<li>$\mathbf{E}$, $\mathbf{H}$, $\mathbf{J}_a$, $\mathbf{M}_a$ are all rescaled by
typical electric current strength $J_0$, i.e., the strength of the
prescribed dipole source at location $a$ in the $e_i$ direction in Cartesian
-coordinates.
+coordinates (here, $\delta$ is the Dirac delta operator).
@f[
\mathbf{J}_a = J_0 e_i\delta(x-a)
@f]
\int_\Omega (\mu_r^{-1}\nabla\times\mathbf{E})\cdot (\nabla\times\bar{\varphi})\;\text{d}x
- \int_\Omega \varepsilon_r\mathbf{E} \cdot \bar{\varphi}\;\text{d}x
- \int_\Sigma [\nu \times (\mu_r^{-1}\nabla\times\mathbf{E} +
-\mu^{-1}\mathbf{M}_a)]_{\Sigma}\cdot \bar{\varphi}_T\;\text{d}o_x
-- \int_{\partial\Omega} (\nu \times (\mu_r^{-1}\nabla\times\mathbf{E} +
+\mu^{-1}\mathbf{M}_a)]_{\Sigma}\cdot \bar{\varphi}_T\;\text{d}o_x\\
+\qquad - \int_{\partial\Omega} (\nu \times (\mu_r^{-1}\nabla\times\mathbf{E} +
\mu^{-1}\mathbf{M}_a)) \cdot \bar{\varphi}_T\;\text{d}o_x =
i\int_\Omega \mathbf{J}_a \cdot \bar{\varphi}\;\text{d}x
- \int_\Omega \mu_r^{-1}\mathbf{M}_a \cdot (\nabla \times \bar{\varphi})\;\text{d}x.
We use the subscript $T$ to denote the tangential part of the given vector
and $[\cdot]_{\Sigma}$ to denote a jump over $\Sigma$, i.e.,
@f[
- F_T = (\nu\times F)\times\nu
+ \mathbf{F}_T = (\mathbf{\nu}\times \mathbf{F})\times\mathbf{\nu}
\text{ and }
- [F]_{\Sigma}(x) = \lim\limits_{s\searrow 0}(F(x+s\nu)-F(x-s\nu))
+ [\mathbf{F}]_{\Sigma}(\mathbf{x}) = \lim\limits_{s\searrow 0}(\mathbf{F}(\mathbf{x}+s\mathbf{\nu})-\mathbf{F}(\mathbf{x}-s\mathbf{\nu}))
@f]
-for $x\in \Sigma$.
+for $\mathbf{x}\in \Sigma$.
For the computational domain $\Omega$, we introduce the absorbing boundary condition
at $\partial\Omega$, which is obtained by using a first-order approximation of
-the Silver-Müller radiation condition, truncated at $\partial\Omega$.
+the Silver-Müller radiation condition, truncated at $\partial\Omega$ @cite Monk2003.
@f[
\nu\times\mathbf{H}+\sqrt{\mu_r^{-1}\varepsilon_r}\mathbf{E}=0\qquad x\in\partial\Omega
@f]
@f[
\int_\Omega (\mu_r^{-1}\nabla\times\mathbf{E})\cdot (\nabla\times\bar{\varphi})\;\text{d}x
- \int_\Omega \varepsilon_r\mathbf{E} \cdot \bar{\varphi}\;\text{d}x
-- i\int_\Sigma (\sigma_r^{\Sigma}\mathbf{E}_T) \cdot \bar{\varphi}_T\;\text{d}o_x
-- i\int_{\partial\Omega} (\sqrt{\mu_r^{-1}\varepsilon}\mathbf{E}_T) \cdot
+- i\int_\Sigma (\sigma_r^{\Sigma}\mathbf{E}_T) \cdot \bar{\varphi}_T\;\text{d}o_x\\
+\qquad - i\int_{\partial\Omega} (\sqrt{\mu_r^{-1}\varepsilon}\mathbf{E}_T) \cdot
(\nabla\times\bar{\varphi}_T)\;\text{d}o_x.=
i\int_\Omega \mathbf{J}_a \cdot \bar{\varphi}\;\text{d}x
- \int_\Omega \mu_r^{-1}\mathbf{M}_a \cdot (\nabla \times \bar{\varphi})\;\text{d}x.
The variational form is discretized on a non-uniform quadrilateral mesh with
higher-order, curl-conforming Nédélec elements implemented by the FE_NedelecSZ
-class. This way the interface with a weak discontinuity can be aligned with or
-away from the mesh and the convergence rate is high. Specifically, we use
-second-order Nédélec elements, which under our conditions will have a
-convergence rate $\mathcal{O}(\#\text{dofs})$.
+class. This way the interface with a weak discontinuity is optimal, and we get optimal convergence rates.
