BIE for Maxwell

BIE for Maxwell#

Energy Spaces and Trace Spaces

\[\begin{split} \begin{array}{rcccccc} \textnormal{trace spaces:} &H^{\frac12}(\Gamma) & \xrightarrow{\nabla_{\Gamma}} & \boldsymbol{H}^{-\frac12}(\mathrm{curl}_{\Gamma},{\Gamma}) & \xrightarrow{\mathrm{curl}_{\Gamma}}& H^{-\frac12}({\Gamma})& \\[1ex] &\gamma_0 \Big\uparrow && \gamma_R \Big\uparrow && \gamma_{\boldsymbol n} \Big\uparrow &\\[1ex] \textnormal{energy spaces:} &H^1({\Omega}) & \xrightarrow{\nabla} & H(\mathbf{curl},{\Omega}) & \xrightarrow{\mathbf{curl}}& H(\mathrm{div},{\Omega}) & \xrightarrow{\mathrm{div}} \; L_2(\Omega) \\[1ex] &\gamma_0 \Big\downarrow && \gamma_D \Big\downarrow && \gamma_{\boldsymbol n} \Big\downarrow &\\[1ex] \textnormal{dual trace spaces:} &H^{\frac12}(\Gamma) & \xrightarrow{\mathbf{curl}_{\Gamma}} & \boldsymbol{H}^{-\frac12}(\mathrm{div}_{\Gamma},{\Gamma}) & \xrightarrow{\mathrm{div}_{\Gamma}}& H^{-\frac12}({\Gamma})& \end{array} \end{split}\]

Trace Operators

\[\begin{split} \begin{array}{r rcl } \textnormal{Dirichlet trace} \quad & \gamma_R \boldsymbol u &=& \left( \boldsymbol n \times \boldsymbol u \right) \times \boldsymbol n \\[1ex] \textnormal{rotated Dirichlet trace} \quad & \gamma_D \boldsymbol u &=& \boldsymbol n \times \boldsymbol u \\ \textnormal{Neumann trace} \quad & \gamma_N \boldsymbol u &=& \dfrac{1}{\kappa} \boldsymbol n \times \mathbf{curl}\, \boldsymbol u\,,\quad \kappa = \omega\, \sqrt{\varepsilon\,\mu} \\ \textnormal{normal trace} \quad & \gamma_{\boldsymbol n} \boldsymbol u &=& \big\langle \boldsymbol n, \boldsymbol u \big\rangle\,. \end{array} \end{split}\]

  • tangential edge projection of densities in \(H^{-\frac12}\left( \mathrm{curl}_\Gamma,\Gamma\right)\) are weakly continous

  • normal edge projection of densities in \(H^{-\frac12}\left( \mathrm{div}_\Gamma,\Gamma\right)\) are weakly continous

  • densities in \(H^{-\frac12}\left( \Gamma\right)\) are not continous

Layer Potentials

\[\begin{split} \begin{array}{rcl} \mathrm{SL}(\boldsymbol j) &=& \kappa \, \displaystyle {\int\limits_\Gamma \frac{1}{4\,\pi} \, \frac{e^{i\,\kappa\,\|x-y\|}}{\| x-y\|} \, \boldsymbol j(y)\, \mathrm{d}\sigma_y + \frac{1}{\kappa} \nabla \int\limits_\Gamma \frac{1}{4\,\pi}\, \frac{e^{i\,\kappa\,\|x-y\|}}{\| x-y\|} \, \mathrm{div}_\Gamma \boldsymbol j(y)\, \mathrm{d}\sigma_y } \\ \mathrm{DL}(\boldsymbol n \times \boldsymbol m) &=& \nabla \times \displaystyle {\int\limits_\Gamma \displaystyle{ \frac{1}{4\,\pi} \, \frac{e^{i\,\kappa\,\|x-y\|}}{\| x-y\|} } \, \boldsymbol n(y) \times \boldsymbol{m}(y)\, \mathrm{d}\sigma_y }\end{array} \end{split}\]

