340 lines
10 KiB
Python
340 lines
10 KiB
Python
from typing import List, Callable, Tuple, Dict
|
|
import numpy
|
|
|
|
from . import dx_lists_t, field_t
|
|
|
|
#TODO fix pmls
|
|
|
|
__author__ = 'Jan Petykiewicz'
|
|
|
|
|
|
functional_matrix = Callable[[field_t], field_t]
|
|
|
|
|
|
def curl_h(dxes: dx_lists_t = None) -> functional_matrix:
|
|
"""
|
|
Curl operator for use with the H field.
|
|
|
|
:param dxes: Grid parameters [dx_e, dx_h] as described in fdfd_tools.operators header
|
|
:return: Function for taking the discretized curl of the H-field, F(H) -> curlH
|
|
"""
|
|
if dxes:
|
|
dxyz_b = numpy.meshgrid(*dxes[1], indexing='ij')
|
|
|
|
def dh(f, ax):
|
|
return (f - numpy.roll(f, 1, axis=ax)) / dxyz_b[ax]
|
|
else:
|
|
def dh(f, ax):
|
|
return f - numpy.roll(f, 1, axis=ax)
|
|
|
|
def ch_fun(h: field_t) -> field_t:
|
|
output = numpy.empty_like(h)
|
|
output[0] = dh(h[2], 1)
|
|
output[1] = dh(h[0], 2)
|
|
output[2] = dh(h[1], 0)
|
|
output[0] -= dh(h[1], 2)
|
|
output[1] -= dh(h[2], 0)
|
|
output[2] -= dh(h[0], 1)
|
|
return output
|
|
|
|
return ch_fun
|
|
|
|
|
|
def curl_e(dxes: dx_lists_t = None) -> functional_matrix:
|
|
"""
|
|
Curl operator for use with the E field.
|
|
|
|
:param dxes: Grid parameters [dx_e, dx_h] as described in fdfd_tools.operators header
|
|
:return: Function for taking the discretized curl of the E-field, F(E) -> curlE
|
|
"""
|
|
if dxes is not None:
|
|
dxyz_a = numpy.meshgrid(*dxes[0], indexing='ij')
|
|
|
|
def de(f, ax):
|
|
return (numpy.roll(f, -1, axis=ax) - f) / dxyz_a[ax]
|
|
else:
|
|
def de(f, ax):
|
|
return numpy.roll(f, -1, axis=ax) - f
|
|
|
|
def ce_fun(e: field_t) -> field_t:
|
|
output = numpy.empty_like(e)
|
|
output[0] = de(e[2], 1)
|
|
output[1] = de(e[0], 2)
|
|
output[2] = de(e[1], 0)
|
|
output[0] -= de(e[1], 2)
|
|
output[1] -= de(e[2], 0)
|
|
output[2] -= de(e[0], 1)
|
|
return output
|
|
|
|
return ce_fun
|
|
|
|
|
|
def maxwell_e(dt: float, dxes: dx_lists_t = None) -> functional_matrix:
|
|
curl_h_fun = curl_h(dxes)
|
|
|
|
def me_fun(e: field_t, h: field_t, epsilon: field_t):
|
|
e += dt * curl_h_fun(h) / epsilon
|
|
return e
|
|
|
|
return me_fun
|
|
|
|
|
|
def maxwell_h(dt: float, dxes: dx_lists_t = None) -> functional_matrix:
|
|
curl_e_fun = curl_e(dxes)
|
|
|
|
def mh_fun(e: field_t, h: field_t):
|
|
h -= dt * curl_e_fun(e)
|
|
return h
|
|
|
|
return mh_fun
|
|
|
|
|
|
def conducting_boundary(direction: int,
|
|
polarity: int
|
|
) -> Tuple[functional_matrix, functional_matrix]:
|
|
dirs = [0, 1, 2]
|
|
if direction not in dirs:
|
|
raise Exception('Invalid direction: {}'.format(direction))
|
