From: David Wells Date: Fri, 25 Mar 2022 14:57:21 +0000 (-0400) Subject: Tidy up some more equations. X-Git-Tag: v9.4.0-rc1~136^2~13 X-Git-Url: https://gitweb.dealii.org/cgi-bin/gitweb.cgi?a=commitdiff_plain;h=f4f55a89af39f5a108e105e3976838338749ecb1;p=dealii.git Tidy up some more equations. --- diff --git a/examples/step-81/doc/intro.dox b/examples/step-81/doc/intro.dox index 769a94900a..74e996aa93 100644 --- a/examples/step-81/doc/intro.dox +++ b/examples/step-81/doc/intro.dox @@ -42,17 +42,15 @@ is described by Maxwell's equations @cite Schwartz1972, @cite Monk2003 : -@f[ -\begin{cases} -\frac{\partial}{\partial t} \mathbf{H} + \nabla \times \mathbf{E} = -\mathbf{M}_a, -\\ -\nabla \cdot \mathbf{H} = \rho, -\\ -\frac{\partial}{\partial t} (\varepsilon\mathbf{E}) - \nabla\times(\mu^{-1}\mathbf{H}) = - \mathbf{J}_a, -\\ -\nabla\cdot(\varepsilon\mathbf{E}) = \rho_m, -\end{cases} -@f] +@f{align*} + \frac{\partial}{\partial t} \mathbf{H} + \nabla \times \mathbf{E} &= -\mathbf{M}_a, + \\ + \nabla \cdot \mathbf{H} &= \rho, + \\ + \frac{\partial}{\partial t} (\varepsilon\mathbf{E}) - \nabla\times(\mu^{-1}\mathbf{H}) &= - \mathbf{J}_a, + \\ + \nabla\cdot(\varepsilon\mathbf{E}) &= \rho_m, +@f} in which $\nabla\times$ is the curl operator, $\nabla\cdot$ is the divergence operator, $\varepsilon$ is the electric permittivity, @@ -84,18 +82,19 @@ $\tilde{\mathbf{F}}(\mathbf{x})$ is a corresponding complex-valued vector field (or density). Inserting this ansatz into Maxwell's equations, substituting the charge conservation equations and some minor algebra then yields the so-called time-harmonic Maxwell's equations: -@f[ -\begin{cases} --i\omega \tilde{\mathbf{H}} + \nabla \times \tilde{\mathbf{E}} = --\tilde{\mathbf{M}}_a,\\ -\nabla \cdot \tilde{\mathbf{H}} = \frac{1}{i\omega}\nabla \cdot -\tilde{\mathbf{M}}_a,\\ -i\omega\varepsilon\tilde{\mathbf{E}} + -\nabla\times(\mu^{-1}\tilde{\mathbf{H}}) = \tilde{\mathbf{J}}_a,\\ -\nabla\cdot(\varepsilon\tilde{\mathbf{E}}) = -\frac{1}{i\omega}\nabla\cdot\tilde{\mathbf{J}}_a. -\end{cases} -@f] +@f{align*} + -i\omega \tilde{\mathbf{H}} + \nabla \times \tilde{\mathbf{E}} &= + -\tilde{\mathbf{M}}_a, + \\ + \nabla \cdot \tilde{\mathbf{H}} &= \frac{1}{i\omega}\nabla \cdot + \tilde{\mathbf{M}}_a, + \\ + i\omega\varepsilon\tilde{\mathbf{E}} + + \nabla\times(\mu^{-1}\tilde{\mathbf{H}}) &= \tilde{\mathbf{J}}_a, + \\ + \nabla\cdot(\varepsilon\tilde{\mathbf{E}}) &= + \frac{1}{i\omega}\nabla\cdot\tilde{\mathbf{J}}_a. +@f} For the sake of better readability we will now drop the tilde and simply write $\mathbf{E}(\mathbf{x})$, $\mathbf{H}(\mathbf{x})$, etc., when @@ -139,14 +138,13 @@ law: and then taking the limit of the upper and lower part of the line integral approaching the sheet. In contrast, the tangential part of the electric field is continuous. By fixing a unit normal $\mathbf{\nu}$ on the hypersurface -$\Sigma$ both jump conditions read, -@f[ -\begin{cases} +$\Sigma$ both jump conditions are +@f{align*} \mathbf{\nu} \times \left[(\mu^{-1}\mathbf{H})^+ - (\mu^{-1}\mathbf{H})^-\right]|_{\Sigma} -= \sigma^{\Sigma}\left[(\mathbf{\nu}\times \mathbf{E}\times \mathbf{\nu})\right]|_{\Sigma},\\ -\mathbf{\nu} \times \left[\mathbf{E}^+ - \mathbf{E}^-\right]|_{\Sigma} = 0. -\end{cases} -@f] +&= \sigma^{\Sigma}\left[(\mathbf{\nu}\times \mathbf{E}\times \mathbf{\nu})\right]|_{\Sigma}, +\\ +\mathbf{\nu} \times \left[\mathbf{E}^+ - \mathbf{E}^-\right]|_{\Sigma} &= 0. +@f} The notation $\mathbf{F}^\pm$ indicates the limit values of the field when approaching the interface from above or below the interface: $\mathbf{F}^\pm(\mathbf{x})=\lim_{\delta\to0,\delta>0}\mathbf{F}(\mathbf{x}\pm\delta\mathbf{\nu})$. @@ -198,18 +196,19 @@ Finally, the interface conductivity is rescaled as @f] Accordingly, our rescaled equations are -@f[ -\begin{cases} --i\mu_r \hat{\mathbf{H}} + \hat{\nabla} \times \hat{\mathbf{E}} -= -\hat{\mathbf{M}}_a,\\ -\hat{\nabla} \cdot (\mu_r\hat{\mathbf{H}}) = \frac{1}{i}\hat{\nabla} -\cdot \hat{\mathbf{M}}_a,\\ -i\varepsilon_r\hat{\mathbf{E}} + \nabla\times(\mu^{-1}\mathbf{H}) -= \mathbf{J}_a,\\ -\nabla\cdot(\varepsilon\mathbf{E}) = \frac{1}{i\omega}\hat{\nabla} -\cdot\hat{\mathbf{J}}_a. -\end{cases} -@f] +@f{align*} + -i\mu_r \hat{\mathbf{H}} + \hat{\nabla} \times \hat{\mathbf{E}} + &= -\hat{\mathbf{M}}_a, + \\ + \hat{\nabla} \cdot (\mu_r\hat{\mathbf{H}}) &= \frac{1}{i}\hat{\nabla} + \cdot \hat{\mathbf{M}}_a, + \\ + i\varepsilon_r\hat{\mathbf{E}} + \nabla\times(\mu^{-1}\mathbf{H}) + &= \mathbf{J}_a, + \\ + \nabla\cdot(\varepsilon\mathbf{E}) &= \frac{1}{i\omega}\hat{\nabla} + \cdot\hat{\mathbf{J}}_a. +@f} We will omit the hat in further discussion for ease of notation. @@ -314,11 +313,13 @@ F(\varphi) \dealcoloneq i\int_\Omega \mathbf{J}_a \cdot \bar{\varphi}\;\text{d}x @f] Then, our rescaled weak formulation is: -

-Find a unique $\mathbf{E} \in X(\Omega)$ such that, for all $\varphi \in X(\Omega)$, -

+ +@f[ + \text{Find a unique } \mathbf{E} \in X(\Omega) \text{ such that, for all } \varphi \in X(\Omega), +@f] + @f[ -A(\mathbf{E},\varphi) = F(\varphi). + A(\mathbf{E},\varphi) = F(\varphi). @f] @@ -368,8 +369,11 @@ F_i = i\int_\Omega \mathbf{J}_a \cdot \bar{\varphi_i}\;\text{d}x @f] Then under the assumption of a sufficiently refined initial mesh -the discretized variational problem is: Find a $\varphi_j \in X_h(\Omega)$ such -that for all $\varphi_i \in X_h(\Omega)$: +the discretized variational problem is: + +@f[ + \text{Find a unique } \varphi_j \in X_h(\Omega) \text{ such that, for all } \varphi_i \in X_h(\Omega), +@f] @f[ A_{ij} = F_i @f] @@ -407,36 +411,40 @@ x\;\;\;\;\;\;\;\;\;\text{otherwise} \end{cases} @f] -where $r = e_r \cdot x$ and $s(\tau)$ is an appropriately chosen, nonnegative -scaling function.
