Source code for trace_matrices

"""
Here we compute the trace matrices :math:`\\mathbb{T}_{\\text{N}}`,
:math:`\\mathbb{T}_{\\text{E}}` and :math:`\\mathbb{T}_{\\text{F}}`.


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:download:`[trace_matrices.py]</contents/LIBRARY/ptc/mathischeap_ptc/trace_matrices.py>`.
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"""

import numpy as np
from scipy.sparse import csr_matrix

from incidence_matrices import local_numbering_NP
from incidence_matrices import local_numbering_EP
from incidence_matrices import local_numbering_FP



[docs]def TN(N_xi, N_et, N_sg): """The trace matrix :math:`\\mathbb{T}_{\\text{N}}` for mimetic polynomials constructed on three sets of nodes, :math:`\\left\\lbrace\\xi_0,\\xi_1,\\cdots, \\xi_{N0} \\right\\rbrace`, :math:`\\left\\lbrace\\eta_0,\\eta_1,\\cdots, \\eta_{N1}\\right\\rbrace` and :math:`\\left\\lbrace\\varsigma_0, \\varsigma_1,\\cdots, \\varsigma_{N2}\\right\\rbrace`. :param N_xi: ``N_xi + 1`` nodes in set :math:`\\left\\lbrace\\xi_0, \\xi_1,\\cdots, \\xi_{N_\\xi} \\right\\rbrace`. :param N_et: ``N_et + 1`` nodes in set :math:`\\left\\lbrace\\eta_0, \\eta_1,\\cdots, \\eta_{N_\\eta} \\right\\rbrace`. :param N_sg: ``N_sg + 1`` nodes in set :math:`\\left\\lbrace \\varsigma_0, \\varsigma_1,\\cdots, \\varsigma_{N_\\varsigma} \\right\\rbrace`. :type N_xi: positive integer :type N_et: positive integer :type N_sg: positive integer :return: A csr_matrix :math:`\\mathbb{T}_{\\text{N}}`. :example: """ raise NotImplementedError(f"Could you code it?")
[docs]def TE(N_xi, N_et, N_sg): """The trace matrix :math:`\\mathbb{T}_{\\text{E}}` for mimetic polynomials constructed on three sets of nodes, :math:`\\left\\lbrace\\xi_0,\\xi_1,\\cdots, \\xi_{N0} \\right\\rbrace`, :math:`\\left\\lbrace\\eta_0,\\eta_1,\\cdots, \\eta_{N1}\\right\\rbrace` and :math:`\\left\\lbrace\\varsigma_0, \\varsigma_1,\\cdots, \\varsigma_{N2}\\right\\rbrace`. :param N_xi: ``N_xi + 1`` nodes in set :math:`\\left\\lbrace\\xi_0, \\xi_1,\\cdots, \\xi_{N_\\xi} \\right\\rbrace`. :param N_et: ``N_et + 1`` nodes in set :math:`\\left\\lbrace\\eta_0, \\eta_1,\\cdots, \\eta_{N_\\eta} \\right\\rbrace`. :param N_sg: ``N_sg + 1`` nodes in set :math:`\\left\\lbrace \\varsigma_0, \\varsigma_1,\\cdots, \\varsigma_{N_\\varsigma} \\right\\rbrace`. :type N_xi: positive integer :type N_et: positive integer :type N_sg: positive integer :return: A csc_matrix :math:`\\mathbb{T}_{\\text{E}}`. :example: """ raise NotImplementedError(f"Could you code it?")
[docs]def TF(N_xi, N_et, N_sg): """The trace matrix :math:`\\mathbb{T}_{\\text{F}}` for mimetic polynomials constructed on three sets of nodes, :math:`\\left\\lbrace\\xi_0,\\xi_1,\\cdots, \\xi_{N0} \\right\\rbrace`, :math:`\\left\\lbrace\\eta_0,\\eta_1,\\cdots, \\eta_{N1}\\right\\rbrace` and :math:`\\left\\lbrace\\varsigma_0, \\varsigma_1,\\cdots, \\varsigma_{N2}\\right\\rbrace`. :param N_xi: ``N_xi + 1`` nodes in set :math:`\\left\\lbrace\\xi_0, \\xi_1,\\cdots, \\xi_{N_\\xi} \\right\\rbrace`. :param N_et: ``N_et + 1`` nodes in set :math:`\\left\\lbrace\\eta_0, \\eta_1,\\cdots, \\eta_{N_\\eta} \\right\\rbrace`. :param N_sg: ``N_sg + 1`` nodes in set :math:`\\left\\lbrace \\varsigma_0, \\varsigma_1,\\cdots, \\varsigma_{N_\\varsigma} \\right\\rbrace`. :type N_xi: positive integer :type N_et: positive integer :type N_sg: positive integer :return: A csr_matrix :math:`\\mathbb{T}_{\\text{F}}`. :example: >>> T = TF(2, 2, 2) >>> T.indices array([ 0, 3, 6, 9, 2, 5, 8, 11, 12, 13, 18, 19, 16, 17, 22, 23, 24, 25, 26, 27, 32, 33, 34, 35], dtype=int32) >>> T.data array([-1, -1, -1, -1, 1, 1, 1, 1, -1, -1, -1, -1, 1, 1, 1, 1, -1, -1, -1, -1, 1, 1, 1, 1], dtype=int32) """ num_dofs_FP = (N_xi+1)*N_et*N_sg + \ N_xi*(N_et+1)*N_sg + \ N_xi*N_et*(N_sg+1) NF = local_numbering_FP(N_xi, N_et, N_sg) N = NF[0][0, :, :].ravel('F') S = NF[0][-1, :, :].ravel('F') W = NF[1][:, 0, :].ravel('F') E = NF[1][:, -1, :].ravel('F') B = NF[2][:, :, 0].ravel('F') F = NF[2][:, :, -1].ravel('F') Tn = np.zeros((N_et*N_sg, num_dofs_FP), dtype='int') Ts = np.zeros((N_et*N_sg, num_dofs_FP), dtype='int') Tw = np.zeros((N_xi*N_sg, num_dofs_FP), dtype='int') Te = np.zeros((N_xi*N_sg, num_dofs_FP), dtype='int') Tb = np.zeros((N_xi*N_et, num_dofs_FP), dtype='int') Tf = np.zeros((N_xi*N_et, num_dofs_FP), dtype='int') for i, j in enumerate(N): Tn[i, j] = -1 for i, j in enumerate(S): Ts[i, j] = 1 for i, j in enumerate(W): Tw[i, j] = -1 for i, j in enumerate(E): Te[i, j] = 1 for i, j in enumerate(B): Tb[i, j] = -1 for i, j in enumerate(F): Tf[i, j] = 1 T = np.vstack((Tn, Ts, Tw, Te, Tb, Tf)) return csr_matrix(T)
if __name__ == '__main__': import doctest doctest.testmod() N = 1 TF(N, N, N) print(TF(N, N, N).toarray())