basix::FiniteElement Class Reference

A finite element. More...

#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, 2 > &wcoeffs, 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, bool discontinuous, int highest_complete_degree, int highest_degree, element::lagrange_variant lvariant, element::dpc_variant dvariant, std::vector< std::tuple< std::vector< FiniteElement >, std::vector< int >>> tensor_factors={})
	FiniteElement (element::family family, cell::type cell_type, int degree, const std::vector< std::size_t > &value_shape, const xt::xtensor< double, 2 > &wcoeffs, 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, bool discontinuous, int highest_complete_degree, int highest_degree, element::lagrange_variant lvariant, std::vector< std::tuple< std::vector< FiniteElement >, std::vector< int >>> tensor_factors={})
	Overload.
	FiniteElement (element::family family, cell::type cell_type, int degree, const std::vector< std::size_t > &value_shape, const xt::xtensor< double, 2 > &wcoeffs, 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, bool discontinuous, int highest_complete_degree, int highest_degree, element::dpc_variant dvariant, std::vector< std::tuple< std::vector< FiniteElement >, std::vector< int >>> tensor_factors={})
	Overload.
	FiniteElement (element::family family, cell::type cell_type, int degree, const std::vector< std::size_t > &value_shape, const xt::xtensor< double, 2 > &wcoeffs, 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, bool discontinuous, int highest_complete_degree, int highest_degree, std::vector< std::tuple< std::vector< FiniteElement >, std::vector< int >>> tensor_factors={})
	Overload.
	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.
bool	operator== (const FiniteElement &e) const
std::array< std::size_t, 4 >	tabulate_shape (std::size_t nd, std::size_t num_points) const
xt::xtensor< double, 4 >	tabulate (int nd, const xt::xtensor< double, 2 > &x) const
void	tabulate (int nd, const xt::xtensor< double, 2 > &x, xt::xtensor< double, 4 > &basis) const
cell::type	cell_type () const
int	degree () const
int	highest_degree () const
int	highest_complete_degree () const
const std::vector< int > &	value_shape () const
int	dim () const
element::family	family () const
element::lagrange_variant	lagrange_variant () const
element::dpc_variant	dpc_variant () const
maps::type	map_type () const
bool	discontinuous () const
bool	dof_transformations_are_permutations () const
bool	dof_transformations_are_identity () const
xt::xtensor< double, 3 >	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
xt::xtensor< double, 3 >	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 R >
std::function< void(O &, const P &, const Q &, double, const R &)>	map_fn () 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::vector< int > > > &	entity_dofs () const
const std::vector< std::vector< std::vector< int > > > &	entity_closure_dofs () const
xt::xtensor< double, 3 >	base_transformations () const
std::map< cell::type, xt::xtensor< double, 3 > >	entity_transformations () const
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
const xt::xtensor< double, 2 > &	interpolation_matrix () const
const xt::xtensor< double, 2 > &	dual_matrix () const
const xt::xtensor< double, 2 > &	wcoeffs () const
const std::array< std::vector< xt::xtensor< double, 2 > >, 4 > &	x () const
const std::array< std::vector< xt::xtensor< double, 3 > >, 4 > &	M () const
const xt::xtensor< double, 2 > &	coefficient_matrix () const
bool	has_tensor_product_factorisation () const
std::vector< std::tuple< std::vector< FiniteElement >, std::vector< int > > >	get_tensor_product_representation () const
bool	interpolation_is_identity () const

Detailed Description

A finite element.

The basis of a finite element 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, 2 > &	wcoeffs,
		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,
		bool	discontinuous,
		int	highest_complete_degree,
		int	highest_degree,
		element::lagrange_variant	lvariant,
		element::dpc_variant	dvariant,
		std::vector< std::tuple< std::vector< FiniteElement >, std::vector< int >>>	tensor_factors = {}
	)

A finite element

Initialising a finite element calculates the basis functions of the finite element, in terms of the polynomial basis.

The below explanation uses Einstein notation.

