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398 lines
13 KiB
Python
398 lines
13 KiB
Python
"""
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Sparse matrix operators for use with electromagnetic wave equations.
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These functions return sparse-matrix (scipy.sparse.spmatrix) representations of
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a variety of operators, intended for use with E and H fields vectorized using the
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fdfd_tools.vec() and .unvec() functions (column-major/Fortran ordering).
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E- and H-field values are defined on a Yee cell; epsilon values should be calculated for
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cells centered at each E component (mu at each H component).
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Many of these functions require a 'dxes' parameter, of type fdfd_tools.dx_lists_type,
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which contains grid cell width information in the following format:
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[[[dx_e_0, dx_e_1, ...], [dy_e_0, ...], [dz_e_0, ...]],
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[[dx_h_0, dx_h_1, ...], [dy_h_0, ...], [dz_h_0, ...]]]
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where dx_e_0 is the x-width of the x=0 cells, as used when calculating dE/dx,
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and dy_h_0 is the y-width of the y=0 cells, as used when calculating dH/dy, etc.
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The following operators are included:
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- E-only wave operator
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- H-only wave operator
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- EH wave operator
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- Curl for use with E, H fields
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- E to H conversion
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- M to J conversion
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- Poynting cross products
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Also available:
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- Circular shifts
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- Discrete derivatives
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- Averaging operators
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- Cross product matrices
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"""
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from typing import List, Tuple
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import numpy
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import scipy.sparse as sparse
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from . import vec, dx_lists_t, vfield_t
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__author__ = 'Jan Petykiewicz'
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def e_full(omega: complex,
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dxes: dx_lists_t,
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epsilon: vfield_t,
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mu: vfield_t = None
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) -> sparse.spmatrix:
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"""
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Wave operator del x (1/mu * del x) - omega**2 * epsilon, for use with E-field,
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with wave equation
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(del x (1/mu * del x) - omega**2 * epsilon) E = -i * omega * J
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To make this matrix symmetric, use the preconditions from e_full_preconditioners().
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:param omega: Angular frequency of the simulation
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:param dxes: Grid parameters [dx_e, dx_h] as described in fdfd_tools.operators header
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:param epsilon: Vectorized dielectric constant
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:param mu: Vectorized magnetic permeability (default 1 everywhere)..
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:return: Sparse matrix containing the wave operator
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"""
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ce = curl_e(dxes)
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ch = curl_h(dxes)
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e = sparse.diags(epsilon)
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if numpy.any(numpy.equal(mu, None)):
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m_div = sparse.eye(epsilon.size)
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else:
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m_div = sparse.diags(1 / mu)
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op = ch @ m_div @ ce - omega**2 * e
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return op
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def e_full_preconditioners(dxes: dx_lists_t
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) -> Tuple[sparse.spmatrix, sparse.spmatrix]:
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"""
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Left and right preconditioners (Pl, Pr) for symmetrizing the e_full wave operator.
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The preconditioned matrix A_symm = (Pl @ A @ Pr) is complex-symmetric
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(non-Hermitian unless there is no loss or PMLs).
