IFGF-accelerated integral equation solvers for acoustic scattering

Abstract

We present an accelerated and hardware parallelized integral-equation solver for the problem of acoustic scattering by a two-dimensional surface in three-dimensional space. The approach is based, in part, on the novel Interpolated Factored Green Function acceleration method (IFGF) that, without recourse to the Fast Fourier Transform (FFT), evaluates the action of Green function-based integral operators for an $N$-point surface discretization at a complexity of $\Ord(N\log N)$ operations instead of the $\Ord(N^2)$ cost associated with nonaccelerated methods. The IFGF algorithm exploits the slow variations of factored Green functions to enable the fast evaluation of fields generated by groups of sources on the basis of a recursive interpolation scheme. In the proposed approach, the IFGF method is used to account for the vast majority of the computations, while, for the relatively few singular, nearly-singular and neighboring non-singular integral operator evaluations, a high-order rectangular-polar quadrature approach is employed instead. Since the overall approach does not rely on the FFT, it is amenable to efficient shared- and distributed-memory parallelization; this paper demonstrates such a capability by means of an OpenMP parallel implementation of the method. A variety of numerical examples presented in this paper demonstrate that the proposed methods enable the efficient solution of large problems over complex geometries on small parallel hardware infrastructures. Numerical examples include acoustic scattering by a sphere of up to $128$ wavelengths, an $80$-wavelength submarine, and a turbofan nacelle that is more than $80$ wavelengths in size, requiring, on a 28-core computer, computing times of the order of a few minutes per iteration and a few tens of iterations of the GMRES iterative solver.