Interpolation Nodes

Generic Shape-Based Interface

modepy.node_tuples_for_space(space: FunctionSpace) → Sequence[tuple[int]]

modepy.node_tuples_for_space(space: PN)

modepy.node_tuples_for_space(space: TensorProductSpace)

modepy.equispaced_nodes_for_space(space: FunctionSpace, shape: Shape) → ndarray

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

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

modepy.edge_clustered_nodes_for_space(space: FunctionSpace, shape: Shape) → ndarray

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

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

modepy.random_nodes_for_shape(shape: Shape, nnodes: int, rng: Generator | None = None) → ndarray

modepy.random_nodes_for_shape(shape: Simplex, nnodes: int, rng=None)

modepy.random_nodes_for_shape(shape: TensorProductShape, nnodes: int, rng=None)

Parameters:

rng – a numpy.random.Generator.

Returns:

a numpy.ndarray of shape (dim, nnodes) of random nodes in the reference shape.

Simplices

modepy.equidistant_nodes(dims: int, n: int, node_tuples: Sequence[tuple[int, ...]] | None = None) → ndarray

Parameters:

• dims – dimensionality of desired simplex (e.g. 1, 2 or 3, for interval, triangle or tetrahedron).

• n – Desired maximum total polynomial degree to interpolate.

• node_tuples – a list of tuples of integers indicating the node order. Use default order if None, see pytools.generate_nonnegative_integer_tuples_summing_to_at_most().

Returns:

An array of shape (dims, nnodes) containing bi-unit coordinates of the interpolation nodes. (see Coordinates on the triangle and Coordinates on the tetrahedron)

modepy.warp_and_blend_nodes(dims: int, n: int, node_tuples: Sequence[tuple[int, ...]] | None = None) → ndarray

Return interpolation nodes as described in [warburton-nodes]

[warburton-nodes]

Warburton, T. “An Explicit Construction of Interpolation Nodes on the Simplex.” Journal of Engineering Mathematics 56, no. 3 (2006): 247-262. http://dx.doi.org/10.1007/s10665-006-9086-6

Parameters:

• dims – dimensionality of desired simplex (1, 2 or 3, i.e. interval, triangle or tetrahedron).

• n – Desired maximum total polynomial degree to interpolate.

• node_tuples – a list of tuples of integers indicating the node order. Use default order if None, see pytools.generate_nonnegative_integer_tuples_summing_to_at_most().

Returns:

An array of shape (dims, nnodes) containing unit coordinates of the interpolation nodes. (see Coordinates on the triangle and Coordinates on the tetrahedron)

The generated nodes have benign Lebesgue constants. (See also modepy.tools.estimate_lebesgue_constant())

In addition to the node sets implemented here, recursivenodes by Tobin Isaac provides a selection of high-quality node sets for interpolation.

modepy.VioreanuRokhlinSimplexQuadrature also provides an interpolatory node set (without nodes on the boundary).

Hypercubes

modepy.tensor_product_nodes(dims_or_nodes: int | Sequence[ndarray], nodes_1d: ndarray | None = None) → ndarray

Returns:

an array of shape (dims, nnodes_1d**dims).

Added in version 2017.1.

Changed in version 2020.3: The node ordering has changed and is no longer documented.

Changed in version 2021.3: dims_or_nodes can contain nodes of general size (dims, nnodes), not only one dimensional nodes.

Changed in version 2022.1: The node ordering changed once again, and it is now accessible, via modepy.tools.reshape_array_for_tensor_product_space().

modepy.legendre_gauss_tensor_product_nodes(dims: int, n: int) → ndarray

Parameters:

n – the degree of polynomial exactly interpolated by the nodes. The one-dimensional base quadrature has n+1 nodes.

Added in version 2024.2.

modepy.legendre_gauss_lobatto_tensor_product_nodes(dims: int, n: int) → ndarray

Parameters:

n – the degree of polynomial exactly interpolated by the nodes. The one-dimensional base quadrature has n+1 nodes.