#include <finite-element.h>

Public Member Functions
	FiniteElement (element::family family, cell::type cell_type, int degree, const std::vector< std::size_t > &value_shape, const xt::xtensor< double, 3 > &coeffs, const std::map< cell::type, xt::xtensor< double, 3 >> &entity_transformations, const std::array< std::vector< xt::xtensor< double, 2 >>, 4 > &x, const std::array< std::vector< xt::xtensor< double, 3 >>, 4 > &M, maps::type map_type=maps::type::identity)

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

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

	~FiniteElement ()=default
	Destructor.

FiniteElement &	operator= (const FiniteElement &element)=default
	Assignment operator.

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

xt::xtensor< double, 4 >	tabulate (int nd, const xt::xarray< double > &x) const

void	tabulate (int nd, const xt::xarray< double > &x, xt::xtensor< double, 4 > &basis_data) const

cell::type	cell_type () const

int	degree () const

int	value_size () const

const std::vector< int > &	value_shape () const

int	dim () const

element::family	family () const

maps::type	mapping_type () const

bool	dof_transformations_are_permutations () const

bool	dof_transformations_are_identity () const

xt::xtensor< double, 3 >	map_push_forward (const xt::xtensor< double, 3 > &U, const xt::xtensor< double, 3 > &J, const xtl::span< const double > &detJ, const xt::xtensor< double, 3 > &K) const

template<typename O , typename P , typename Q , typename S , typename T >
void	map_push_forward_m (const O &U, const P &J, const Q &detJ, const S &K, T &&u) const

xt::xtensor< double, 3 >	map_pull_back (const xt::xtensor< double, 3 > &u, const xt::xtensor< double, 3 > &J, const xtl::span< const double > &detJ, const xt::xtensor< double, 3 > &K) const

template<typename O , typename P , typename Q , typename S , typename T >
void	map_pull_back_m (const O &u, const P &J, const Q &detJ, const S &K, T &&U) const

const std::vector< std::vector< int > > &	num_entity_dofs () const

const std::vector< std::vector< int > > &	num_entity_closure_dofs () const

const std::vector< std::vector< std::set< int > > > &	entity_dofs () const

const std::vector< std::vector< std::set< int > > > &	entity_closure_dofs () const

xt::xtensor< double, 3 >	base_transformations () const

std::map< cell::type, xt::xtensor< double, 3 > >	entity_transformations () const
	Return the entity dof transformation matricess.

void	permute_dofs (const xtl::span< std::int32_t > &dofs, std::uint32_t cell_info) const

void	unpermute_dofs (const xtl::span< std::int32_t > &dofs, std::uint32_t cell_info) const

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

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

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

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

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

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

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

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

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

int	num_points () const
	Return the number of interpolation points.

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

Public Attributes
maps::type	map_type
	Element map type.

Detailed Description

Finite Element The basis is stored as a set of coefficients, which are applied to the underlying expansion set for that cell type, when tabulating.

Constructor & Destructor Documentation

◆ FiniteElement()

FiniteElement::FiniteElement	(	element::family	family,
		cell::type	cell_type,
		int	degree,
		const std::vector< std::size_t > &	value_shape,
		const xt::xtensor< double, 3 > &	coeffs,
		const std::map< cell::type, xt::xtensor< double, 3 >> &	entity_transformations,
		const std::array< std::vector< xt::xtensor< double, 2 >>, 4 > &	x,
		const std::array< std::vector< xt::xtensor< double, 3 >>, 4 > &	M,
		maps::type	map_type = `maps::type::identity`
	)

Todo:: Document A finite element

Parameters

[in]	family
[in]	cell_type
[in]	degree
[in]	value_shape
[in]	coeffs	Expansion coefficients. The shape is (num_dofs, value_size, basis_dim)
[in]	entity_transformations	Entity transformations
[in]	x	Interpolation points. Shape is (tdim, entity index, point index, dim)
[in]	M	The interpolation matrices. Indices are (tdim, entity index, dof, vs, point_index)
[in]	map_type

Member Function Documentation

◆ apply_dof_transformation()

template<typename T >

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

Apply DOF transformations to some data

Parameters

[in,out]	data	The data
	block_size	The number of data points per DOF
	cell_info	The permutation info for the cell

◆ apply_dof_transformation_to_transpose()

template<typename T >

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

Apply DOF transformations to some transposed data

Parameters

[in,out]	data	The data
	block_size	The number of data points per DOF
	cell_info	The permutation info for the cell

◆ apply_inverse_dof_transformation()

template<typename T >

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

Apply inverse DOF transformations to some data

Parameters

[in,out]	data	The data
	block_size	The number of data points per DOF
	cell_info	The permutation info for the cell

