Two-Scale Finite Element Approximation of a Homogenized Plate Model

TL;DR

This paper develops and analyzes a finite element method for a homogenized plate model, demonstrating convergence and validating results through numerical tests and experiments on microstructured sheets.

Contribution

It introduces a novel finite element discretization approach for a homogenized thin plate model with proven convergence and numerical validation.

Findings

Convergence of the discretization is rigorously proven.

Numerical tests confirm the theoretical convergence.

Results qualitatively match deformation experiments.

Abstract

This paper studies the discretization of a homogenization and dimension reduction model for the elastic deformation of microstructured thin plates proposed by Hornung, Neukamm, and Vel\v{c}i\'c in 2014. Thereby, a nonlinear bending energy is based on a homogenized quadratic form which acts on the second fundamental form associated with the elastic deformation. Convergence is proven for a multi-affine finite element discretization of the involved three-dimensional microscopic cell problems and a discrete Kirchhoff triangle discretization of the two-dimensional isometry-constrained macroscopic problem. Finally, the convergence properties are numerically verified in selected test cases and qualitatively compared with deformation experiments for microstructured sheets of paper.