Modes (Basis functions)

Function Spaces

class modepy.FunctionSpace

An opaque object representing a finite-dimensional function space of functions \(\mathbb{R}^n \to \mathbb{R}\).

spatial_dim

\(n\) in the above definition, the number of spatial dimensions in which the functions in the space operate.

space_dim

The number of dimensions of the function space.

class modepy.PN(spatial_dim: int, order: int)

The function space of polynomials with total degree \(N\) = order.

\[P^N:=\operatorname{span}\left\{\prod_{i=1}^d x_i^{n_i}:\sum n_i\le N\right\}.\]

order

__init__(spatial_dim: int, order: int) → None

class modepy.QN(spatial_dim: int, order: int)

The function space of polynomials with maximum degree \(N\) = order:

\[Q^N:=\operatorname{span} \left \{\prod_{i=1}^d x_i^{n_i}:\max n_i\le N\right\}.\]

order

__init__(spatial_dim: int, order: int) → None

class modepy.TensorProductSpace(bases: Tuple[FunctionSpace, ...])

bases

A tuple of the base spaces that take part in the tensor product.

__init__(bases: Tuple[FunctionSpace, ...]) → None

To recover the tensor product structure of degree-of-freedom arrays (nodal or modal) associated with this type of space, see reshape_array_for_tensor_product_space().

modepy.space_for_shape(shape: Shape, order: int | Tuple[int, ...]) → FunctionSpace

modepy.space_for_shape(shape: TensorProductShape, order: int | Tuple[int, ...]) → TensorProductSpace

modepy.space_for_shape(shape: Simplex, order: int | Tuple[int, ...]) → PN

Return an unspecified instance of FunctionSpace suitable for approximation on shape attaining interpolation error of \(O(h^{\text{order}+1})\).

Parameters:

order – an integer interpolation order or a tuple of orders. Taking a tuple of orders is not supported by all function spaces. A notable exception being TensorProductSpace, which allows defining different orders for each base space.

This functionality provides sets of basis functions for the reference elements in modepy.shapes.

class modepy.modes.RealValueT

TypeVar for basis function inputs and outputs, can be one of numpy.ndarray, pymbolic.primitives.Expression or a numbers.Number.

Basis Retrieval

exception modepy.BasisNotOrthonormal

class modepy.Basis

abstract orthonormality_weight() → float

Raises:

BasisNotOrthonormal if the basis does not have a weight, i.e. it is not orthogonal.

mode_ids

A tuple of of mode (basis function) identifiers, one for each basis function. Mode identifiers should generally be viewed as opaque. They are hashable. For some bases, they are tuples of length matching the number of dimensions and indicating the order along each reference axis.

functions

A tuple of (callable) basis functions of length matching mode_ids. Each function takes a vector of \((r, s, t)\) reference coordinates (depending on dimension) as input.

gradients

A tuple of (callable) basis functions of length matching mode_ids. Each function takes a vector of \((r, s, t)\) reference coordinates (depending on dimension) as input. Each function returns a tuple of derivatives, one per reference axis.

modepy.basis_for_space(space: FunctionSpace, shape: Shape) → Basis

modepy.basis_for_space(space: PN, shape: Simplex)

modepy.basis_for_space(space: TensorProductSpace, shape: TensorProductShape)

modepy.orthonormal_basis_for_space(space: FunctionSpace, shape: Shape) → Basis

modepy.orthonormal_basis_for_space(space: PN, shape: Simplex)

modepy.orthonormal_basis_for_space(space: TensorProductSpace, shape: TensorProductShape)

modepy.monomial_basis_for_space(space: FunctionSpace, shape: Shape) → Basis

modepy.monomial_basis_for_space(space: PN, shape: Simplex)

modepy.monomial_basis_for_space(space: TensorProductSpace, shape: TensorProductShape)

Jacobi polynomials

modepy.jacobi(alpha: float, beta: float, n: int, x: RealValueT) → RealValueT

Evaluate Jacobi polynomials of type \((\alpha, \beta)\), with \(\alpha, \beta > -1\), and order n at a vector of points x. The points x must lie on the interval \([-1, 1]\).

The polynomials are normalized to be orthonormal with respect to the Jacobi weight \((1 - x)^\alpha (1 + x)^\beta\).

Observe that choosing \(\alpha = \beta = 0\) will yield the Legendre polynomials.

Returns:

a vector of \(P^{(\alpha, \beta)}_n\) evaluated at all x.

modepy.grad_jacobi(alpha: float, beta: float, n: int, x: RealValueT) → RealValueT

Evaluate the derivative of jacobi(), with the same meanings and restrictions for all arguments.

Conversion to Symbolic

modepy.symbolicize_function(f: Callable[[RealValueT], RealValueT | Tuple[RealValueT, ...]], dim: int, ref_coord_var_name: str = 'r') → RealValueT | Tuple[RealValueT, ...]

For a function f (basis or gradient) returned by one of the functions in this module, return a pymbolic expression representing the same function.

Parameters:

dim – the number of dimensions of the reference element on which basis is defined.

New in version 2020.2.

Tensor product adapter

class modepy.TensorProductBasis(bases: Sequence[Sequence[Callable[[ndarray], ndarray]]], grad_bases: Sequence[Sequence[Callable[[ndarray], Tuple[ndarray, ...]]]], orth_weight: float | None, dims_per_basis: Tuple[int, ...] | None = None)

Adapts multiple bases into a tensor product basis.