|
dirs.remove(direction)
|
|
u, v = dirs
|
|
|
|
if polarity < 0:
|
|
boundary_slice = [slice(None)] * 3
|
|
shifted1_slice = [slice(None)] * 3
|
|
boundary_slice[direction] = 0
|
|
shifted1_slice[direction] = 1
|
|
|
|
def en(e: field_t):
|
|
e[direction][boundary_slice] = 0
|
|
e[u][boundary_slice] = e[u][shifted1_slice]
|
|
e[v][boundary_slice] = e[v][shifted1_slice]
|
|
return e
|
|
|
|
def hn(h: field_t):
|
|
h[direction][boundary_slice] = h[direction][shifted1_slice]
|
|
h[u][boundary_slice] = 0
|
|
h[v][boundary_slice] = 0
|
|
return h
|
|
|
|
return en, hn
|
|
|
|
elif polarity > 0:
|
|
boundary_slice = [slice(None)] * 3
|
|
shifted1_slice = [slice(None)] * 3
|
|
shifted2_slice = [slice(None)] * 3
|
|
boundary_slice[direction] = -1
|
|
shifted1_slice[direction] = -2
|
|
shifted2_slice[direction] = -3
|
|
|
|
def ep(e: field_t):
|
|
e[direction][boundary_slice] = -e[direction][shifted2_slice]
|
|
e[direction][shifted1_slice] = 0
|
|
e[u][boundary_slice] = e[u][shifted1_slice]
|
|
e[v][boundary_slice] = e[v][shifted1_slice]
|
|
return e
|
|
|
|
def hp(h: field_t):
|
|
h[direction][boundary_slice] = h[direction][shifted1_slice]
|
|
h[u][boundary_slice] = -h[u][shifted2_slice]
|
|
h[u][shifted1_slice] = 0
|
|
h[v][boundary_slice] = -h[v][shifted2_slice]
|
|
h[v][shifted1_slice] = 0
|
|
return h
|
|
|
|
return ep, hp
|
|
|
|
else:
|
|
raise Exception('Bad polarity: {}'.format(polarity))
|
|
|
|
|
|
def cpml(direction:int,
|
|
polarity: int,
|
|
dt: float,
|
|
epsilon: field_t,
|
|
thickness: int = 8,
|
|
ln_R_per_layer: float = -1.6,
|
|
epsilon_eff: float = 1,
|
|
mu_eff: float = 1,
|
|
m: float = 3.5,
|
|
ma: float = 1,
|
|
cfs_alpha: float = 0,
|
|
dtype: numpy.dtype = numpy.float32,
|
|
) -> Tuple[Callable, Callable, Dict[str, field_t]]:
|
|
|
|
if direction not in range(3):
|
|
raise Exception('Invalid direction: {}'.format(direction))
|
|
|
|
if polarity not in (-1, 1):
|
|
raise Exception('Invalid polarity: {}'.format(polarity))
|
|
|
|
if thickness <= 2:
|
|
raise Exception('It would be wise to have a pml with 4+ cells of thickness')
|
|
|
|
if epsilon_eff <= 0:
|
|
raise Exception('epsilon_eff must be positive')
|
|
|
|
sigma_max = -ln_R_per_layer / 2 * (m + 1)
|
|
kappa_max = numpy.sqrt(epsilon_eff * mu_eff)
|
|
alpha_max = cfs_alpha
|
|
transverse = numpy.delete(range(3), direction)
|
|
u, v = transverse
|
|
|
|
xe = numpy.arange(1, thickness+1, dtype=float)
|
|
xh = numpy.arange(1, thickness+1, dtype=float)
|
|
if polarity > 0:
|
|
xe -= 0.5
|
|
elif polarity < 0:
|
|
xh -= 0.5
|
|
xe = xe[::-1]
|
|
xh = xh[::-1]
|
|
else:
|
|
raise Exception('Bad polarity!')
