Acoustical scattering from N spheres using a multilevel fast multipole method

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(NlogN) 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.]