Projection-Free Method for the Full Frank-Oseen Model of Liquid Crystals

TL;DR

This paper introduces a novel projection-free finite element method for the full Frank-Oseen model of liquid crystals, effectively handling the nonconvex unit-length constraint and proving convergence properties.

Contribution

It develops a finite element discretization and a projection-free gradient flow algorithm for the full Frank-Oseen model, with rigorous convergence analysis under minimal assumptions.

Findings

Gamma-convergence of discrete to continuous problem

Energy decrease property of the gradient flow algorithm

Handles physical constants positivity without restrictions

Abstract

Liquid crystals are materials that experience an intermediate phase where the material can flow like a liquid, but the molecules maintain an orientation order. The Frank-Oseen model is a continuum model of a liquid crystal. The model represents the liquid crystal orientation as a vector field and posits that the vector field minimizes some elastic energy subject to a pointwise unit length constraint, which is a nonconvex constraint. Previous numerical methods in the literature assumed restrictions on the physical constants or had regularity assumptions that ruled out point defects, which are important physical phenomena to model. We present a finite element discretization of the full Frank-Oseen model and a projection free gradient flow algorithm for the discrete problem in the spirit of Bartels (2016). We prove Gamma-convergence of the discrete to the continuous problem: weak convergence of subsequences of discrete minimizers and convergence of energies. We also prove that the gradient flow algorithm has a desirable energy decrease property. Our analysis only requires that the physical constants are positive, which presents challenges due to the additional nonlinearities from the elastic energy.