Maxwell equations PEC#

Let \(\Omega \in \mathbb R^3\) denote a perfect electric conductor and \(\gamma_R \boldsymbol E^i = -\boldsymbol m\) the given Dirichlet trace of an incoming time-harmonic electromagnetic signal \(\boldsymbol E^i\). The scattered electromagnetic field with components \(\boldsymbol E\) and \(\boldsymbol H\) solves the following equations in the exterior domain \(\Omega^c\):

\[\begin{split} \left\{ \begin{array}{ccl} \mathbf{curl} \, \boldsymbol H &=& -i\omega\varepsilon \boldsymbol E\,, \\ \mathbf{curl} \, \boldsymbol E &=& i\omega\mu \boldsymbol H\,, \\ \gamma_R\, \boldsymbol E &=& \boldsymbol m \end{array} \right. \end{split}\]

From here we can derive two second order equations: one for the electric field \(\boldsymbol E\) and one for the magnetic field \(\boldsymbol H\).

Maxwell Dirichlet BVP#

The electric field \(\boldsymbol E\) solves the second order equation with Dirichlet boundary conditions:

\(\left\{ \begin{array}{rcl l} \mathbf{curl} \, \mathbf{curl}\, \boldsymbol E - \kappa^2 \, \boldsymbol E &=& \boldsymbol 0, \quad &\textnormal{in } \Omega^c \subset \mathbb R^3\,,\\ \gamma_R \,\boldsymbol E &=& \boldsymbol m, \quad & \textnormal{on }\Gamma \\ \left| \mathbf{curl} \, \boldsymbol E( x) - i\,\omega\,\epsilon \, \boldsymbol E( x)\right| &=& \mathcal O\left( \displaystyle \frac{1}{| x|^2}\right), &\textnormal{for} \; |x| \to \infty\,.\end{array} \right. \)

\(\quad\quad\quad\)

1. Direct Method

\[\begin{split} \begin{array}{r rcl } \textnormal{direct ansatz } & \boldsymbol E(x) &=& \mathrm{SL}\left( \gamma_N \, \boldsymbol E\right)(x) + \mathrm{DL}\left( \gamma_D \,\boldsymbol E\right)(x) \\ \textnormal{variational formulation } & \forall \, \boldsymbol v\in H^{-\frac12}(\mathrm{div}_\Gamma, \Gamma): \; \big\langle \gamma_R \, \mathrm{SL} (\boldsymbol j), \boldsymbol v \big\rangle_{-\frac12} &=& \big\langle \boldsymbol m, \boldsymbol v\big\rangle_{-\frac12} - \big\langle \gamma_R\,\mathrm{DL}(\boldsymbol n \times \boldsymbol{m}), \boldsymbol v\big\rangle_{-\frac12} \\ \textnormal{discretisation} & \mathrm{V} \, \mathbf{j} &=& \left( \dfrac12 \mathrm{M} - \mathrm{K}\right) \,\mathbf{m} \end{array}\end{split}\]

2. Indirect Method

\[\begin{split} \begin{array}{r rcl } \textnormal{indirect ansatz} & \boldsymbol E(x) &=& \mathrm{SL}\left(\boldsymbol j^t\right)(x)\\ \textnormal{variational formulation } & \forall \, \boldsymbol v\in H^{-\frac12}(\mathrm{div}_\Gamma, \Gamma): \; \big\langle \gamma_R \, \mathrm{SL} (\boldsymbol j^t), \boldsymbol v \big\rangle_{-\frac12} &=& \big\langle \boldsymbol m, \boldsymbol v\big\rangle_{-\frac12} \\ \textnormal{discretisation} & \mathrm{V} \, \mathbf{j^t} &=& \mathrm{M} \,\mathbf{m} \end{array}\end{split}\]

Notes:

  • \(\boldsymbol j^t\) is the Neumann trace of the total electric field \(\boldsymbol E^t = \boldsymbol E + \boldsymbol E^i\).