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The preconditioner matrices are diagonal and complex, with Pr = 1 / Pl
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:param dxes: Grid parameters [dx_e, dx_h] as described in fdfd_tools.operators header
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:return: Preconditioner matrices (Pl, Pr)
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"""
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p_squared = [dxes[0][0][:, None, None] * dxes[1][1][None, :, None] * dxes[1][2][None, None, :],
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dxes[1][0][:, None, None] * dxes[0][1][None, :, None] * dxes[1][2][None, None, :],
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dxes[1][0][:, None, None] * dxes[1][1][None, :, None] * dxes[0][2][None, None, :]]
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p_vector = numpy.sqrt(vec(p_squared))
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P_left = sparse.diags(p_vector)
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P_right = sparse.diags(1 / p_vector)
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return P_left, P_right
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def h_full(omega: complex,
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dxes: dx_lists_t,
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epsilon: vfield_t,
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mu: vfield_t = None
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) -> sparse.spmatrix:
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"""
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Wave operator del x (1/epsilon * del x) - omega**2 * mu, for use with H-field,
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with wave equation
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(del x (1/epsilon * del x) - omega**2 * mu) H = i * omega * M
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:param omega: Angular frequency of the simulation
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:param dxes: Grid parameters [dx_e, dx_h] as described in fdfd_tools.operators header
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:param epsilon: Vectorized dielectric constant
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:param mu: Vectorized magnetic permeability (default 1 everywhere)
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:return: Sparse matrix containing the wave operator
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"""
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ec = curl_e(dxes)
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hc = curl_h(dxes)
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e_div = sparse.diags(1 / epsilon)
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if numpy.any(numpy.equal(mu, None)):
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m = sparse.eye(epsilon.size)
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else:
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m = sparse.diags(mu)
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A = ec @ e_div @ hc - omega**2 * m
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return A
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def eh_full(omega, dxes, epsilon, mu=None):
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"""
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Wave operator for [E, H] field representation. This operator implements Maxwell's
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equations without cancelling out either E or H. The operator is
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[[-i * omega * epsilon, del x],
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[del x, i * omega * mu]]
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for use with a field vector of the form hstack(vec(E), vec(H)).
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:param omega: Angular frequency of the simulation
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:param dxes: Grid parameters [dx_e, dx_h] as described in fdfd_tools.operators header
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:param epsilon: Vectorized dielectric constant
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:param mu: Vectorized magnetic permeability (default 1 everywhere)
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:return: Sparse matrix containing the wave operator
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"""
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A2 = curl_e(dxes)
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A1 = curl_h(dxes)
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iwe = 1j * omega * sparse.diags(epsilon)
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iwm = 1j * omega
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if not numpy.any(numpy.equal(mu, None)):
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iwm *= sparse.diags(mu)
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A = sparse.bmat([[-iwe, A1],
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[A2, +iwm]])
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return A
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def curl_h(dxes: dx_lists_t) -> sparse.spmatrix:
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"""
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Curl operator for use with the H field.
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:param dxes: Grid parameters [dx_e, dx_h] as described in fdfd_tools.operators header
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:return: Sparse matrix for taking the discretized curl of the H-field
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"""
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return cross(deriv_back(dxes[1]))
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def curl_e(dxes: dx_lists_t) -> sparse.spmatrix:
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"""
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Curl operator for use with the E field.
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:param dxes: Grid parameters [dx_e, dx_h] as described in fdfd_tools.operators header
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:return: Sparse matrix for taking the discretized curl of the E-field
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"""
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return cross(deriv_forward(dxes[0]))
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def e2h(omega: complex,
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dxes: dx_lists_t,
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mu: vfield_t = None,
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) -> sparse.spmatrix:
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"""
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Utility operator for converting the E field into the H field.
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For use with e_full -- assumes that there is no magnetic current M.
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:param omega: Angular frequency of the simulation
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:param dxes: Grid parameters [dx_e, dx_h] as described in fdfd_tools.operators header
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:param mu: Vectorized magnetic permeability (default 1 everywhere)
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:return: Sparse matrix for converting E to H
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"""
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op = curl_e(dxes) / (-1j * omega)
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if not numpy.any(numpy.equal(mu, None)):
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op = sparse.diags(1 / mu) @ op
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return op
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def m2j(omega: complex,
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dxes: dx_lists_t,
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mu: vfield_t = None):
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"""
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Utility operator for converting M field into J.
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Converts a magnetic current M into an electric current J.
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For use with eg. e_full.