|
|
|
|
expand_slice = [None] * 3
|
|
expand_slice[direction] = slice(None)
|
|
|
|
def par(x):
|
|
scaling = (x / thickness) ** m
|
|
sigma = scaling * sigma_max
|
|
kappa = 1 + scaling * (kappa_max - 1)
|
|
alpha = ((1 - x / thickness) ** ma) * alpha_max
|
|
p0 = numpy.exp(-(sigma / kappa + alpha) * dt)
|
|
p1 = sigma / (sigma + kappa * alpha) * (p0 - 1)
|
|
p2 = 1 / kappa
|
|
return p0[expand_slice], p1[expand_slice], p2[expand_slice]
|
|
|
|
p0e, p1e, p2e = par(xe)
|
|
p0h, p1h, p2h = par(xh)
|
|
|
|
region = [slice(None)] * 3
|
|
if polarity < 0:
|
|
region[direction] = slice(None, thickness)
|
|
elif polarity > 0:
|
|
region[direction] = slice(-thickness, None)
|
|
else:
|
|
raise Exception('Bad polarity!')
|
|
|
|
se = 1 if direction == 1 else -1
|
|
|
|
# TODO check if epsilon is uniform in pml region?
|
|
shape = list(epsilon[0].shape)
|
|
shape[direction] = thickness
|
|
psi_e = [numpy.zeros(shape, dtype=dtype), numpy.zeros(shape, dtype=dtype)]
|
|
psi_h = [numpy.zeros(shape, dtype=dtype), numpy.zeros(shape, dtype=dtype)]
|
|
|
|
fields = {
|
|
'psi_e_u': psi_e[0],
|
|
'psi_e_v': psi_e[1],
|
|
'psi_h_u': psi_h[0],
|
|
'psi_h_v': psi_h[1],
|
|
}
|
|
|
|
# Note that this is kinda slow -- would be faster to reuse dHv*p2h for the original
|
|
# H update, but then you have multiple arrays and a monolithic (field + pml) update operation
|
|
def pml_e(e: field_t, h: field_t, epsilon: field_t) -> Tuple[field_t, field_t]:
|
|
dHv = h[v][region] - numpy.roll(h[v], 1, axis=direction)[region]
|
|
dHu = h[u][region] - numpy.roll(h[u], 1, axis=direction)[region]
|
|
psi_e[0] *= p0e
|
|
psi_e[0] += p1e * dHv * p2e
|
|
psi_e[1] *= p0e
|
|
psi_e[1] += p1e * dHu * p2e
|
|
e[u][region] += se * dt / epsilon[u][region] * (psi_e[0] + (p2e - 1) * dHv)
|
|
e[v][region] -= se * dt / epsilon[v][region] * (psi_e[1] + (p2e - 1) * dHu)
|
|
return e, h
|
|
|
|
def pml_h(e: field_t, h: field_t) -> Tuple[field_t, field_t]:
|
|
dEv = (numpy.roll(e[v], -1, axis=direction)[region] - e[v][region])
|
|
dEu = (numpy.roll(e[u], -1, axis=direction)[region] - e[u][region])
|
|
psi_h[0] *= p0h
|
|
psi_h[0] += p1h * dEv * p2h
|
|
psi_h[1] *= p0h
|
|
psi_h[1] += p1h * dEu * p2h
|
|
h[u][region] -= se * dt * (psi_h[0] + (p2h - 1) * dEv)
|
|
h[v][region] += se * dt * (psi_h[1] + (p2h - 1) * dEu)
|
|
return e, h
|
|
|
|
return pml_e, pml_h, fields
|
|
|
|
|
|
def poynting(e, h):
|
|
s = (numpy.roll(e[1], -1, axis=0) * h[2] - numpy.roll(e[2], -1, axis=0) * h[1],
