#include <CoordinateElement.h>

Public Member Functions
	CoordinateElement (std::shared_ptr< const basix::FiniteElement > element)
	Create a coordinate element from a Basix element. More...

	CoordinateElement (mesh::CellType celltype, int degree, basix::element::lagrange_variant type=basix::element::lagrange_variant::equispaced)
	Create a Lagrange coordinate element. More...

virtual	~CoordinateElement ()=default
	Destructor.

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

int	degree () const
	The polynomial degree of the element.

int	dim () const
	The dimension of the geometry element space. More...

basix::element::lagrange_variant	variant () const
	The variant of the element.

std::array< std::size_t, 4 >	tabulate_shape (std::size_t nd, std::size_t num_points) const
	Shape of array to fill when calling `FiniteElement::tabulate` More...

xt::xtensor< double, 4 >	tabulate (int nd, const xt::xtensor< double, 2 > &X) const
	Evaluate basis values and derivatives at set of points. More...

void	tabulate (int nd, const xt::xtensor< double, 2 > &X, xt::xtensor< double, 4 > &basis) const
	Evaluate basis values and derivatives at set of points. More...

ElementDofLayout	create_dof_layout () const
	Compute and return the dof layout.

void	pull_back_nonaffine (xt::xtensor< double, 2 > &X, const xt::xtensor< double, 2 > &x, const xt::xtensor< double, 2 > &cell_geometry, double tol=1.0e-8, int maxit=10) const
	Compute reference coordinates X for physical coordinates x for a non-affine map. More...

void	permute_dofs (const xtl::span< std::int32_t > &dofs, std::uint32_t cell_perm) const
	Permutes a list of DOF numbers on a cell.

void	unpermute_dofs (const xtl::span< std::int32_t > &dofs, std::uint32_t cell_perm) const
	Reverses a DOF permutation.

bool	needs_dof_permutations () const
	Indicates whether the geometry DOF numbers on each cell need permuting. More...

bool	is_affine () const noexcept
	Check is geometry map is affine. More...

Static Public Member Functions
template<typename U , typename V , typename W >
static void	compute_jacobian (const U &dphi, const V &cell_geometry, W &&J)
	Compute Jacobian for a cell with given geometry using the basis functions and first order derivatives. More...

template<typename U , typename V >
static void	compute_jacobian_inverse (const U &J, V &&K)
	Compute the inverse of the Jacobian. More...

template<typename U >
static double	compute_jacobian_determinant (const U &J)
	Compute the determinant of the Jacobian. More...

static void	push_forward (xt::xtensor< double, 2 > &x, const xt::xtensor< double, 2 > &cell_geometry, const xt::xtensor< double, 2 > &phi)
	Compute physical coordinates x for points X in the reference configuration. More...

static std::array< double, 3 >	x0 (const xt::xtensor< double, 2 > &cell_geometry)
	Compute the physical coordinate of the reference point X=(0 , 0, 0) More...

static void	pull_back_affine (xt::xtensor< double, 2 > &X, const xt::xtensor< double, 2 > &K, const std::array< double, 3 > &x0, const xt::xtensor< double, 2 > &x)
	Compute reference coordinates X for physical coordinates x for an affine map. For the affine case, `x = J X + x0`, and this function computes `X = K(x -x0)` where `K = J^{-1}`. More...

Detailed Description

Todo:: A dof layout on a reference cell needs to be defined.

A CoordinateElement manages coordinate mappings for isoparametric cells.

Constructor & Destructor Documentation

◆ CoordinateElement() [1/2]

CoordinateElement ( std::shared_ptr< const basix::FiniteElement > element )

explicit

Create a coordinate element from a Basix element.

Parameters

[in] element Element from Basix

◆ CoordinateElement() [2/2]

CoordinateElement	(	mesh::CellType	celltype,
		int	degree,
		basix::element::lagrange_variant	type = `basix::element::lagrange_variant::equispaced`
	)

Create a Lagrange coordinate element.

Parameters

[in]	celltype	The cell shape
[in]	degree	Polynomial degree of the map
[in]	type	The type of Lagrange element (see Basix documentation for possible types)

Member Function Documentation

◆ cell_shape()

mesh::CellType cell_shape ( ) const

Cell shape.

Returns: The cell shape

◆ compute_jacobian()

static void compute_jacobian	(	const U &	dphi,
		const V &	cell_geometry,
		W &&	J
	)

inlinestatic

Compute Jacobian for a cell with given geometry using the basis functions and first order derivatives.

Parameters

[in]	dphi	Derivatives of the basis functions (shape=(tdim, num geometry nodes))
[in]	cell_geometry	The cell nodes coordinates (shape=(num geometry nodes, gdim))
[out]	J	The Jacobian. It must have shape=(gdim, tdim) and must initialized to zero

◆ compute_jacobian_determinant()

static double compute_jacobian_determinant ( const U & J )

inlinestatic

Compute the determinant of the Jacobian.

