三维光滑表面波散射的FFT加速边界积分方程方法

中文摘要 AI 辅助

对于轴对称表面的波散射问题，快速傅里叶变换（FFT）方法为加速标准边界积分方程（BIE）求解器提供了有效工具。由于积分算子具有类卷积特性，表面积分方程可被解耦为生成曲线上的一系列曲线积分方程；三维基本核的傅里叶系数可通过基于Miller算法的三项递推关系快速计算。这类成熟技术对于非轴对称表面不再适用。本文提出一种新型FFT加速边界积分方法，用于求解任意形状光滑表面的波散射问题。奇异核的傅里叶系数现在满足高阶递推关系，虽然可通过标准Olver算法以最优线性复杂度求解，但实际应用中，一种奇点交换方法（将每个核重写为光滑函数与轴对称相关奇异因子的乘积）速度快得多。因此，Miller算法结合标准FFT卷积，可得到一种复杂度为O(M log M)的方法，用于计算O(M)个核的傅里叶模式，其复杂度与轴对称表面的方法完全相同！借助这类基于FFT的高效程序，我们将表面积分方程重写为O(M)个弱奇异曲线积分，通过基于面板的广义高斯求积法对其进行离散，得到高精度线性系统以近似波场。开展了大量数值实验以验证该新方法的有效性。

英文摘要

For wave scattering by axisymmetric surfaces, the fast Fourier transform (FFT) method provides an effective tool to accelerate standard boundary integral equation (BIE) solvers. Surface BIEs can be decoupled into a series of curve integral equations on the generating curve, due to the convolution-like integral operators. The Fourier coefficients of the three-dimensional fundamental kernels can be rapidly computed through three-term recurrence relations based on Miller's algorithm. Such well-established techniques break down for nonaxisymmetric surfaces. This paper proposes a novel FFT-accelerated boundary integral method for wave scattering by smooth surfaces of arbitrary shapes. The Fourier coefficients of the singular kernels now satisfy higher-order recurrence relations. Although they can be solved with an optimal linear complexity by the standard Olver's algorithm, it turns out that a singularity swapping approach that rewrites each kernel as the product of a smooth function and an axisymmetric-related singular factor is realistically much faster. Consequently, Miller's algorithm together with the standard FFT convolution yields an ${\cal O}(M\log M)$ approach for evaluating the ${\cal O}(M)$ Fourier coefficients of the kernels, attaining exactly the same order of complexity for axisymmetric surfaces! With such FFT-based efficient procedures, we rewrite the surface BIEs in terms of ${\cal O}(M)$ curve integrals, which are proved to exhibit logarithmic singularities, discretize them by panel-based generalized Gaussian quadratures, and obtain spectrally accurate linear systems to approximate the wavefields. Extensive numerical experiments are carried out to demonstrate the effectiveness and spectral accuracy of the new approach.

发表机构

• Academy of Mathematics and Systems Science, Chinese Academy of Sciences（中国科学院数学与系统科学研究院）
• School of Mathematical Sciences, Zhejiang University（浙江大学数学科学学院）

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