A finite element method for nematic liquid crystals with variable degree of orientation

TL;DR

This paper develops a finite element method for modeling nematic liquid crystals with variable orientation, accurately capturing defects and ensuring convergence and stability of the numerical solutions.

Contribution

It introduces a structure-preserving discretization for the Ericksen model that handles degenerate elliptic equations without regularization, with proven convergence and a monotone energy scheme.

Findings

The method effectively captures line and plane defects with finite energy.

Discrete minimizers converge to continuous solutions as mesh refines.

Simulations demonstrate the method's ability to handle complex defect structures.

Abstract

We consider the simplest one-constant model, put forward by J. Ericksen, for nematic liquid crystals with variable degree of orientation. The equilibrium state is described by a director field $\mathbf{n}$ and its degree of orientation $s$, where the pair $(s, \mathbf{n})$ minimizes a sum of Frank-like energies and a double well potential. In particular, the Euler-Lagrange equations for the minimizer contain a degenerate elliptic equation for $\mathbf{n}$, which allows for line and plane defects to have finite energy. We present a structure preserving discretization of the liquid crystal energy with piecewise linear finite elements that can handle the degenerate elliptic part without regularization, and show that it is consistent and stable. We prove $\Gamma$-convergence of discrete global minimizers to continuous ones as the mesh size goes to zero. We develop a quasi-gradient flow scheme for computing discrete equilibrium solutions and prove it has a strictly monotone energy decreasing property. We present simulations in two and three dimensions to illustrate the method's ability to handle non-trivial defects. A music video summary of the paper is available on YouTube: "Mathematical Modeling and Simulation of Nematic Liquid Crystals (A Montage)," http://www.youtube.com/watch?v=pWWw7\_6cQ-U