[docs] correct Yee calculus, source polarity, and phasor conventions
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3 changed files with 58 additions and 53 deletions
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@ -19,38 +19,39 @@ Submodules:
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================================================================
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From the "Frequency domain" section of `meanas.fdmath`, we have
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Following the "Frequency domain" section of `meanas.fdmath`, we use
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$e^{+\imath \omega t}$ time dependence. Phasors are referenced to the
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actual sample times of the staggered fields and sources:
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$$
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\begin{aligned}
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\tilde{E}_{l, \vec{r}} &= \tilde{E}_{\vec{r}} e^{-\imath \omega l \Delta_t} \\
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\tilde{H}_{l - \frac{1}{2}, \vec{r} + \frac{1}{2}} &= \tilde{H}_{\vec{r} + \frac{1}{2}} e^{-\imath \omega (l - \frac{1}{2}) \Delta_t} \\
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\tilde{J}_{l, \vec{r}} &= \tilde{J}_{\vec{r}} e^{-\imath \omega (l - \frac{1}{2}) \Delta_t} \\
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\tilde{M}_{l - \frac{1}{2}, \vec{r} + \frac{1}{2}} &= \tilde{M}_{\vec{r} + \frac{1}{2}} e^{-\imath \omega l \Delta_t} \\
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\hat{\nabla} \times (\mu^{-1}_{\vec{r} + \frac{1}{2}} \cdot \tilde{\nabla} \times \tilde{E}_{\vec{r}})
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-\Omega^2 \epsilon_{\vec{r}} \cdot \tilde{E}_{\vec{r}} &= -\imath \Omega \tilde{J}_{\vec{r}} e^{\imath \omega \Delta_t / 2} \\
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\tilde{E}_{l, \vec{r}} &= \tilde{E}_{\vec{r}} e^{+\imath \omega l \Delta_t} \\
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\hat{H}_{l - \frac{1}{2}, \vec{r} + \frac{1}{2}} &= \hat{H}_{\vec{r} + \frac{1}{2}} e^{+\imath \omega (l - \frac{1}{2}) \Delta_t} \\
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\tilde{J}_{l - \frac{1}{2}, \vec{r}} &= \tilde{J}_{\vec{r}} e^{+\imath \omega (l - \frac{1}{2}) \Delta_t} \\
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\hat{M}_{l, \vec{r} + \frac{1}{2}} &= \hat{M}_{\vec{r} + \frac{1}{2}} e^{+\imath \omega l \Delta_t} \\
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\Omega &= 2 \sin(\omega \Delta_t / 2) / \Delta_t
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\end{aligned}
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$$
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resulting in
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The difference multipliers for an integer-indexed sequence are
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$$
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\begin{aligned}
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\tilde{\partial}_t &\Rightarrow -\imath \Omega e^{-\imath \omega \Delta_t / 2}\\
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\hat{\partial}_t &\Rightarrow -\imath \Omega e^{ \imath \omega \Delta_t / 2}\\
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\tilde{\partial}_t &\Rightarrow \imath \Omega e^{ \imath \omega \Delta_t / 2}\\
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\hat{\partial}_t &\Rightarrow \imath \Omega e^{-\imath \omega \Delta_t / 2}\\
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\end{aligned}
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$$
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Maxwell's equations are then
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The half-step factors cancel when the staggered sample times are included.
