An accelerated direct solver for scalar wave scattering by multiple transmissive inclusions in two dimensions

Abstract

This paper discusses a fast direct solver using boundary integral equations for Helmholtz transmission problems involving multiple inclusions in two dimensions. Efficiently addressing scattering problems in the presence of numerous inclusions remains a key challenge for various practical applications. For problems involving a large number of scatterers, the number of iterations in Krylov subspace methods is known to increase significantly. This occurs even when using second-kind boundary integral equations, which are typically recognized for their rapid convergence. We consider a fast direct solver as an alternative, an approach that has been less commonly explored for transmission problems with disjoint multiple inclusions. The low-rank approximation based on the proxy method achieve speedup by calculating interactions between disjoint scatterers without the terms derived from the internal integral representation. Notably, this advantage applies to the Poggio--Miller--Chang--Harrington--Wu--Tsai (PMCHWT) formulation but breaks down in the Burton--Miller case. Numerical examples demonstrate that the proposed solver can compress the system of linear algebraic equations to a size of $O(ωD)$, where $ω$ is the frequency of the incident wave and $D$ is the diameter of the (smallest) bounding box enclosing the multiple inclusions. The total computational cost scales as $O(N^{1.5})$ $(= O(\sqrt{N}^3))$ at most for a fixed $ω$ when the inclusions are arranged on a grid. Moreover, the PMCHWT formulation, that omits the interior term in the proxy method, is approximately six times faster than the Burton--Miller formulation when treating each inclusion as a cell. Furthermore, in the same setting, the former can compress the size of the system of linear algebraic equations by half compared to the latter.

Figures (11)

• Figure 1: Multiple star-shaped scatterers. This figure corresponds to $4 \times 4 = 16$ scatterers case.
• Figure 2: Comparison of the absolute value of the numerical solution $|u(x)| = \sqrt{(\Re(u_j (x)))^2 + (\Im(u_j (x)))^2}$ at quadrature points on the star-shaped boundary $\Gamma_j$ for $j = 1, 2, \ldots, 16$. The solid line stands for the reference solution obtained using the BEM module of FreeFEM. The markers $+$ and $\times$ correspond to the solutions obtained by the proposed fast direct solver using PMCHWT (Omit) and BM formulations, respectively. Both solutions are in good agreement with the reference solution.
• Figure 3: Comparison of the numerical solution on the boundary $\Gamma_1$ of the inclusion centered at $(x_1, x_2) = (0, 0)$. The magnitude $|u (x(\theta))| = \sqrt{\qty{\Re(u_1 (x(\theta)))}^2 + \qty{\Im(u_1 (x(\theta)))}^2}$ is plotted at quadrature points, where $\theta$ is the polar angle and $x(\theta)$ represents the parametric boundary. The solid line stands for the reference solution obtained using the BEM module of FreeFEM. The markers $+$ and $\times$ correspond to the solutions obtained by the proposed fast direct solver using PMCHWT (Omit) and BM formulations, respectively. Both solutions are in good agreement with the reference solution.
• Figure 4: Comparison of the numerical solution on the boundary $\Gamma_1$ of the inclusion centered at $(x_1, x_2) = (0, 0)$. The magnitude $|q (x(\theta))| = \sqrt{\qty{\Re(q_1 (x(\theta)))}^2 + \qty{\Im(q_1 (x(\theta)))}^2}$ is plotted at quadrature points, where $\theta$ is the polar angle and $x(\theta)$ represents the parametric boundary. The solid line stands for the reference solution obtained using the BEM module of FreeFEM. The markers $+$ and $\times$ correspond to the solutions obtained by the proposed fast direct solver using PMCHWT (Omit) and BM formulations, respectively. Both solutions are in good agreement with the reference solution.
• Figure 5: Relative 2-norm error defined in \ref{['eq:relative_norm']}. Plots are sorted in descending order of their relative errors. The labels "PMCHWT (Omit)", "PMCHWT (All)" and "BM" correspond to the fast direct solver based on \ref{['eq:pmchwt_omit_bie']}, \ref{['eq:pmchwt_all_bie']} and \ref{['eq:bm_bie']}, respectively. The labels "PMCHWT (Conventional)" correspond to \ref{['eq:pmchwt_omit_bie']}, and is solved by a standard partial pivoted LU decomposition. In the labels, $\epsilon$ represents the threshold of the column-pivoted QR decomposition in the fast direct solver. The results of the PMCHWT (Omit) formulation are represented by solid lines, while those of the other formulations are shown as dashed lines. The result of Burton--Miller formulated conventional BIEM is used as the reference. In this figure, the angular frequency $\omega$ and the material constants $\varepsilon^\pm$ are fixed. It can be observed that PMCHWT (Omit) formulation has negligible impact on accuracy.