The basis functions \({\phi_i}\) of a finite element are represented as a linear combination of polynomials \(\{p_j\}\) in an underlying polynomial basis that span the space of all d-dimensional polynomials up to order \(k \ (P_k^d)\):

\[ \phi_i = c_{ij} p_j \]

In some cases, the basis functions \(\{\phi_i\}\) do not span the full space \(P_k\), in which case we denote space spanned by the basis functions by \(\{q_k\}\), which can be represented by:

\[ q_i = b_{ij} p_j. \]

This leads to

\[ \phi_i = c^{\prime}_{ij} q_j = c^{\prime}_{ij} b_{jk} p_k, \]

and in matrix form:

\[ \phi = C^{\prime} B p \]

If the basis functions span the full space, then \( B \) is simply the identity.

The basis functions \(\phi_i\) are defined by a dual set of functionals \(\{f_i\}\). The basis functions are the functions in span{ \(q_k\)} such that

\[ f_i(\phi_j) = \delta_{ij} \]

and inserting the expression for \(\phi_{j}\):

\[ f_i(c^{\prime}_{jk}b_{kl}p_{l}) = c^{\prime}_{jk} b_{kl} f_i \left( p_{l} \right) \]

Defining a matrix D given by applying the functionals to each polynomial \(p_j\):

\[ [D] = d_{ij},\mbox{ where } d_{ij} = f_i(p_j), \]

we have:

\[ C^{\prime} B D^{T} = I \]

and

\[ C^{\prime} = (B D^{T})^{-1}. \]

Recalling that \(C = C^{\prime} B\), where \(C\) is the matrix form of \(c_{ij}\),

\[ C = (B D^{T})^{-1} B \]

This function takes the matrices \(B\) (wcoeffs) and \(D\) (M) as inputs and will internally compute \(C\).

The matrix \(BD^{T}\) can be obtained from an element by using the function dual_matrix(). The matrix \(C\) can be obtained from an element by using the function coefficient_matrix().

Example: Order 1 Lagrange elements on a triangle

On a triangle, the scalar expansion basis is:

\[ p_0 = \sqrt{2}/2 \qquad p_1 = \sqrt{3}(2x + y - 1) \qquad p_2 = 3y - 1 \]

These span the space \(P_1\).

Lagrange order 1 elements span the space P_1, so in this example, B (span_coeffs) is the identity matrix:

\[ B = \begin{bmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{bmatrix} \]

The functionals defining the Lagrange order 1 space are point evaluations at the three vertices of the triangle. The matrix D (dual) given by applying these to p_0 to p_2 is:

\[ \mbox{dual} = \begin{bmatrix} \sqrt{2}/2 & -\sqrt{3} & -1 \\ \sqrt{2}/2 & \sqrt{3} & -1 \\ \sqrt{2}/2 & 0 & 2 \end{bmatrix} \]

For this example, this function outputs the matrix:

\[ C = \begin{bmatrix} \sqrt{2}/3 & -\sqrt{3}/6 & -1/6 \\ \sqrt{2}/3 & \sqrt{3}/6 & -1/6 \\ \sqrt{2}/3 & 0 & 1/3 \end{bmatrix} \]

The basis functions of the finite element can be obtained by applying the matrix C to the vector \([p_0, p_1, p_2]\), giving:

\[ \begin{bmatrix} 1 - x - y \\ x \\ y \end{bmatrix} \]

Example: Order 1 Raviart-Thomas on a triangle

On a triangle, the 2D vector expansion basis is:

\[ \begin{matrix} p_0 & = & (\sqrt{2}/2, 0) \\ p_1 & = & (\sqrt{3}(2x + y - 1), 0) \\ p_2 & = & (3y - 1, 0) \\ p_3 & = & (0, \sqrt{2}/2) \\ p_4 & = & (0, \sqrt{3}(2x + y - 1)) \\ p_5 & = & (0, 3y - 1) \end{matrix} \]

These span the space \( P_1^2 \).

