Homogenization of discrete thin structures

Abstract

We consider graphs parameterized on a portion $X\subset {\mathbb{Z}}^d\times \{1,\dots ,M{\}}^k$ of a cylindrical subset of the lattice ${\mathbb{Z}}^d\times {\mathbb{Z}}^k$, and perform a discrete-to-continuum dimension-reduction process for energies defined on $X$ of quadratic type. Our only assumptions are that $X$ be connected as a graph and periodic in the first $d$-directions. We show that, upon scaling of the domain and of the energies by a small parameter $\epsilon$, the scaled energies converge to a $d$-dimensional limit energy. The main technical points are a dimension-lowering coarse-graining process and a discrete version of the $p$-connectedness approach by Zhikov.