basix::polyset Namespace Reference

Functions
xt::xtensor< double, 3 >	tabulate (cell::type celltype, int d, int n, const xt::xarray< double > &x)
int	dim (cell::type cell, int d)

Detailed Description

Orthonormal polynomial basis on reference cell

These are the underlying "expansion sets" for all finite elements, which when multiplied by a set of "coefficients" give the FE basis functions.

The polynomials (and their derivatives) can be tabulated on unit interval, triangle, tetrahedron, quadrilateral, hexahedron, prism and pyramids.

Function Documentation

dim()

int basix::polyset::dim	(	cell::type	cell,
		int	d
	)

Dimension of a polynomial space

Parameters

[in]	cell	The cell type
[in]	d	The polynomial degree

Returns

The number terms in the basis spanning a space of polynomial degree d

tabulate()

xt::xtensor< double, 3 > basix::polyset::tabulate	(	cell::type	celltype,
		int	d,
		int	n,
		const xt::xarray< double > &	x
	)

Tabulate the orthonormal polynomial basis, and derivatives, at points on the reference cell.

All derivatives up to the given order are computed. If derivatives are not required, use n = 0. For example, order n = 2 for a 2D cell, will compute the basis \(\psi, d\psi/dx, d\psi/dy, d^2 \psi/dx^2, d^2\psi/dxdy, d^2\psi/dy^2\) in that order (0, 0), (1, 0), (0, 1), (2, 0), (1, 1), (0 ,2).

For an interval cell there are nderiv + 1 derivatives, for a 2D cell, there are (nderiv + 1)(nderiv + 2)/2 derivatives, and in 3D, there are (nderiv + 1)(nderiv + 2)(nderiv + 3)/6. The ordering is 'triangular' with the lower derivatives appearing first.

Parameters

[in]	celltype	Cell type
[in]	d	Polynomial degree
[in]	n	Maximum derivative order. Use n = 0 for the basis only.
[in]	x	Points at which to evaluate the basis. The shape is (number of points, geometric dimension).

Returns

Polynomial sets, for each derivative, tabulated at points. The shape is (number of derivatives computed, number of points, basis index).

• The first index is the derivative. The first entry is the basis itself. Derivatives are stored in triangular (2D) or tetrahedral (3D) ordering, e.g. if (p, q) denotes p order dervative with repsect to x and q order derivative with respect to y, [0] -> (0, 0), [1] -> (1, 0), [2] -> (0, 1), [3] -> (2, 0), [4] -> (1, 1), [5] -> (0, 2), [6] -> (3, 0),... The function basix::idx maps tuples (p, q, r) to the array index.
• The second index is the point, with index i correspondign to the point in row i of x.
• The third index is the basis function index.

  Todo:

  Does the order for the third index need to be documented?