Multi-Model Multi-Domain Computational Methods: Helmholtz

Introduction

The Helmholtz problem is the basic model of propagation of waves in the frequency domain. It is typically used as the farfield model in a full description of sound radiation from an unsteady flow-body interaction, in which the forcing for the Helmholtz problem is parameterized from the solution of the higher fidelity nearfield problem. It also arises in electromagnetic scattering contexts, and in internal as well as external contexts. Two prime difficulties of Helmholtz problems are the truncation of the computational domain in the farfield and the resolution requirements of the oscillatory solution everywhere in the domain. These difficulties combine to give rise to rather large-scale discrete problems, with an indefinite system matrix that is not always sparse, depending upon the type of nonreflecting closure selected for the farfield boundary.

Research Overview

We have assembled the best of recent innovations in farfield boundary and internal interface discretizations for Helmholtz problems into a pair of demonstration codes. For nonreflecting farfield conditions we use the exact (but nonlocal) Dirichlet-to-Neumann map inside a Krylov iterative technique. For internal interfaces we use Sommerfeld conditions on overlapped subdomains components of the additive Schwarz preconditioner. Several model problems involving point and planar sources in 2D and in 3D have been considered. We present here some solutions for the classic "Givoli" problem in an eccentric annulus.

For parallel implementation, we employ the Krylov-Schwarz domain decomposition method, using one subdomain per processor. Though the iteration count remains sensitive to parameters of the resolution, parallel scalability per iteration scales well in the Gustafson sense.

Papers