Acoustical scattering from N spheres using a multilevel fast multipole method

Abstract

We develop an efficient computational method for wave scattering from a large number of spherical objects that are characterized by their radii, locations, and complex acoustical impedances of the surfaces. The direct T-matrix method for solution of the problem that was developed and tested in [Gumerov and Duraiswami, J. Acoust. Soc. Am. 112, 2688–2701 (2002)] is inefficient for computations involving a large number of scatterers. Here, we implement and test a multilevel fast multipole method for speeding up the solution and achieving better memory complexity. The method is based on hierarchical space subdivision with oct-trees using optimal space-partitioning, on a theory for fast translation and re-expansion of multipole solutions of the Helmholtz equation, and employs an iterative technique for the solution of large dense systems of linear equations with reflection-based iterations. For N scatterers the method provides O(N<th>log<th>N) asymptotic complexity opposed to O(N3) complexity of the direct T-matrix approach. The results of computations were tested against solutions obtained by other methods, such as the boundary element method and the direct T-matrix method tested in our early study, and show the computational efficiency and accuracy of the solution technique. [Work supported by NSF Awards 0086075 and 0219681 is gratefully acknowledged.]