◆ apply_inverse_dof_transformation_to_transpose()

template<typename T >

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

Apply inverse DOF transformations to some transposed data

Parameters

[in,out]	data	The data
	block_size	The number of data points per DOF
	cell_info	The permutation info for the cell

◆ apply_inverse_transpose_dof_transformation()

template<typename T >

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

Apply inverse transpose DOF transformations to some data

Parameters

[in,out]	data	The data
	block_size	The number of data points per DOF
	cell_info	The permutation info for the cell

◆ apply_inverse_transpose_dof_transformation_to_transpose()

template<typename T >

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

Apply inverse transpose DOF transformations to some transposed data

Parameters

[in,out]	data	The data
	block_size	The number of data points per DOF
	cell_info	The permutation info for the cell

◆ apply_transpose_dof_transformation()

template<typename T >

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

Apply transpose DOF transformations to some data

Parameters

[in,out]	data	The data
	block_size	The number of data points per DOF
	cell_info	The permutation info for the cell

◆ apply_transpose_dof_transformation_to_transpose()

template<typename T >

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

Apply transpose DOF transformations to some transposed data

Parameters

[in,out]	data	The data
	block_size	The number of data points per DOF
	cell_info	The permutation info for the cell

◆ base_transformations()

xt::xtensor< double, 3 > FiniteElement::base_transformations ( ) const

Get the base transformations The base transformations represent the effect of rotating or reflecting a subentity of the cell on the numbering and orientation of the DOFs. This returns a list of matrices with one matrix for each subentity permutation in the following order: Reversing edge 0, reversing edge 1, ... Rotate face 0, reflect face 0, rotate face 1, reflect face 1, ...

Example: Order 3 Lagrange on a triangle

This space has 10 dofs arranged like:

2
|\
6 4
|  \
5 9 3
|    \
0-7-8-1

For this element, the base transformations are: [Matrix swapping 3 and 4, Matrix swapping 5 and 6, Matrix swapping 7 and 8] The first row shows the effect of reversing the diagonal edge. The second row shows the effect of reversing the vertical edge. The third row shows the effect of reversing the horizontal edge.

Example: Order 1 Raviart-Thomas on a triangle

This space has 3 dofs arranged like:

  |\
  | \
  |  \
<-1   0
  |  / \
  | L ^ \
  |   |  \
   ---2---

These DOFs are integrals of normal components over the edges: DOFs 0 and 2 are oriented inward, DOF 1 is oriented outwards. For this element, the base transformation matrices are:

0: [[-1, 0, 0],
    [ 0, 1, 0],
    [ 0, 0, 1]]
1: [[1,  0, 0],
    [0, -1, 0],
    [0,  0, 1]]
2: [[1, 0,  0],
    [0, 1,  0],
    [0, 0, -1]]

The first matrix reverses DOF 0 (as this is on the first edge). The second matrix reverses DOF 1 (as this is on the second edge). The third matrix reverses DOF 2 (as this is on the third edge).

Example: DOFs on the face of Order 2 Nedelec first kind on a tetrahedron

On a face of this tetrahedron, this space has two face tangent DOFs:

|\        |\
| \       | \
|  \      | ^\
|   \     | | \
| 0->\    | 1  \
|     \   |     \
 ------    ------

For these DOFs, the subblocks of the base transformation matrices are:

rotation: [[-1, 1],
           [ 1, 0]]
reflection: [[0, 1],
             [1, 0]]

◆ cell_type()

cell::type FiniteElement::cell_type ( ) const

Get the element cell type

Returns: The cell type

◆ degree()

int FiniteElement::degree ( ) const

Get the element polynomial degree

Returns: Polynomial degree

◆ dim()

int FiniteElement::dim ( ) const

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

Returns: Number of degrees of freedom

◆ dof_transformations_are_identity()

bool FiniteElement::dof_transformations_are_identity ( ) const

Indicates whether the dof transformations are all the identity

Returns: True or False

◆ dof_transformations_are_permutations()

bool FiniteElement::dof_transformations_are_permutations ( ) const

Indicates whether the dof transformations are all permutations

Returns: True or False

◆ entity_closure_dofs()

const std::vector< std::vector< std::set< int > > > & FiniteElement::entity_closure_dofs ( ) const

Get the dofs on the closure of each topological entity: (vertices, edges, faces, cell) in that order. For example, Lagrange degree 2 on a triangle has vertices: [[0], [1], [2]], edges: [[1, 2, 3], [0, 2, 4], [0, 1, 5]], cell: [[0, 1, 2, 3, 4, 5]]

Returns: Dofs associated with the closre of an entity of a given topological dimension. The shape is (tdim + 1, num_entities, num_dofs).

