Convergent FEM for a membrane model of liquid crystal polymer networks

TL;DR

This paper introduces a finite element method for modeling liquid crystal polymer networks, addressing non-convex energy minimization, and demonstrates convergence and stability through numerical simulations.

Contribution

It presents a novel FEM discretization with regularization for a non-convex membrane model of LCNs, including an iterative scheme with proven stability and convergence.

Findings

The proposed scheme converges to discrete minimizers.

Numerical simulations illustrate the model's features.

The method handles non-convex energy functional effectively.

Abstract

We design a finite element method (FEM) for a membrane model of liquid crystal polymer networks (LCNs). This model consists of a minimization problem of a non-convex stretching energy. We discuss properties of this energy functional such as lack of weak lower semicontinuity. We devise a discretization with regularization, propose a novel iterative scheme to solve the non-convex discrete minimization problem, and prove stability of the scheme and convergence of discrete minimizers. We present numerical simulations to illustrate convergence properties of our algorithm and features of the model.