We consider graphs parameterized on a portion X⊂Zd×{1,…,M}k of a cylindrical subset of the lattice Zd×Zk, 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.
Homogenization of discrete thin structures / Braides, Andrea; D’Elia, Lorenza. - In: NONLINEAR ANALYSIS. - ISSN 0362-546X. - 231:(2023), pp. 1-26. [10.1016/j.na.2022.112951]
Homogenization of discrete thin structures
Andrea Braides
;
2023-01-01
Abstract
We consider graphs parameterized on a portion X⊂Zd×{1,…,M}k of a cylindrical subset of the lattice Zd×Zk, 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.| File | Dimensione | Formato | |
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Discrete thin objects_revised.pdf
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