◆ entity_dofs()

const std::vector< std::vector< std::set< int > > > & FiniteElement::entity_dofs ( ) const

Get the dofs on each topological entity: (vertices, edges, faces, cell) in that order. For example, Lagrange degree 2 on a triangle has vertices: [[0], [1], [2]], edges: [[3], [4], [5]], cell: [[]]

Returns: Dofs associated with an entity of a given topological dimension. The shape is (tdim + 1, num_entities, num_dofs).

◆ family()

element::family FiniteElement::family ( ) const

Get the finite element family

Returns: The family

◆ interpolation_matrix()

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

Return a matrix of weights interpolation To interpolate a function in this finite element, the functions should be evaluated at each point given by FiniteElement::points(). These function values should then be multiplied by the weight matrix to give the coefficients of the interpolated function.

◆ map_pull_back()

xt::xtensor< double, 3 > FiniteElement::map_pull_back	(	const xt::xtensor< double, 3 > &	u,
		const xt::xtensor< double, 3 > &	J,
		const xtl::span< const double > &	detJ,
		const xt::xtensor< double, 3 > &	K
	)		const

Map function values from a physical cell to the reference

Parameters

[in]	u	The function values on the cell
[in]	J	The Jacobian of the mapping
[in]	detJ	The determinant of the Jacobian of the mapping
[in]	K	The inverse of the Jacobian of the mapping

Returns: The function values on the reference

◆ map_pull_back_m()

template<typename O , typename P , typename Q , typename S , typename T >

void basix::FiniteElement::map_pull_back_m	(	const O &	u,
		const P &	J,
		const Q &	detJ,
		const S &	K,
		T &&	U
	)		const

inline

Map function values from a physical cell back to to the reference

Parameters

[in]	u	Data defined on the physical element. It must have dimension 3. The first index is for the geometric/map data, the second is the point index for points that share map data, and the third index is (vector) component, e.g. `u[i,:,:]` are points that are mapped by `J[i,:,:]`.
[in]	J	The Jacobians. It must have dimension 3. The first index is for the ith Jacobian, i.e. J[i,:,:] is the ith Jacobian.
[in]	detJ	The determinant of J. `detJ[i]` is equal to `det(J[i,:,:])`. It must have dimension 1.
[in]	K	The inverse of J, `K[i,:,:] = J[i,:,:]^-1`. It must have dimension 3.
[out]	U	The input `u` mapped to the reference element. It must have dimension 3.

◆ map_push_forward()

xt::xtensor< double, 3 > FiniteElement::map_push_forward	(	const xt::xtensor< double, 3 > &	U,
		const xt::xtensor< double, 3 > &	J,
		const xtl::span< const double > &	detJ,
		const xt::xtensor< double, 3 > &	K
	)		const

Map function values from the reference to a physical cell. This function can perform the mapping for multiple points, grouped by points that share a common Jacobian.

Parameters

U	The function values on the reference. The indices are [Jacobian index, point index, components].
J	The Jacobian of the mapping. The indices are [Jacobian index, J_i, J_j].
detJ	The determinant of the Jacobian of the mapping. It has length `J.shape(0)`
K	The inverse of the Jacobian of the mapping. The indices are [Jacobian index, K_i, K_j].

Returns: The function values on the cell. The indices are [Jacobian index, point index, components].

◆ map_push_forward_m()

template<typename O , typename P , typename Q , typename S , typename T >

void basix::FiniteElement::map_push_forward_m	(	const O &	U,
		const P &	J,
		const Q &	detJ,
		const S &	K,
		T &&	u
	)		const

inline

Direct to memory push forward

Parameters

[in]	U	Data defined on the reference element. It must have dimension 3. The first index is for the geometric/map data, the second is the point index for points that share map data, and the third index is (vector) component, e.g. `u[i,:,:]` are points that are mapped by `J[i,:,:]`.
[in]	J	The Jacobians. It must have dimension 3. The first index is for the ith Jacobian, i.e. J[i,:,:] is the ith Jacobian.
[in]	detJ	The determinant of J. `detJ[i]` is equal to `det(J[i,:,:])`. It must have dimension 1.
[in]	K	The inverse of J, `K[i,:,:] = J[i,:,:]^-1`. It must have dimension 3.
[out]	u	The input `U` mapped to the physical. It must have dimension 3.