Maxwell Neumann BVP#

The magnetic field \(\boldsymbol H\) solves the second order equation with Neumann boundary conditions:

\( \left\{ \begin{array}{rcl l} \mathbf{curl} \, \mathbf{curl}\, \boldsymbol H - \kappa^2 \, \boldsymbol H &=& \boldsymbol 0, \quad &\textnormal{in } \Omega^c \subset \mathbb R^3\,,\\ \gamma_N \,\boldsymbol H &=& -\dfrac{i\omega\varepsilon}{\kappa} \, \boldsymbol n\times \boldsymbol m, \quad & \textnormal{on }\Gamma \\[1ex] \left| \mathbf{curl} \, \boldsymbol H( x) + i\,\omega\,\mu \, \boldsymbol H( x)\right| &=& \mathcal O\left( \displaystyle \frac{1}{| x|^2}\right), &\textnormal{for} \; |x| \to \infty\end{array} \right. \)

\(\quad\quad\quad\)

1. Direct Method

Representation formula:

\[\begin{split}\begin{array}{rcl} \boldsymbol H(x) &=& \mathrm{SL}\left( \gamma_N \, \boldsymbol H\right)(x) +\mathrm{DL}\left( \gamma_D\,\boldsymbol H\right)(x) \\[1ex] &=& -\dfrac{i\omega\varepsilon}{\kappa} \, \Big(\underbrace{ \kappa\, \displaystyle\int\limits_\Gamma \displaystyle{ \frac{1}{4\,\pi} \, \frac{e^{i\,\kappa\,\|x-y\|}}{\| x-y\|} } \, \boldsymbol n(y)\times \boldsymbol m(y)\, \mathrm{d}\sigma_y + \frac{1}{\kappa} \nabla \displaystyle\int\limits_\Gamma \displaystyle{ \frac{1}{4\,\pi}\, \frac{e^{i\,\kappa\,\|x-y\|}}{\| x-y\|} } \, \mathrm{curl}_\Gamma \,\boldsymbol m(y)\, \mathrm{d}\sigma_y }_{\displaystyle{\mathrm{SL}(\boldsymbol n \times \boldsymbol m)} } \Big) \\ && + \dfrac{ \kappa }{ i\omega\mu} \underbrace{ \nabla \times \displaystyle\int\limits_\Gamma \displaystyle{ \frac{1}{4\,\pi} \, \frac{e^{i\,\kappa\,\|x-y\|}}{\| x-y\|} } \, \boldsymbol{j}(y) \, \mathrm{d}\sigma_y }_{\displaystyle{ \mathrm{DL} (\boldsymbol{j}) } }\end{array}\end{split}\]

Apply rotated Dirichlet trace:

\[\begin{split}\begin{array}{c rcl} & \gamma_D \,\boldsymbol H &=& -\dfrac{i\omega\varepsilon}{\kappa} \gamma_D \,\mathrm{SL}(\boldsymbol n\times \boldsymbol m) + \dfrac{\kappa}{i\omega\mu} \gamma_D \,\mathrm{DL}( \boldsymbol j) \\[1ex] \Rightarrow & \dfrac{\kappa}{i\omega\mu}\boldsymbol j &=& -\dfrac{i\omega\varepsilon}{\kappa} \gamma_D \, \mathrm{SL}(\boldsymbol n\times \boldsymbol m) + \dfrac{\kappa}{i\omega\mu} \gamma_D \,\mathrm{DL}( \boldsymbol j) \quad \Rightarrow \quad \boldsymbol j = \gamma_D \,\mathrm{SL}( \boldsymbol n\times \boldsymbol m) + \gamma_D\, \mathrm{DL}(\boldsymbol j) \end{array}\end{split}\]