Raviart-Thomas order 1 elements span a space smaller than \( P_1^2 \), so B (span_coeffs) is not the identity. It is given by:

\[ B = \begin{bmatrix} 1 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 & 0 \\ 1/12 & \sqrt{6}/48 & -\sqrt{2}/48 & 1/12 & 0 & \sqrt{2}/24 \end{bmatrix} \]

Applying the matrix B to the vector \([p_0, p_1, ..., p_5]\) gives the basis of the polynomial space for Raviart-Thomas:

\[ \begin{bmatrix} \sqrt{2}/2 & 0 \\ 0 & \sqrt{2}/2 \\ \sqrt{2}x/8 & \sqrt{2}y/8 \end{bmatrix} \]

The functionals defining the Raviart-Thomas order 1 space are integral of the normal components along each edge. The matrix D (dual) given by applying these to \(p_0\) to \(p_5\) is:

\[ D = \begin{bmatrix} -\sqrt{2}/2 & -\sqrt{3}/2 & -1/2 & -\sqrt{2}/2 & -\sqrt{3}/2 & -1/2 \\ -\sqrt{2}/2 & \sqrt{3}/2 & -1/2 & 0 & 0 & 0 \\ 0 & 0 & 0 & \sqrt{2}/2 & 0 & -1 \end{bmatrix} \]

In this example, this function outputs the matrix:

\[ C = \begin{bmatrix} -\sqrt{2}/2 & -\sqrt{3}/2 & -1/2 & -\sqrt{2}/2 & -\sqrt{3}/2 & -1/2 \\ -\sqrt{2}/2 & \sqrt{3}/2 & -1/2 & 0 & 0 & 0 \\ 0 & 0 & 0 & \sqrt{2}/2 & 0 & -1 \end{bmatrix} \]

The basis functions of the finite element can be obtained by applying the matrix C to the vector \([p_0, p_1, ..., p_5]\), giving:

\[ \begin{bmatrix} -x & -y \\ x - 1 & y \\ -x & 1 - y \end{bmatrix} \]

Parameters

[in]	family	The element family
[in]	cell_type	The cell type
[in]	degree	The degree of the element
[in]	value_shape	The value shape of the element
[in]	wcoeffs	Matrices for the kth value index containing the expansion coefficients defining a polynomial basis spanning the polynomial space for this element
[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	The type of map to be used to map values from the reference to a cell
[in]	discontinuous	Indicates whether or not this is the discontinuous version of the element
[in]	highest_complete_degree	The highest degree n such that a Lagrange (or vector Lagrange) element of degree n is a subspace of this element
[in]	highest_degree	The highest degree n such that at least one polynomial of degree n is included in this element's polymonial set
[in]	lvariant	The Lagrange variant of the element
[in]	dvariant	The DPC variant of the element
[in]	tensor_factors	The factors in the tensor product representation of this element

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

Note

This function is designed to be called at runtime, so its performance is critical.

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

Note

This function is designed to be called at runtime, so its performance is critical.

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

Note

This function is designed to be called at runtime, so its performance is critical.

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

Note

This function is designed to be called at runtime, so its performance is critical.

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

Note

This function is designed to be called at runtime, so its performance is critical.

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

Note

This function is designed to be called at runtime, so its performance is critical.

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

Note

This function is designed to be called at runtime, so its performance is critical.

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

Note

This function is designed to be called at runtime, so its performance is critical.

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]]

Returns

The base transformations for this element

cell_type()

cell::type FiniteElement::cell_type

(

)

const

Get the element cell type

Returns

The cell type

coefficient_matrix()

const xt::xtensor< double, 2 > & FiniteElement::coefficient_matrix

(

)

const

Get the matrix of coefficients.

This is the matrix \(C\), as described in the documentation of the FiniteElement() constructor.

Returns

The dual matrix

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

discontinuous()

bool FiniteElement::discontinuous

(

)

const

Indicates whether this element is the discontinuous variant

Returns

True if this element is a discontinuous version of the element

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

dpc_variant()

element::dpc_variant FiniteElement::dpc_variant

(

)

const

Get the DPC variant of the element.

Returns

The DPC variant

dual_matrix()

const xt::xtensor< double, 2 > & FiniteElement::dual_matrix

(

)

const

Get the dual matrix.

This is the matrix \(BD^{T}\), as described in the documentation of the FiniteElement() constructor.

