150 lines
6.3 KiB
Python
150 lines
6.3 KiB
Python
"""
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Utilities for running finite-difference time-domain (FDTD) simulations
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See the discussion of `Maxwell's Equations` in `meanas.fdmath` for basic
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mathematical background.
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Timestep
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========
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From the discussion of "Plane waves and the Dispersion relation" in `meanas.fdmath`,
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we have
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$$ c^2 \\Delta_t^2 = \\frac{\\Delta_t^2}{\\mu \\epsilon} < 1/(\\frac{1}{\\Delta_x^2} + \\frac{1}{\\Delta_y^2} + \\frac{1}{\\Delta_z^2}) $$
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or, if \\( \\Delta_x = \\Delta_y = \\Delta_z \\), then \\( c \\Delta_t < \\frac{\\Delta_x}{\\sqrt{3}} \\).
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Based on this, we can set
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dt = sqrt(mu.min() * epsilon.min()) / sqrt(1/dx_min**2 + 1/dy_min**2 + 1/dz_min**2)
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The `dx_min`, `dy_min`, `dz_min` should be the minimum value across both the base and derived grids.
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Poynting Vector and Energy Conservation
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=======================================
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Let
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$$ \\begin{align*}
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\\tilde{S}_{l, l', \\vec{r}} &=& &\\tilde{E}_{l, \\vec{r}} \\otimes \\hat{H}_{l', \\vec{r} + \\frac{1}{2}} \\\\
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&=& &\\vec{x} (\\tilde{E}^y_{l,m+1,n,p} \\hat{H}^z_{l',\\vec{r} + \\frac{1}{2}} - \\tilde{E}^z_{l,m+1,n,p} \\hat{H}^y_{l', \\vec{r} + \\frac{1}{2}}) \\\\
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& &+ &\\vec{y} (\\tilde{E}^z_{l,m,n+1,p} \\hat{H}^x_{l',\\vec{r} + \\frac{1}{2}} - \\tilde{E}^x_{l,m,n+1,p} \\hat{H}^z_{l', \\vec{r} + \\frac{1}{2}}) \\\\
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& &+ &\\vec{z} (\\tilde{E}^x_{l,m,n,p+1} \\hat{H}^y_{l',\\vec{r} + \\frac{1}{2}} - \\tilde{E}^y_{l,m,n,p+1} \\hat{H}^z_{l', \\vec{r} + \\frac{1}{2}})
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\\end{align*}
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$$
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where \\( \\vec{r} = (m, n, p) \\) and \\( \\otimes \\) is a modified cross product
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in which the \\( \\tilde{E} \\) terms are shifted as indicated.
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By taking the divergence and rearranging terms, we can show that
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$$
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\\begin{align*}
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\\hat{\\nabla} \\cdot \\tilde{S}_{l, l', \\vec{r}}
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&= \\hat{\\nabla} \\cdot (\\tilde{E}_{l, \\vec{r}} \\otimes \\hat{H}_{l', \\vec{r} + \\frac{1}{2}}) \\\\
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&= \\hat{H}_{l', \\vec{r} + \\frac{1}{2}} \\cdot \\tilde{\\nabla} \\times \\tilde{E}_{l, \\vec{r}} -
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\\tilde{E}_{l, \\vec{r}} \\cdot \\hat{\\nabla} \\times \\hat{H}_{l', \\vec{r} + \\frac{1}{2}} \\\\
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&= \\hat{H}_{l', \\vec{r} + \\frac{1}{2}} \\cdot
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(-\\tilde{\\partial}_t \\mu_{\\vec{r} + \\frac{1}{2}} \\hat{H}_{l - \\frac{1}{2}, \\vec{r} + \\frac{1}{2}} -
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\\hat{M}_{l-1, \\vec{r} + \\frac{1}{2}}) -
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\\tilde{E}_{l, \\vec{r}} \\cdot (\\hat{\\partial}_t \\tilde{\\epsilon}_\\vec{r} \\tilde{E}_{l'+\\frac{1}{2}, \\vec{r}} +
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\\tilde{J}_{l', \\vec{r}}) \\\\
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&= \\hat{H}_{l'} \\cdot (-\\mu / \\Delta_t)(\\hat{H}_{l + \\frac{1}{2}} - \\hat{H}_{l - \\frac{1}{2}}) -
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\\tilde{E}_l \\cdot (\\epsilon / \\Delta_t )(\\tilde{E}_{l'+\\frac{1}{2}} - \\tilde{E}_{l'-\\frac{1}{2}})
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- \\hat{H}_{l'} \\cdot \\hat{M}_{l-1} - \\tilde{E}_l \\cdot \\tilde{J}_{l'} \\\\
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\\end{align*}
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$$
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where in the last line the spatial subscripts have been dropped to emphasize
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the time subscripts \\( l, l' \\), i.e.
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$$
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\\begin{align*}
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\\tilde{E}_l &= \\tilde{E}_{l, \\vec{r}} \\\\
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\\hat{H}_l &= \\tilde{H}_{l, \\vec{r} + \\frac{1}{2}} \\\\
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\\tilde{\\epsilon} &= \\tilde{\\epsilon}_\\vec{r} \\\\
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\\end{align*}
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$$
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etc.
