Efficient computation of acoustical scattering from N spheres via the fast multipole method accelerated flexible generalized minimal residual method

Many problems require computation of acoustic fields in systems consisting of a large number of scatterers, which can be modeled as spheres (or enclosed by them). These spheres can have different sizes, can be arbitrarily distributed in three dimensional space, and can have different surface impedance. Solution of this problem via direct T‐matrix approach [Gumerov and Duraiswami, J. Acoust. Soc. Am., 112, 2688–2701 (2002)] is practical only for relatively low number of scatterers, N, since its computational complexity grows as O(N3). We developed and implemented an efficient computational technique, based on an iterative solver employing a flexible generalized minimal residual method with a right preconditioner. Matrix‐vector multiplications involving a large system matrix and the preconditioner are sped up with the aid of the multilevel fast multipole method. We tested the accuracy, convergence and complexity of the method on example problems with N∼104 (millions of unknowns). These tests showed that the method is accurate for a range of frequencies, and experimentally scales as O(N1.25). The method has substantial advantages in speed and convergence compared to the reflection method reported earlier [Gumerov and Duraiswami, J. Acoust. Soc. Am., 113, 2334 (2002)]. [Work supported NSF Awards 0086075 and 0219681, which are gratefully acknowledged.]