Robust fast direct integral equation solver for three-dimensional quasi-periodic scattering problems with a large number of layers

TL;DR

This paper introduces a robust and efficient boundary integral equation solver for 3D multilayered quasi-periodic scattering problems, capable of handling large-scale structures with high accuracy and linear complexity.

Contribution

The paper presents a novel high-order quadrature and a periodizing scheme that significantly improve robustness and efficiency over traditional methods for multilayered quasi-periodic scattering.

Findings

Capable of solving 100-interface structures with over 960,000 unknowns

Achieves $10^{-5}$ accuracy in less than 2 hours on a desktop workstation

Maintains linear computational complexity with respect to the number of layers

Abstract

A boundary integral equation method for the 3-D Helmholtz equation in multilayered media with many quasi-periodic layers is presented. Compared with conventional quasi-periodic Green's function method, the new method is robust at all scattering parameters. A periodizing scheme is used to decompose the solution into near- and far-field contributions. The near-field contribution uses the free-space Green's function in an integral equation on the interface in the unit cell and its immediate eight neighbors; the far-field contribution uses proxy point sources that enclose the unit cell. A specialized high-order quadrature is developed to discretize the underlying surface integral operators to keep the number of unknowns per layer small. We achieve overall linear computational complexity in the number of layers by reducing the linear system into block tridiagonal form and then solving the system directly via block LU decomposition. The new solver is capable of handling a 100-interface structure with 961.3k unknowns to $10^{-5}$ accuracy in less than 2 hours on a desktop workstation.