A structure-preserving FEM for the uniaxially constrained $\mathbf{Q}$-tensor model of nematic liquid crystals

TL;DR

This paper introduces a structure-preserving finite element method for simulating uniaxially constrained Q-tensor models of nematic liquid crystals, ensuring stability, consistency, and energy convergence.

Contribution

It develops a novel finite element approach that maintains the uniaxial constraint and proves its stability, consistency, and energy convergence without regularization.

Findings

Method is stable and consistent.

Discrete energies converge to continuous energies.

Numerical simulations demonstrate effectiveness in 2D and 3D.

Abstract

We consider the one-constant Landau - de Gennes model for nematic liquid crystals. The order parameter is a traceless tensor field $\mathbf{Q}$, which is constrained to be uniaxial: $\mathbf{Q} = s (\mathbf{n}\otimes\mathbf{n} - d^{-1} \mathbf{I})$ where $\mathbf{n}$ is a director field, $s\in\mathbb{R}$ is the degree of orientation, and $d\ge2$ is the dimension. Building on similarities with the one-constant Ericksen energy, we propose a structure-preserving finite element method for the computation of equilibrium configurations. We prove stability and consistency of the method without regularization, and $\Gamma$-convergence of the discrete energies towards the continuous one as the mesh size goes to zero. We design an alternating direction gradient flow algorithm for the solution of the discrete problems, and we show that such a scheme decreases the energy monotonically. Finally, we illustrate the method's capabilities by presenting some numerical simulations in two and three dimensions including non-orientable line fields.