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### Proceedings Paper

Variational methods in phononics
Author(s): Yan Lu; Ankit Srivastava
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Paper 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.

Paper Details

Date Published: 23 March 2015
PDF: 14 pages
Proc. SPIE 9438, Health Monitoring of Structural and Biological Systems 2015, 94381K (23 March 2015); doi: 10.1117/12.2083746
Show Author Affiliations
Yan Lu, Illinois Institute of Technology (United States)
Ankit Srivastava, Illinois Institute of Technology (United States)

Published in SPIE Proceedings Vol. 9438:
Health Monitoring of Structural and Biological Systems 2015
Tribikram Kundu, Editor(s)