◆ mapping_type()

maps::type FiniteElement::mapping_type ( ) const

Get the mapping type used for this element

Returns: The mapping

◆ num_entity_closure_dofs()

const std::vector< std::vector< int > > & FiniteElement::num_entity_closure_dofs ( ) const

Get the number of dofs on the closure of each topological entity: (vertices, edges, faces, cell) in that order. For example, Lagrange degree 2 on a triangle has vertices: [1, 1, 1], edges: [3, 3, 3], cell: [6]

Returns: Number of dofs associated with the closure of an entity of a given topological dimension. The shape is (tdim + 1, num_entities).

◆ num_entity_dofs()

const std::vector< std::vector< int > > & FiniteElement::num_entity_dofs ( ) const

Get the number of dofs on each topological entity: (vertices, edges, faces, cell) in that order. For example, Lagrange degree 2 on a triangle has vertices: [1, 1, 1], edges: [1, 1, 1], cell: [0] The sum of the entity dofs must match the total number of dofs reported by FiniteElement::dim,

const std::vector<std::vector<int>>& dofs = e.entity_dofs();
int num_dofs0 = dofs[1][3]; // Number of dofs associated with edge 3
int num_dofs1 = dofs[2][0]; // Number of dofs associated with face 0

Returns: Number of dofs associated with an entity of a given topological dimension. The shape is (tdim + 1, num_entities).

◆ permute_dofs()

void FiniteElement::permute_dofs	(	const xtl::span< std::int32_t > &	dofs,
		std::uint32_t	cell_info
	)		const

Permute the dof numbering on a cell

Parameters

[in,out]	dofs	The dof numbering for the cell
	cell_info	The permutation info for the cell

◆ points()

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

Return the interpolation points, i.e. the coordinates on the reference element where a function need to be evaluated in order to interpolate it in the finite element space.

Returns: Array of coordinate with shape (num_points, tdim)

◆ tabulate() [1/2]

xt::xtensor< double, 4 > FiniteElement::tabulate	(	int	nd,
		const xt::xarray< double > &	x
	)		const

Compute basis values and derivatives at set of points.

Parameters

[in]	nd	The order of derivatives, up to and including, to compute. Use 0 for the basis functions only.
[in]	x	The points at which to compute the basis functions. The shape of x is (number of points, geometric dimension).

Returns

The basis functions (and derivatives). The shape is (derivative, point, basis fn index, value index).

The first index is the derivative, with higher derivatives are stored in triangular (2D) or tetrahedral (3D) ordering, i.e. for the (x,y) derivatives in 2D: (0,0), (1,0), (0,1), (2,0), (1,1), (0,2), (3,0)... The function basix::idx can be used to find the appropriate derivative.
The second index is the point index
The third index is the basis function index
The fourth index is the basis function component. Its has size one for scalar basis functions.

◆ tabulate() [2/2]

void FiniteElement::tabulate	(	int	nd,
		const xt::xarray< double > &	x,
		xt::xtensor< double, 4 > &	basis_data
	)		const

Direct to memory block tabulation

Parameters

nd	Number of derivatives
x	Points
basis_data	Memory location to fill

◆ unpermute_dofs()

void FiniteElement::unpermute_dofs	(	const xtl::span< std::int32_t > &	dofs,
		std::uint32_t	cell_info
	)		const

Unpermute the dof numbering on a cell

Parameters

[in,out]	dofs	The dof numbering for the cell
	cell_info	The permutation info for the cell

◆ value_shape()

const std::vector< int > & FiniteElement::value_shape ( ) const

Get the element value tensor shape, e.g. returning [1] for scalars.

Returns: Value shape

◆ value_size()

int FiniteElement::value_size ( ) const

Get the element value size This is just a convenience function returning product(value_shape)

Returns: Value size

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

/home/runner/work/basix/basix/cpp/basix/finite-element.h
/home/runner/work/basix/basix/cpp/basix/finite-element.cpp

Public Member Functions

Public Attributes

Detailed Description

Constructor & Destructor Documentation

◆ FiniteElement()

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()

◆ base_transformations()

Example: Order 3 Lagrange on a triangle

Example: Order 1 Raviart-Thomas on a triangle

Example: DOFs on the face of Order 2 Nedelec first kind on a tetrahedron

◆ cell_type()

◆ degree()

◆ dim()

◆ dof_transformations_are_identity()

◆ dof_transformations_are_permutations()

◆ entity_closure_dofs()

◆ entity_dofs()

◆ family()

◆ interpolation_matrix()

◆ map_pull_back()

◆ map_pull_back_m()

◆ map_push_forward()

◆ map_push_forward_m()

◆ mapping_type()

◆ num_entity_closure_dofs()

◆ num_entity_dofs()

◆ permute_dofs()

◆ points()

◆ tabulate() [1/2]

◆ tabulate() [2/2]

◆ unpermute_dofs()

◆ value_shape()

◆ value_size()