Numerical solution of the dynamics of director fields in nematic liquid crystals

Since their discovery in the late 1800s, liquid crystals have become an important part of the technology of the modern world. As a consequence the study of anisotropic liquids in general, and liquid crystals in particular, has grown into a large interdisciplinary field involving physics, mathematics, chemistry and biology to name a few. In a series of papers we consider numerical solution of the evolution of the director, a vector valued field giving the local average orientation of the long axis of molecules in nematic liquid crystals. The flow field is assumed to be stationary throughout this work. We consider both the free elastic dynamics of the director as well as the case with applied electric fields on a finite domain. We study the dynamics of the 1D Fréedericksz transition, where an applied electric field forces reorientation in the director field. The director is assumed strongly anchored and the boundaries. Herein, we study the role of inertia and dissipation on the time evolution of the director eld during the reorientation. In particular, we show through simulations that inertia will introduce standing waves that might e ect transition time of the reorientation, but only for very small time scales or extremely high molecular inertia. The Fréedericksz transition is also numerically studied with weak boundary anchoring. For this problem it has been shown analytically that there exists a hierarchy of meta-stable equilibrium con gurations. This is in sharp contrast to the strongly anchored case, where the equilibrium is globally well defined. We derive an implicit numerical scheme for this problem and show the well-posedness of the discrete equation system. The method can be used for the fully nonlinear model with coupled electric field. Through simulations we show that the director can transition into different meta-stable states given different small perturbations to the initial data. The numerical solution of variational wave equations describing the elastic dynamics of nematic liquid crystals is considered in both 1D and 2D. Using energy respecting Runge{Kutta Discontinuous Galerkin methods we show that numerical solutions that either conserve or dissipate a discrete version of the energy can be obtained by efficient time marching. The dissipative scheme uses a dissipative up-winding at the cell interfaces combined with a shock-capturing method. Finally, we consider the application of nonintrusive sampling methods for uncertainty quantification for the elastic problem with uncertain Frank constants. The multi-level Monte Carlo (MLMC) method has been successfully applied to systems of hyperbolic conservation laws, but its applicability to other nonlinear problems is unclear. We show that MLMC is 5-10 times more efficient in approximating the mean compared to regular Monte Carlo sampling, when applied to variational wave equations in both 1D and 2D