An efficient algorithm for the simultaneous solution of exterior acoustic problems over a frequency band

Abstract

A method for solving time-harmonic exterior acoustics problems in a frequency band over selected regions of the computational domain is presented. The partial fields of interest include surfaces enclosing the sound source, as well as distinct points in the near-field of the source. The discretization of the boundary-value problem is based on finite elements. Replacing the infinite domain problem with an equivalent formulation in a bounded domain leads to the incorporation of a Dirichlet-to-Neumann (DtN) map, which accounts for the analytical asymptotic behavior of the solution. By interpreting the discretized DtN operator as a low-rank update of the system matrix, a standard shifted form of the inverse of this matrix, which is incorporated in the expression for the partial fields, is obtained. Due to the complex frequency dependence of this inverse, a direct evaluation is prohibitive. Hence, a matrix-valued Padé approximation of the inverse operator is employed via a two-sided block Lanczos algorithm, which provides a stable and efficient representation of the Padé approximation. A numerical example illustrates the performance and advantages of the outlined method.