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This paper outlines an energy-minimization finite-element approach to themodeling of equilibrium configurations for nematic liquid crystals in thepresence of internal and external electric fields. The method targetsminimization of system free energy based on the electrically andflexoelectrically augmented Frank-Oseen free energy models. The Hessian,resulting from the linearization of the first-order optimality conditions, isshown to be invertible for both models when discretized by a mixedfinite-element method under certain assumptions. This implies that theintermediate discrete linearizations are well-posed. A coupled multigrid solverwith Vanka-type relaxation is proposed and numerically vetted for approximationof the solution to the linear systems arising in the linearizations. Twoelectric model numerical experiments are performed with the proposed iterativesolver. The first compares the algorithm's solution of a classical Freedericksztransition problem to the known analytical solution and demonstrates theconvergence of the algorithm to the true solution. The second experimenttargets a problem with more complicated boundary conditions, simulating anano-patterned surface. In addition, numerical simulations incorporating thesenano-patterned boundaries for a flexoelectric model are run with the iterativesolver. These simulations verify expected physical behavior predicted by aperturbation model. The algorithm accurately handles heterogeneous coefficientsand efficiently resolves configurations resulting from classical andcomplicated boundary conditions relevant in ongoing research.

Energy Minimization for Liquid Crystal Equilibrium with Electric and Flexoelectric Effects