Homogenization of discrete thin structures. (June 2023)

• Record Type: Journal Article
• Authors: Braides, Andrea; D'Elia, Lorenza
• Is Part Of: Nonlinear analysis. Volume 231(2023)
• Publication Date: 2023-06
• Subjects: Thin structures -- Dimension reduction -- Discrete-to-continuum -- Lattice systems -- Homogenization; Mathematical analysis -- Periodicals; Functional analysis -- Periodicals; Nonlinear theories -- Periodicals; Analyse mathématique -- Périodiques; Analyse fonctionnelle -- Périodiques; Théories non linéaires -- Périodiques; Functional analysis; Mathematical analysis; Nonlinear theories; Periodicals; Electronic journals; 515.7248
• DOI: 10.1016/j.na.2022.112951
• Languages: English
• ISSNs: 0362-546X

Abstract

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.