Maxwell Double Layer Operator

The Maxwell double layer operator is encountered in many time-harmonic BEM scattering formulations in electromagnetics. So far, only the 3D variant is implemented.

Definition

The operator is defined as (see, e.g., ...)

\[\bm{\mathcal{K}} \bm b = α \int_\Gamma ∇_{\!x} g_γ(\bm x,\bm y) \times \bm b(\bm y) \,\mathrm{d}\bm y\]

for a vector field $\bm{b}$ and a parameter $α$ with the free-space Green's function

\[g_{γ}(\bm x,\bm y) = \dfrac{\mathrm{e}^{-γ|x-y|}}{4π|x-y|} \,.\]

The parameters are typically $α=1$ and $γ = \mathrm{j}k$ with $k$ denoting the wavenumber and $\mathrm{j}$ the imaginary unit. As variation, the rotaded double layer operator

\[\bar{\bm{\mathcal{K}}} = \bm{n} \times \bm{\mathcal{K}} \]

can be used which simply involves the cross product with the normal vector $\bm{n}$ of the surface $\Gamma$.

As Bilinear Form

When handed to the assemble function, the operators are interpreted as the corresponding bilinear forms

\[a(\bm t, \bm b) = α ∬_{\Gamma \times \Gamma} \bm t(\bm x) ⋅ \,( ∇_{\!x} g_γ(\bm x,\bm y) \times \bm b(\bm y) ) \,\mathrm{d}\bm y \mathrm{d}\bm x\]

and

\[\bar{a}(\bm t, \bm b) = α ∬_{\Gamma \times \Gamma} \bm t(\bm x) ⋅ \, (\bm{n} \times ( ∇_{\!x} g_γ(\bm x,\bm y) \times \bm b(\bm y) )) \,\mathrm{d}\bm y \mathrm{d}\bm x\]

resulting in the matrix

\[[\bm A]_{mn} = a(\bm t_m, \bm b_n) \,.\]

API

doublelayer(;gamma)
doublelayer(;wavenumber)

As Linear Map

When handed to the potential function, the operator can be evaluated at provided points in space. Commonly, this is used in post-processing.

Far-Field

In the limit that the observation point $\bm x \rightarrow \infty$, the operator simplifies to the far-field (FF) version

\[(\bm{\mathcal{K}}_\mathrm{FF} \bm b)(\bm x) = α \bm{u}_r \times \int_\Gamma \bm{b}(\bm y) \mathrm{e}^{\mathrm{j}\bm{u}_r \cdot\, \bm{y}} \,\mathrm{d}\bm{y}\]

API

MWDoubleLayerField3D(; gamma, wavenumber)