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Maxwell's equations for the phasors are then
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$$
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\begin{aligned}
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\tilde{\nabla} \times \tilde{E}_{\vec{r}} &=
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\imath \Omega e^{-\imath \omega \Delta_t / 2} \hat{B}_{\vec{r} + \frac{1}{2}}
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-\imath \Omega \hat{B}_{\vec{r} + \frac{1}{2}}
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- \hat{M}_{\vec{r} + \frac{1}{2}} \\
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\hat{\nabla} \times \hat{H}_{\vec{r} + \frac{1}{2}} &=
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-\imath \Omega e^{ \imath \omega \Delta_t / 2} \tilde{D}_{\vec{r}}
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\imath \Omega \tilde{D}_{\vec{r}}
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+ \tilde{J}_{\vec{r}} \\
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\tilde{\nabla} \cdot \hat{B}_{\vec{r} + \frac{1}{2}} &= 0 \\
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\hat{\nabla} \cdot \tilde{D}_{\vec{r}} &= \rho_{\vec{r}}
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@ -61,13 +62,9 @@ With $\Delta_t \to 0$, this simplifies to
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$$
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\begin{aligned}
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\tilde{E}_{l, \vec{r}} &\to \tilde{E}_{\vec{r}} \\
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\tilde{H}_{l - \frac{1}{2}, \vec{r} + \frac{1}{2}} &\to \tilde{H}_{\vec{r} + \frac{1}{2}} \\
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\tilde{J}_{l, \vec{r}} &\to \tilde{J}_{\vec{r}} \\
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\tilde{M}_{l - \frac{1}{2}, \vec{r} + \frac{1}{2}} &\to \tilde{M}_{\vec{r} + \frac{1}{2}} \\
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\Omega &\to \omega \\
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\tilde{\partial}_t &\to -\imath \omega \\
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\hat{\partial}_t &\to -\imath \omega \\
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\tilde{\partial}_t &\to \imath \omega \\
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\hat{\partial}_t &\to \imath \omega \\
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\end{aligned}
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$$
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@ -76,14 +73,17 @@ and then
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$$
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\begin{aligned}
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\tilde{\nabla} \times \tilde{E}_{\vec{r}} &=
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\imath \omega \hat{B}_{\vec{r} + \frac{1}{2}}
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-\imath \omega \hat{B}_{\vec{r} + \frac{1}{2}}
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- \hat{M}_{\vec{r} + \frac{1}{2}} \\
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\hat{\nabla} \times \hat{H}_{\vec{r} + \frac{1}{2}} &=
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-\imath \omega \tilde{D}_{\vec{r}}
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\imath \omega \tilde{D}_{\vec{r}}
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+ \tilde{J}_{\vec{r}} \\
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\end{aligned}
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$$
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For zero magnetic current, eliminating $H$ gives the electric wave equation
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used by `operators.e_full` and `functional.e_full`:
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$$
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\hat{\nabla} \times (\mu^{-1}_{\vec{r} + \frac{1}{2}} \cdot \tilde{\nabla} \times \tilde{E}_{\vec{r}})
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-\omega^2 \epsilon_{\vec{r}} \cdot \tilde{E}_{\vec{r}} = -\imath \omega \tilde{J}_{\vec{r}} \\
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@ -116,9 +116,9 @@ Expanding to three dimensions, we can define two gradients
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$$ [\tilde{\nabla} f]_{m,n,p} = \vec{x} [\tilde{\partial}_x f]_{m + \frac{1}{2},n,p} +
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\vec{y} [\tilde{\partial}_y f]_{m,n + \frac{1}{2},p} +
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\vec{z} [\tilde{\partial}_z f]_{m,n,p + \frac{1}{2}} $$
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$$ [\hat{\nabla} f]_{m,n,p} = \vec{x} [\hat{\partial}_x f]_{m + \frac{1}{2},n,p} +
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\vec{y} [\hat{\partial}_y f]_{m,n + \frac{1}{2},p} +
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\vec{z} [\hat{\partial}_z f]_{m,n,p + \frac{1}{2}} $$
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$$ [\hat{\nabla} f]_{m,n,p} = \vec{x} [\hat{\partial}_x f]_{m - \frac{1}{2},n,p} +
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\vec{y} [\hat{\partial}_y f]_{m,n - \frac{1}{2},p} +
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\vec{z} [\hat{\partial}_z f]_{m,n,p - \frac{1}{2}} $$
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or
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@ -232,7 +232,7 @@ The two curls are then
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[\tilde{\nabla} \times \tilde{g}]_{m + \frac{1}{2}, n + \frac{1}{2}, p + \frac{1}{2}} &=
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\vec{x} (\tilde{\partial}_y g^z_{m,n,p + \frac{1}{2}} - \tilde{\partial}_z g^y_{m,n + \frac{1}{2},p}) \\
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&+ \vec{y} (\tilde{\partial}_z g^x_{m + \frac{1}{2},n,p} - \tilde{\partial}_x g^z_{m,n,p + \frac{1}{2}}) \\
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&+ \vec{z} (\tilde{\partial}_x g^y_{m,n + \frac{1}{2},p} - \tilde{\partial}_y g^z_{m + \frac{1}{2},n,p})
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&+ \vec{z} (\tilde{\partial}_x g^y_{m,n + \frac{1}{2},p} - \tilde{\partial}_y g^x_{m + \frac{1}{2},n,p})