Consider the finite element subspace $X_h(\Omega) \subset X(\Omega)$. Define
the matrices
@f[
-A_{ij} = \int_\Omega (\mu_r^{-1}\nabla \times \varphi_i) \cdot
- (\nabla\times\bar{\varphi}_j)\;\text{d}x
- - \int_\Omega \varepsilon_r\varphi_i \cdot \bar{\varphi}_j\;\text{d}x
- - i\int_\Sigma (\sigma_r^{\Sigma}\varphi_{i_T}) \cdot
- \bar{\varphi}_{j_T}\;\text{d}o_x
- - i\int_{\partial\Omega} (\sqrt{\mu_r^{-1}\varepsilon}\varphi_{i_T}
- \cdot (\nabla\times \bar{\varphi}_{j_T})\;\text{d}o_x,
+A_{ij} = \int_\Omega (\mu_r^{-1}\nabla \times \varphi_j) \cdot
+ (\nabla\times\bar{\varphi}_i)\;\text{d}x
+ - \int_\Omega \varepsilon_r\varphi_j \cdot \bar{\varphi}_i\;\text{d}x
+ - i\int_\Sigma (\sigma_r^{\Sigma}\varphi_{j_T}) \cdot
+ \bar{\varphi}_{i_T}\;\text{d}o_x
+ - i\int_{\partial\Omega} (\sqrt{\mu_r^{-1}\varepsilon}\varphi_{j_T})
+ \cdot (\nabla\times \bar{\varphi}_{i_T})\;\text{d}o_x,
@f]
@f[
F_i = i\int_\Omega \mathbf{J}_a \cdot \bar{\varphi_i}\;\text{d}x
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),
+ \text{Find a unique } \varphi_i \in X_h(\Omega) \text{ such that, for all } \varphi_j \in X_h(\Omega),
@f]
@f[
A_{ij} = F_i
$\nu$ of $\Sigma$ is orthogonal to the radial direction $e_r$, which makes
$\mathbf{J}_a \equiv 0$ and $\mathbf{M}_a \equiv 0$ within the PML.
+@htmlonly
<p align="center">
<img src = "https://www.dealii.org/images/steps/developer/step-81-PML.png">
</p>
+@endhtmlonly
Introduce a change of coordinates
@f[
\bar{\sigma}^{\Sigma}_r &= C^{-1} \sigma^{\Sigma}_r B^{-1}.
@f}
-These PML transformations are implemented in our <tt>PerfectlyPatchedLayer</tt>
+These PML transformations are implemented in our <tt>PerfectlyMatchedLayer</tt>
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
// the interface between two materials. If we are at an interface, we assign
// the i^th diagonal element of the tensor to the private sigma_ value.
- // J_a is the strength and orientation of the dipole. It is a rank 1 tensor
+ // J_a is the strength and orientation of the dipole. As mentioned in the rescaling,
+ // @f[
+ // \mathbf{J}_a = J_0 e_i\delta(x-a)
+ // @f]
+ // It is a rank 1 tensor
// that depends on the private dipole_position_, dipole_radius_,
// dipole_strength_, dipole_orientation_ variables.
// Assemble the stiffness matrix and the right-hand side:
//\f{align*}{
- // A_{ij} = \int_\Omega (\mu_r^{-1}\nabla \times \varphi_i) \cdot
- // (\nabla\times\bar{\varphi}_j)\text{d}x
- // - \int_\Omega \varepsilon_r\varphi_i \cdot \bar{\varphi}_j\text{d}x
- // - i\int_\Sigma (\sigma_r^{\Sigma}(\varphi_i)_T) \cdot
- // (\bar{\varphi}_j)_T\text{do}x
- // - i\int_{\partial\Omega} (\sqrt{\mu_r^{-1}\varepsilon}(\varphi_i)_T) \cdot
- // (\nabla\times(\bar{\varphi}_j)_T)\text{d}x, \f} \f{align}{
+ // A_{ij} = \int_\Omega (\mu_r^{-1}\nabla \times \varphi_j) \cdot
+ // (\nabla\times\bar{\varphi}_i)\text{d}x
+ // - \int_\Omega \varepsilon_r\varphi_j \cdot \bar{\varphi}_i\text{d}x
+ // - i\int_\Sigma (\sigma_r^{\Sigma}(\varphi_j)_T) \cdot
+ // (\bar{\varphi}_i)_T\text{do}x
+ // - i\int_{\partial\Omega} (\sqrt{\mu_r^{-1}\varepsilon}(\varphi_j)_T) \cdot
+ // (\nabla\times(\bar{\varphi}_i)_T)\text{d}x, \f} \f{align}{
// F_i = i\int_\Omega J_a \cdot \bar{\varphi_i}\text{d}x - \int_\Omega
// \mu_r^{-1} \cdot (\nabla \times \bar{\varphi_i}) \text{d}x.
// \f}
// This is assembling the interior of the domain on the left hand side.
// So we are assembling
- // //\f{align*}{
+ // \f{align*}{
// \int_\Omega (\mu_r^{-1}\nabla \times \varphi_i) \cdot
// (\nabla\times\bar{\varphi}_j)\text{d}x
// - \int_\Omega \varepsilon_r\varphi_i \cdot \bar{\varphi}_j\text{d}x
// i\int_\Omega J_a \cdot \bar{\varphi_i}\text{d}x
// - \int_\Omega \mu_r^{-1} \cdot (\nabla \times \bar{\varphi_i}) \text{d}x.
// \f}
- // In doing so, we need test functions $\phi_i$ and $\phi_j$, and the curl
+ // In doing so, we need test functions $\varphi_i$ and $\varphi_j$, and the curl
// of these test variables. We must be careful with the signs of the
// imaginary parts of these complex test variables. Moreover, we have a
// conditional that changes the parameters if the cell is in the PML region.
projects / dealii.git / commitdiff