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An FFT-accelerated PML-BIE Solver for Three-Dimensional Acoustic Wave Scattering in Layered Media

Hangya Wang, Wangtao Lu

TL;DR

The paper develops a fast, high-order solver for 3D acoustic scattering in layered media with an axially symmetric perturbation by combining a boundary integral formulation with a cylindrical PML and FFT-based azimuthal decoupling. The method derives PML-transformed Green's functions and boundary operators, reduces the problem to mode-wise NtD maps on a generating curve via Fourier transforms, and stabilizes the computation with a kernel-splitting strategy and graded-mesh Alpert quadrature. Key contributions include an efficient NtD map assembly for each Fourier mode, a robust treatment of complex-distance kernels, and an explicit construction that yields exponential PML truncation error decay. Numerical experiments confirm high accuracy and significant speedups, validating the approach for large-scale 3D scattering problems in layered media with axisymmetric perturbations.

Abstract

This paper is concerned with three-dimensional acoustic wave scattering in two-layer media, where the two homogeneous layers are separated by a locally perturbed plane featuring an axially symmetric perturbation. A fast novel boundary integral equation (BIE) method is proposed to solve the scattering problem within a cylindrical perfectly matched layer (PML) truncation. We use PML-transformed Green's functions to derive BIEs in terms of single- and double-layer potentials for the wave field and its normal derivative on the boundary of each truncated homogeneous region. These BIEs, combined with interface and PML boundary conditions, form a complete system that accurately approximates the scattering problem. An FFT-based approach is introduced to efficiently and accurately discretize the surface integral operators in the BIEs, where a new kernel splitting technique is developed to resolve instabilities arising from the complex arguments in Green's functions. Numerical experiments demonstrate the efficiency and accuracy of the proposed method, as well as the exponential decay of truncation errors introduced by the PML.

An FFT-accelerated PML-BIE Solver for Three-Dimensional Acoustic Wave Scattering in Layered Media

TL;DR

The paper develops a fast, high-order solver for 3D acoustic scattering in layered media with an axially symmetric perturbation by combining a boundary integral formulation with a cylindrical PML and FFT-based azimuthal decoupling. The method derives PML-transformed Green's functions and boundary operators, reduces the problem to mode-wise NtD maps on a generating curve via Fourier transforms, and stabilizes the computation with a kernel-splitting strategy and graded-mesh Alpert quadrature. Key contributions include an efficient NtD map assembly for each Fourier mode, a robust treatment of complex-distance kernels, and an explicit construction that yields exponential PML truncation error decay. Numerical experiments confirm high accuracy and significant speedups, validating the approach for large-scale 3D scattering problems in layered media with axisymmetric perturbations.

Abstract

This paper is concerned with three-dimensional acoustic wave scattering in two-layer media, where the two homogeneous layers are separated by a locally perturbed plane featuring an axially symmetric perturbation. A fast novel boundary integral equation (BIE) method is proposed to solve the scattering problem within a cylindrical perfectly matched layer (PML) truncation. We use PML-transformed Green's functions to derive BIEs in terms of single- and double-layer potentials for the wave field and its normal derivative on the boundary of each truncated homogeneous region. These BIEs, combined with interface and PML boundary conditions, form a complete system that accurately approximates the scattering problem. An FFT-based approach is introduced to efficiently and accurately discretize the surface integral operators in the BIEs, where a new kernel splitting technique is developed to resolve instabilities arising from the complex arguments in Green's functions. Numerical experiments demonstrate the efficiency and accuracy of the proposed method, as well as the exponential decay of truncation errors introduced by the PML.

Paper Structure

This paper contains 11 sections, 3 theorems, 89 equations, 11 figures.

Table of Contents

  1. Introduction
  2. Problem formulation
  3. The PML truncation and boundary integral equations.
  4. Boundary integral equations for the PML problem
  5. Truncation of boundary integral equations onto $\Gamma_{AB}$
  6. Fourier transformation of boundary integral equations
  7. Numerical implementation
  8. Approximating $\tilde{\mathcal{N}}_n$ on $\gamma_{AB}$
  9. Wave field evaluations
  10. Instability issues
  11. Numerical examples

Key Result

Lemma 3.1

For any $\boldsymbol{r} \in \Gamma^a$, we have where $\phi_0$ is half the angle of the curve $\gamma_b$, see $\gamma_b$ in Figure fg2(b).

Figures (11)

  • Figure 1: Profile of a 3D layered medium.
  • Figure 2: (a) The 3D illustration of the PML truncation. (b) The 2D cross-section of the PML truncation.
  • Figure 3: Geometric configuration for computing ${\mathcal{K}}_{+}^{0}1 (\boldsymbol{r})$
  • Figure 4: (a) Accuracy of the NtD matrix ${\bm N}^{\text{sub}}_n$ for varying $L$. (b) Accuracy of the two NtD matrices ${\bm N}^{\text{ori}}_n$ and ${\bm N}^{\text{sub}}_n$, for $L=5$.
  • Figure 5: Result for Example 1 with $N=800$. (a) Relative errors of $c\boldsymbol{u}^{s,j}$ with respect to $N_f$. (b) Relative errors of $u^{\text{tot}}$ against $T$ for $S=2$. (c) Relative errors of $u^{\text{tot}}$ against $S$ for $T=2$. SWI: Spherical wave incidence, PWI: Plane wave incidence.
  • ...and 6 more figures

Theorems & Definitions (7)

  • Lemma 3.1
  • proof
  • Lemma 3.2
  • proof
  • Lemma 3.3
  • proof
  • Remark 3.1