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

这一建议主要涉及与液晶有关问题的数学和数值分析。还考虑了奇摄动常微分方程的数值分析。液晶问题的分析涉及与肯特州立大学Glenn H. Brown液晶研究所（LCI）的跨学科工作；它主要关注分析和数值模拟聚合物分散液晶（PDLC）中液晶分子在小范围内的平衡构型，特别关注缺陷结构以及几何形状、边界条件和外部应用领域的影响。这些解场被建模，使用一个连续统数学模型，作为最小的朗道-德-热纳自由能泛函，这源于朗道相变理论。这些问题具有大量的自由度（一个张量序参数）和空间维度（三个），导致具有挑战性的大规模科学计算问题，需要强大而有效的有限元离散化、大稀疏优化和对称破缺存在下的数值分岔技术。奇异摄动领域的工作涉及到对任意阶线性和非线性微分方程的一般一致稳定性理论的进一步发展、改进和推广。这些项目的主要意义和重要性源于液晶工作的跨学科方面。研究者将对LCI中理论和实验物理学家感兴趣的当前问题进行数值模拟。他将运用现代数值分析和科学计算的最新技术来补充物理理论和实验。LCI的研究人员正处于他们的极限，使用复杂的核磁共振技术来探测目前正在开发的某些显示技术中非常小（亚微米）尺寸容器中的液晶结构。希望数值模拟能在此发挥重要作用。

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.