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

数学科学：液晶问题和奇异扰动的数值分析

• 批准号: 9107434
• 负责人: Eugene Gartland
• 金额: $ 4.7万
• 依托单位: Kent State University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 1991
• 资助国家: 美国
• 起止时间: 1991-09-15 至 1994-02-28
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9107434&HistoricalAwards=false
• 关键词: Mathematical; Sciences; Numerical; Analysis; Problems

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.

这一建议主要涉及数学 和液晶相关问题的数值分析。 还考虑了奇异摄动的数值分析 常微分方程 液晶分析 问题涉及跨学科的工作与格伦H。布朗 肯特州立大学的液晶研究所（LCI）， 主要涉及分析和数值模拟 液晶分子的平衡构型 限制，如在聚合物分散液晶中出现的 (PDLC的），特别注意缺陷结构和 几何形状、边界条件和外部应用的影响 领域的 这些解决方案领域的建模，使用连续 数学模型，作为Landau-de Gennes自由的极小化器 能量泛函，它源于相位的朗道理论 过渡。 这些问题，由于它们的学位数量众多， 自由度（张量序参数）和空间维度 （三）、引领挑战大规模科学计算 问题需要强大的和有效的技术有限 单元离散化、大稀疏优化和数值 对称性破缺时的分叉。 的工作 奇异摄动的区域与进一步的 一般制服的发展、改进和延伸 线性和非线性微分方程稳定性理论 任意的秩序。 的主要意义和重要性 这些项目源于跨学科的方面， 研究液晶 研究者将进行数值计算 模拟当前的问题感兴趣的理论 和实验物理学家。 他会让我们 现代数值分析的最新技术， 科学计算，以补充物理理论， 实验 LCI的调查人员已经达到了极限， 先进的核磁共振技术， 晶体在非常小的（亚微米）大小的包容， 目前正在开发的一些显示技术。 是 希望数值模拟能发挥重要作用 这里.