Advanced Discretizations and Multigrid Methods for Liquid Crystal Configurations.
Abstract: Liquid crystals are substances that possess mesophases with
properties intermediate between liquids and crystals. Here, we consider nematic liquid
crystals, which consist of rod-like molecules whose average pointwise orientation is
represented by a unit-length vector, n(x,y,z) = (n_1, n_2, n_3)^T. In addition to their
self-structuring properties, nematics are dielectrically active ... read moreand birefringent. These
traits continue to lead to many important applications and discoveries. Numerical
simulations of liquid crystal configurations are used to suggest the presence of new
physical phenomena, analyze experiments, and optimize devices. This thesis develops a
constrained energy-minimization finite-element method for the efficient computation of
nematic liquid crystal equilibrium configurations based on a Lagrange multiplier
formulation and the Frank-Oseen free-elastic energy model. First-order optimality
conditions are derived and linearized via a Newton approach, yielding a linear system of
equations. Due to the nonlinear unit-length constraint, novel well-posedness theory for the
variational systems, as well as error analysis, is conducted. The approach is shown to
constitute a convergent and well-posed approach, absent typical simplifying assumptions.
Moreover, the energy-minimization method and well-posedness theory developed for the
free-elastic case are extended to include the effects of applied electric fields and
flexoelectricity. In the computational algorithm, nested iteration is applied and proves
highly effective at reducing computational costs. Additionally, an alternative technique is
studied, where the unit-length constraint is imposed by a penalty method. The performance
of the penalty and Lagrange multiplier methods is compared. Furthermore, tailored
trust-region strategies are introduced to improve robustness and efficiency. While both
approaches yield effective algorithms, the Lagrange multiplier method demonstrates superior
accuracy per unit cost. In addition, we present two novel, optimally scaling, multigrid
approaches for these systems based on Vanka- and Braess-Sarazin-type relaxation. Both
approaches outperform direct methods and represent highly efficient and scalable iterative
solvers. Finally, a three-dimensional problem considering the effects of geometrically
patterned surfaces is presented, which gives rise to a nonlinear anisotropic
reaction-diffusion equation. Well-posedness is shown for the intermediate linearization
systems of the proposed Newton linearization. The configurations under consideration are
part of ongoing physics research seeking new bistable configurations induced by geometric
Thesis (Ph.D.)--Tufts University, 2015.
Submitted to the Dept. of Mathematics.
Advisors: James Adler, and Scott MacLachlan.
Committee: James Adler, Scott MacLachlan, Timothy Atherton, Thomas Manteuffel, and Xiaozhe Hu.
Keywords: Applied mathematics, Mathematics, and Physics.read less