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Jan Petykiewicz 4 years ago
parent f69b8c9f11
commit f9d90378b4

@ -392,13 +392,108 @@ $$
-\\tilde{\\partial}_t \\hat{\\nabla} \\times \\hat{H}_{l-\\frac{1}{2}, \\vec{r} + \\frac{1}{2}} \\\\
\\hat{\\nabla} \\times (\\mu^{-1}_{\\vec{r} + \\frac{1}{2}} \\cdot \\tilde{\\nabla} \\times \\tilde{E}_{l,\\vec{r}}) &=
-\\tilde{\\partial}_t \\hat{\\partial}_t \\epsilon_\\vec{r} \\tilde{E}_{l, \\vec{r}} + \\hat{\\partial}_t \\tilde{J}_{l-\\frac{1}{2},\\vec{r}} \\\\
\\hat{\\nabla} \\times (\\mu^{-1}_{\\vec{r} + \\frac{1}{2}} \\cdot \\tilde{\\nabla} \\times \\tilde{E}_{l, \\vec{r}})
\\hat{\\nabla} \\times (\\mu^{-1}_{\\vec{r} + \\frac{1}{2}} \\cdot \\tilde{\\nabla} \\times \\tilde{E}_{l,\\vec{r}})
+ \\tilde{\\partial}_t \\hat{\\partial}_t \\epsilon_\\vec{r} \\cdot \\tilde{E}_{l, \\vec{r}}
&= \\tilde{\\partial}_t \\tilde{J}_{l - \\frac{1}{2}, \\vec{r}}
\\end{align*}
$$
Frequency domain
----------------
We can substitute in a time-harmonic fields
$$
\\begin{align*}
\\tilde{E}_\\vec{r} &= \\tilde{E}_\\vec{r} e^{-\\imath \\omega l \\Delta_t} \\\\
\\tilde{J}_\\vec{r} &= \\tilde{J}_\\vec{r} e^{-\\imath \\omega (l - \\frac{1}{2}) \\Delta_t}
\\end{align*}
$$
resulting in
$$
\\begin{align*}
\\tilde{\\partial}_t &\\Rightarrow (e^{ \\imath \\omega \\Delta_t} - 1) / \\Delta_t = \\frac{-2 \\imath}{\\Delta_t} \\sin(\\omega \\Delta_t / 2) e^{-\\imath \\omega \\Delta_t / 2} = -\\imath \\Omega e^{-\\imath \\omega \\Delta_t / 2}\\\\
\\hat{\\partial}_t &\\Rightarrow (1 - e^{-\\imath \\omega \\Delta_t}) / \\Delta_t = \\frac{-2 \\imath}{\\Delta_t} \\sin(\\omega \\Delta_t / 2) e^{ \\imath \\omega \\Delta_t / 2} = -\\imath \\Omega e^{ \\imath \\omega \\Delta_t / 2}\\\\
\\Omega &= 2 \\sin(\\omega \\Delta_t / 2) / \\Delta_t
\\end{align*}
$$
This gives the frequency-domain wave equation,
$$
\\hat{\\nabla} \\times (\\mu^{-1}_{\\vec{r} + \\frac{1}{2}} \\cdot \\tilde{\\nabla} \\times \\tilde{E}_\\vec{r})
-\\Omega^2 \\epsilon_\\vec{r} \\cdot \\tilde{E}_\\vec{r} = \\imath \\Omega \\tilde{J}_\\vec{r}
$$
Plane waves and Dispersion relation
------------------------------------
With uniform material distribution and no sources
$$
\\begin{align*}
\\mu_{\\vec{r} + \\frac{1}{2}} &= \\mu \\\\
\\epsilon_\\vec{r} &= \\epsilon \\\\
\\tilde{J}_\\vec{r} &= 0 \\\\
\\end{align*}
$$
the frequency domain wave equation simplifies to
$$ \\hat{\\nabla} \\times \\tilde{\\nabla} \\times \\tilde{E}_\\vec{r} - \\Omega^2 \\epsilon \\mu \\tilde{E}_\\vec{r} = 0 $$
Since \\( \\hat{\\nabla} \\cdot \\tilde{E}_\\vec{r} = 0 \\), we can simplify
$$
\\begin{align*}
\\hat{\\nabla} \\times \\tilde{\\nabla} \\times \\tilde{E}_\\vec{r}
&= \\tilde{\\nabla}(\\hat{\\nabla} \\cdot \\tilde{E}_\\vec{r}) - \\hat{\\nabla} \\cdot \\tilde{\\nabla} \\tilde{E}_\\vec{r} \\\\
&= - \\hat{\\nabla} \\cdot \\tilde{\\nabla} \\tilde{E}_\\vec{r} \\\\
&= - \\tilde{\\nabla}^2 \\tilde{E}_\\vec{r}
\\end{align*}
$$
and we get
$$ \\tilde{\\nabla}^2 \\tilde{E}_\\vec{r} + \\Omega^2 \\epsilon \\mu \\tilde{E}_\\vec{r} = 0 $$
We can convert this to three scalar-wave equations of the form
