Energy Minimization for Liquid Crystal Equilibrium with Electric and Flexoelectric Effects

TL;DR

This paper presents an energy-minimization finite-element method for modeling equilibrium liquid crystal configurations with electric and flexoelectric effects, validated through numerical experiments demonstrating accuracy and efficiency.

Contribution

It introduces a novel finite-element approach with a coupled multigrid solver for liquid crystal equilibrium modeling including flexoelectric effects, ensuring well-posedness and computational efficiency.

Findings

Solver accurately reproduces analytical solutions for classical problems.

Method effectively handles complex boundary conditions and heterogeneous coefficients.

Numerical simulations align with physical predictions for flexoelectric behavior.

Abstract

This paper outlines an energy-minimization finite-element approach to the modeling of equilibrium configurations for nematic liquid crystals in the presence of internal and external electric fields. The method targets minimization of system free energy based on the electrically and flexoelectrically augmented Frank-Oseen free energy models. The Hessian, resulting from the linearization of the first-order optimality conditions, is shown to be invertible for both models when discretized by a mixed finite-element method under certain assumptions. This implies that the intermediate discrete linearizations are well-posed. A coupled multigrid solver with Vanka-type relaxation is proposed and numerically vetted for approximation of the solution to the linear systems arising in the linearizations. Two electric model numerical experiments are performed with the proposed iterative solver. The first compares the algorithm's solution of a classical Freedericksz transition problem to the known analytical solution and demonstrates the convergence of the algorithm to the true solution. The second experiment targets a problem with more complicated boundary conditions, simulating a nano-patterned surface. In addition, numerical simulations incorporating these nano-patterned boundaries for a flexoelectric model are run with the iterative solver. These simulations verify expected physical behavior predicted by a perturbation model. The algorithm accurately handles heterogeneous coefficients and efficiently resolves configurations resulting from classical and complicated boundary conditions relevant in ongoing research.