mimetic_basis_polynomials.py

Node, edge, face and volume polynomials in the 3d reference domain .

grid

Let A=(a1, a2), B=(b1,b2,b3), C=(c1,c2). Then grid (A, B, C) refers to a sequence of coordinates, (a1, b1, c1), (a2, b1, c1), (a1, b2, c1), (a2, b2, c1), (a1, b3, c1), (a2, b3, c1), (a1, b1, c2), …… Namely, we first do a meshgrid (A, B, C), then put the coordinates into a sequence one by one picking along A direction firstly, B direction secondly and finally C direction. Also see grid().

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class mimetic_basis_polynomials.MimeticBasisPolynomials(nodes_xi, nodes_et, nodes_sg)

A wrapper of basis node, edge, face and volume polynomials in the 3d reference domain .

Parameters:

• nodes_xi (1d np.array) – The nodes on which the 1D mimetic polynomials are built along the first axis ().

• nodes_et (1d np.array) – The nodes on which the 1D mimetic polynomials are built along the second axis ().

• nodes_sg (1d np.array) – The nodes on which the 1D mimetic polynomials are built along the third axis ().

Example:

>>> bf = MimeticBasisPolynomials('Lobatto-3', 'Lobatto-3', 'Lobatto-3')
>>> bf.degree # N = N_xi = N_eta = N_sigma = 3
[3, 3, 3]
>>> xi = np.linspace(-1, 1, 5)
>>> et = np.linspace(-1, 1, 6)
>>> sg = np.linspace(-1, 1, 7)
>>> NP = bf.node_polynomials(xi, et, sg)
>>> NP.shape # 4^3=64 node polynomials evaluated at 5*6*7=210 points
(64, 210)
>>> EP_xi, EP_et, EP_sg = bf.edge_polynomials(xi, et, sg)
>>> EP_xi.shape # 3*4*4=48 edge polynomials
(48, 210)
>>> FP_xi, FP_et, FP_sg = bf.face_polynomials(xi, et, sg)
>>> FP_et.shape # 3*4*3=36 face polynomials
(36, 210)
>>> VP = bf.volume_polynomials(xi, et, sg)
>>> VP.shape # 3^3=27 volume polynomials
(27, 210)

edge_polynomials(xi, et, sg)

Evaluate the edge polynomials at grid(xi, et, sg).

Parameters:

• xi (1d np.array) – grid (xi, eta, sigma) is the grid to evaluate the polynomials.

• et (1d np.array) – grid (xi, eta, sigma) is the grid to evaluate the polynomials.

• sg (1d np.array) – grid (xi, eta, sigma) is the grid to evaluate the polynomials.

Returns:

A 2d np.array whose 0-dimension refers to the number of node polynomials and 1-dimension refers to the values evaluated at grid (xi, eta, sigma).

face_polynomials(xi, et, sg)

Evaluate the face polynomials at grid(xi, et, sg).

Parameters:

• xi (1d np.array) – grid (xi, eta, sigma) is the grid to evaluate the polynomials.

• et (1d np.array) – grid (xi, eta, sigma) is the grid to evaluate the polynomials.

• sg (1d np.array) – grid (xi, eta, sigma) is the grid to evaluate the polynomials.

Returns:

A 2d np.array whose 0-dimension refers to the number of node polynomials and 1-dimension refers to the values evaluated at grid (xi, eta, sigma).

node_polynomials(xi, et, sg)

Evaluate the node polynomials at grid(xi, et, sg).

Parameters:

• xi (1d np.array) – grid (xi, eta, sigma) is the grid to evaluate the polynomials.

• et (1d np.array) – grid (xi, eta, sigma) is the grid to evaluate the polynomials.

• sg (1d np.array) – grid (xi, eta, sigma) is the grid to evaluate the polynomials.

Returns:

A 2d np.array whose 0-dimension refers to the number of node polynomials and 1-dimension refers to the values evaluated at grid (xi, eta, sigma).

volume_polynomials(xi, et, sg)

Evaluate the volume polynomials at grid(xi, et, sg).

Parameters:

• xi (1d np.array) – grid (xi, eta, sigma) is the grid to evaluate the polynomials.

• et (1d np.array) – grid (xi, eta, sigma) is the grid to evaluate the polynomials.

• sg (1d np.array) – grid (xi, eta, sigma) is the grid to evaluate the polynomials.

Returns:

A 2d np.array whose 0-dimension refers to the number of node polynomials and 1-dimension refers to the values evaluated at grid (xi, eta, sigma).

mimetic_basis_polynomials.grid(A, B, C)

The function to compute the grid of three sets of nodes.

Parameters:

• A (1d data object) – The first (along ) set of nodes.

• B (1d data object) – The second (along ) set of nodes.

• C (1d data object) – The third (along ) set of nodes.

Returns:

A tuple of three outputs:

1. (1d np.array) The grided coordinates.

2. (1d np.array) The grided coordinates.

3. (1d np.array) The grided coordinates.

Example:

>>> A = np.array([1, 2])
>>> B = np.array([3, 4, 5])
>>> C = np.array([6, 7])
>>> x, y, z = grid(A, B, C)
>>> D = np.vstack((x,y,z)).T
>>> D
array([[1, 3, 6],
       [2, 3, 6],
       [1, 4, 6],
       [2, 4, 6],
       [1, 5, 6],
       [2, 5, 6],
       [1, 3, 7],
       [2, 3, 7],
       [1, 4, 7],
       [2, 4, 7],
       [1, 5, 7],
       [2, 5, 7]])

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