Parameters

[in] J Jacobian (shape=(gdim, tdim))

Returns: Determinant of J

◆ compute_jacobian_inverse()

static void compute_jacobian_inverse	(	const U &	J,
		V &&	K
	)

inlinestatic

Compute the inverse of the Jacobian.

Parameters

[in]	J	The Jacobian (shape=(gdim, tdim))
[out]	K	The Jacobian (shape=(tdim, gdim))

◆ dim()

int dim ( ) const

The dimension of the geometry element space.

The number of basis function is returned. E.g., for a linear triangle cell the dimension will be 3.

Returns: The coordinate element dimension.

◆ is_affine()

bool is_affine ( ) const

inlinenoexcept

Check is geometry map is affine.

Returns: True is geometry map is affine

◆ needs_dof_permutations()

bool needs_dof_permutations ( ) const

Indicates whether the geometry DOF numbers on each cell need permuting.

For higher order geometries (where there is more than one DOF on a subentity of the cell), this will be true.

◆ pull_back_affine()

void pull_back_affine	(	xt::xtensor< double, 2 > &	X,
		const xt::xtensor< double, 2 > &	K,
		const std::array< double, 3 > &	x0,
		const xt::xtensor< double, 2 > &	x
	)

static

Compute reference coordinates X for physical coordinates x for an affine map. For the affine case, x = J X + x0, and this function computes X = K(x -x0) where K = J^{-1}.

Parameters

[out]	X	The reference coordinates to compute (shape=(num_points, tdim))
[in]	K	The inverse of the geometry Jacobian (shape=(tdim, gdim))
[in]	x0	The physical coordinate of reference coordinate X0=(0, 0, 0).
[in]	x	The physical coordinates (shape=(num_points, gdim))

◆ pull_back_nonaffine()

void pull_back_nonaffine	(	xt::xtensor< double, 2 > &	X,
		const xt::xtensor< double, 2 > &	x,
		const xt::xtensor< double, 2 > &	cell_geometry,
		double	tol = `1.0e-8`,
		int	maxit = `10`
	)		const

Compute reference coordinates X for physical coordinates x for a non-affine map.

Parameters

[in,out]	X	The reference coordinates to compute (shape=(num_points, tdim))
[in]	x	The physical coordinates (shape=(num_points, gdim))
[in]	cell_geometry	The cell nodes coordinates (shape=(num geometry nodes, gdim))
[in]	tol	Tolerance for termination of Newton method.
[in]	maxit	Maximum number of Newton iterations

Note: If convergence is not achieved within maxit, the function throws a run-time error.

◆ push_forward()

void push_forward	(	xt::xtensor< double, 2 > &	x,
		const xt::xtensor< double, 2 > &	cell_geometry,
		const xt::xtensor< double, 2 > &	phi
	)

static

Compute physical coordinates x for points X in the reference configuration.

Parameters

[in,out]	x	The physical coordinates of the reference points X
[in]	cell_geometry	The cell node coordinates (physical)
[in]	phi	Tabulated basis functions at reference points X

◆ tabulate() [1/2]

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

Evaluate 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, number point, number of basis fn, value size).

◆ tabulate() [2/2]

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

Evaluate 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).
[out]	basis	The array to fill with the basis function values. The shape can be computed using `FiniteElement::tabulate_shape`

◆ tabulate_shape()

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

Shape of array to fill when calling FiniteElement::tabulate

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 at which to evaluate the basis functions

Returns: The shape of the array to be filled by FiniteElement::tabulate

◆ x0()

std::array< double, 3 > x0 ( const xt::xtensor< double, 2 > & cell_geometry )

static

Compute the physical coordinate of the reference point X=(0 , 0, 0)

Parameters

[in] cell_geometry The cell nodes coordinates (shape=(num geometry nodes, gdim))

Returns: Physical coordinate of the X=(0, 0, 0)

Note: Assumes that x0 is given by cell_geometry(0, i)

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

/__w/dolfinx/dolfinx/cpp/dolfinx/fem/CoordinateElement.h
/__w/dolfinx/dolfinx/cpp/dolfinx/fem/CoordinateElement.cpp

Public Member Functions

Static Public Member Functions

Detailed Description

Constructor & Destructor Documentation

◆ CoordinateElement() [1/2]

◆ CoordinateElement() [2/2]

Member Function Documentation

◆ cell_shape()

◆ compute_jacobian()

◆ compute_jacobian_determinant()

◆ compute_jacobian_inverse()

◆ dim()

◆ is_affine()

◆ needs_dof_permutations()

◆ pull_back_affine()

◆ pull_back_nonaffine()

◆ push_forward()

◆ tabulate() [1/2]

◆ tabulate() [2/2]

◆ tabulate_shape()

◆ x0()