Bilayer Plates: Model Reduction, $\Gamma$-Convergent Finite Element Approximation and Discrete Gradient Flow

TL;DR

This paper develops a mathematical model and numerical methods for simulating large deformations of bilayer plates caused by lattice mismatches, including a $ ext{Gamma}$-convergent finite element discretization and an energy-decreasing iterative solver.

Contribution

It introduces a $ ext{Gamma}$-convergent finite element discretization for bilayer plate bending and analyzes an energy-decreasing iterative method for finding stationary states.

Findings

Finite element discretization converges to the continuous model.

The iterative method effectively finds energy-minimizing configurations.

Numerical experiments demonstrate large deformation capabilities.

Abstract

The bending of bilayer plates is a mechanism which allows for large deformations via small externally induced lattice mismatches of the underlying materials. Its mathematical modeling, discussed herein, consists of a nonlinear fourth order problem with a pointwise isometry constraint. A discretization based on Kirchhoff quadrilaterals is devised and its $\Gamma$-convergence is proved. An iterative method that decreases the energy is proposed and its convergence to stationary configurations is investigated. Its performance, as well as reduced model capabilities, are explored via several insightful numerical experiments involving large (geometrically nonlinear) deformations.