549 lines
19 KiB
Python
549 lines
19 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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meanas.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 meanas.dx_lists_type; see
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the meanas.types submodule for details.
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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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pec: vfield_t = None,
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pmc: 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 meanas.types
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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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:param pec: Vectorized mask specifying PEC cells. Any cells where pec != 0 are interpreted
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as containing a perfect electrical conductor (PEC).
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The PEC is applied per-field-component (ie, pec.size == epsilon.size)
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:param pmc: Vectorized mask specifying PMC cells. Any cells where pmc != 0 are interpreted
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as containing a perfect magnetic conductor (PMC).
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The PMC is applied per-field-component (ie, pmc.size == epsilon.size)
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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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if numpy.any(numpy.equal(pec, None)):
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pe = sparse.eye(epsilon.size)
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else:
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pe = sparse.diags(numpy.where(pec, 0, 1)) # Set pe to (not PEC)
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if numpy.any(numpy.equal(pmc, None)):
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pm = sparse.eye(epsilon.size)
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else:
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pm = sparse.diags(numpy.where(pmc, 0, 1)) # set pm to (not PMC)
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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 = pe @ (ch @ pm @ m_div @ ce - omega**2 * e) @ pe
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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 meanas.types
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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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pec: vfield_t = None,
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pmc: 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 meanas.types
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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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:param pec: Vectorized mask specifying PEC cells. Any cells where pec != 0 are interpreted
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as containing a perfect electrical conductor (PEC).
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The PEC is applied per-field-component (ie, pec.size == epsilon.size)
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:param pmc: Vectorized mask specifying PMC cells. Any cells where pmc != 0 are interpreted
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as containing a perfect magnetic conductor (PMC).
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The PMC is applied per-field-component (ie, pmc.size == epsilon.size)
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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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if numpy.any(numpy.equal(pec, None)):
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pe = sparse.eye(epsilon.size)
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else:
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pe = sparse.diags(numpy.where(pec, 0, 1)) # set pe to (not PEC)
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if numpy.any(numpy.equal(pmc, None)):
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pm = sparse.eye(epsilon.size)
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else:
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pm = sparse.diags(numpy.where(pmc, 0, 1)) # Set pe to (not PMC)
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e_div = sparse.diags(1 / epsilon)
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if mu is 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 = pm @ (ec @ pe @ e_div @ hc - omega**2 * m) @ pm
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return A
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def eh_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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pec: vfield_t = None,
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pmc: vfield_t = None
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) -> sparse.spmatrix:
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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 meanas.types
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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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:param pec: Vectorized mask specifying PEC cells. Any cells where pec != 0 are interpreted
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as containing a perfect electrical conductor (PEC).
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The PEC is applied per-field-component (i.e., pec.size == epsilon.size)
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:param pmc: Vectorized mask specifying PMC cells. Any cells where pmc != 0 are interpreted
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as containing a perfect magnetic conductor (PMC).
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The PMC is applied per-field-component (i.e., pmc.size == epsilon.size)
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:return: Sparse matrix containing the wave operator
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"""
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if numpy.any(numpy.equal(pec, None)):
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pe = sparse.eye(epsilon.size)
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else:
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pe = sparse.diags(numpy.where(pec, 0, 1)) # set pe to (not PEC)
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if numpy.any(numpy.equal(pmc, None)):
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pm = sparse.eye(epsilon.size)
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else:
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pm = sparse.diags(numpy.where(pmc, 0, 1)) # set pm to (not PMC)
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iwe = pe @ (1j * omega * sparse.diags(epsilon)) @ pe
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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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iwm = pm @ iwm @ pm
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A1 = pe @ curl_h(dxes) @ pm
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A2 = pm @ curl_e(dxes) @ pe
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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 meanas.types
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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 meanas.types
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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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pmc: 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 meanas.types
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:param mu: Vectorized magnetic permeability (default 1 everywhere)
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:param pmc: Vectorized mask specifying PMC cells. Any cells where pmc != 0 are interpreted
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as containing a perfect magnetic conductor (PMC).
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The PMC is applied per-field-component (ie, pmc.size == epsilon.size)
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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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if not numpy.any(numpy.equal(pmc, None)):
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op = sparse.diags(numpy.where(pmc, 0, 1)) @ 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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) -> sparse.spmatrix:
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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 meanas.types
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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], shift_distance: int=1) -> sparse.spmatrix:
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"""
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Utility operator for performing a circular shift along a specified axis by a
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specified number of elements.
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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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:param shift_distance: Number of cells to shift by. May be negative. Default 1.
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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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shifts = [abs(shift_distance) if a == axis else 0 for a 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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n = numpy.prod(shape)
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i_ind = numpy.arange(n)
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j_ind = numpy.ravel_multi_index(ijk, shape, order='C')
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vij = (numpy.ones(n), (i_ind, j_ind.ravel(order='C')))
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d = sparse.csr_matrix(vij, shape=(n, n))
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if shift_distance < 0:
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d = d.T
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return d
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def shift_with_mirror(axis: int, shape: List[int], shift_distance: int=1) -> sparse.spmatrix:
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"""
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Utility operator for performing an n-element shift along a specified axis, with mirror
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boundary conditions applied to the cells beyond the receding edge.
