An FFT-Accelerated Boundary Integral Equation Method for Wave Scattering by Smooth Surfaces in Three Dimensions
Abstract
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(Mlog 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.
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BibTeX
@misc{hua2026fftaccelerated,
title = {{An FFT-Accelerated Boundary Integral Equation Method for Wave Scattering by Smooth Surfaces in Three Dimensions}},
author = {Wenmao Hua and Jun Lai and Huiyi Li and Wangtao Lu},
howpublished = {arXiv},
year = {2026},
month = aug,
doi = {10.48550/arxiv.2608.16208},
eprint = {2608.16208},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2608.16208},
}
RIS
TY - UNPB
TI - An FFT-Accelerated Boundary Integral Equation Method for Wave Scattering by Smooth Surfaces in Three Dimensions
AU - Wenmao Hua
AU - Jun Lai
AU - Huiyi Li
AU - Wangtao Lu
PY - 2026
DA - 2026/08/31
PB - arXiv
DO - 10.48550/arxiv.2608.16208
UR - https://arxiv.org/abs/2608.16208
AB - 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(Mlog 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.
ER -