A low-rank iteration scheme for multi-frequency acoustic problems

The boundary element method for the discretization of the Kirchhoff-Helmholtz integral equation is a popular numerical tool for solving linear time-harmonic acoustic problems. However, the implicit frequency dependence of the boundary element matrices necessitates an efficient treatment for problems requiring solutions in a frequency range. While several approaches based on approximations of the Green's function, the matrix, or even the solution itself exist in the literature, this paper presents an alternative method that provides a simultaneous solution over a frequency range within a single iterative scheme. A frequency approximation of the boundary element system in conjunction with a low-rank approximation of the solution enables efficient matrix vector multiplications. The algorithm can be incorporated into iterative solvers, such as BiCGstab in order to obtain the frequency range solution. The proposed scheme is applied to an acoustic interior problem subject to different boundary conditions. The influence of both the approximation order and the accuracy of the low-rank truncations on the convergence behavior of the solution are studied. The results verify the effectiveness of the proposed iterative scheme. It opens up possibilities for the efficient evaluation of structural-acoustic interactions and associated phenomena in the future.