Homogenization of discrete thin structures

TL;DR

This paper develops a method to derive continuum models from discrete lattice structures by analyzing the asymptotic behavior of quadratic energies on periodic, connected graphs as the scale parameter tends to zero.

Contribution

It introduces a discrete-to-continuum dimension reduction technique for quadratic energies on periodic graphs, utilizing a novel coarse-graining process and a discrete $p$-connectedness approach.

Findings

Scaled energies converge to a $d$-dimensional limit energy.

The method applies to graphs parameterized on cylindrical subsets of lattices.

The approach combines coarse-graining with discrete $p$-connectedness techniques.

Abstract

We consider graphs parameterized on a portion $X\subset\mathbb Z^d\times \{1,\ldots, M\}^k$ 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 $\varepsilon$, 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.