A Preconditioned Nullspace Method for Liquid Crystal Director Modeling

Abstract

Abstract. We present a preconditioned nullspace method for the numerical solution of large sparse linear systems that arise from discretizations of continuum models for the orientational properties of liquid crystals. The approach effectively deals with pointwise unit-vector constraints, which are prevalent in such models. The indefinite, saddle-point nature of such problems, which can arise from either or both of two sources (pointwise unit-vector constraints, coupled electric fields), is illustrated. Both analytical and numerical results are given for a model problem.Key words. nullspace method, liquid crystals, saddle-point problems, unit-vector constraints AMS subject classifications. 65F08, 65F10, 65F50, 65H10, 65N22 DOI. 10.1137/120870219 1. Introduction. Many continuum models for the orientational properties of liquid crystals involve one or more state variables that are vector fields of unit length. The pointwise unit-vector constraints associated with discretizations of such models give rise to indefinite linear systems of saddle-point form, when these constraints are imposed via Lagrange multipliers. In problems such as these, indefiniteness also frequently manifests itself due to another influence (coupling with applied electric fields), and this leads to a double saddle-point structure. We are interested in the efficient numerical solution by iterative methods of large sparse linear systems of algebraic equations associated with such problems. We begin by presenting some background on these materials and models.

Energy Minimization for Liquid Crystal Equilibrium with Electric and Flexoelectric Effects

“…Such a domain will be considered in numerical experiments below. A number of methods involving computation of liquid crystal equilibria or dynamics utilize the so called one-constant approximation that K 1 = K 2 = K 3 and K 4 = 0 [10,42,48,52], in order to significantly simplify the free elastic energy density toŵ…”

“…Due to such difficulties, efficient, theoretically supported, numerical approaches to the modeling of nematic liquid crystals under free elastic and augmented electric effects are of great importance. A number of computational techniques for liquid crystal equilibrium and dynamics problems exist [31,32,48,52], including least-squares finite-element methods [3] and discrete Lagrange multiplier approaches with simplifying assumptions [27,42]. In addition, numerical experiments involving finite-element methods with Lagrange multipliers, applied to the equilibrium equations, have been successful in capturing certain liquid crystal characteristics [41].…”

“…In this paper, we propose a method that directly targets energy minimization in the continuum, via Lagrange multiplier theory on Banach spaces, to resolve liquid crystal equilibrium configurations in the presence of applied electric fields and internally induced electric fields due to flexoelectric effects. The approach is derived absent the often used one-constant approximation [10,42,48,52]; that is, the method described here, and the accompanying theory, are applicable for a wide range of physical parameters. This allows for significantly improved modeling of physical phenomena not captured in many models.…”

“…These first-order conditions contain highly nonlinear terms and are, therefore, linearized with a generalized Newton's method. The resulting Newton iteration inherently contains a complicated saddle-point structure [7,42]. The discrete Hessians associated with finite-element discretization of the Newton linearizations are shown to be invertible, for both the electric and flexoelectric models, when employing certain finite-element spaces.…”

A Renormalized Newton Method for Liquid Crystal Director Modeling

“…In such cases, it is common to enforce the constraint |n| = 1 either by Lagrange multipliers or by penalty methods. Several other liquid crystal models involve unit-length vector fields and constraints-see [8] for more discussion. Standard references on liquid crystals include [2,3,9,10].…”