Thus,

\[\begin{split} \begin{array}{r rcl } \textnormal{direct ansatz} & \boldsymbol H(x) &=& -\dfrac{i\omega\varepsilon}{\kappa} \mathrm{SL}(\boldsymbol n \times \boldsymbol m) + \dfrac{\kappa}{i\omega\mu} \mathrm{DL}(\boldsymbol j) \\ \textnormal{variational formulation } & \forall \, \boldsymbol v\in H^{-\frac12}(\mathrm{curl}_\Gamma, \Gamma): \; \big\langle \boldsymbol v, \boldsymbol j\big\rangle_{-\frac12} - \big\langle \boldsymbol v, \gamma_D \,\mathrm{DL} (\boldsymbol j) \big\rangle_{-\frac12} &=& \big\langle \boldsymbol v, \gamma_D\, \mathrm{SL}(\boldsymbol n \times \boldsymbol{m}) \big\rangle_{-\frac12} \\ \textnormal{discretisation} & \left( \dfrac12 \mathrm{M}^\intercal + \mathrm{K}^\intercal\right) \,\mathbf{j} &=& -\mathrm D \, \mathbf m \end{array}\end{split}\]

Indiret Method

Representation formula:

\[ \boldsymbol H(x) = \mathrm{DL}\left( \gamma_R\,\boldsymbol H^t\right)(x) = \dfrac{ \kappa }{ i\omega\mu} \underbrace{ \nabla \times \displaystyle\int\limits_\Gamma \displaystyle{ \frac{1}{4\,\pi} \, \frac{e^{i\,\kappa\,\|x-y\|}}{\| x-y\|} } \, \boldsymbol{j}^t(y) \, \mathrm{d}\sigma_y }_{\displaystyle{ \mathrm{DL} (\boldsymbol{j}^t) } }\]

Apply rotated Dirichlet trace and use \(\boldsymbol j = \boldsymbol j^t - \boldsymbol j^i\):

\[\begin{split}\begin{array}{ l c rcl} \gamma_D \,\boldsymbol H = \dfrac{\kappa}{i\omega\mu} \gamma_D \,\mathrm{DL}( \boldsymbol j^t) \quad &\Rightarrow & \dfrac{\kappa}{i\omega\mu}\boldsymbol j &=& \dfrac{\kappa}{i\omega\mu} \gamma_D \,\mathrm{DL}( \boldsymbol j^t) \\[2ex] &\Rightarrow &\boldsymbol j^t &=& \gamma_D \,\mathrm{DL}(\boldsymbol j^t) + \boldsymbol j^i \end{array}\end{split}\]

Thus,

\[\begin{split} \begin{array}{r rcl } \textnormal{indirect ansatz } & \boldsymbol H(x) &=& \dfrac{\kappa}{i\omega\mu} \mathrm{DL}\left(\boldsymbol j^t\right) \\ \textnormal{variational formulation } & \forall \, \boldsymbol v\in H^{-\frac12}(\mathrm{curl}_\Gamma, \Gamma): \; \left\langle \boldsymbol v, \boldsymbol j^t\right\rangle_{-\frac12} - \left\langle \boldsymbol v, \gamma_D \,\mathrm{DL} (\boldsymbol j^t) \right\rangle_{-\frac12} &=& \left\langle \boldsymbol v, \boldsymbol{j}^i \right\rangle_{-\frac12}\\ \textnormal{discretisation} & \left( \dfrac12 \mathrm{M}^\intercal + \mathrm{K}^\intercal\right) \,\mathbf{j}^t &=& \mathrm M \, \mathbf j^i \end{array}\end{split}\]

NG-BEM Operators#

The discretiszation of the boundary integral equations leads to following layer potential operators:

trial space

test space

single layer potential operator

\(H^{-\frac12}(\mathrm{div}_\Gamma,\Gamma)\)

\(H^{-\frac12}(\mathrm{div}_\Gamma,\Gamma)\)

double layer potential operator

\(H^{-\frac12}(\mathrm{curl}_\Gamma,\Gamma)\)

\(H^{-\frac12}(\mathrm{div}_\Gamma,\Gamma)\)

hypersingular operator

\(H^{-\frac12}(\mathrm{curl}_\Gamma,\Gamma)\)

\(H^{-\frac12}(\mathrm{curl}_\Gamma,\Gamma)\)

adjoint double layer potential operator

\(H^{-\frac12}(\mathrm{div}_\Gamma,\Gamma)\)

\(H^{-\frac12}(\mathrm{curl}_\Gamma,\Gamma)\)

  • NG-BEM implements the layper potential operators based on conforming finite element spaces.

