An efficient boundary integral equation solution technique for solving aperiodic scattering problems from two-dimensional, periodic boundaries

TL;DR

The paper addresses aperiodic scattering from two-dimensional periodic boundaries by recasting the problem with a Floquet--Bloch transform into a contour integral of quasiperiodic problems. It introduces a periodizing boundary integral formulation with proxy-based representations and a new efficient direct solver that reuses Bloch-phase independent computations and employs low-rank factorizations and corner compression to handle geometries with corners. The approach yields substantial speedups (roughly 20–30×) for evaluating the Floquet--Bloch integral, particularly on stair-like geometries, and maintains high accuracy even near Wood anomalies and branch points. This method enhances computational viability for aperiodic scattering in periodic structures and shows promise for extension to three-dimensional problems where precomputation will be even more crucial.

Abstract

This manuscript presents an efficient boundary integral equation technique for solving two-dimensional Helmholtz problems defined in the half-plane bounded by an infinite, periodic curve with Neumann boundary conditions and an aperiodic point source. The technique is designed for boundaries where one period does not require a large number of discretization points to achieve high accuracy. The Floquet--Bloch transform turns the problem into evaluating a contour integral where the integrand is the solution of quasiperiodic boundary value problems. To approximate the integral, one must solve a collection of these problems. This manuscript uses a variant of the periodizing scheme by Cho and Barnett which alleviates the need for evaluating the quasiperiodic Green's function and is amenable to a large amount of precomputation that can be reused for all of the necessary solves. The solution technique is accelerated by the use of low rank linear algebra. The numerical results illustrate that the presented method is 20-30 faster than the technique utilizing the quasiperiodic Green's function for a stair-like geometry.

Figures (6)

• Figure 2.1: Deformation of the integral contour for the Floquet--Bloch transform. (a) The black, sinusoidal curve shows the deformed contour that goes around the poles corresponding to trapped modes ($\kappa_{\mathrm{tr}}$, black dots) and branch cuts corresponding to Wood anomalies ($\kappa_{\mathrm{W}}$, blue dots and wavy lines). (b) Shows the real part of the integrand $u_{\kappa}$ in the inverse Floquet--Bloch transform \ref{['eq:fb-integral']}, with the branch cuts and poles visible, evaluated at the target point $\mathbf{x} = [0.22, 0.34]$, with $\omega = 2.4$.
• Figure 3.1: Illustration of the (a) geometry and the (b) unit cell. (a) A portion of the domain $\Omega$ and boundary $\Gamma$ are shown for the cosine geometry. The point source $\bm{x}_0$ and the periodicity $d$ are labeled. (b) The unit cell is shown with its boundary $\Gamma_0$ in black, left and right walls in blue, and upper wall in red. The unit cell domain $\Omega_0$ is labeled. The left and right neighbors, $\Gamma_{-1}$ and $\Gamma_1$, are shown in gray. The proxy circle $\mathcal{P}$ with charges $\{y_j\}_{j=1}^{N_{\mathrm{proxy}}}$ is shown in orange encompassing the unit cell and its two neighbors.
• Figure 4.1: Illustrations of the proxy surfaces used to reduce the cost of creating the low-rank factorization of $\mathsf{A}_{-1}$. The unit cell boundary $\Gamma_0$ is shown in black, the near left neighbor boundary is shown in blue, and the far left neighbor boundary is shown in gray. The proxy surface $\mathcal{P}_{-1}$ used in place of the far nodes is shown in orange. (a) The proxy surface is a full circle. (b) The proxy surface is a half circle in the direction of the neighbor boundary. (c) The skeleton points found from the interpolatory decomposition index matrix $J(1:l)$ are denoted on the unit cell boundary with black crosses.
• Figure 4.2: (a) Illustration of the partitioning of the staircase geometry with three periods. (b) Illustration of $\Gamma_0$ with the corner partitioning. $\Gamma_c$ in black denotes the part of the geometry near the corner that needs refinement, $\Gamma_s = \Gamma_0 \setminus \Gamma_c$ in blue. Shown are two levels of refinement into the corner.
• Figure 4.3: Illustration of a staircase geometry where the corners have two levels of refinement. The refined panels at each corner are shown in black. The near part of the remaining boundary is shown in blue, and the far part in gray. The proxy surfaces used for compression of each corner are shown in orange. The inset shows the top corner compression. The black crosses denote the skeleton points.

Theorems & Definitions (2)

• Definition 4.1
• Remark 4.1