167 lines
6.9 KiB
Python
167 lines
6.9 KiB
Python
"""
|
|
Utilities for running finite-difference time-domain (FDTD) simulations
|
|
|
|
See the discussion of `Maxwell's Equations` in `meanas.fdmath` for basic
|
|
mathematical background.
|
|
|
|
|
|
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 and Energy Conservation
|
|
=======================================
|
|
|
|
Let
|
|
|
|
$$ \\begin{align*}
|
|
\\tilde{S}_{l, l', \\vec{r}} &=& &\\tilde{E}_{l, \\vec{r}} \\otimes \\hat{H}_{l', \\vec{r} + \\frac{1}{2}} \\\\
|
|
&=& &\\vec{x} (\\tilde{E}^y_{l,m+1,n,p} \\hat{H}^z_{l',\\vec{r} + \\frac{1}{2}} - \\tilde{E}^z_{l,m+1,n,p} \\hat{H}^y_{l', \\vec{r} + \\frac{1}{2}}) \\\\
|
|
& &+ &\\vec{y} (\\tilde{E}^z_{l,m,n+1,p} \\hat{H}^x_{l',\\vec{r} + \\frac{1}{2}} - \\tilde{E}^x_{l,m,n+1,p} \\hat{H}^z_{l', \\vec{r} + \\frac{1}{2}}) \\\\
|
|
& &+ &\\vec{z} (\\tilde{E}^x_{l,m,n,p+1} \\hat{H}^y_{l',\\vec{r} + \\frac{1}{2}} - \\tilde{E}^y_{l,m,n,p+1} \\hat{H}^z_{l', \\vec{r} + \\frac{1}{2}})
|
|
\\end{align*}
|
|
$$
|
|
|
|
where \\( \\vec{r} = (m, n, p) \\) and \\( \\otimes \\) is a modified cross product
|
|
in which the \\( \\tilde{E} \\) terms are shifted as indicated.
|
|
|
|
By taking the divergence and rearranging terms, we can show that
|
|
|
|
$$
|
|
\\begin{align*}
|
|
\\hat{\\nabla} \\cdot \\tilde{S}_{l, l', \\vec{r}}
|
|
&= \\hat{\\nabla} \\cdot (\\tilde{E}_{l, \\vec{r}} \\otimes \\hat{H}_{l', \\vec{r} + \\frac{1}{2}}) \\\\
|
|
&= \\hat{H}_{l', \\vec{r} + \\frac{1}{2}} \\cdot \\tilde{\\nabla} \\times \\tilde{E}_{l, \\vec{r}} -
|
|
\\tilde{E}_{l, \\vec{r}} \\cdot \\hat{\\nabla} \\times \\hat{H}_{l', \\vec{r} + \\frac{1}{2}} \\\\
|
|
&= \\hat{H}_{l', \\vec{r} + \\frac{1}{2}} \\cdot
|
|
(-\\tilde{\\partial}_t \\mu_{\\vec{r} + \\frac{1}{2}} \\hat{H}_{l - \\frac{1}{2}, \\vec{r} + \\frac{1}{2}} -
|
|
\\hat{M}_{l-1, \\vec{r} + \\frac{1}{2}}) -
|
|
\\tilde{E}_{l, \\vec{r}} \\cdot (\\hat{\\partial}_t \\tilde{\\epsilon}_\\vec{r} \\tilde{E}_{l'+\\frac{1}{2}, \\vec{r}} +
|
|
\\tilde{J}_{l', \\vec{r}}) \\\\
|
|
&= \\hat{H}_{l'} \\cdot (-\\mu / \\Delta_t)(\\hat{H}_{l + \\frac{1}{2}} - \\hat{H}_{l - \\frac{1}{2}}) -
|
|
\\tilde{E}_l \\cdot (\\epsilon / \\Delta_t )(\\tilde{E}_{l'+\\frac{1}{2}} - \\tilde{E}_{l'-\\frac{1}{2}})
|
|
- \\hat{H}_{l'} \\cdot \\hat{M}_{l-1} - \\tilde{E}_l \\cdot \\tilde{J}_{l'} \\\\
|
|
\\end{align*}
|
|
$$
|
|
|
|
where in the last line the spatial subscripts have been dropped to emphasize
|
|
the time subscripts \\( l, l' \\), i.e.
|
|
|
|
$$
|
|
\\begin{align*}
|
|
\\tilde{E}_l &= \\tilde{E}_{l, \\vec{r}} \\\\
|
|
\\hat{H}_l &= \\tilde{H}_{l, \\vec{r} + \\frac{1}{2}} \\\\
|
|
\\tilde{\\epsilon} &= \\tilde{\\epsilon}_\\vec{r} \\\\
|
|
\\end{align*}
|
|
$$
|
|
|
|
etc.
