Periodicity of FEM Discrete Models and Its Influence on Solutions to the 1-D Wave Equation

This paper discusses the influence of the periodicity of finite-element (FE) discrete models and its influence on solutions to the one-dimensional (1-D) wave equation. Numerical solutions to wave-propagation problems obtained via the displacement-based formulations of the finite-element method (FEM) often exhibit high-frequency behavior, which is frequently dismissed in the literature as undesired, spurious, and/or having no physical meaning. In this paper, we verify this notion by demonstrating that this behavior is not merely a computational anomaly but is due to the inherent periodic properties of discrete numerical models. Using Bloch’s theorem, we reveal and demonstrate how, at high frequencies, the discrete nature of FEM numerical models leads to the prevailing behavior governed by the periodic nature of the computational models. In order to illustrate this phenomenon, we investigate 1-D wave propagation in rods, leveraging the non-dispersive nature of the governing equation as a benchmark. In addition to the classical and specialized FEM, we analyze two alternative formulations: the time-domain spectral finite-element method (TD-SFEM) and a novel spline-based finite-element method (spFEM) proposed by the authors. The results obtained and presented explain qualitatively the origins of these numerical anomalies and suggest strategies to mitigate their effects, effectively shifting the periodicity-induced behavior beyond the range of physically relevant frequencies by appropriate selection of approximation polynomials. The authors demonstrate that this can be fully achieved only in the case of spFEM, for which the usable percentage of the available spectra of eigenfrequencies reaches 67%, while in the case of other FEM approaches discussed is significantly smaller as determined by numerical dispersion and the presence of frequency band gaps.