Multifrequency Analysis using Matrix Padé–via–Lanczos

Abstract

In many problems in acoustics, the frequency response is required over a large range of frequencies in order to characterize the behavior of the system. The simplest approach based on solving the matrix system of equations arising from a finite element discretization of the problem across a frequency range using a direct or iterative solver at each frequency can be prohibitively expensive. However, in modeling a physical problem, the analyst may be interested only in the solution on a subset of the computational domain, called the partial field. This smaller field of interest may be used to form a Pad´e approximation for the matrix equations in the frequency domain, constructing a reduced–order model that can be solved efficiently over multiple frequencies to compute the response of the partial field. In this chapter, the matrix Pad´e–via–Lanczos algorithm is applied to structural acoustics and for problems with non–reflecting boundary conditions. The approach is extended by the introduction of an adaptive scheme which can automatically span a frequency range of interest, producing an efficient and robust algorithm for multifrequency analysis.