$$ (\\tilde{\\nabla}^2 + K^2) \\phi_\\vec{r} = 0 $$
with \\( K^2 = \\Omega^2 \\mu \\epsilon \\). Now we let
$$ \\phi_\\vec{r} = A e^{\\imath (k_x m \\Delta_x + k_y n \\Delta_y + k_z p \\Delta_z)} $$
resulting in
$$
\\begin{align*}
\\tilde{\\partial}_x &\\Rightarrow (e^{ \\imath k_x \\Delta_x} - 1) / \\Delta_t = \\frac{-2 \\imath}{\\Delta_x} \\sin(k_x \\Delta_x / 2) e^{ \\imath k_x \\Delta_x / 2} = \\imath K_x e^{ \\imath k_x \\Delta_x / 2}\\\\
\\hat{\\partial}_x &\\Rightarrow (1 - e^{-\\imath k_x \\Delta_x}) / \\Delta_t = \\frac{-2 \\imath}{\\Delta_x} \\sin(k_x \\Delta_x / 2) e^{-\\imath k_x \\Delta_x / 2} = \\imath K_x e^{-\\imath k_x \\Delta_x / 2}\\\\
K_x &= 2 \\sin(k_x \\Delta_x / 2) / \\Delta_x \\\\
\\end{align*}
$$
with similar expressions for the y and z dimnsions (and \\( K_y, K_z \\)).
This implies
$$
\\tilde{\\nabla}^2 = -(K_x^2 + K_y^2 + K_z^2) \\phi_\\vec{r} \\\\
K_x^2 + K_y^2 + K_z^2 = \\Omega^2 \\mu \\epsilon = \\Omega^2 / c^2
$$
Assuming real \\( (k_x, k_y, k_z), \\omega \\) will be real only if
$$ c^2 \\Delta_t^2 = \\frac{\\Delta_t^2}{\\mu \\epsilon} < 1/(\\frac{1}{\\Delta_x^2} + \\frac{1}{\\Delta_y^2} + \\frac{1}{\\Delta_z^2}) $$
If \\( \\Delta_x = \\Delta_y = \\Delta_z \\), this simplifies to \\( c \\Delta_t < \\frac{\\Delta_x}{\\sqrt{3}} \\).
Grid description
================
@ -598,6 +693,46 @@ H-field back-vectors:
where `dx_e[0]` is the x-width of the `m=0` cells, as used when calculating dE/dx,
and `dy_h[0]` is the y-width of the `n=0` cells, as used when calculating dH/dy, etc.
Permittivity and Permeability
=============================
Since each vector component of E and H is defined in a different location and represents
a different volume, the value of the spatially-discrete `epsilon` and `mu` can also be
different for all three field components, even when representing a simple planar interface
between two isotropic materials.
As a result, `epsilon` and `mu` are taken to have the same dimensions as the field, and
composed of the three diagonal tensor components:
[equations: epsilon_and_mu]
epsilon = [epsilon_xx, epsilon_yy, epsilon_zz]
mu = [mu_xx, mu_yy, mu_zz]
or
$$
\\epsilon = \\begin{bmatrix} \\epsilon_{xx} & 0 & 0 \\\\
0 & \\epsilon_{yy} & 0 \\\\
0 & 0 & \\epsilon_{zz} \\end{bmatrix}
$$
$$
\\mu = \\begin{bmatrix} \\mu_{xx} & 0 & 0 \\\\
0 & \\mu_{yy} & 0 \\\\
0 & 0 & \\mu_{zz} \\end{bmatrix}
$$
where the off-diagonal terms (e.g. `epsilon_xy`) are assumed to be zero.
High-accuracy volumetric integration of shapes on multiple grids can be performed
by the [gridlock](https://mpxd.net/code/jan/gridlock) module.
The values of the vacuum permittivity and permability effectively become scaling
factors that appear in several locations (e.g. between the E and H fields). In
order to limit floating-point inaccuracy and simplify calculations, they are often
set to 1 and relative permittivities and permeabilities are used in their places;
the true values can be multiplied back in after the simulation is complete if non-
normalized results are needed.
"""
from .types import fdfield_t, vfdfield_t, dx_lists_t, fdfield_updater_t

