Well-conditioned asymptotic waveform evaluation for finite elements

The frequency-domain finite-element method (FEM) results in matrix equations that have polynomial dependence on the frequency of excitation. For a wide-band fast frequency sweep technique based on a moment-matching model order reduction (MORe) process, researchers generally take one of two approaches. The first is to linearize the polynomial dependence (which will either limit the bandwidth of accuracy or require the introduction of extra degrees of freedom) and then use a well-conditioned Krylov subspace technique. The second approach is to work directly with the polynomial matrix equation and use one of the available, but ill-conditioned, asymptotic waveform evaluation (AWE) methods. For large-scale FEM simulations, introducing extra degrees of freedom, and therefore increasing the length of the MORe vectors and the amount of memory required, is not desirable; therefore, the first approach is not alluring. On the other hand, an ill-conditioned AWE process is unattractive. This paper presents a novel MORe technique for polynomial matrix equations that circumvents these problematic issues. First, this novel process does not require any additional unknowns. Second, this process is well-conditioned. Along with the presentation of the novel algorithm, which is called well-conditioned AWE (WCAWE), numerical examples modeled using the FEM are given to illustrate its accuracy.