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Poynting vector doc updates

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Jan Petykiewicz 2020-01-12 22:50:01 -08:00
commit 034f79eae6
3 changed files with 131 additions and 16 deletions
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meanas/fdmath/__init__.py
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@ -288,9 +288,9 @@ If we discretize both space (m,n,p) and time (l), Maxwell's equations become
$$ \\begin{align*}
\\tilde{\\nabla} \\times \\tilde{E}_{l,\\vec{r}} &= -\\tilde{\\partial}_t \\hat{B}_{l-\\frac{1}{2}, \\vec{r} + \\frac{1}{2}}
+ \\hat{M}_{l-1, \\vec{r} + \\frac{1}{2}} \\\\
\\hat{\\nabla} \\times \\hat{H}_{l,\\vec{r} + \\frac{1}{2}} &= \\hat{\\partial}_t \\tilde{D}_{l, \\vec{r}}
+ \\tilde{J}_{l-\\frac{1}{2},\\vec{r}} \\\\
- \\hat{M}_{l, \\vec{r} + \\frac{1}{2}} \\\\
\\hat{\\nabla} \\times \\hat{H}_{l-\\frac{1}{2},\\vec{r} + \\frac{1}{2}} &= \\hat{\\partial}_t \\tilde{D}_{l, \\vec{r}}
+ \\tilde{J}_{l-\\frac{1}{2},\\vec{r}} \\\\
\\tilde{\\nabla} \\cdot \\hat{B}_{l-\\frac{1}{2}, \\vec{r} + \\frac{1}{2}} &= 0 \\\\
\\hat{\\nabla} \\cdot \\tilde{D}_{l,\\vec{r}} &= \\rho_{l,\\vec{r}}
\\end{align*} $$
@ -311,9 +311,9 @@ and \\( \\epsilon \\) and \\( \\mu \\) are the dielectric permittivity and magne
The above is Yee's algorithm, written in a form analogous to Maxwell's equations.
The time derivatives can be expanded to form the update equations:
[code: Maxwell's equations]
H[i, j, k] -= (curl_forward(E[t])[i, j, k] - M[t, i, j, k]) / mu[i, j, k]
E[i, j, k] += (curl_back( H[t])[i, j, k] + J[t, i, j, k]) / epsilon[i, j, k]
[code: Maxwell's equations updates]
H[i, j, k] -= dt * (curl_forward(E)[i, j, k] + M[t, i, j, k]) / mu[i, j, k]
E[i, j, k] += dt * (curl_back( H)[i, j, k] + J[t, i, j, k]) / epsilon[i, j, k]
Note that the E-field fore-vector and H-field back-vector are offset by a half-cell, resulting
in distinct locations for all six E- and H-field components:
@ -383,7 +383,7 @@ $$
\\begin{align*}
\\tilde{\\nabla} \\times \\tilde{E}_{l,\\vec{r}} &=
-\\tilde{\\partial}_t \\hat{B}_{l-\\frac{1}{2}, \\vec{r} + \\frac{1}{2}}
+ \\hat{M}_{l-1, \\vec{r} + \\frac{1}{2}} \\\\
- \\hat{M}_{l-1, \\vec{r} + \\frac{1}{2}} \\\\
\\mu^{-1}_{\\vec{r} + \\frac{1}{2}} \\cdot \\tilde{\\nabla} \\times \\tilde{E}_{l,\\vec{r}} &=
-\\tilde{\\partial}_t \\hat{H}_{l-\\frac{1}{2}, \\vec{r} + \\frac{1}{2}} \\\\
\\hat{\\nabla} \\times (\\mu^{-1}_{\\vec{r} + \\frac{1}{2}} \\cdot \\tilde{\\nabla} \\times \\tilde{E}_{l,\\vec{r}}) &=
@ -488,11 +488,16 @@ $$
K_x^2 + K_y^2 + K_z^2 = \\Omega^2 \\mu \\epsilon = \\Omega^2 / c^2
$$
where \\( c = \\sqrt{\\mu \\epsilon} \\).
Assuming real \\( (k_x, k_y, k_z), \\omega \\) will be real only if
$$ 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}} \\).
If \\( \\Delta_x = \\Delta_y = \\Delta_z \\), this simplifies to \\( c \\Delta_t < \\Delta_x / \\sqrt{3} \\).
This last form can be interpreted as enforcing causality; the distance that light
travels in one timestep (i.e., \\( c \\Delta_t \\)) must be less than the diagonal
of the smallest cell ( \\( \\Delta_x / \\sqrt{3} \\) when on a uniform cubic grid).
Grid description