Returns

The dual matrix

entity_closure_dofs()

const std::vector< std::vector< std::vector< 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 closure 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::vector< 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).

entity_transformations()

std::map< cell::type, xt::xtensor< double, 3 > > FiniteElement::entity_transformations

(

)

const

Return the entity dof transformation matrices

Returns

The entity transformations for the subentities of this element

family()

element::family FiniteElement::family

(

)

const

Get the finite element family

Returns

The family

get_tensor_product_representation()

std::vector< std::tuple< std::vector< FiniteElement >, std::vector< int > > > FiniteElement::get_tensor_product_representation

(

)

const

Get the tensor product representation of this element, or throw an error if no such factorisation exists.

The tensor product representation will be a vector of tuples. Each tuple contains a vector of finite elements, and a vector on integers. The vector of finite elements gives the elements on an interval that appear in the tensor product representation. The vector of integers gives the permutation between the numbering of the tensor product DOFs and the number of the DOFs of this Basix element.

Returns

The tensor product representation

has_tensor_product_factorisation()

bool FiniteElement::has_tensor_product_factorisation

(

)

const

Indicates whether or not this element has a tensor product representation.

highest_complete_degree()

int FiniteElement::highest_complete_degree

(

)

const

Get the highest degree n such that a Lagrange (or vector Lagrange) element of degree n is a subspace of this element

Returns

Polynomial degree

highest_degree()

int FiniteElement::highest_degree

(

)

const

Get the lowest degree n such that the highest degree polynomial in this element is contained in a Lagrange (or vector Lagrange) element of degree n

Returns

Polynomial degree

interpolation_is_identity()

bool FiniteElement::interpolation_is_identity

(

)

const

Indicates whether or not the interpolation matrix for this element is an identity matrix

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.

The shape of the returned matrix will be (dim, num_points * value_size), where dim is the number of DOFs in the finite element, num_points is the number of points returned by points(), and value_size is the value size of the finite element.

For example, to interpolate into a Lagrange space, the following should be done:

i_m = element.interpolation_matrix()
pts = element.points()
values = vector(pts.shape(0))
FOR i, p IN ENUMERATE(pts):
    values[i] = f.evaluate_at(p)
coefficients = i_m * values

To interpolate into a Raviart-Thomas space, the following should be done:

i_m = element.interpolation_matrix()
pts = element.points()
vs = prod(element.value_shape())
values = VECTOR(pts.shape(0) * vs)
FOR i, p IN ENUMERATE(pts):
    values[i::pts.shape(0)] = f.evaluate_at(p)
coefficients = i_m * values

To interpolate into a Lagrange space with a block size, the following should be done:

i_m = element.interpolation_matrix()
pts = element.points()
coefficients = VECTOR(element.dim() * block_size)
FOR b IN RANGE(block_size):
    values = vector(pts.shape(0))
    FOR i, p IN ENUMERATE(pts):
        values[i] = f.evaluate_at(p)[b]
    coefficients[::block_size] = i_m * values

Returns

The interpolation matrix

lagrange_variant()

element::lagrange_variant FiniteElement::lagrange_variant

(

)

const

Get the Lagrange variant of the element.

Returns

The Lagrange variant

M()

const std::array< std::vector< xt::xtensor< double, 3 > >, 4 > & FiniteElement::M

(

)

const

Get the interpolation matrices for each subentity.

The shape of this data is (tdim, entity index, dof, value size, point_index).

These matrices define how to evaluate the DOF functionals accociated with each sub-entity of the cell. Given a functional f, the functionals associated with the e-th entity of dimension d can be computed as follows:

matrix = element.M()[d][e]
pts = element.x()[d][e]
values = f(pts)
result = ZEROS(matrix.shape(0))
FOR i IN RANGE(matrix.shape(0)):
    FOR j IN RANGE(matrix.shape(1)):
        FOR k IN RANGE(matrix.shape(2)):
            result[i] += matrix[i, j, k] * values[k][j]

For example, for a degree 1 Raviart-Thomas (RT) element on a triangle, the DOF functionals are integrals over the edges of the dot product of the function with the normal to the edge. In this case, x() would contain quadrature points for each edge, and M() would by a 1 by 2 by npoints array for each edge. For each point, the [1, :, point] slice of this would be the quadrature weight multiplied by the normal. For all entities that are not edges, the entries in x() and M() for a degree 1 RT element would have size 0.

