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%T Advanced Discretizations and Multigrid Methods for Liquid Crystal Configurations.
%A Emerson, David.
%8 2017-04-20
%R http://localhost/files/0c483w579
%X 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 and 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 nano-patterning.; 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.
%[ 2018-10-09
%9 Text
%~ Tufts Digital Library
%W Institution