update fdtd and add some fdtd tests
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@ -3,6 +3,8 @@ import numpy
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from . import dx_lists_t, field_t
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#TODO fix pmls
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__author__ = 'Jan Petykiewicz'
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@ -71,7 +73,7 @@ def maxwell_e(dt: float, dxes: dx_lists_t = None) -> functional_matrix:
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curl_h_fun = curl_h(dxes)
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def me_fun(e: field_t, h: field_t, epsilon: field_t):
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e += dt * curl_h_fun(h)/ epsilon
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e += dt * curl_h_fun(h) / epsilon
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return e
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return me_fun
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@ -150,7 +152,12 @@ def cpml(direction:int,
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dt: float,
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epsilon: field_t,
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thickness: int = 8,
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ln_R_per_layer: float = -1.6,
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epsilon_eff: float = 1,
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mu_eff: float = 1,
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m: float = 3.5,
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ma: float = 1,
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cfs_alpha: float = 0,
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dtype: numpy.dtype = numpy.float32,
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) -> Tuple[Callable, Callable, Dict[str, field_t]]:
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@ -166,9 +173,9 @@ def cpml(direction:int,
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if epsilon_eff <= 0:
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raise Exception('epsilon_eff must be positive')
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m = (3.5, 1)
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sigma_max = 0.8 * (m[0] + 1) / numpy.sqrt(epsilon_eff)
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alpha_max = 0 # TODO: Decide what to do about non-zero alpha
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sigma_max = -ln_R_per_layer / 2 * (m + 1)
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kappa_max = numpy.sqrt(epsilon_eff * mu_eff)
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alpha_max = cfs_alpha
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transverse = numpy.delete(range(3), direction)
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u, v = transverse
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@ -187,14 +194,17 @@ def cpml(direction:int,
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expand_slice[direction] = slice(None)
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def par(x):
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sigma = ((x / thickness) ** m[0]) * sigma_max
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alpha = ((1 - x / thickness) ** m[1]) * alpha_max
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p0 = numpy.exp(-(sigma + alpha) * dt)
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p1 = sigma / (sigma + alpha) * (p0 - 1)
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return p0[expand_slice], p1[expand_slice]
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scaling = (x / thickness) ** m
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sigma = scaling * sigma_max
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kappa = 1 + scaling * (kappa_max - 1)
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alpha = ((1 - x / thickness) ** ma) * alpha_max
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p0 = numpy.exp(-(sigma / kappa + alpha) * dt)
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p1 = sigma / (sigma + kappa * alpha) * (p0 - 1)
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p2 = 1 / kappa
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return p0[expand_slice], p1[expand_slice], p2[expand_slice]
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p0e, p1e = par(xe)
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p0h, p1h = par(xh)
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p0e, p1e, p2e = par(xe)
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p0h, p1h, p2h = par(xh)
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region = [slice(None)] * 3
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if polarity < 0:
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@ -204,12 +214,9 @@ def cpml(direction:int,
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else:
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raise Exception('Bad polarity!')
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if direction == 1:
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se = 1
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else:
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se = -1
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se = 1 if direction == 1 else -1
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# TODO check if epsilon is uniform?
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# TODO check if epsilon is uniform in pml region?
