Introduction to boundary integral equations in BEM

Boundary element method (BEM) is an effective tool compared to finite element method (FEM) for resolving those electromagnetic field problems including open domain and/or complex models with geometric details, especially those having large dimensional scale difference. Its basic idea is to construct the solution of a partial differential equation (PDE), like the 2nd order Laplace equation, by using a representation formula derived from the Green's 2nd identity. By approaching this representation formula to the domain boundary with some presumption on potential continuity, boundary integral equation can be obtained. This article explains how this equation is derived and introduces four integral operators thereof.

Fundamental solution

Let $\Omega$ be an open domain in $\mathbb{R}{^n}$ with boundary $\pdiff\Omega = \Gamma = \Gamma_D \cup \Gamma_N$ and $u$ be the electric potential such that

\begin{equation} \begin{aligned} -\Delta u(x) &= 0 \quad \forall x \in \Omega \\ u(x) &= g \quad \forall x \in \Gamma_D \\ \pdiff_{\vect{n}} u(x) &= 0 \quad \forall x \in \Gamma_N \end{aligned}. \label{eq:laplace-problem} \end{equation}

The fundamental solution to the above Laplace operator is

\begin{equation} \gamma(x) = \begin{cases} -\frac{1}{2\pi}\log\lvert x \rvert & (n = 2) \\ \frac{\lvert x \rvert^{2-n}}{(n-2)\omega_{n}} & (n > 2) \end{cases}, \label{eq:fundamental-solution} \end{equation}

where $n$ is the space dimension and $\omega_n = \frac{2\pi^{n/2}}{\Gamma(n/2)}$. The fundamental solution is the potential response caused by a source charge density with unit Dirac distribution centered at the origin.

Representation formula

The electric potential distribution $u$ in the domain $\Omega$ can be represented as a combination of double and single layer potentials as

\begin{equation} u(x) = \int_{\Gamma} \pdiff_{\vect{n}(y)}[\gamma(x,y)] \left[ u(y) \right]_{\Gamma} \intd o(y) - \int_{\Gamma} \gamma(x,y) \left[ \pdiff_{\vect{n}(y)} u(y) \right]_{\Gamma} \intd o(y) \quad (x \in \Omega), \label{eq:representation-formula} \end{equation}

where $\gamma(x, y) = \gamma(x - y)$, $\vect{n}(y)$ is the outward unit normal vector at $y \in \Gamma$, $\intd o(y)$ is the surface integral element with respect to coordinate $y$ and $[\cdot]_{\Gamma}$ represents the jump across the boundary $\Gamma$, which is defined as

$$ [u(x)]_{\Gamma} = u\big\vert^{+}_{\vect{n}(x)} - u\big\vert^{-}_{\vect{n}(x)}. $$

Remark

It can be seen that the electric potential $u$ in the domain $\Omega$ is represented as a convolution between the fundamental solution $\gamma(x)$ and source layer charges configured on the domain boundary $\Gamma$, which is the same as the convolution between an unit impulse response function and source excitation exhibited in electric circuit theory. The difference is for the electrostatic Laplace problem, the convolution is carried out in space domain, while in circuit theory it is in time domain.
Convolution implies that a system's response should be linearly dependent on the source excitation. Therefore, the total response can be given as a linear superposition of the contributions from continuously distributed sources.
Accordingly, the medium described by the PDE should be linear, homogeneous (spatial invariant) and time invariant. We should also note that if the medium's parameter is inhomogeneous but time invariant, hence the response linearly depends on a source located at a specified position. Then the fundamental solution changes its form when the source changes position. This is because the space loses symmetry.

Because the representation formula is a corner stone for BEM, BEM can only be used for linear and homogeneous medium. In addition, BEM can handle open domain problem. These two factors render BEM quite suitable for solving electromagnetic field problems with a large air box, which are usually difficult for FEM.