@ -37,7 +37,7 @@ dx_lists_t = List[List[numpy.ndarray]]
'''
fdfield_updater_t = Callable[[fdfield_t], fdfield_t]
fdfield_updater_t = Callable[..., fdfield_t]
'''
Convenience type for functions which take and return an fdfield_t
'''

@ -1,5 +1,35 @@
"""
Utilities for running finite-difference time-domain (FDTD) simulations
Timestep
========
From the discussion of "Plane waves and the Dispersion relation" in `meanas.fdmath`,
we have
$$ c^2 \\Delta_t^2 = \\frac{\\Delta_t^2}{\\mu \\epsilon} < 1/(\\frac{1}{\\Delta_x^2} + \\frac{1}{\\Delta_y^2} + \\frac{1}{\\Delta_z^2}) $$
or, if \\( \\Delta_x = \\Delta_y = \\Delta_z \\), then \\( c \\Delta_t < \\frac{\\Delta_x}{\\sqrt{3}} \\).
Based on this, we can set
dt = sqrt(mu.min() * epsilon.min()) / sqrt(1/dx_min**2 + 1/dy_min**2 + 1/dz_min**2)
The `dx_min`, `dy_min`, `dz_min` should be the minimum value across both the base and derived grids.
Poynting Vector
===============
# TODO
Energy conservation
===================
# TODO
Boundary conditions
===================
# TODO notes about boundaries / PMLs
"""
from .base import maxwell_e, maxwell_h

@ -1,5 +1,7 @@
"""
Basic FDTD field updates
"""
from typing import List, Callable, Tuple, Dict
import numpy
@ -12,12 +14,51 @@ __author__ = 'Jan Petykiewicz'
def maxwell_e(dt: float, dxes: dx_lists_t = None) -> fdfield_updater_t:
"""
Build a function which performs a portion the time-domain E-field update,
E += curl_back(H[t]) / epsilon
The full update should be
E += (curl_back(H[t]) + J) / epsilon
which requires an additional step of `E += J / epsilon` which is not performed
by the generated function.
See `meanas.fdmath` for descriptions of
- This update step: "Maxwell's equations" section
- `dxes`: "Datastructure: dx_lists_t" section
- `epsilon`: "Permittivity and Permeability" section
Also see the "Timestep" section of `meanas.fdtd` for a discussion of
the `dt` parameter.
Args:
dt: Timestep. See `meanas.fdtd` for details.
dxes: Grid description; see `meanas.fdmath`.
Returns:
Function `f(E_old, H_old, epsilon) -> E_new`.
"""
if dxes is not None:
curl_h_fun = curl_back(dxes[1])
else:
curl_h_fun = curl_back()
def me_fun(e: fdfield_t, h: fdfield_t, epsilon: fdfield_t):
"""
Update the E-field.
Args:
e: E-field at time t=0
h: H-field at time t=0.5
epsilon: Dielectric constant distribution.
Returns:
E-field at time t=1
"""
e += dt * curl_h_fun(h) / epsilon
return e
@ -25,13 +66,57 @@ def maxwell_e(dt: float, dxes: dx_lists_t = None) -> fdfield_updater_t:
def maxwell_h(dt: float, dxes: dx_lists_t = None) -> fdfield_updater_t:
"""
Build a function which performs part of the time-domain H-field update,
H -= curl_forward(E[t]) / mu
The full update should be
H -= (curl_forward(E[t]) - M) / mu
which requires an additional step of `H += M / mu` which is not performed
by the generated function; this step can be omitted if there is no magnetic
current `M`.
See `meanas.fdmath` for descriptions of
- This update step: "Maxwell's equations" section
- `dxes`: "Datastructure: dx_lists_t" section
- `mu`: "Permittivity and Permeability" section
Also see the "Timestep" section of `meanas.fdtd` for a discussion of
the `dt` parameter.
Args:
dt: Timestep. See `meanas.fdtd` for details.
dxes: Grid description; see `meanas.fdmath`.
Returns:
Function `f(E_old, H_old, epsilon) -> E_new`.
"""
if dxes is not None:
curl_e_fun = curl_forward(dxes[0])
else:
curl_e_fun = curl_forward()
def mh_fun(e: fdfield_t, h: fdfield_t):
h -= dt * curl_e_fun(e)
def mh_fun(e: fdfield_t, h: fdfield_t, mu: fdfield_t = None):
"""
Update the H-field.
Args:
e: E-field at time t=1
h: H-field at time t=0.5
mu: Magnetic permeability. Default is 1 everywhere.
Returns:
H-field at time t=1.5
"""
if mu is not None:
h -= dt * curl_e_fun(e) / mu
else:
h -= dt * curl_e_fun(e)
return h
return mh_fun

@ -1,5 +1,7 @@
"""
Boundary conditions
#TODO conducting boundary documentation
"""
from typing import List, Callable, Tuple, Dict

@ -5,10 +5,14 @@ import numpy
from ..fdmath import dx_lists_t, fdfield_t, fdfield_updater_t
from ..fdmath.functional import deriv_back, deriv_forward
def poynting(e: fdfield_t,
h: fdfield_t,
dxes: dx_lists_t = None,
) -> fdfield_t:
"""
Calculate the poynting vector
"""
if dxes is None:
dxes = tuple(tuple(numpy.ones(1) for _ in range(3)) for _ in range(2))
@ -32,6 +36,9 @@ def poynting_divergence(s: fdfield_t = None,
h: fdfield_t = None,
dxes: dx_lists_t = None,
) -> fdfield_t:
"""
Calculate the divergence of the poynting vector
"""
if s is None:
s = poynting(e, h, dxes=dxes)

@ -1,6 +1,9 @@
"""
PML implementations
#TODO discussion of PMLs
#TODO cpml documentation
"""
# TODO retest pmls!

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