Iterative solution of exterior Helmholtz equation with exact non-reflecting boundary conditions on distributed memory systems

This work is focused on efficient numerical solution of acoustic scattering problems on unbounded domains in two and three dimensions. A separable boundary is applied at an arbitrary distance from the scatterer, to define a computational domain and thus enable application of domain based methods (e.g. Finite Element Method). The non-reflecting Dirichlet-to-Neumann (DtN) boundary condition is imposed along the truncation boundary as an infinite harmonic series. For elongated scatterers (e.g. a submarine or a ship), suitable formulations are derived and implemented on elliptical and prolate spheroidal coordinates, reducing the size of computational domain without affecting the accuracy of the numerical solution. The outer-product structure of the boundary operator is exploited to efficiently handle non-locality in an iterative process. Computational performance of the DtN condition is analyzed and compared with local boundary conditions producing similar accuracy, in particular with Bayliss-Turkel B1 and B2 operators. A second objective in this work is to demonstrate the effective use of non-local DtN condition in a parallel distributed memory environment. By exploiting the outer-product structure, it is shown that extra communication costs due to non-locality depend solely on the number of harmonics in the DtN operator, which typically is a small number. Scalability studies show insignificant losses in speedup and parallel efficiency due to non-locality of DtN, versus local B1 or B2 conditions, when decomposing global problems into small to medium number of subdomains. Finally, a hybrid Jacobi-SSOR preconditioner is also applied, to accelerate convergence rates of the parallel iterative algorithm, without compromising communication costs.