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\end{aligned} $$
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and
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@ -314,7 +314,7 @@ The time derivatives can be expanded to form the update equations:
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[code: Maxwell's equations updates]
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H[i, j, k] -= dt * (curl_forward(E)[i, j, k] + M[t, i, j, k]) / mu[i, j, k]
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E[i, j, k] += dt * (curl_back( H)[i, j, k] + J[t, i, j, k]) / epsilon[i, j, k]
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E[i, j, k] += dt * (curl_back( H)[i, j, k] - J[t, i, j, k]) / epsilon[i, j, k]
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Note that the E-field fore-vector and H-field back-vector are offset by a half-cell, resulting
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in distinct locations for all six E- and H-field components:
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@ -384,7 +384,7 @@ $$
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\begin{aligned}
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\tilde{\nabla} \times \tilde{E}_{l,\vec{r}} &=
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-\tilde{\partial}_t \hat{B}_{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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- \hat{M}_{l, \vec{r} + \frac{1}{2}} \\
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\mu^{-1}_{\vec{r} + \frac{1}{2}} \cdot \tilde{\nabla} \times \tilde{E}_{l,\vec{r}} &=
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-\tilde{\partial}_t \hat{H}_{l-\frac{1}{2}, \vec{r} + \frac{1}{2}} \\
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\hat{\nabla} \times (\mu^{-1}_{\vec{r} + \frac{1}{2}} \cdot \tilde{\nabla} \times \tilde{E}_{l,\vec{r}}) &=
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@ -392,10 +392,10 @@ $$
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\hat{\nabla} \times (\mu^{-1}_{\vec{r} + \frac{1}{2}} \cdot \tilde{\nabla} \times \tilde{E}_{l,\vec{r}}) &=
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-\tilde{\partial}_t \hat{\nabla} \times \hat{H}_{l-\frac{1}{2}, \vec{r} + \frac{1}{2}} \\
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\hat{\nabla} \times (\mu^{-1}_{\vec{r} + \frac{1}{2}} \cdot \tilde{\nabla} \times \tilde{E}_{l,\vec{r}}) &=
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-\tilde{\partial}_t \hat{\partial}_t \epsilon_{\vec{r}} \tilde{E}_{l, \vec{r}} + \hat{\partial}_t \tilde{J}_{l-\frac{1}{2},\vec{r}} \\
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-\tilde{\partial}_t \hat{\partial}_t \epsilon_{\vec{r}} \tilde{E}_{l, \vec{r}} - \tilde{\partial}_t \tilde{J}_{l-\frac{1}{2},\vec{r}} \\
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\hat{\nabla} \times (\mu^{-1}_{\vec{r} + \frac{1}{2}} \cdot \tilde{\nabla} \times \tilde{E}_{l,\vec{r}})
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+ \tilde{\partial}_t \hat{\partial}_t \epsilon_{\vec{r}} \cdot \tilde{E}_{l, \vec{r}}
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&= \tilde{\partial}_t \tilde{J}_{l - \frac{1}{2}, \vec{r}}
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&= -\tilde{\partial}_t \tilde{J}_{l - \frac{1}{2}, \vec{r}}
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\end{aligned}
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$$
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@ -403,37 +403,42 @@ $$
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Frequency domain
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----------------
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We can substitute in a time-harmonic fields
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We use the $e^{+\imath \omega t}$ convention, matching the FDFD operators.
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Substitute time-harmonic fields with phases referenced to their actual sample times:
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$$
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\begin{aligned}
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\tilde{E}_{l, \vec{r}} &= \tilde{E}_{\vec{r}} e^{-\imath \omega l \Delta_t} \\
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\tilde{J}_{l, \vec{r}} &= \tilde{J}_{\vec{r}} e^{-\imath \omega (l - \frac{1}{2}) \Delta_t}
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\tilde{E}_{l, \vec{r}} &= \tilde{E}_{\vec{r}} e^{+\imath \omega l \Delta_t} \\
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\tilde{J}_{l-\frac{1}{2}, \vec{r}} &= \tilde{J}_{\vec{r}} e^{+\imath \omega (l - \frac{1}{2}) \Delta_t}
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\end{aligned}
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$$
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resulting in
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For an integer-indexed sequence proportional to $e^{+\imath \omega l \Delta_t}$,
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the forward and backward difference multipliers are
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$$
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\begin{aligned}
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\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}\\
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\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}\\
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\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}\\
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\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}\\
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\Omega &= 2 \sin(\omega \Delta_t / 2) / \Delta_t
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\end{aligned}
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$$
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This gives the frequency-domain wave equation,
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The half-step factors account for the shift in the derivative's sample time.
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In particular, $\tilde{\partial}_t J_{l-\frac{1}{2}} = \imath \Omega J e^{\imath \omega l \Delta_t}$.