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:param omega: Angular frequency of the simulation
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:param dxes: Grid parameters [dx_e, dx_h] as described in fdfd_tools.operators header
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:param mu: Vectorized magnetic permeability (default 1 everywhere)
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:return: Sparse matrix for converting E to H
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"""
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op = curl_h(dxes) / (1j * omega)
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if not numpy.any(numpy.equal(mu, None)):
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op = op @ sparse.diags(1 / mu)
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return op
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def rotation(axis: int, shape: List[int]) -> sparse.spmatrix:
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"""
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Utility operator for performing a circular shift along a specified axis by 1 element.
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:param axis: Axis to shift along. x=0, y=1, z=2
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:param shape: Shape of the grid being shifted
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:return: Sparse matrix for performing the circular shift
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"""
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if len(shape) not in (2, 3):
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raise Exception('Invalid shape: {}'.format(shape))
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if axis not in range(len(shape)):
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raise Exception('Invalid direction: {}, shape is {}'.format(axis, shape))
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n = numpy.prod(shape)
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shifts = [1 if k == axis else 0 for k in range(3)]
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shifted_diags = [(numpy.arange(n) + s) % n for n, s in zip(shape, shifts)]
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ijk = numpy.meshgrid(*shifted_diags, indexing='ij')
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i_ind = numpy.arange(n)
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j_ind = ijk[0] + ijk[1] * shape[0]
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if len(shape) == 3:
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j_ind += ijk[2] * shape[0] * shape[1]
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vij = (numpy.ones(n), (i_ind, j_ind.flatten(order='F')))
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D = sparse.csr_matrix(vij, shape=(n, n))
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return D
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def deriv_forward(dx_e: List[numpy.ndarray]) -> List[sparse.spmatrix]:
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"""
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Utility operators for taking discretized derivatives (forward variant).
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:param dx_e: Lists of cell sizes for all axes [[dx_0, dx_1, ...], ...].
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:return: List of operators for taking forward derivatives along each axis.
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"""
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shape = [s.size for s in dx_e]
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n = numpy.prod(shape)
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dx_e_expanded = numpy.meshgrid(*dx_e, indexing='ij')
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def deriv(axis):
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return rotation(axis, shape) - sparse.eye(n)
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Ds = [sparse.diags(+1 / dx.flatten(order='F')) @ deriv(a)
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for a, dx in enumerate(dx_e_expanded)]
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return Ds
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def deriv_back(dx_h: List[numpy.ndarray]) -> List[sparse.spmatrix]:
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"""
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Utility operators for taking discretized derivatives (backward variant).
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:param dx_h: Lists of cell sizes for all axes [[dx_0, dx_1, ...], ...].
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:return: List of operators for taking forward derivatives along each axis.
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"""
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shape = [s.size for s in dx_h]
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n = numpy.prod(shape)
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dx_h_expanded = numpy.meshgrid(*dx_h, indexing='ij')
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def deriv(axis):
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return rotation(axis, shape) - sparse.eye(n)
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Ds = [sparse.diags(-1 / dx.flatten(order='F')) @ deriv(a).T
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for a, dx in enumerate(dx_h_expanded)]
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return Ds
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def cross(B: List[sparse.spmatrix]) -> sparse.spmatrix:
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"""
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Cross product operator
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:param B: List [Bx, By, Bz] of sparse matrices corresponding to the x, y, z
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portions of the operator on the left side of the cross product.
