Constrained Optimization for Liquid Crystal Equilibria: Extended Results

TL;DR

This paper compares energy-minimization finite-element methods for liquid crystal equilibrium, focusing on constraints enforcement, and introduces techniques to improve efficiency and robustness in complex simulations.

Contribution

It extends previous work by deriving a penalty method, analyzing linearizations, and evaluating the performance of various algorithms for liquid crystal equilibrium problems.

Findings

Lagrange multiplier method outperforms penalty method in several measures.

Trust-region methods enhance robustness of the algorithms.

Nested iteration significantly reduces computational costs.

Abstract

This paper investigates energy-minimization finite-element approaches for the computation of nematic liquid crystal equilibrium configurations. We compare the performance of these methods when the necessary unit-length constraint is enforced by either continuous Lagrange multipliers or a penalty functional. Building on previous work in [1,2], the penalty method is derived and the linearizations within the nonlinear iteration are shown to be well-posed under certain assumptions. In addition, the paper discusses the effects of tailored trust-region methods and nested iteration for both formulations. Such methods are aimed at increasing the efficiency and robustness of each algorithms' nonlinear iterations. Three representative, free-elastic, equilibrium problems are considered to examine each method's performance. The first two configurations have analytical solutions and, therefore, convergence to the true solution is considered. The third problem considers more complicated boundary conditions, relevant in ongoing research, simulating surface nano-patterning. A multigrid approach is introduced and tested for a flexoelectrically coupled model to establish scalability for highly complicated applications. The Lagrange multiplier method is found to outperform the penalty method in a number of measures, trust regions are shown to improve robustness, and nested iteration proves highly effective at reducing computational costs.