Boundary integral equation and integral operators

If we assume a constant zero field condition outside the domain $\Omega$, i.e. $u(x) \big\vert_{\mathbb{R}^n\backslash\Omega} \equiv 0$, which is called direct method, the representation formula becomes

\begin{equation} u(x) = -\int_{\Gamma} \pdiff_{\vect{n}(y)} \left[\gamma(x,y)\right] u(y) \intd o(y) + \int_{\Gamma} \gamma(x,y) \pdiff_{\vect{n}(y)} u(y) \intd o(y) \quad (x \in \Omega). \label{eq:representation-formula-zero-field-cond} \end{equation}

Its normal derivative is

\begin{equation} \pdiff_{\vect{n}(x)} u(x) = -\int_{\Gamma} \pdiff_{\vect{n}(x)} \left\{ \pdiff_{\vect{n}(y)}[\gamma(x,y)] \right\} u(y) \intd o(y) + \int_{\Gamma} \pdiff_{\vect{n}(x)} \left[ \gamma(x,y) \right] \pdiff_{\vect{n}(y)} u(y) \intd o(y) \quad (x \in \Omega). \label{eq:normal-derivative-formula-zero-field-cond} \end{equation}

When $u(x)$ and $\pdiff_{\vect{n}(x)} u(x)$ approach to the boundary $\Gamma_D$ and $\Gamma_N$ respectively, the Cauchy data ¹ are obtained, which specify both the function value and normal derivative on the boundary of the domain. They can be used to match the already given Dirichlet and homogeneous Neumann boundary conditions in \eqref{eq:laplace-problem} and hence the boundary integral equation can be obtained. However, before presenting its formulation, we need to clarify the behavior of single and double layer potentials near the boundary.

When approaching to the boundary, the single layer potential $$ \int_{\Gamma} \gamma(x,y) \pdiff_{\vect{n}(y)} u(y) \intd o(y) \quad (x \in \Omega) $$ in \eqref{eq:representation-formula-zero-field-cond} is continuous across the boundary $\Gamma$. For simplicity, let $t(y) = \pdiff_{\vect{n}(y)} u(y)$ and define an integral operator $V$ to represent this component as $$ Vt = (Vt(y))(x) = \int_{\Gamma} \gamma(x,y) \pdiff_{\vect{n}(y)} u(y) \intd o(y). $$

The double layer potential $$ \int_{\Gamma} \pdiff_{\vect{n}(y)} \left[\gamma(x,y)\right] u(y) \intd o(y) $$ in \eqref{eq:representation-formula-zero-field-cond} depends on from which direction, i.e. interior or exterior, it approaches to the boundary. This discontinuous behavior is governed by the following theorem.

Theorem Let $\phi \in C(\Gamma)$ be the double layer charge density and $u(x)$ be the double layer potential, which is given as $$ u(x) = \int_{\Gamma} K(x, y) \phi(y) \intd o(y) \quad (x \in \Omega), $$ where $K(x, y) = \pdiff_{\vect{n}(y)} \left[\gamma(x,y)\right]$. The restrictions of $u$ to $\Omega$ and $\Omega' = \mathbb{R}^n\backslash\Omega$ both have continuous extension to $\overline{\Omega}$ and $\overline{\Omega}'$ respectively. Then $u_{\varepsilon}(x) = u(x + \varepsilon \vect{n}(x))$ with $x \in \Gamma$ converges uniformly to $u_{-}$ and $u_{+}$ when $\varepsilon \longrightarrow 0^{-}$ and $\varepsilon \longrightarrow 0^{+}$, where

\begin{equation} \begin{aligned} u_{-}(x) &= -\frac{1}{2} \phi(x) + \int_{\Gamma} K(x, y) \phi(y) \intd o(y) \\ u_{+}(x) &= \frac{1}{2} \phi(x) + \int_{\Gamma} K(x, y) \phi(y) \intd o(y) \end{aligned} \quad (x \in \Gamma). \end{equation}

We then define the compact integral operator $T_K$ as follows, which maps a bounded function to continuous function:

\begin{equation} T_K\phi(x) = (T_K\phi(y))(x) = \int_{\Gamma} K(x, y) \phi(y) \intd o(y) \quad (x \in \Gamma). \label{eq:tk-operator} \end{equation}

