Variational Methods For Phononic Calculations

TL;DR

This paper compares three variational methods for solving elastodynamic eigenvalue problems in phononics, analyzing their convergence based on material property variations and highlighting the efficiency of the mixed quotient.

Contribution

It provides a detailed analysis of the convergence behavior of displacement, stress, and mixed variational quotients in phononic eigenvalue problems, guiding method selection.

Findings

Mixed quotient converges faster than displacement and stress quotients in general.

Convergence rates depend on the smoothness of material property variations.

Results suggest potential for more efficient algorithms in phononics and photonics.

Abstract

Three fundamental variational principles used for solving elastodynamic eigenvalue problems are studied within the context of elastic wave propagation in periodic composites (phononics). We study the convergence of the eigenvalue problems resulting from the displacement Rayleigh quotient, the stress Rayleigh quotient and the mixed quotient. The convergence rates of the three quotients are found to be related to the continuity and differentiability of the density and compliance variation over the unit cell. In general, the mixed quotient converges faster than both the displacement Rayleigh and the stress Rayleigh quotients, however, there exist special cases where either the displacement Rayleigh or the stress Rayleigh quotient shows the exact same convergence as the mixed-method. We show that all methods converge faster for smoother material property variations, but when density variation is rough, the difference between the mixed quotient and stress Rayleigh quotient is higher and similarly, when compliance variation is rough, the difference between the mixed quotient and displacement Rayleigh quotient is higher. Since eigenvalue problems such as those considered in this paper tend to be highly computationally intensive, it is expected that these results will lead to fast and efficient algorithms in the areas of phononics and photonics.