+in which $r = e_r \cdot x$ and $s(\tau)$ is an appropriately chosen, nonnegative +scaling function. We introduce the following $2\times2$ matrices -@f[ -A = T_{e_xe_r}^{-1} \text{diag}\left(\frac{1}{\bar{d}^2}, -\frac{1}{d\bar{d}}\right)T_{e_xe_r},\qquad -B = T_{e_xe_r}^{-1} \text{diag}\left(d,\bar{d}\right)T_{e_xe_r},\qquad -C = T_{e_xe_r}^{-1} \text{diag}\left(\frac{1}{\bar{d}},\frac{1}{d}\right) -T_{e_xe_r}.\qquad -@f] - -where - -@f[ -d = 1 + is(r),\qquad -\bar{d} = 1 + i/r \int\limits_{\rho}^{r}s(\tau)\text{d}\tau. -@f] - -and $T_{e_xe_r}$ is the rotation matrix that rotates $e_r$ onto $e_x$.
- -Thus, after applying the rescaling we get the following modified parameters -@f[ -\bar{\mu}_r^{-1} = \frac{\mu_r^{-1}}{d},\qquad -\bar{\varepsilon}_r = A^{-1} \varepsilon_r B^{-1},\qquad -\bar{\sigma}^{\Sigma}_r = C^{-1} \sigma^{\Sigma}_r B^{-1}. -@f] - -These PML transformations are implement in our PMLParameters class. After the PML +@f{align*} + A &= T_{e_xe_r}^{-1} \text{diag}\left(\frac{1}{\bar{d}^2}, + \frac{1}{d\bar{d}}\right)T_{e_xe_r} + \\ + B &= T_{e_xe_r}^{-1} \text{diag}\left(d,\bar{d}\right)T_{e_xe_r} + \\ + C &= T_{e_xe_r}^{-1} \text{diag}\left(\frac{1}{\bar{d}},\frac{1}{d}\right) + T_{e_xe_r} +@f} + +in which + +@f{align*} + d &= 1 + is(r) \\ + \bar{d} &= 1 + i/r \int\limits_{\rho}^{r}s(\tau)\text{d}\tau +@f} + +and $T_{e_xe_r}$ is the rotation matrix which rotates $e_r$ onto $e_x$. +Thus, after applying the rescaling, we get the following modified parameters +@f{align*} + \bar{\mu}_r^{-1} &= \frac{\mu_r^{-1}}{d}, + \\ + \bar{\varepsilon}_r &= A^{-1} \varepsilon_r B^{-1}, \text{ and } + \\ + \bar{\sigma}^{\Sigma}_r &= C^{-1} \sigma^{\Sigma}_r B^{-1}. +@f} + +These PML transformations are implemented in our PerfectlyPatchedLayer +class. After the PML is implemented, the electromagnetic wave essentially decays exponentially within the PML region near the boundary, therefore reducing reflection from the boundary of our finite domain. The decay function also depends on the strength of our PML, -which can be adjusted to see more or less visible decaying in the PML region.
+which can be adjusted to see more or less visible decaying in the PML region.