A comparison between two-scale asymptotic expansions and Bloch wave expansions for the homogenization of periodic structures

Abstract

In this paper we make a comparison between the two-scale asymptotic expansion method for periodic homogenization and the so-called Bloch wave method. It is well-known that the homogenized tensor coincides with the Hessian matrix of the first Bloch eigenvalue when the Bloch parameter vanishes. In the context of the two-scale asymptotic expansion method, there is the notion of high order homogenized equation (Bakhvalov and Panasenko in Homogenization: averaging processes in periodic media. Kluwer, Dordrecht, 1989) where the homogenized equation can be improved by adding small additional higher order differential terms. The next non-zero high order term is a fourth-order term, accounting for dispersion effects (see e.g. Santosa and Symes in SIAM J Appl Math 51:984–1005, 1991; Lamacz in Math Models Methods Appl Sci 21(9):1871–1899, 2011; Dohnal et al. Multiscale Model Simul 12(2):488–513, 2014). Surprisingly, this homogenized fourth-order tensor is not equal to the fourth-order tensor arising in the Taylor expansion of the first Bloch eigenvalue, which is often called Burnett tensor. Here, we establish an exact relation between the homogenized fourth-order tensor and the Burnett fourth-order tensor. It was proved in Conca et al. (J Math Phys 47(3):11, 2006) that the Burnett fourth-order tensor has a sign. For the special case of a simple laminate we prove that the homogenized fourth-order tensor may change sign. In the elliptic case we explain the difference between the homogenized and Burnett fourth-order tensors by a difference in the source term which features an additional corrector term. Finally, for the wave equation, the two fourth-order tensors coincide again, so dispersion is unambiguously defined, and only the source terms differ as in the elliptic case.