|
|
For \\( l' = l + \\frac{1}{2} \\) we get
|
|
|
|
$$
|
|
\\begin{align*}
|
|
\\hat{\\nabla} \\cdot \\tilde{S}_{l, l + \\frac{1}{2}}
|
|
&= \\hat{H}_{l + \\frac{1}{2}} \\cdot
|
|
(-\\mu / \\Delta_t)(\\hat{H}_{l + \\frac{1}{2}} - \\hat{H}_{l - \\frac{1}{2}}) -
|
|
\\tilde{E}_l \\cdot (\\epsilon / \\Delta_t)(\\tilde{E}_{l+1} - \\tilde{E}_l)
|
|
- \\hat{H}_{l'} \\cdot \\hat{M}_l - \\tilde{E}_l \\cdot \\tilde{J}_{l + \\frac{1}{2}} \\\\
|
|
&= (-\\mu / \\Delta_t)(\\hat{H}^2_{l + \\frac{1}{2}} - \\hat{H}_{l + \\frac{1}{2}} \\cdot \\hat{H}_{l - \\frac{1}{2}}) -
|
|
(\\epsilon / \\Delta_t)(\\tilde{E}_{l+1} \\cdot \\tilde{E}_l - \\tilde{E}^2_l)
|
|
- \\hat{H}_{l'} \\cdot \\hat{M}_l - \\tilde{E}_l \\cdot \\tilde{J}_{l + \\frac{1}{2}} \\\\
|
|
&= -(\\mu \\hat{H}^2_{l + \\frac{1}{2}}
|
|
+\\epsilon \\tilde{E}_{l+1} \\cdot \\tilde{E}_l) / \\Delta_t \\ \\
|
|
+(\\mu \\hat{H}_{l + \\frac{1}{2}} \\cdot \\hat{H}_{l - \\frac{1}{2}}
|
|
+\\epsilon \\tilde{E}^2_l) / \\Delta_t \\ \\
|
|
- \\hat{H}_{l+\\frac{1}{2}} \\cdot \\hat{M}_l \\ \\
|
|
- \\tilde{E}_l \\cdot \\tilde{J}_{l+\\frac{1}{2}} \\\\
|
|
\\end{align*}
|
|
$$
|
|
|
|
and for \\( l' = l - \\frac{1}{2} \\),
|
|
|
|
$$
|
|
\\begin{align*}
|
|
\\hat{\\nabla} \\cdot \\tilde{S}_{l, l - \\frac{1}{2}}
|
|
&= (\\mu \\hat{H}^2_{l - \\frac{1}{2}}
|
|
+\\epsilon \\tilde{E}_{l-1} \\cdot \\tilde{E}_l) / \\Delta_t \\ \\
|
|
-(\\mu \\hat{H}_{l + \\frac{1}{2}} \\cdot \\hat{H}_{l - \\frac{1}{2}}
|
|
+\\epsilon \\tilde{E}^2_l) / \\Delta_t \\ \\
|
|
- \\hat{H}_{l-\\frac{1}{2}} \\cdot \\hat{M}_l \\ \\
|
|
- \\tilde{E}_l \\cdot \\tilde{J}_{l-\\frac{1}{2}} \\\\
|
|
\\end{align*}
|
|
$$
|
|
|
|
These two results form the discrete time-domain analogue to Poynting's theorem.
|
|
They hint at the expressions for the energy, which can be calculated at the same
|
|
time-index as either the E or H field:
|
|
|
|
$$
|
|
\\begin{align*}
|
|
U_l &= \\epsilon \\tilde{E}^2_l + \\mu \\hat{H}_{l + \\frac{1}{2}} \\cdot \\hat{H}_{l - \\frac{1}{2}} \\\\
|
|
U_{l + \\frac{1}{2}} &= \\epsilon \\tilde{E}_l \\cdot \\tilde{E}_{l + 1} + \\mu \\hat{H}^2_{l + \\frac{1}{2}} \\\\
|
|
\\end{align*}
|
|
$$
|
|
|
|
Rewriting the Poynting theorem in terms of the energy expressions,
|
|
|
|
$$
|
|
\\begin{align*}
|
|
(U_{l+\\frac{1}{2}} - U_l) / \\Delta_t
|
|
&= -\\hat{\\nabla} \\cdot \\tilde{S}_{l, l + \\frac{1}{2}} \\ \\
|
|
- \\hat{H}_{l+\\frac{1}{2}} \\cdot \\hat{M}_l \\ \\
|
|
- \\tilde{E}_l \\cdot \\tilde{J}_{l+\\frac{1}{2}} \\\\
|
|
(U_l - U_{l-\\frac{1}{2}}) / \\Delta_t
|
|
&= -\\hat{\\nabla} \\cdot \\tilde{S}_{l, l - \\frac{1}{2}} \\ \\
|
|
- \\hat{H}_{l-\\frac{1}{2}} \\cdot \\hat{M}_l \\ \\
|
|
- \\tilde{E}_l \\cdot \\tilde{J}_{l-\\frac{1}{2}} \\\\
|
|
\\end{align*}
|
|
$$
|
|
|
|
This result is exact an should practically hold to within numerical precision. No time-
|
|
or spatial-averaging is necessary.
|
|
|
|
Note that each value of \\( J \\) contributes to the energy twice (i.e. once per field update)
|
|
despite only causing the value of \\( E \\) to change once (same for \\( M \\) and \\( H \\)).
|
|
|
|
|
|
Sources
|
|
=============
|
|
|
|
It is often useful to excite the simulation with an arbitrary broadband pulse and then
|
|
extract the frequency-domain response by performing an on-the-fly Fourier transform
|
|
of the time-domain fields.
|
|
|
|
The Ricker wavelet (normalized second derivative of a Gaussian) is commonly used for the pulse
|
|
shape. It can be written
|
|
|
|
$$ f_r(t) = (1 - \\frac{1}{2} (\\omega (t - \\tau))^2) e^{-(\\frac{\\omega (t - \\tau)}{2})^2} $$
|
|
|
|
with \\( \\tau > \\frac{2 * \\pi}{\\omega} \\) as a minimum delay to avoid a discontinuity at
|
|
t=0 (assuming the source is off for t<0 this gives \\( \\sim 10^{-3} \\) error at t=0).
|
|
|
|
|
|
|
|
Boundary conditions
|
|
===================
|
|
# TODO notes about boundaries / PMLs
|
|
"""
|
|
|
|
from .base import maxwell_e, maxwell_h
|
|
from .pml import cpml
|
|
from .energy import (poynting, poynting_divergence, energy_hstep, energy_estep,
|
|
delta_energy_h2e, delta_energy_h2e, delta_energy_j)
|
|
from .boundaries import conducting_boundary
|