A new approach to efficiently compute the low-frequency eigenmodes of a time-domain model approximating a linear physical system is proposed. The method is based on principles known from digital signal processing, in particular from parametric spectrum estimation, so it is not surprising that the achievable accuracy is much higher than the accuracy of the usual non-parametric discrete Fourier transform approach. The algorithm works in the general lossy case even for a very large number of unknowns and can easily be extended to calculate steady-state solutions under sinusoidal excitations for several different frequencies simultaneously.
1. Introduction
Time-domain modelling of partial differential equations (PDEs) has become popular in the last few years for several reasons. In particular, time-domain methods offer great flexibility and are quite easy to program without the need for huge analytical precalculations. Furthermore, modern computers can handle the large amounts of data often necessary for the solution of practical problems and the efficiency of implementation can substantially be increased by exploiting the high degree of parallelism available in common time-domain algorithms.
The detailed time-domain behaviour of the solution, however, is often not of primary concern, but one is more interested in global information such as in the eigenvalues and eigenfunctions of a system or its steady-state response to a sinusoidal excitation, so there is some demand for methods to extract frequency domain data out of time-domain models. A new approach to solving problems of these kinds will be proposed in section 3, after a brief overview over common methods in section 2. The main advantage of the new algorithm is that it is able to compute the relevant eigenvalues and eigenvectors of a time-domain model efficiently and very accurately, even in the general lossy case and for very large systems, e.g. 100,000 unknowns cause no difficulties on a common workstation. The method is quite easily applicable, since the only information needed about the special problem under consideration is a procedure performing the calculations leading from one timestep to the next, and such a procedure is always available because it is the base of all time-domain algorithms.
After a slight modification, steady-state responses to sinusoidal excitations, including the static case, i.e. elliptic problems, can also be treated efficiently as will be shown in section 4. This approach, i.e. first to discretize a time-dependent PDE (usually of hyperbolic type) and then to perform some postprocessing to get the steady-state solution, is an interesting alternative to the direct solution of elliptic PDEs. A nice property of the method is the possibility to compute the steady-state responses for several different frequencies simultaneously in one program run. After a brief comparison to the related Lanczos method in section 5, numerical examples will be given in section 6.