Efficient computation of multi-frequency far-field solutions of the Helmholtz equation using Pade approximation

For many problems in exterior structural acoustics! the solution is require d to be computed over multiple frequencies. For some classes of these probl ems, however, it may be sufficient to evaluate the multiple frequency solut ions over restricted regions of the spatial domain. Examples include optimi zation and inverse problems based on the minimization of a functional defin ed over a specified surface or sub-region. For such problems, which include both near-held and far-field computations, we recently proposed an efficie nt algorithm to compute the partial-held solutions at multiple frequencies simultaneously. In this paper, we consider the particular case of far-field computations and simplify the recently proposed algorithm by exploiting th e symmetry of linear operators. The approach involves a reformulation of th e Dirichlet-to-Neumann (DtN) map based finite-element matrix problem into a transfer-function form that can efficiently describe the far-field solutio n. A multi-frequency approximation of the transfer function is developed by constructing matrix-valued Pade approximation of the transfer function via a symmetric, banded Lanczos process. Numerical tests illustrate the accura cy of the approach for a nide range of frequencies and cost reductions of a n order of magnitude when compared to commonly used factorization based met hods.