$\Gamma$-convergent LDG method for large bending deformations of bilayer plates

TL;DR

This paper introduces a $ ext{Γ}$-convergent local discontinuous Galerkin method for modeling large bending deformations in bilayer plates, ensuring energy stability and accurate constraint control.

Contribution

It develops a novel LDG method with relaxed isometry constraints, proves $ ext{Γ}$-convergence, and demonstrates its effectiveness through simulations of complex large deformations.

Findings

Proves $ ext{Γ}$-convergence of the LDG method.

Designs a practical gradient flow with linear steps.

Shows accurate simulation of large deformations including curved creases.

Abstract

Bilayer plates are slender structures made of two thin layers of different materials. They react to environmental stimuli and undergo large bending deformations with relatively small actuation. The reduced model is a constrained minimization problem for the second fundamental form, with a given spontaneous curvature that encodes material properties, subject to an isometry constraint. We design a local discontinuous Galerkin (LDG) method which imposes a relaxed discrete isometry constraint and controls deformation gradients at barycenters of elements. We prove $\Gamma$-convergence of LDG, design a fully practical gradient flow, which gives rise to a linear scheme at every step, and show energy stability and control of the isometry defect. We extend the $\Gamma$-convergence analysis to piecewise quadratic creases. We also illustrate the performance of the LDG method with several insightful simulations of large deformations, one including a curved crease.