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Thus, when the staggered fields are expressed using the phasors above, the
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frequency-domain wave equation has no extra half-step phase:
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$$
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\hat{\nabla} \times (\mu^{-1}_{\vec{r} + \frac{1}{2}} \cdot \tilde{\nabla} \times \tilde{E}_{\vec{r}})
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-\Omega^2 \epsilon_{\vec{r}} \cdot \tilde{E}_{\vec{r}} = -\imath \Omega \tilde{J}_{\vec{r}} e^{\imath \omega \Delta_t / 2} \\
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-\Omega^2 \epsilon_{\vec{r}} \cdot \tilde{E}_{\vec{r}} = -\imath \Omega \tilde{J}_{\vec{r}} \\
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$$
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Plane waves and Dispersion relation
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------------------------------------
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With uniform material distribution and no sources
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With uniform grid spacings, a uniform material distribution, and no sources
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$$
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\begin{aligned}
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@ -474,31 +479,31 @@ resulting in
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$$
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\begin{aligned}
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\tilde{\partial}_x &\Rightarrow (e^{ \imath k_x \Delta_x} - 1) / \Delta_t = \frac{-2 \imath}{\Delta_x} \sin(k_x \Delta_x / 2) e^{ \imath k_x \Delta_x / 2} = \imath K_x e^{ \imath k_x \Delta_x / 2}\\
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\hat{\partial}_x &\Rightarrow (1 - e^{-\imath k_x \Delta_x}) / \Delta_t = \frac{-2 \imath}{\Delta_x} \sin(k_x \Delta_x / 2) e^{-\imath k_x \Delta_x / 2} = \imath K_x e^{-\imath k_x \Delta_x / 2}\\
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\tilde{\partial}_x &\Rightarrow (e^{ \imath k_x \Delta_x} - 1) / \Delta_x = \frac{2 \imath}{\Delta_x} \sin(k_x \Delta_x / 2) e^{ \imath k_x \Delta_x / 2} = \imath K_x e^{ \imath k_x \Delta_x / 2}\\
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\hat{\partial}_x &\Rightarrow (1 - e^{-\imath k_x \Delta_x}) / \Delta_x = \frac{2 \imath}{\Delta_x} \sin(k_x \Delta_x / 2) e^{-\imath k_x \Delta_x / 2} = \imath K_x e^{-\imath k_x \Delta_x / 2}\\
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K_x &= 2 \sin(k_x \Delta_x / 2) / \Delta_x \\
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\end{aligned}
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$$
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with similar expressions for the y and z dimnsions (and $K_y, K_z$).
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with similar expressions for the y and z dimensions (and $K_y, K_z$).
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This implies
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$$
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\tilde{\nabla}^2 = -(K_x^2 + K_y^2 + K_z^2) \phi_{\vec{r}} \\
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\tilde{\nabla}^2 \phi_{\vec{r}} = -(K_x^2 + K_y^2 + K_z^2) \phi_{\vec{r}} \\
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K_x^2 + K_y^2 + K_z^2 = \Omega^2 \mu \epsilon = \Omega^2 / c^2
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$$
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where $c = \sqrt{\mu \epsilon}$.
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where $c = 1 / \sqrt{\mu \epsilon}$.
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Assuming real $(k_x, k_y, k_z), \omega$ will be real only if
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To keep $\omega$ real for every real grid wavevector $(k_x, k_y, k_z)$,
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we use the conservative stability bound
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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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If $\Delta_x = \Delta_y = \Delta_z$, this simplifies to $c \Delta_t < \Delta_x / \sqrt{3}$.
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This last form can be interpreted as enforcing causality; the distance that light
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travels in one timestep (i.e., $c \Delta_t$) must be less than the diagonal
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of the smallest cell ( $\Delta_x / \sqrt{3}$ when on a uniform cubic grid).
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This limits the distance that light travels in one timestep to less than
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$\Delta_x / \sqrt{3}$ on a uniform cubic grid.
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Grid description
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@ -17,13 +17,13 @@ def maxwell_e(
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"""
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Build a function which performs a portion the time-domain E-field update,
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E += curl_back(H[t]) / epsilon
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E += dt * curl_back(H[t]) / epsilon
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The full update should be
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E += (curl_back(H[t]) + J) / epsilon
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E += dt * (curl_back(H[t]) - J) / epsilon
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which requires an additional step of `E += J / epsilon` which is not performed
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which requires an additional step of `E -= dt * J / epsilon` which is not performed
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by the generated function.
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See `meanas.fdmath` for descriptions of
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@ -72,13 +72,13 @@ def maxwell_h(
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"""
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Build a function which performs part of the time-domain H-field update,
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H -= curl_forward(E[t]) / mu
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H -= dt * curl_forward(E[t]) / mu
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The full update should be
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H -= (curl_forward(E[t]) + M) / mu
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H -= dt * (curl_forward(E[t]) + M) / mu
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which requires an additional step of `H -= M / mu` which is not performed
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which requires an additional step of `H -= dt * M / mu` which is not performed
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by the generated function; this step can be omitted if there is no magnetic
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current `M`.
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