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Fast boundary integral method for acoustic wave scattering in two-dimensional layered media

Linfeng Xia, Heng Yuan, Bo Wang, Wei Cai

TL;DR

The paper tackles acoustic scattering in 2D layered media by recasting the problem as a boundary integral equation using a layered Green's function that inherently enforces transmission across interfaces. It advances Nyström discretization for this BIE and introduces a layered-media FMM with enhanced polarization coordinates and effective target locations to achieve near-linear scalability, complemented by an overlapping domain decomposition preconditioner to stabilize GMRES iterations. The method demonstrates robust accuracy and efficiency for complex configurations, achieving $O(N)$ complexity at low frequencies and strong performance across multiple incident angles. This results in a scalable, high-accuracy solver for large-scale layered-medium scattering problems with potential applications in subsurface imaging and acoustic design.

Abstract

In this paper, we present a fast boundary integral method accelerated by the fast multipole method (FMM) for acoustic wave scattering governed by the scalar Helmholtz equation in multi-layered two-dimensional media. Multiple scatterers are randomly distributed in the multi-layered medium with some scatterers possibly intersecting layer interfaces. The boundary integral formulation employs a layered-medium Green's function to enforce transmission conditions across interfaces, thus avoiding unknowns on the interfaces and significantly reducing the size of the discretized problem compared to approaches that use a free-space Green's function. To demonstrate the FMM speedup, a low-order Nyström method is used to discretize the boundary integral equation and then the resulting dense linear system is solved by GMRES iterative solver accelerated by an improved layered media FMM and a overlapping domain decomposition preconditioning. In the low frequency regime, the proposed algorithm achieves an $\mathcal O(N)$ complexity. Numerical results validate the accuracy, efficiency and robustness of the method under complex settings and various incident angles. The proposed framework provides a scalable and efficient solver for acoustic wave scattering in layered media.

Fast boundary integral method for acoustic wave scattering in two-dimensional layered media

TL;DR

The paper tackles acoustic scattering in 2D layered media by recasting the problem as a boundary integral equation using a layered Green's function that inherently enforces transmission across interfaces. It advances Nyström discretization for this BIE and introduces a layered-media FMM with enhanced polarization coordinates and effective target locations to achieve near-linear scalability, complemented by an overlapping domain decomposition preconditioner to stabilize GMRES iterations. The method demonstrates robust accuracy and efficiency for complex configurations, achieving complexity at low frequencies and strong performance across multiple incident angles. This results in a scalable, high-accuracy solver for large-scale layered-medium scattering problems with potential applications in subsurface imaging and acoustic design.

Abstract

In this paper, we present a fast boundary integral method accelerated by the fast multipole method (FMM) for acoustic wave scattering governed by the scalar Helmholtz equation in multi-layered two-dimensional media. Multiple scatterers are randomly distributed in the multi-layered medium with some scatterers possibly intersecting layer interfaces. The boundary integral formulation employs a layered-medium Green's function to enforce transmission conditions across interfaces, thus avoiding unknowns on the interfaces and significantly reducing the size of the discretized problem compared to approaches that use a free-space Green's function. To demonstrate the FMM speedup, a low-order Nyström method is used to discretize the boundary integral equation and then the resulting dense linear system is solved by GMRES iterative solver accelerated by an improved layered media FMM and a overlapping domain decomposition preconditioning. In the low frequency regime, the proposed algorithm achieves an complexity. Numerical results validate the accuracy, efficiency and robustness of the method under complex settings and various incident angles. The proposed framework provides a scalable and efficient solver for acoustic wave scattering in layered media.

Paper Structure

This paper contains 12 sections, 1 theorem, 86 equations, 14 figures, 3 tables.

Table of Contents

  1. Introduction
  2. Acoustic wave scattering problem in two-dimensional layered media
  3. Model problem
  4. Background scattering field
  5. Boundary integral equation and Nyström discretization
  6. Green's function of the Helmholtz equation in layered media
  7. Effective transmission distance of reaction fields
  8. Indirect boundary integral equation and Nyström method
  9. Fast multipole acceleration and preconditioning
  10. Fast multipole method for reaction components
  11. An overlapping domain decomposition preconditioner
  12. Numerical Examples

Key Result

Proposition 4.1

Given four points $\boldsymbol{r}=(x, y), \boldsymbol{r}'=(x', y')$, $\boldsymbol{r}_c=(x_c, y_c)$ and $\boldsymbol{r}_c'=(x'_c, y_c')$ in $\mathbb R^2$ such that $|\boldsymbol{r}-\boldsymbol{r}_c|<|\boldsymbol{r}'-\boldsymbol{r}_c|$ and $|\boldsymbol{r}-\boldsymbol{r}_c'|>|\boldsymbol{r}'-\boldsymb then there holds the following expansions where $(\rho_{c}', \theta_{c}')$ and $(\rho_{c}, \theta

Figures (14)

  • Figure 2.1: Configurations of the scattering problem in layered media.
  • Figure 3.1: Equivalent polarization source coordinates and effective target locations in the case of $\ell=\ell'$.
  • Figure 3.2: Equivalent polarization source coordinates and effective target locations in the case of $\ell<\ell'$.
  • Figure 3.3: Equivalent polarization source coordinates and effective target locations in the case of $\ell>\ell'$.
  • Figure 4.1: Unknowns in the local system for preconditioning.
  • ...and 9 more figures

Theorems & Definitions (2)

  • Remark 3.1
  • Proposition 4.1