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For \\( l' = l + \\frac{1}{2} \\) we get
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$$
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\\begin{align*}
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\\hat{\\nabla} \\cdot \\tilde{S}_{l, l + \\frac{1}{2}}
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&= \\hat{H}_{l + \\frac{1}{2}} \\cdot
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(-\\mu / \\Delta_t)(\\hat{H}_{l + \\frac{1}{2}} - \\hat{H}_{l - \\frac{1}{2}}) -
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\\tilde{E}_l \\cdot (\\epsilon / \\Delta_t)(\\tilde{E}_{l+1} - \\tilde{E}_l)
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- \\hat{H}_{l'} \\cdot \\hat{M}_l - \\tilde{E}_l \\cdot \\tilde{J}_{l + \\frac{1}{2}} \\\\
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&= (-\\mu / \\Delta_t)(\\hat{H}^2_{l + \\frac{1}{2}} - \\hat{H}_{l + \\frac{1}{2}} \\cdot \\hat{H}_{l - \\frac{1}{2}}) -
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(\\epsilon / \\Delta_t)(\\tilde{E}_{l+1} \\cdot \\tilde{E}_l - \\tilde{E}^2_l)
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- \\hat{H}_{l'} \\cdot \\hat{M}_l - \\tilde{E}_l \\cdot \\tilde{J}_{l + \\frac{1}{2}} \\\\
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&= -(\\mu \\hat{H}^2_{l + \\frac{1}{2}}
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+\\epsilon \\tilde{E}_{l+1} \\cdot \\tilde{E}_l) / \\Delta_t \\ \\
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+(\\mu \\hat{H}_{l + \\frac{1}{2}} \\cdot \\hat{H}_{l - \\frac{1}{2}}
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+\\epsilon \\tilde{E}^2_l) / \\Delta_t \\ \\
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- \\hat{H}_{l+\\frac{1}{2}} \\cdot \\hat{M}_l \\ \\
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- \\tilde{E}_l \\cdot \\tilde{J}_{l+\\frac{1}{2}} \\\\
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\\end{align*}
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$$
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and for \\( l' = l - \\frac{1}{2} \\),
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$$
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\\begin{align*}
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\\hat{\\nabla} \\cdot \\tilde{S}_{l, l - \\frac{1}{2}}
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&= (\\mu \\hat{H}^2_{l - \\frac{1}{2}}
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+\\epsilon \\tilde{E}_{l-1} \\cdot \\tilde{E}_l) / \\Delta_t \\ \\
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-(\\mu \\hat{H}_{l + \\frac{1}{2}} \\cdot \\hat{H}_{l - \\frac{1}{2}}
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+\\epsilon \\tilde{E}^2_l) / \\Delta_t \\ \\
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- \\hat{H}_{l-\\frac{1}{2}} \\cdot \\hat{M}_l \\ \\
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- \\tilde{E}_l \\cdot \\tilde{J}_{l-\\frac{1}{2}} \\\\
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\\end{align*}
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$$
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These two results form the discrete time-domain analogue to Poynting's theorem.
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They hint at the expressions for the energy, which can be calculated at the same
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time-index as either the E or H field:
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$$
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\\begin{align*}
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U_l &= \\epsilon \\tilde{E}^2_l + \\mu \\hat{H}_{l + \\frac{1}{2}} \\cdot \\hat{H}_{l - \\frac{1}{2}} \\\\
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U_{l + \\frac{1}{2}} &= \\epsilon \\tilde{E}_l \\cdot \\tilde{E}_{l + 1} + \\mu \\hat{H}^2_{l + \\frac{1}{2}} \\\\
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\\end{align*}
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$$
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Rewriting the Poynting theorem in terms of the energy expressions,
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$$
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\\begin{align*}
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(U_{l+\\frac{1}{2}} - U_l) / \\Delta_t
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&= -\\hat{\\nabla} \\cdot \\tilde{S}_{l, l + \\frac{1}{2}} \\ \\
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- \\hat{H}_{l+\\frac{1}{2}} \\cdot \\hat{M}_l \\ \\
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- \\tilde{E}_l \\cdot \\tilde{J}_{l+\\frac{1}{2}} \\\\
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(U_l - U_{l-\\frac{1}{2}}) / \\Delta_t
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&= -\\hat{\\nabla} \\cdot \\tilde{S}_{l, l - \\frac{1}{2}} \\ \\
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- \\hat{H}_{l-\\frac{1}{2}} \\cdot \\hat{M}_l \\ \\
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- \\tilde{E}_l \\cdot \\tilde{J}_{l-\\frac{1}{2}} \\\\
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\\end{align*}
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$$
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This result is exact an should practically hold to within numerical precision. No time-
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or spatial-averaging is necessary.
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Note that each value of \\( J \\) contributes to the energy twice (i.e. once per field update)
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despite only causing the value of \\( E \\) to change once (same for \\( M \\) and \\( H \\)).
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Boundary conditions
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===================
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# TODO notes about boundaries / PMLs
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"""
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from .base import maxwell_e, maxwell_h
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from .pml import cpml
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from .energy import (poynting, poynting_divergence, energy_hstep, energy_estep,
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delta_energy_h2e, delta_energy_h2e, delta_energy_j)
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from .boundaries import conducting_boundary
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