__init__(bases: Sequence[Sequence[Callable[[ndarray], ndarray]]], grad_bases: Sequence[Sequence[Callable[[ndarray], Tuple[ndarray, ...]]]], orth_weight: float | None, dims_per_basis: Tuple[int, ...] | None = None) → None

Parameters:

• bases – a sequence of sequences (representing the basis) of functions representing the approximation basis.

• grad_bases – a sequence of sequences representing the derivatives of bases.

• orth_weight – if bases forms an orthogonal basis, this should be the normalizing weight. If None, then the basis is assumed to not be orthogonal (this is not checked).

PKDO basis functions

modepy.modes.pkdo_2d(order: Tuple[int, int], rs: ndarray) → RealValueT

Evaluate a 2D orthonormal (with weight 1) polynomial on the unit simplex.

Parameters:

• order – A tuple (i, j) representing the order of the polynomial.

• rs – rs[0], rs[1] are arrays of \((r,s)\) coordinates. (See Coordinates on the triangle)

Returns:

a vector of values of the same length as the rs arrays.

See the following publications:

• Proriol, Joseph. “Sur une famille de polynomes á deux variables orthogonaux dans un triangle.” CR Acad. Sci. Paris 245 (1957): 2459-2461.

• Koornwinder, T. “Two-variable analogues of the classical orthogonal polynomials.” Theory and Applications of Special Functions. 1975, pp. 435-495.

• Dubiner, Moshe. “Spectral Methods on Triangles and Other Domains.” Journal of Scientific Computing 6, no. 4 (December 1, 1991): 345–390. http://dx.doi.org/10.1007/BF01060030

modepy.modes.grad_pkdo_2d(order: Tuple[int, int], rs: ndarray) → Tuple[RealValueT, RealValueT]

Evaluate the derivatives of pkdo_2d().

Parameters:

• order – A tuple (i, j) representing the order of the polynomial.

• rs – rs[0], rs[1] are arrays of \((r, s)\) coordinates. (See Coordinates on the triangle)

Returns:

a tuple of vectors (dphi_dr, dphi_ds), each of the same length as the rs arrays.

See the following publications:

• Proriol, Joseph. “Sur une famille de polynomes á deux variables orthogonaux dans un triangle.” CR Acad. Sci. Paris 245 (1957): 2459-2461.

• Koornwinder, T. “Two-variable analogues of the classical orthogonal polynomials.” Theory and Applications of Special Functions. 1975, pp. 435-495.

• Dubiner, Moshe. “Spectral Methods on Triangles and Other Domains.” Journal of Scientific Computing 6, no. 4 (December 1, 1991): 345–390. http://dx.doi.org/10.1007/BF01060030

modepy.modes.pkdo_3d(order: Tuple[int, int, int], rst: ndarray) → RealValueT

Evaluate a 2D orthonormal (with weight 1) polynomial on the unit simplex.

Parameters:

• order – A tuple (i, j, k) representing the order of the polynomial.

• rs – rst[0], rst[1], rst[2] are arrays of \((r, s, t)\) coordinates. (See Coordinates on the tetrahedron)

Returns:

a vector of values of the same length as the rst arrays.

See the following publications:

• Proriol, Joseph. “Sur une famille de polynomes á deux variables orthogonaux dans un triangle.” CR Acad. Sci. Paris 245 (1957): 2459-2461.

• Koornwinder, T. “Two-variable analogues of the classical orthogonal polynomials.” Theory and Applications of Special Functions. 1975, pp. 435-495.

• Dubiner, Moshe. “Spectral Methods on Triangles and Other Domains.” Journal of Scientific Computing 6, no. 4 (December 1, 1991): 345–390. http://dx.doi.org/10.1007/BF01060030

modepy.modes.grad_pkdo_3d(order: Tuple[int, int, int], rst: ndarray) → Tuple[RealValueT, RealValueT, RealValueT]

Evaluate the derivatives of pkdo_3d().

Parameters:

• order – A tuple (i, j, k) representing the order of the polynomial.

• rs – rs[0], rs[1], rs[2] are arrays of \((r,s,t)\) coordinates. (See Coordinates on the tetrahedron)

Returns:

a tuple of vectors (dphi_dr, dphi_ds, dphi_dt), each of the same length as the rst arrays.

See the following publications:

• Proriol, Joseph. “Sur une famille de polynomes á deux variables orthogonaux dans un triangle.” CR Acad. Sci. Paris 245 (1957): 2459-2461.

• Koornwinder, T. “Two-variable analogues of the classical orthogonal polynomials.” Theory and Applications of Special Functions. 1975, pp. 435-495.

• Dubiner, Moshe. “Spectral Methods on Triangles and Other Domains.” Journal of Scientific Computing 6, no. 4 (December 1, 1991): 345–390. http://dx.doi.org/10.1007/BF01060030

Monomials

modepy.modes.monomial(order: Tuple[int, ...], rst: ndarray) → RealValueT

Evaluate the monomial of order order at the points rst.

Parameters:

• order – A tuple (i, j,…) representing the order of the polynomial.

• rst – rst[0], rst[1] are arrays of \((r, s, ...)\) coordinates. (See Coordinates on the triangle)

modepy.modes.grad_monomial(order: Tuple[int, ...], rst: ndarray) → Tuple[RealValueT, ...]

Evaluate the derivative of the monomial of order order at the points rst.

Parameters:

• order – A tuple (i, j,…) representing the order of the polynomial.

• rst – rst[0], rst[1] are arrays of \((r, s, ...)\) coordinates. (See Coordinates on the triangle)

Returns:

a tuple of vectors (dphi_dr, dphi_ds, dphi_dt, ….), each of the same length as the rst arrays.

New in version 2016.1.