add fdtd and test

fdfd_tools/fdtd.py

@@ -0,0 +1,239 @@
+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