Mathematical Sciences: Numerical Analysis of Problems in Liquid Crystals and Singular Perturbations

点击翻译按钮获取中文摘要

This proposal is concerned principally with the mathematical and numerical analysis of problems related to liquid crystals. Also considered is the numerical analysis of singularly-perturbed ordinary differential equations. The analysis of liquid-crystal problems involves interdisciplinary work with the Glenn H. Brown Liquid Crystal Institute (LCI) at Kent State University; it is primarily concerned with analyzing and numerically simulating the equilibrium configurations of liquid crystal molecules in small confinements, as arise in polymer-dispersed liquid crystals (PDLC's), with special attention to defect structures and the influence of geometry, boundary conditions, and external applied fields. These solution fields are modeled, using a continuum mathematical model, as minimizers of the Landau-de Gennes free energy functional, which derives from the Landau theory of phase transitions. These problems, with their large number of degrees of freedom (a tensor order parameter) and spatial dimensions (three), lead to challenging large-scale scientific computing problems requiring robust and efficient techniques of finite element discretization, large-sparse optimization, and numerical bifurcation in the presence of symmetry breaking. The work in the area of singular perturbations is concerned with the further development, refinement, and extension of a general uniform stability theory for linear and nonlinear differential equations of arbitrary order. The main significance and importance of these projects derives from the interdisciplinary aspect of the work on liquid crystals. The investigator will perform numerical simulations on current problems of interest to the theoretical and experimental physicists in the LCI. He will bring to bear state-of-the-art techniques from modern numerical analysis and scientific computing to complement physical theory and experiment. Investigators in the LCI are at their limits, using sophisticated NMR techniques, to probe the structure of liquid crystals in the very small (submicron) size containments in certain display technologies currently under development. It is hoped that numerical simulations will play an important role here.