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:return: Sparse matrix corresponding to (B x), where x is the cross product
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"""
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n = B[0].shape[0]
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zero = sparse.csr_matrix((n, n))
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return sparse.bmat([[zero, -B[2], B[1]],
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[B[2], zero, -B[0]],
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[-B[1], B[0], zero]])
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def vec_cross(b: vfield_t) -> sparse.spmatrix:
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"""
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Vector cross product operator
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:param b: Vector on the left side of the cross product
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:return: Sparse matrix corresponding to (b x), where x is the cross product
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"""
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B = [sparse.diags(c) for c in numpy.split(b, 3)]
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return cross(B)
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def avgf(axis: int, shape: List[int]) -> sparse.spmatrix:
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"""
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Forward average operator (x4 = (x4 + x5) / 2)
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:param axis: Axis to average along (x=0, y=1, z=2)
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:param shape: Shape of the grid to average
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:return: Sparse matrix for forward average operation
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"""
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if len(shape) not in (2, 3):
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raise Exception('Invalid shape: {}'.format(shape))
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n = numpy.prod(shape)
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return 0.5 * (sparse.eye(n) + rotation(axis, shape))
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def avgb(axis: int, shape: List[int]) -> sparse.spmatrix:
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"""
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Backward average operator (x4 = (x4 + x3) / 2)
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:param axis: Axis to average along (x=0, y=1, z=2)
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:param shape: Shape of the grid to average
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:return: Sparse matrix for backward average operation
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"""
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return avgf(axis, shape).T
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def poynting_e_cross(e: vfield_t, dxes: dx_lists_t) -> sparse.spmatrix:
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"""
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Operator for computing the Poynting vector, contining the (E x) portion of the Poynting vector.
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:param e: Vectorized E-field for the ExH cross product
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:param dxes: Grid parameters [dx_e, dx_h] as described in fdfd_tools.operators header
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:return: Sparse matrix containing (E x) portion of Poynting cross product
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"""
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shape = [len(dx) for dx in dxes[0]]
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fx, fy, fz = [avgf(i, shape) for i in range(3)]
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bx, by, bz = [avgb(i, shape) for i in range(3)]
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dxag = [dx.flatten(order='F') for dx in numpy.meshgrid(*dxes[0], indexing='ij')]
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dbgx, dbgy, dbgz = [sparse.diags(dx.flatten(order='F'))
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for dx in numpy.meshgrid(*dxes[1], indexing='ij')]
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Ex, Ey, Ez = [sparse.diags(ei * da) for ei, da in zip(numpy.split(e, 3), dxag)]
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n = numpy.prod(shape)
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zero = sparse.csr_matrix((n, n))
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P = sparse.bmat(
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[[ zero, -fx @ Ez @ bz @ dbgy, fx @ Ey @ by @ dbgz],
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[ fy @ Ez @ bz @ dbgx, zero, -fy @ Ex @ bx @ dbgz],
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[-fz @ Ey @ by @ dbgx, fz @ Ex @ bx @ dbgy, zero]])
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return P
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def poynting_h_cross(h: vfield_t, dxes: dx_lists_t) -> sparse.spmatrix:
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"""
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Operator for computing the Poynting vector, containing the (H x) portion of the Poynting vector.
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:param h: Vectorized H-field for the HxE cross product
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:param dxes: Grid parameters [dx_e, dx_h] as described in fdfd_tools.operators header
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:return: Sparse matrix containing (H x) portion of Poynting cross product
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"""
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shape = [len(dx) for dx in dxes[0]]
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fx, fy, fz = [avgf(i, shape) for i in range(3)]
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bx, by, bz = [avgb(i, shape) for i in range(3)]
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dxbg = [dx.flatten(order='F') for dx in numpy.meshgrid(*dxes[1], indexing='ij')]
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dagx, dagy, dagz = [sparse.diags(dx.flatten(order='F'))
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for dx in numpy.meshgrid(*dxes[0], indexing='ij')]
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Hx, Hy, Hz = [sparse.diags(hi * db) for hi, db in zip(numpy.split(h, 3), dxbg)]
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n = numpy.prod(shape)
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zero = sparse.csr_matrix((n, n))
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P = sparse.bmat(
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[[ zero, -by @ Hz @ fx @ dagy, bz @ Hy @ fx @ dagz],
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[ bx @ Hz @ fy @ dagx, zero, -bz @ Hx @ fy @ dagz],
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[-bx @ Hy @ fz @ dagx, by @ Hx @ fz @ dagy, zero]])
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return P
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