Homogenization of discrete thin structures. (June 2023) - Braides, Andrea; D'Elia, Lorenza
Abstract: We consider graphs parameterized on a portion X ⊂ Z d × { 1, …, M } k of a cylindrical subset of the lattice Z d × 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 ɛ, the scaled energies converge to a d -dimensional limit energy. The main technical points are a dimension-reducing coarse-graining process and a discrete version of the p -connectedness approach by Zhikov.