  • The finite element spaces are either natural traces of energy spaces:

    • The trace space \(H^{-\frac12}(\mathrm{curl}_\Gamma, \Gamma)\) is naturally given by \(\gamma_R\) Hcurl.

    • The trace space \(H^{-\frac12}(\mathrm{div}_\Gamma, \Gamma)\) is explicitely implemented as finite element (FE) space HDivSurface.

The Python interface functions that provide the assembly of the resulting matrices for a given mesh are given in the following table:

Python interface

symbol

FE trial space

FE test space

MaxwellSingleLayerPotentialOperator

\(\mathrm V\)

HDivSurface

HDivSurface

MaxwellDoubleLayerPotentialOperator

\(\mathrm K\)

\(\gamma_R\) HCurl

HDivSurface

MaxwellSingleLayerPotentialOperatorCurl

\(\mathrm D\)

\(\gamma_R\) HCurl

\(\gamma_R\) HCurl

MaxwellDoublelayerPotentialOperator

\(\mathrm K'\)

HDivSurface

\(\gamma_R\) HCurl

Notes:

  • The indirect ansatz for Dirichlet problem is often called EFIE (electric field integral equation).

  • The indirect ansatz for Neumann problem is often called MFIE (magnetic field integral equation).

  • MFIE equation is only valid on closed boundaries whereas EFIE holds in a generalized form on open screens

  • Also the magnetic field leads to boundary integral equations for \(\boldsymbol j\) and \(\boldsymbol j^t\). The boundary integral equations are integral equations of second type.

  • For low frequencie problems it is necessary to introduce explicitly the electrostatic potential. This leads to an additional equation which is a weak form of the continuity equation relating traces on the boundary holds. Here the normal trace, i.e. the Neuman trace of the electrostatic potential pops up.

  • Scattering at arbitrary dielectrics and pec bodies leads to a system of equations which is coupled by corresponding interface conditions.

  • In case the wave number \(\kappa\) corresponds to an interior eigenvalue of \(\mathbf{curl}\mathbf{curl}\) the BIE is not uniquely solvable. Instead of EFIE one consideres the combined electric field integral equation (CFIE).

  • consider \(H^{-\frac12}(\mathrm{curl}_\Gamma,\Gamma)\) conforming finite elements for test and trial space. Here is the hypersingular operator entry \(lk\)

\[ \mathrm{D}_{lk} = \int\limits_\Gamma \int\limits_\Gamma \displaystyle{ \frac{1}{4\,\pi} \, \frac{e^{i\,\kappa\,\|x-y\|}}{\| x-y\|} } \, \langle \boldsymbol n(y)\times \boldsymbol v_l(y), \boldsymbol n(x) \times \boldsymbol \varphi_k(x)\rangle\, \mathrm{d}\sigma_y \, \mathrm{d}\sigma_x - \int\limits_\Gamma \displaystyle{ \frac{1}{4\,\pi}\, \frac{e^{i\,\kappa\,\|x-y\|}}{\| x-y\|} } \, \mathrm{curl}_\Gamma \,\boldsymbol v_l(y)\, \mathrm{curl}_\Gamma\,\boldsymbol \varphi_k(x) \mathrm{d}\sigma_y \mathrm{d}\sigma_x \]

  • consider a trial function \(\boldsymbol \psi_k \in H^{-\frac12}(\mathrm{div}_\Gamma,\Gamma)\) and a test function \(\boldsymbol v_l \in H^{-\frac12}(\mathrm{curl}_\Gamma,\Gamma)\). Here is the adjoint double layer potential operator entry \(lk\)

\[ \mathrm{K}'_{lk} = \int\limits_\Gamma \int\limits_\Gamma \displaystyle{ \frac{1}{4\,\pi} \, \big\langle \boldsymbol n(y)\times \boldsymbol v_l(y), \nabla_{x} \frac{e^{i\,\kappa\,\|x-y\|}}{\| x-y\|} } \times \boldsymbol \psi_k(y) \big\rangle\, \mathrm{d}\sigma_y \, \mathrm{d}\sigma_x \]
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