A fast frequency sweep approach using Padé approximations for solving Helmholtz finite element models

In various engineering applications, the solution of the Helmholtz equation is required over a broad frequency range. The simplest approach, which consists in solving the system of equations obtained from a finite element discretization for each frequency, becomes computationally prohibitive for fine increments, particularly when dealing with large systems, like those encountered when addressing mid-frequency problems. Alternative approaches involving reduced-order models built via Padé approximations are now well established for systems exhibiting polynomial frequency dependency of second-order kind and for frequency independent excitations. This paper treats systems of more complicated wavenumber dependency, likely to be encountered when applying frequency dependent boundary conditions and/or loadings. The well-conditioned asymptotic waveform evaluation (WCAWE) is selected as the method of choice and the approximated Taylor coefficients are computed by differentiating the continuous frequency dependent models obtained through a fitting process of the system entries. The method is benchmarked first against the Second-Order Arnoldi (SOAR) algorithm on a simple second-order system. Then it is applied to realistic large scale interior and exterior Helmholtz problems exhibiting high-order polynomial or rational frequency behavior. In either case, the proposed methodology is shown to reduce the computational time of the frequency sweep by an order of magnitude when compared to the direct approach.