一种求解二维周期边界非周期散射问题的高效边界积分方程解法
An efficient boundary integral equation solution technique for solving aperiodic scattering problems from two-dimensional, periodic boundaries
AI总结：
本文针对二维周期边界上的非周期 Helmholtz 散射问题，提出一种基于 Floquet–Bloch 变换和 Cho–Barnett 周期化方案变体的高效边界积分方程方法，结合低秩线性代数加速，在阶梯状几何上比传统准周期 Green 函数方法快 20–30 倍。
摘要：
本文提出了一种高效的边界积分方程技术，用于求解定义在半平面上的二维 Helmholtz 问题，该半平面由一条具有 Neumann 边界条件的无限周期曲线和非周期点源所界定。该技术适用于一个周期内不需要大量离散点即可达到高精度的边界。Floquet–Bloch 变换将该问题转化为计算一个围道积分，其中被积函数是准周期边值问题的解。为了近似该积分，必须求解一组这样的问题。本文采用了 Cho 和 Barnett 的周期化方案的一种变体，该变体避免了计算准周期 Green 函数的需要，并且适合进行大量可复用于所有必要求解的预计算。该求解技术通过使用低秩线性代数得到加速。数值结果表明，对于阶梯状几何结构，所提出的方法比利用准周期 Green 函数的技术快 20–30 倍。
英文摘要：
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 with 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 inverse 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 avoids using the quasiperiodic Green's function in the integral representation and is amenable to a large amount of precomputation that can be reused for all the necessary solves. The solution technique is accelerated using low rank linear algebra and corner compression. The performance of the method is measured on a range of simple and complex boundary geometries, including in the presence of one or more corners with straight or curved edges present in the unit cell. The numerical results illustrate that the presented method is competitive with techniques using the quasiperiodic Green's function kernel for smooth boundaries, and up to 20--30 faster when corner compression and high accuracy is required.
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