-major layout, shape=(num_points, ref value_size)
u [in] The data on the physical cell that should be pulled back , flattened with row-major layout, shape=(num_points, value_size)
K [in] The inverse oif the Jacobian matrix of the map, shape=(tdim, gdim)
detJ_inv [in] 1/det(J)
J [in] The Jacobian matrix, shape=(gdim, tdim)

◆ map_ident()

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()

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 subentity 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()

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 subentity, 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()

int num_sub_elements ( ) const

noexcept

Get the number of sub elements (for a mixed or blocked element)

Returns: The number of sub elements

◆ operator!=()

bool operator!= ( const FiniteElement & 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==()

bool operator== ( const FiniteElement & 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_dofs()

void permute_dofs	(	const xtl::span< std::int32_t > &	doflist,
		std::uint32_t	cell_permutation
	)		const

Permute the DOFs of the element.

Parameters

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

◆ reference_value_size()

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()

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()

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

◆ tabulate()

void tabulate	(	xt::xtensor< double, 4 > &	values,
		const xt::xtensor< double, 2 > &	X,
		int	order
	)		const

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

Parameters

[in,out]	values	Four dimensional xtensor that will be filled with the tabulated values. Should be of shape {num_derivatives, num_points, num_dofs, reference_value_size}
[in]	X	Two dimensional xtensor of shape [num_points, geometric dimension] containing the points at the reference element
[in]	order	The number of derivatives (up to and including this order) to tabulate for

◆ unpermute_dofs()

void unpermute_dofs	(	const xtl::span< std::int32_t > &	doflist,
		std::uint32_t	cell_permutation
	)		const

Unpermute the DOFs of the element.

Parameters

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

◆ value_size()

int 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.

Note: The return value of this function is equal to std::accumulate(value_shape().begin(), value_shape().end(), 1, std::multiplies<int>()).

Returns: The value size

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 Member Functions

Detailed Description

Constructor & Destructor Documentation

◆ FiniteElement() [1/2]

◆ FiniteElement() [2/2]

Member Function Documentation

◆ apply_dof_transformation()

◆ apply_dof_transformation_to_transpose()

◆ apply_inverse_dof_transformation()

◆ apply_inverse_dof_transformation_to_transpose()

◆ apply_inverse_transpose_dof_transformation()

◆ apply_inverse_transpose_dof_transformation_to_transpose()

◆ apply_transpose_dof_transformation()

◆ apply_transpose_dof_transformation_to_transpose()

◆ block_size()

◆ cell_shape()

◆ create_interpolation_operator()

◆ family()

◆ get_dof_permutation_function()

◆ get_dof_transformation_function()

◆ get_dof_transformation_to_transpose_function()

◆ interpolation_ident()

◆ interpolation_operator()

◆ interpolation_points()

◆ is_mixed()

◆ map_fn()

◆ map_ident()

◆ needs_dof_permutations()

◆ needs_dof_transformations()

◆ num_sub_elements()

◆ operator!=()

◆ operator==()

◆ permute_dofs()

◆ reference_value_size()

◆ signature()

◆ space_dimension()

◆ tabulate()

◆ unpermute_dofs()

◆ value_size()