Shapes

modepy.shapes provides a generic description of the supported shapes (i.e. reference elements).

class modepy.Shape(dim: int)

dim: int

Spatial dimension of the shape.

abstract property nvertices: int

Number of vertices.

abstract property nfaces: int

Number of faces.

class modepy.Face(volume_shape: Shape, face_index: int, volume_vertex_indices: Tuple[int, ...], map_to_volume: Callable[[ndarray], ndarray])

Mix-in to be used with a concrete Shape subclass to represent geometry information about a face of a shape.

volume_shape: Shape

The volume Shape to which the face belongs.

face_index: int

The face index in volume_shape of this face.

volume_vertex_indices: Tuple[int, ...]

A tuple of indices into the vertices returned by unit_vertices_for_shape() for the volume_shape.

map_to_volume: Callable[[ndarray], ndarray]

A Callable that takes an array of size (dim, nnodes) of unit nodes on the face represented by face_vertices and maps them to the volume_shape.

modepy.unit_vertices_for_shape(shape: Shape) → ndarray

modepy.unit_vertices_for_shape(shape: TensorProductShape)

modepy.unit_vertices_for_shape(shape: Simplex)

Returns:

an ndarray of shape (dim, nvertices).

modepy.faces_for_shape(shape: Shape) → Tuple[Face, ...]

modepy.faces_for_shape(shape: Simplex)

modepy.faces_for_shape(shape: Hypercube)

Returns:

a tuple of Faces representing the faces of shape.

modepy.face_normal(face: Face, normalize: bool = True) → ndarray

Added in version 2021.2.1.

Tensor Product Shapes

class modepy.TensorProductShape(bases: Tuple[Shape, ...])

Bases: Shape

A shape formed as a tensor product of other shapes.

For example, the tensor product of a line segment and a triangle (2D simplex) results in a prism shape. Special cases include the Hypercube.

bases: Tuple[Shape, ...]

A tuple of base shapes that form the tensor product.

property nvertices: int

Number of vertices.

property nfaces: int

Number of faces.

Simplices

class modepy.Simplex(dim: int)

Bases: Shape

An n-dimensional simplex (lines, triangle, tetrahedron, etc.).

property nfaces: int

Number of faces.

property nvertices: int

Number of vertices.

Coordinates on the triangle

Bi-unit coordinates \((r, s)\) (also called ‘unit’ coordinates):

^ s
|
C
|\
| \
|  O
|   \
|    \
A-----B--> r

Vertices in bi-unit coordinates:

O = ( 0,  0)
A = (-1, -1)
B = ( 1, -1)
C = (-1,  1)

Equilateral coordinates \((x, y)\):

      C
     / \
    /   \
   /     \
  /   O   \
 /         \
A-----------B

Vertices in equilateral coordinates:

O = ( 0,          0)
A = (-1, -1/sqrt(3))
B = ( 1, -1/sqrt(3))
C = ( 0,  2/sqrt(3))

Coordinates on the tetrahedron

Bi-unit coordinates \((r, s, t)\) (also called ‘unit’ coordinates):

           ^ s
           |
           C
          /|\
         / | \
        /  |  \
       /   |   \
      /   O|    \
     /   __A-----B---> r
    /_--^ ___--^^
   ,D--^^^
t L

(squint, and it might start making sense…)

Vertices in bi-unit coordinates \((r, s, t)\):

O = ( 0,  0,  0)
A = (-1, -1, -1)
B = ( 1, -1, -1)
C = (-1,  1, -1)
D = (-1, -1,  1)

Vertices in equilateral coordinates \((x, y, z)\):

O = ( 0,          0,          0)
A = (-1, -1/sqrt(3), -1/sqrt(6))
B = ( 1, -1/sqrt(3), -1/sqrt(6))
C = ( 0,  2/sqrt(3), -1/sqrt(6))
D = ( 0,          0,  3/sqrt(6))

Hypercubes

class modepy.Hypercube(dim: int)

Bases: TensorProductShape

An n-dimensional hypercube (line, square, hexahedron, etc.).

Coordinates on the square

Bi-unit coordinates on \((r, s)\) (also called ‘unit’ coordinates):

^ s
|
C---------D
|         |
|         |
|    O    |
|         |
|         |
A---------B --> r

Vertices in bi-unit coordinates:

O = ( 0,  0)
A = (-1, -1)
B = ( 1, -1)
C = (-1,  1)
D = ( 1,  1)

Coordinates on the cube

Bi-unit coordinates on \((r, s, t)\) (also called ‘unit’ coordinates):

t
^
|
E----------G
|\         |\
| \        | \
|  \       |  \
|   F------+---H
|   |  O   |   |
A---+------C---|--> s
 \  |       \  |
  \ |        \ |
   \|         \|
    B----------D
     \
      v r

Vertices in bi-unit coordinates:

O = ( 0,  0,  0)
A = (-1, -1, -1)
B = ( 1, -1, -1)
C = (-1,  1, -1)
D = ( 1,  1, -1)
E = (-1, -1,  1)
F = ( 1, -1,  1)
G = (-1,  1,  1)
H = ( 1,  1,  1)

The order of the vertices in the hypercubes follows binary counting in tsr (i.e. in reverse axis order). For example, in 3D, A, B, C, D, ... is 000, 001, 010, 011, ....

Submeshes

modepy.submesh_for_shape(shape: Shape, node_tuples: Sequence[Tuple[int, ...]]) → Sequence[Tuple[int, ...]]

modepy.submesh_for_shape(shape: Simplex, node_tuples)

modepy.submesh_for_shape(shape: TensorProductShape, node_tuples)

Return a list of tuples of indices into the node list that generate a tessellation of the reference element.

Parameters:

node_tuples –

A list of tuples (i, j, …) of integers indicating node positions inside the unit element. The returned list references indices in this list.

modepy.node_tuples_for_space() may be used to generate node_tuples.

Added in version 2020.3.