Multiple multipole expansions for acoustic scattering

Abstract

The paper presents a new approach to solve multiple scattering of acoustic waves in two dimensions. Traditionally wave fields are expanded into an orthogonal set of basis functions. Often these functions build a multipole. Unfortunately, these multipoles converge rather slowly for complex geometries. The new approach enhances convergence by including multiple multipoles into each region, allowing irregularities of the boundary to be resolved locally. The wave fields are expanded into a set of nonorthogonal basis functions. The incident wave field and the fields induces by the scatterers are matched in the least-square sense by evaluating the boundary conditions at discrete matching points along the domain boundaries. Due to the nonorthogonal expansions we choose more matching points than actually needed resulting in an overdetermined system which is solved in the least-squares sense. This allows an estimate of how well the expansion converges and can help to tune the scheme to enhance accuracy or reduce runtime. The idea of the multiple multipole can be extended by using more complicated basis functions which are closer to the solution sought. The resulting algorithm is a very general tool to solve relatively large and complex two-dimensional scattering problems.