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fdfd_tools/fdfd_tools/fdtd.py

240 lines
7.1 KiB
Python

from typing import List, Callable, Tuple, Dict
import numpy
from . import dx_lists_t, field_t
__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:
e = [dh(h[2], 1) - dh(h[1], 2),
dh(h[0], 2) - dh(h[2], 0),
dh(h[1], 0) - dh(h[0], 1)]
return e
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:
h = [de(e[2], 1) - de(e[1], 2),
de(e[0], 2) - de(e[2], 0),
de(e[1], 0) - de(e[0], 1)]
return h
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):
ch = curl_h_fun(h)
for ei, ci, epsi in zip(e, ch, epsilon):
ei += dt * ci / epsi
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):
ce = curl_e_fun(e)
for hi, ci in zip(h, ce):
hi -= dt * ci
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,
epsilon_eff: float = 1,
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')
m = (3.5, 1)
sigma_max = 0.8 * (m[0] + 1) / numpy.sqrt(epsilon_eff)
alpha_max = 0 # TODO: Decide what to do about non-zero 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):
sigma = ((x / thickness) ** m[0]) * sigma_max
alpha = ((1 - x / thickness) ** m[1]) * alpha_max
p0 = numpy.exp(-(sigma + alpha) * dt)
p1 = sigma / (sigma + alpha) * (p0 - 1)
return p0[expand_slice], p1[expand_slice]
p0e, p1e = par(xe)
p0h, p1h = 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!')
if direction == 1:
se = 1
else:
se = -1
# TODO check if epsilon is uniform?
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],
}
def pml_e(e: field_t, h: field_t, epsilon: field_t) -> Tuple[field_t, field_t]:
psi_e[0] *= p0e
psi_e[0] += p1e * (h[v][region] - numpy.roll(h[v], 1, axis=direction)[region])
psi_e[1] *= p0e
psi_e[1] += p1e * (h[u][region] - numpy.roll(h[u], 1, axis=direction)[region])
e[u][region] += se * dt * psi_e[0] / epsilon[u][region]
e[v][region] -= se * dt * psi_e[1] / epsilon[v][region]
return e, h
def pml_h(e: field_t, h: field_t) -> Tuple[field_t, field_t]:
psi_h[0] *= p0h
psi_h[0] += p1h * (numpy.roll(e[v], -1, axis=direction)[region] - e[v][region])
psi_h[1] *= p0h
psi_h[1] += p1h * (numpy.roll(e[u], -1, axis=direction)[region] - e[u][region])
h[u][region] -= se * dt * psi_h[0]
h[v][region] += se * dt * psi_h[1]
return e, h
return pml_e, pml_h, fields