For the components in the normal derivative of the representation formula in Equation \eqref{eq:normal-derivative-formula-zero-field-cond}, we introduce an integral operator $D$ with a hyper-singular kernel as $$ Du = -\int_{\Gamma} \pdiff_{\vect{n}(x)} \left\{ \pdiff_{\vect{n}(y)}[\gamma(x,y)] \right\} u(y) \intd o(y). $$ Then let $K^{*}(x, y) = \pdiff_{\vect{n}(x)} \left[\gamma(x,y)\right] $, which has the following property:

\begin{equation} K^{*}(x, y) = K(y, x) = -K(x, y). \label{eq:symmetry-of-k} \end{equation}

Let $$ \psi(x) = \int_{\Gamma} K^{*}(x, y) \phi(y) \intd o(y) \quad (x \in \Omega) $$ approach to the boundary, we have similar results as the above theorem:

\begin{equation} \begin{aligned} \psi_{-}(x) &= \frac{1}{2} \phi(x) + \int_{\Gamma} K^{*}(x, y) \phi(y) \intd o(y) \\ \psi_{+}(x) &= -\frac{1}{2} \phi(x) + \int_{\Gamma} K^{*}(x, y) \phi(y) \intd o(y) \end{aligned} \quad (x \in \Gamma). \end{equation}

Then a new compact integral operator $T_{K^{*}}$ is defined as

\begin{equation} T_{K^{*}}\phi(x) = (T_{K^{*}}\phi(y))(x) = \int_{\Gamma} K^{*}(x, y) \phi(y) \intd o(y) \quad (x \in \Gamma). \label{eq:tk-star-operator} \end{equation}

Up to now, we have defined four integral operators, $V$, $D$, $T_K$ and $T_{K^{*}}$. We further introduce Calderón projector, i.e. the Dirichlet-trace $\gamma_0$ and the Neumann-trace $\gamma_1$, which are defined as

\begin{equation} \begin{aligned} \gamma_0[u](x) &=\lim_{\varepsilon \rightarrow 0^{-}} u(x + \varepsilon\vect{n}(x)) \\ \gamma_1[u](x) &= \lim_{\varepsilon \rightarrow 0^{-}} t(x + \varepsilon\vect{n}(x)) \end{aligned} \quad (x \in \Gamma). \label{eq:calderon-projector} \end{equation}

Finally, the boundary integral equations can be represented as

\begin{equation} \begin{cases} \gamma_0[u] = \frac{1}{2}\gamma_0[u] - T_K \gamma_0[u] + V\gamma_1[u] \\ \gamma_1[u] = D\gamma_0[u] + \frac{1}{2}\gamma_1[u] + T_{K^{*}} \gamma_1[u] \end{cases} \quad (x \in \Gamma). \label{eq:boundary-integral-equations} \end{equation}

It is more compact if written in matrix form:

\begin{equation} \begin{pmatrix} \gamma_0[u] \\ \gamma_1[u] \end{pmatrix} = \begin{pmatrix} \frac{1}{2}I - T_K & V \\ D & \frac{1}{2}I + T_{K^{*}} \end{pmatrix} \begin{pmatrix} \gamma_0[u] \\ \gamma_1[u] \end{pmatrix} \quad (x \in \Gamma). \label{eq:boundary-integral-equations-in-matrix-form} \end{equation}

Summary

In this article, we introduced the corner stones of BEM, namely fundamental solution, representation formula and boundary integral equations. The convolution concept adopted in the representation formula is explained and clarified. By introducing four integral operators, $V$, $D$, $T_K$ and $T_{K^{*}}$, the boundary integral equations are obtained in a compact matrix form. In our next post, we'll reveal more properties of the two compact operators $T_K$ and $T_{K^{*}}$, which are a pair of adjoint operators in the variational formulation of the boundary integral equations, and are conjugate transpose to each other in the Galerkin discretization.

References

¹ “Cauchy Boundary Condition.” 2017. Wikipedia. https://en.wikipedia.org/w/index.php?title=Cauchy_boundary_condition&oldid=775884091.