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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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:param shift_distance: Number of cells to shift by. May be negative. Default 1.
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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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if shift_distance >= shape[axis]:
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raise Exception('Shift ({}) is too large for axis {} of size {}'.format(
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shift_distance, axis, shape[axis]))
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def mirrored_range(n, s):
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v = numpy.arange(n) + s
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v = numpy.where(v >= n, 2 * n - v - 1, v)
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v = numpy.where(v < 0, - 1 - v, v)
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return v
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shifts = [shift_distance if a == axis else 0 for a in range(3)]
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shifted_diags = [mirrored_range(n, s) for n, s in zip(shape, shifts)]
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ijk = numpy.meshgrid(*shifted_diags, indexing='ij')
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n = numpy.prod(shape)
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i_ind = numpy.arange(n)
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j_ind = numpy.ravel_multi_index(ijk, shape, order='C')
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vij = (numpy.ones(n), (i_ind, j_ind.ravel(order='C')))
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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, 1) - sparse.eye(n)
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Ds = [sparse.diags(+1 / dx.ravel(order='C')) @ 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, -1) - sparse.eye(n)
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Ds = [sparse.diags(-1 / dx.ravel(order='C')) @ deriv(a)
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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, containing 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 meanas.types
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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.ravel(order='C') for dx in numpy.meshgrid(*dxes[0], indexing='ij')]
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dbgx, dbgy, dbgz = [sparse.diags(dx.ravel(order='C'))
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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]])
|
|
return P
|
|
|
|
|
|
def poynting_h_cross(h: vfield_t, dxes: dx_lists_t) -> sparse.spmatrix:
|
|
"""
|
|
Operator for computing the Poynting vector, containing the (H x) portion of the Poynting vector.
|
|
|
|
: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 meanas.types
|
|
:return: Sparse matrix containing (H x) portion of Poynting cross product
|
|
"""
|
|
shape = [len(dx) for dx in dxes[0]]
|
|
|
|
fx, fy, fz = [avgf(i, shape) for i in range(3)]
|
|
bx, by, bz = [avgb(i, shape) for i in range(3)]
|
|
|
|
dxbg = [dx.ravel(order='C') for dx in numpy.meshgrid(*dxes[1], indexing='ij')]
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|
dagx, dagy, dagz = [sparse.diags(dx.ravel(order='C'))
|
|
for dx in numpy.meshgrid(*dxes[0], indexing='ij')]
|
|
|
|
Hx, Hy, Hz = [sparse.diags(hi * db) for hi, db in zip(numpy.split(h, 3), dxbg)]
|
|
|
|
n = numpy.prod(shape)
|
|
zero = sparse.csr_matrix((n, n))
|
|
|
|
P = sparse.bmat(
|
|
[[ zero, -by @ Hz @ fx @ dagy, bz @ Hy @ fx @ dagz],
|
|
[ bx @ Hz @ fy @ dagx, zero, -bz @ Hx @ fy @ dagz],
|
|
[-bx @ Hy @ fz @ dagx, by @ Hx @ fz @ dagy, zero]])
|
|
return P
|
|
|
|
|
|
def e_tfsf_source(TF_region: vfield_t,
|
|
omega: complex,
|
|
dxes: dx_lists_t,
|
|
epsilon: vfield_t,
|
|
mu: vfield_t = None,
|
|
) -> sparse.spmatrix:
|
|
"""
|
|
Operator that turns an E-field distribution into a total-field/scattered-field
|
|
(TFSF) source.
|
|
"""
|
|
# TODO documentation
|
|
A = e_full(omega, dxes, epsilon, mu)
|
|
Q = sparse.diags(TF_region)
|
|
return (A @ Q - Q @ A) / (-1j * omega)
|
|
|
|
|
|
def e_boundary_source(mask: vfield_t,
|
|
omega: complex,
|
|
dxes: dx_lists_t,
|
|
epsilon: vfield_t,
|
|
mu: vfield_t = None,
|
|
periodic_mask_edges: bool = False,
|
|
) -> sparse.spmatrix:
|
|
"""
|
|
Operator that turns an E-field distrubtion into a current (J) distribution
|
|
along the edges (external and internal) of the provided mask. This is just an
|
|
e_tfsf_source with an additional masking step.
|
|
"""
|
|
full = e_tfsf_source(TF_region=mask, omega=omega, dxes=dxes, epsilon=epsilon, mu=mu)
|
|
|
|
shape = [len(dxe) for dxe in dxes[0]]
|
|
jmask = numpy.zeros_like(mask, dtype=bool)
|
|
|
|
if periodic_mask_edges:
|
|
shift = lambda axis, polarity: rotation(axis=axis, shape=shape, shift_distance=polarity)
|
|
else:
|
|
shift = lambda axis, polarity: shift_with_mirror(axis=axis, shape=shape, shift_distance=polarity)
|
|
|
|
for axis in (0, 1, 2):
|
|
for polarity in (-1, +1):
|
|
r = shift(axis, polarity) - sparse.eye(numpy.prod(shape)) # shifted minus original
|
|
r3 = sparse.block_diag((r, r, r))
|
|
jmask = numpy.logical_or(jmask, numpy.abs(r3 @ mask))
|
|
|
|
return sparse.diags(jmask.astype(int)) @ full
|