An Augmented Lagrangian Method for the Microstructure of a Liquid Crystal Model

Abstract

The liquid crystal microstructure is obtained by minimizing the so-called Oseen-Frank energy functional. The liquid crystal molecule is assumed to be of unit length. It is important to deal with the unit length constraint since it is a main reason for orientation singularities of liquid crystal molecules. For a better understanding of complicated orientation singularities associated with the microstructure, simplified models resulted from specific choices of elastic constants are always of interest. In this paper an augmented Lagrangian method together with an explicit-implicit scheme is used to compute the solution of a liquid crystal system based on a simplified Oseen-Frank energy functional. The augmented Lagrangian method is used to deal with the unit-length constraint of liquid crystal molecules, where the penalty parameter need not be small so the resulting system may be more stable/less stiff than the penalty method. Unlike the projection method its energy functional would not go up and down dramatically during the minimization process. The explicit-implicit scheme allows a matrix free implementation in the pseudo-time gradient flow minimization process. Numerical examples in domains of typical shapes (circle, square and rectangle) and with various rotational boundary conditions are computed and computational results are compared with those obtained by the penalty method.