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shape = list(epsilon[0].shape)
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shape[direction] = thickness
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psi_e = [numpy.zeros(shape, dtype=dtype), numpy.zeros(shape, dtype=dtype)]
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@ -222,37 +229,107 @@ def cpml(direction:int,
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'psi_h_v': psi_h[1],
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}
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# Note that this is kinda slow -- would be faster to reuse dHv*p2h for the original
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# H update, but then you have multiple arrays and a monolithic (field + pml) update operation
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def pml_e(e: field_t, h: field_t, epsilon: field_t) -> Tuple[field_t, field_t]:
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dHv = h[v][region] - numpy.roll(h[v], 1, axis=direction)[region]
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dHu = h[u][region] - numpy.roll(h[u], 1, axis=direction)[region]
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psi_e[0] *= p0e
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psi_e[0] += p1e * (h[v][region] - numpy.roll(h[v], 1, axis=direction)[region])
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psi_e[0] += p1e * dHv * p2e
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psi_e[1] *= p0e
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psi_e[1] += p1e * (h[u][region] - numpy.roll(h[u], 1, axis=direction)[region])
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e[u][region] += se * dt * psi_e[0] / epsilon[u][region]
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e[v][region] -= se * dt * psi_e[1] / epsilon[v][region]
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psi_e[1] += p1e * dHu * p2e
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e[u][region] += se * dt / epsilon[u][region] * (psi_e[0] + (p2e - 1) * dHv)
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e[v][region] -= se * dt / epsilon[v][region] * (psi_e[1] + (p2e - 1) * dHu)
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return e, h
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def pml_h(e: field_t, h: field_t) -> Tuple[field_t, field_t]:
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dEv = (numpy.roll(e[v], -1, axis=direction)[region] - e[v][region])
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dEu = (numpy.roll(e[u], -1, axis=direction)[region] - e[u][region])
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psi_h[0] *= p0h
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psi_h[0] += p1h * (numpy.roll(e[v], -1, axis=direction)[region] - e[v][region])
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psi_h[0] += p1h * dEv * p2h
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psi_h[1] *= p0h
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psi_h[1] += p1h * (numpy.roll(e[u], -1, axis=direction)[region] - e[u][region])
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h[u][region] -= se * dt * psi_h[0]
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h[v][region] += se * dt * psi_h[1]
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psi_h[1] += p1h * dEu * p2h
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h[u][region] -= se * dt * (psi_h[0] + (p2h - 1) * dEv)
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h[v][region] += se * dt * (psi_h[1] + (p2h - 1) * dEu)
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return e, h
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return pml_e, pml_h, fields
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def poynting(e, h):
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s = [numpy.roll(e[1], -1, axis=0) * h[2] - numpy.roll(e[2], -1, axis=0) * h[1],
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s = (numpy.roll(e[1], -1, axis=0) * h[2] - numpy.roll(e[2], -1, axis=0) * h[1],
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numpy.roll(e[2], -1, axis=1) * h[0] - numpy.roll(e[0], -1, axis=1) * h[2],
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numpy.roll(e[0], -1, axis=2) * h[1] - numpy.roll(e[1], -1, axis=2) * h[0]]
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numpy.roll(e[0], -1, axis=2) * h[1] - numpy.roll(e[1], -1, axis=2) * h[0])
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return numpy.array(s)
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def div_poyting(dt, dxes, e, h):
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s = poynting(e, h)
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ds = (s[0] - numpy.roll(s[0], 1, axis=0) +
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s[1] - numpy.roll(s[1], 1, axis=1) +
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s[2] - numpy.roll(s[2], 1, axis=2))
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def poynting_divergence(dt, dxes, s=None, *, e=None, h=None): # TODO dxes
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if s is None:
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s = poynting(e, h)
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ds = ((s[0] - numpy.roll(s[0], 1, axis=0)) / numpy.sqrt(dxes[0][0] * dxes[1][0])[:, None, None] +
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(s[1] - numpy.roll(s[1], 1, axis=1)) / numpy.sqrt(dxes[0][1] * dxes[1][1])[None, :, None] +
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(s[2] - numpy.roll(s[2], 1, axis=2)) / numpy.sqrt(dxes[0][2] * dxes[1][2])[None, None, :] )
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return ds
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def energy_hstep(e0, h1, e2, epsilon=None, mu=None, dxes=None):
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u = dxmul(e0 * e2, h1 * h1, epsilon, mu, dxes)
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return u
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def energy_estep(h0, e1, h2, epsilon=None, mu=None, dxes=None):
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u = dxmul(e1 * e1, h0 * h2, epsilon, mu, dxes)
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return u
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def delta_energy_h2e(dt, e0, h1, e2, h3, epsilon=None, mu=None, dxes=None):
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"""
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This is just from (e2 * e2 + h3 * h1) - (h1 * h1 + e0 * e2)
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"""
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de = e2 * (e2 - e0) / dt
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dh = h1 * (h3 - h1) / dt
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du = dt * dxmul(de, dh, epsilon, mu, dxes)
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return du
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def delta_energy_e2h(dt, h0, e1, h2, e3, epsilon=None, mu=None, dxes=None):
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"""
