A computationally efficient k(ω)-spectral form for partial dispersion analyses within the wave finite element framework

This paper addresses the computation of frequency-dependent dispersion curves (i.e., $k\left(\omega \right)$) and wave modes within the framework of the Wave Finite Element Method (WFEM) and in the context of high-dimensional periodic unit cell models. Numerous applications, ranging from phononics to vibroacoustics, now rely on dispersion analyses or wave expansion over a subset of eigensolutions – complex wavenumbers and Bloch waves – resulting from the resolution of an eigenvalue problem with a $T$-palindromic quadratic structure ($T$-PQEP). To exploit the structure of finite element models, various structure-preserving linearizations such as the Zhong-Williams and the $(S+S^{-1})$-transform have already been developed to achieve partial wave resolution of large $T$-PQEP, primarily targeting the dominating (least decaying) waves. In this paper we derive an alternative linearization of the $T$-PQEP for the $k\left(\omega \right)$ problem, which leads to enhanced targeting of the eigenvalues around the unit circle and reduces the inaccuracies induced by root multiplicity. A specific form of the problem is then proposed as an optimal compromise between ease of implementation, numerical stability, convergence and accuracy enhancement. The performance of our proposed linearization is compared against existing ones across various iterative eigensolvers, since the generalized eigenvalue problems involve complex non-hermitian matrices, which are not extensively included in eigensolvers. Results indicate that the proposed linearization should be favored for the WFEM, as it provides numerical enhancements in dispersion and wave vectors computation for large eigenvalue problems, as well as for further wave expansion applications.