#### Description

The relative convergence rates of 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, where the displacement field is varied, the stress Rayleigh quotient, where the stress field is varied, and the mixed quotient, where both the displacement and stress fields are varied. The convergence rates of the three quotients are shown to be related to the continuity and differentiability of the density and stiffness variations over the unit cell. In general, the three quotients converge fastest when the density and stiffness functions are continuous and continuously differentiable. Conversely the rates of convergence of the methods are slowest when the stiffness and -density functions exhibit discontinuities. The rates of convergence are shown to be associated with the function spaces of the density and stiffness functions. We also show that the mixed quotient, in general, converges faster than both the displacement Rayleigh and the stress Rayleigh quotients; however, there exists special cases when either the displacement Rayleigh or the stress Rayleigh quotients shows the exact same convergence as the mixed method. Because eigenvalue problems such as those considered in this article 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.

#### Recommended Citation

Lu, Y.,
&
Srivastava, A.
(2014).
Variational methods for wave propagation in periodic structures.
In A. Bajaj, P. Zavattieri, M. Koslowski, & T. Siegmund (Eds.).
*
Proceedings of the Society of Engineering Science 51st Annual Technical Meeting, October 1-3, 2014
*,
West Lafayette: Purdue University Libraries Scholarly Publishing Services, 2014.
http://docs.lib.purdue.edu/ses2014/mss/damage/3

Variational methods for wave propagation in periodic structures

The relative convergence rates of 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, where the displacement field is varied, the stress Rayleigh quotient, where the stress field is varied, and the mixed quotient, where both the displacement and stress fields are varied. The convergence rates of the three quotients are shown to be related to the continuity and differentiability of the density and stiffness variations over the unit cell. In general, the three quotients converge fastest when the density and stiffness functions are continuous and continuously differentiable. Conversely the rates of convergence of the methods are slowest when the stiffness and -density functions exhibit discontinuities. The rates of convergence are shown to be associated with the function spaces of the density and stiffness functions. We also show that the mixed quotient, in general, converges faster than both the displacement Rayleigh and the stress Rayleigh quotients; however, there exists special cases when either the displacement Rayleigh or the stress Rayleigh quotients shows the exact same convergence as the mixed method. Because eigenvalue problems such as those considered in this article 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.