Homogenization of some elliptic equations with connections to elasticity

University dissertation from Luleå : Luleå tekniska universitet

Abstract: In this PhD thesis we prove some properties of elliptic equations in connection with periodic homogenization. We mainly study the stationary linearized elasticity system. This thesis consists of five papers complemented with a short introduction.
The first paper concerns some elliptic equations on a domain with oscillating boundary. The homogenization is carried out using the method of periodic unfolding, which was recently introduced by D. Cioranescu, A. Damlamian, and G. Griso. We show how the epsilon-sequence can be extended to a stationary domain such that strong convergence is obtained. In this way we justify the homogenization.
In the second paper we address a model of a perforated structure, a honeycomb structure, which is locally and effectively isotropic. We show how to obtain simple numerical formulas which give approximate values of the effective elastic properties for any volume fraction and any choice of local material. We also show that the structure is optimal in the limit as the volume fraction of the connected material goes to zero.
In the third paper we continue the study of linearly elastic honeycombs, by considering a class of effectively isotropic structures. We prove that the model structure, which was considered in the second paper, is not unique in the way that it is optimal in the low-density limit. In particular, we show that this optimality is not shared among all the structures. Discrete lower bounds for the suboptimality of the structures are also given. This extends the work of G. Beeri, D. Lukkassen, and A. Meidell.
The forth paper extends recent results of C. Carstensen and J. Thiele, on a method for a posteriori error estimation for finite element approximations of the linear elasticity system. The error estimate can be used to construct an adaptive method for mesh refinement. We show the details for the cell problems which occur in the second and third papers. In a numerical example we observe an increased rate of convergence when using the adaptive method, when comparing with uniform mesh refinement.
Our final paper partially extends recent work of D. Lukkassen, A. Meidell, S. Vigdergauz, on the use of spatial symmetry to obtain domain reduction for the so-called cell problems in the periodic case. This can be useful, for example, in numerical approximations of the coefficients of the homogenized linearized system of elasticity. We use the method of periodic unfolding as well as the theory developed by A. Bossavit for treatment of symmetry in linear equations.

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