These matrices are only stored for custom elements. This function will throw an exception if called on a non-custom element

map_fn()

template<typename O , typename P , typename Q , typename R >

std::function<void(O&, const P&, const Q&, double, const R&)> basix::FiniteElement::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) mapped data (ndim==2)
P	The type that hold the data to be mapped (ndim==2)
Q	The type that holds the Jacobian (or inverse) matrix (ndim==2)
R	The type that holds the inverse of the Q data (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 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_type()

maps::type FiniteElement::map_type

(

)

const

Get the map type for this element

Returns

The map type

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).

operator==()

bool FiniteElement::operator==

(

const FiniteElement &

e

)

const

Check if two elements are the same

Note

This operator compares the element properties, eg family, degree, etc, and not computed numerical data

Returns

True if elements are the same

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

Note

This function is designed to be called at runtime, so its performance is critical.

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, ie 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)

pull_back()

xt::xtensor< double, 3 > FiniteElement::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

push_forward()

xt::xtensor< double, 3 > FiniteElement::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

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

tabulate() [1/2]

xt::xtensor< double, 4 > FiniteElement::tabulate	(	int	nd,
		const xt::xtensor< double, 2 > &	x
	)		const

Compute basis values and derivatives at set of points.

Note

The version of FiniteElement::tabulate with the basis data as an out argument should be preferred for repeated call where performance is critical

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, ie for the (x,y) derivatives in 2D: (0,0), (1,0), (0,1), (2,0), (1,1), (0,2), (3,0)... The function basix::indexing::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::xtensor< double, 2 > &	x,
		xt::xtensor< double, 4 > &	basis
	)		const

Compute basis values and derivatives at set of points.

Note

This function is designed to be called at runtime, so its performance is critical.

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).
[out]	basis	Memory location to fill. It must be allocated with shape (num_derivatives, num_points, num basis functions, value_size). The function FiniteElement::tabulate_shape can be used to get the required shape. The first index is the derivative, with higher derivatives are stored in triangular (2D) or tetrahedral (3D) ordering, ie for the (x,y) derivatives in 2D: (0,0), (1,0), (0,1), (2,0), (1,1), (0,2), (3,0)... The function basix::indexing::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.

Todo:

Remove all internal dynamic memory allocation, pass scratch space as required

tabulate_shape()

std::array< std::size_t, 4 > FiniteElement::tabulate_shape	(	std::size_t	nd,
		std::size_t	num_points
	)		const

Array shape for tabulate 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]	num_points	Number of points that basis will be computed at

Returns

The shape of the array to will filled when passed to FiniteElement::tabulate

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

Note

This function is designed to be called at runtime, so its performance is critical.

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

The element value tensor shape, eg returning {} for scalars, {3} for vectors in 3D, {2, 2} for a rank-2 tensor in 2D.

Returns

Value shape

wcoeffs()

const xt::xtensor< double, 2 > & FiniteElement::wcoeffs

(

)

const

Get the coefficients that define the polynomial set in terms of the orthonormal polynomials.

The polynomials spanned by each finite element in Basix are represented as a linear combination of the orthonormal polynomials of a given degree on the cell. Each row of this matrix defines a polynomial in the set spanned by the finite element.

For example, the orthonormal polynomials of degree <= 1 on a triangle are (where a, b, c, d are some constants):

• (sqrt(2), 0)
• (a*x - b, 0)
• (c*y - d, 0)
• (0, sqrt(2))
• (0, a*x - b)
• (0, c*y - d)

For a degree 1 Raviart-Thomas element, the first two rows of wcoeffs would be the following, as (1, 0) and (0, 1) are spanned by the element

• [1, 0, 0, 0, 0, 0]
• [0, 0, 0, 1, 0, 0]

The third row of wcoeffs in this example would give coefficients that represent (x, y) in terms of the orthonormal polynomials:

• [-b/(a*sqrt(2)), 1/a, 0, -d/(c*sqrt(2)), 0, 1/c]

These coefficients are only stored for custom elements. This function will throw an exception if called on a non-custom element

x()

const std::array< std::vector< xt::xtensor< double, 2 > >, 4 > & FiniteElement::x

(

)

const

Get the interpolation points for each subentity.

The shape of this data is (tdim, entity index, point index, dim).

---

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