This paper outlines an energy-minimization finite-element approach to the computational modeling of equilibrium configurations for nematic liquid crystals under free elastic effects. The method targets minimization of the system free energy based on the Frank-Oseen free-energy model. Solutions to the intermediate discretized free elastic linearizations are shown to exist generally and are unique under certain assumptions. This requires proving continuity, coercivity, and weak coercivity for the accompanying appropriate bilinear forms within a mixed finite-element framework. Error analysis demonstrates that the method constitutes a convergent scheme. Numerical experiments are performed for problems with a range of physical parameters as well as simple and patterned boundary conditions. The resulting algorithm accurately handles heterogeneous constant coefficients and effectively resolves configurations resulting from complicated boundary conditions relevant in ongoing research.

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.

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.

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This paper derives a posteriori error estimators for the nonlinear first-order optimality conditions associated with the Frank-Oseen elastic free-energy model of nematic and cholesteric liquid crystals, where the required unit-length constraint is imposed via either a Lagrange multiplier or penalty method. Furthermore, theory establishing the reliability of the proposed error estimator for the penalty method is presented, yielding a concrete upper bound on the approximation error of discrete solutions. The error estimators herein are composed of readily computable quantities on each element of a finite-element mesh, allowing the formulation of an efficient adaptive mesh refinement strategy. Four elastic equilibrium problems are considered to examine the performance of the error estimators and corresponding adaptive mesh refinements against that of a simple uniform refinement scheme. The adapted grids successfully provide significant reductions in computational work while producing solutions that are highly competitive with those of uniform mesh in terms of constraint conformance and computed free energies.