Variational Methods For Phononic Calculations

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.

arXiv admin note: text overlap with arXiv:1411.2996

References in corpus (4)

• Ultrawide phononic band gap for combined in-plane and out-of-plane waves
• Reduced Bloch mode expansion for periodic media band structure calculations
• Anti-plane Shear Waves in Layered Composites: Band Structure and Anomalous Wave-refraction
• Mixed-variational formulation for phononic band-structure calculation of arbitrary unit cells

Cited by in corpus (5)

• Level Repulsion and Band Sorting in Phononic Crystals
• On the Properties of Phononic Eigenvalue Problems
• Spectral extended finite element method for band structure calculations in phononic crystals
• Combining plane wave expansion and variational techniques for fast phononic computations
• 3-D phononic crystals with ultra-wide band gaps