|
|
numpy.roll(e[2], -1, axis=1) * h[0] - numpy.roll(e[0], -1, axis=1) * h[2],
|
|
numpy.roll(e[0], -1, axis=2) * h[1] - numpy.roll(e[1], -1, axis=2) * h[0])
|
|
return numpy.array(s)
|
|
|
|
|
|
def poynting_divergence(s=None, *, e=None, h=None, dxes=None): # TODO dxes
|
|
if dxes is None:
|
|
dxes = tuple(tuple(numpy.ones(1) for _ in range(3)) for _ in range(2))
|
|
|
|
if s is None:
|
|
s = poynting(e, h)
|
|
|
|
ds = ((s[0] - numpy.roll(s[0], 1, axis=0)) / numpy.sqrt(dxes[0][0] * dxes[1][0])[:, None, None] +
|
|
(s[1] - numpy.roll(s[1], 1, axis=1)) / numpy.sqrt(dxes[0][1] * dxes[1][1])[None, :, None] +
|
|
(s[2] - numpy.roll(s[2], 1, axis=2)) / numpy.sqrt(dxes[0][2] * dxes[1][2])[None, None, :] )
|
|
return ds
|
|
|
|
|
|
def energy_hstep(e0, h1, e2, epsilon=None, mu=None, dxes=None):
|
|
u = dxmul(e0 * e2, h1 * h1, epsilon, mu, dxes)
|
|
return u
|
|
|
|
|
|
def energy_estep(h0, e1, h2, epsilon=None, mu=None, dxes=None):
|
|
u = dxmul(e1 * e1, h0 * h2, epsilon, mu, dxes)
|
|
return u
|
|
|
|
|
|
def delta_energy_h2e(dt, e0, h1, e2, h3, epsilon=None, mu=None, dxes=None):
|
|
"""
|
|
This is just from (e2 * e2 + h3 * h1) - (h1 * h1 + e0 * e2)
|
|
"""
|
|
de = e2 * (e2 - e0) / dt
|
|
dh = h1 * (h3 - h1) / dt
|
|
du = dt * dxmul(de, dh, epsilon, mu, dxes)
|
|
return du
|
|
|
|
|
|
def delta_energy_e2h(dt, h0, e1, h2, e3, epsilon=None, mu=None, dxes=None):
|
|
"""
|
|
This is just from (h2 * h2 + e3 * e1) - (e1 * e1 + h0 * h2)
|
|
"""
|
|
de = e1 * (e3 - e1) / dt
|
|
dh = h2 * (h2 - h0) / dt
|
|
du = dxmul(de, dh, epsilon, mu, dxes)
|
|
return du
|
|
|
|
|
|
def delta_energy_j(j0, e1, dxes=None):
|
|
if dxes is None:
|
|
dxes = tuple(tuple(numpy.ones(1) for _ in range(3)) for _ in range(2))
|
|
|
|
du = ((j0 * e1).sum(axis=0) *
|
|
dxes[0][0][:, None, None] *
|
|
dxes[0][1][None, :, None] *
|
|
dxes[0][2][None, None, :])
|
|
return du
|
|
|
|
|
|
def dxmul(ee, hh, epsilon=None, mu=None, dxes=None):
|
|
if epsilon is None:
|
|
epsilon = 1
|
|
if mu is None:
|
|
mu = 1
|
|
if dxes is None:
|
|
dxes = tuple(tuple(numpy.ones(1) for _ in range(3)) for _ in range(2))
|
|
|
|
result = ((ee * epsilon).sum(axis=0) *
|
|
dxes[0][0][:, None, None] *
|
|
dxes[0][1][None, :, None] *
|
|
dxes[0][2][None, None, :] +
|
|
(hh * mu).sum(axis=0) *
|
|
dxes[1][0][:, None, None] *
|
|
dxes[1][1][None, :, None] *
|
|
dxes[1][2][None, None, :])
|
|
return result
|
|
|
|
|
|
|