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This is just from (h2 * h2 + e3 * e1) - (e1 * e1 + h0 * h2)
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"""
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de = e1 * (e3 - e1) / dt
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dh = h2 * (h2 - h0) / dt
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du = dxmul(de, dh, epsilon, mu, dxes)
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return du
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def delta_energy_j(j0, e1, dxes=None):
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if dxes is None:
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dxes = tuple(tuple(numpy.ones(1) for _ in range(3)) for _ in range(2))
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du = ((j0 * e1).sum(axis=0) *
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dxes[0][0][:, None, None] *
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dxes[0][1][None, :, None] *
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dxes[0][2][None, None, :])
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return du
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def dxmul(ee, hh, epsilon=None, mu=None, dxes=None):
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if epsilon is None:
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epsilon = 1
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if mu is None:
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mu = 1
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if dxes is None:
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dxes = tuple(tuple(numpy.ones(1) for _ in range(3)) for _ in range(2))
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result = ((ee * epsilon).sum(axis=0) *
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dxes[0][0][:, None, None] *
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dxes[0][1][None, :, None] *
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dxes[0][2][None, None, :] +
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(hh * mu).sum(axis=0) *
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dxes[1][0][:, None, None] *
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dxes[1][1][None, :, None] *
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dxes[1][2][None, None, :])
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return result
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68
fdfd_tools/test_fdtd.py
Normal file
68
fdfd_tools/test_fdtd.py
Normal file
@ -0,0 +1,68 @@
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import unittest
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import numpy
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from fdfd_tools import fdtd
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class TestBasic2D(unittest.TestCase):
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def setUp(self):
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shape = [3, 5, 5, 1]
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dt = 0.5
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epsilon = numpy.ones(shape, dtype=float)
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src_mask = numpy.zeros_like(epsilon, dtype=bool)
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src_mask[1, 2, 2, 0] = True
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e = numpy.zeros_like(epsilon)
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h = numpy.zeros_like(epsilon)
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e[src_mask] = 32
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es = [e]
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hs = [h]
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eh2h = fdtd.maxwell_h(dt=dt)
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eh2e = fdtd.maxwell_e(dt=dt)
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for _ in range(9):
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e = e.copy()
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h = h.copy()
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eh2h(e, h)
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eh2e(e, h, epsilon)
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es.append(e)
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hs.append(h)
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self.es = es
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self.hs = hs
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self.dt = dt
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self.epsilon = epsilon
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self.src_mask = src_mask
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def test_initial_fields(self):
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# Make sure initial fields didn't change
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e0 = self.es[0]
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h0 = self.hs[0]
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self.assertEqual(e0[1, 2, 2, 0], 32)
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self.assertFalse(e0[~self.src_mask].any())
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self.assertFalse(h0.any())
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def test_initial_energy(self):
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e0 = self.es[0]
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h0 = self.hs[0]
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h1 = self.hs[1]
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mask = self.src_mask[1]
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# Make sure initial energy and E dot J are correct
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energy0 = fdtd.energy_estep(h0=h0, e1=e0, h2=self.hs[1])
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e_dot_j_0 = fdtd.delta_energy_j(j0=e0 - 0, e1=e0)
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self.assertEqual(energy0[mask], 32 * 32)
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self.assertFalse(energy0[~mask].any())
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self.assertEqual(e_dot_j_0[mask], 32 * 32)
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self.assertFalse(e_dot_j_0[~mask].any())
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def test_energy_conservation(self):
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for ii in range(1, 8):
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with self.subTest(i=ii):
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u_estep = fdtd.energy_estep(h0=self.hs[ii], e1=self.es[ii], h2=self.hs[ii + 1])
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u_hstep = fdtd.energy_hstep(e0=self.es[ii-1], h1=self.hs[ii], e2=self.es[ii])
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self.assertTrue(numpy.allclose(u_estep.sum(), 32 * 32))
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self.assertTrue(numpy.allclose(u_hstep.sum(), 32 * 32))
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