Efficient Computation of Multi-Frequency Far-Field Solutions of the Helmholtz Equation Using Padé Approximation

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Abstract
For many problems in exterior structural acoustics, the solution is required to be computed over multiple frequencies. For some classes of these problems, however, it may be sufficient to evaluate the multiple frequency solutions over restricted regions of the spatial domain. Examples include optimization and inverse problems based on the minimization of a functional defined over a specified surface or sub-region. For such problems, which include both near-field and far-field computations, we recently proposed an efficient algorithm to compute the partial-field 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 the symmetry of linear operators. The approach involves a reformulation of the Dirichlet-to-Neumann (DtN) map based finite-element matrix problem into a transfer-function form that can efficiently describe the far-field solution. A multi-frequency approximation of the transfer function is developed by constructing matrix-valued Padé approximation of the transfer function via a symmetric, banded Lanczos process. Numerical tests illustrate the accuracy of the approach for a wide range of frequencies and cost reductions of an order of magnitude when compared to commonly used factorization based methods.

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