A stable and fast method for solving multibody scattering problems via the method of fundamental solutions

Abstract

The paper describes a numerical method for solving acoustic multibody scattering problems in two and three dimensions. The idea is to compute a highly accurate approximation to the scattering operator for each body through a local computation, and then use these scattering matrices to form a global linear system. The resulting coefficient matrix is relatively well-conditioned, even for problems involving a very large number of scatterers. The linear system is amenable to iterative solvers, and can readily be accelerated via fast algorithms for the matrix-vector multiplication such as the fast multipole method. The key point of the work is that the local scattering matrices can be constructed using potentially ill-conditioned techniques such as the method of fundamental solutions (MFS), while still maintaining scalability and numerical stability of the global solver. The resulting algorithm is simple, as the MFS is far simpler to implement than alternative techniques based on discretizing boundary integral equations using Nyström or Galerkin.

Figures (9)

• Figure 1: (a) The multibody scattering problem (\ref{['eq:BVP']}) with $T=4$ scattering bodies, (b) The proxy circle introduced in Section \ref{['sec:geometryofSmatrix']}; the scattering matrix $\bm{\mathsf{S}}_{\tau}$ is computed to be accurate for interactions between $\Gamma_{\tau}$ and other objects outside of $\Psi_\tau$
• Figure 2: A single-body scattering problem for (a) a smooth object or (b) an object with a corner
• Figure 3: The 2D 4-starfish geometry
• Figure 4: The 2D 8-cavity geometry
• Figure 5: The 2D 8-teardrop geometry

Theorems & Definitions (6)

• Example 1
• Example 2
• Example 3
• Example 4
• Example 5
• Example 6