CAREER: Numerical Methods For Liquid Crystals And Their Optimal Design

职业：液晶数值方法及其优化设计

• 批准号: 1555222
• 负责人: Shawn Walker
• 金额: $ 40万
• 依托单位: Louisiana State University
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2016
• 资助国家: 美国
• 起止时间: 2016-08-01 至 2022-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1555222&HistoricalAwards=false
• 关键词: CAREER; Numerical; Methods; Liquid; Crystals

Liquid crystals are commonplace in modern technological devices, and are most famously used for their optical properties in electronic displays (for example, LCDs). What makes them so useful is that they are easily manipulated and controlled by applying voltages or magnetic fields. Moreover, liquid crystals can interact with fine-scale "material particles." Thus, liquid crystals enable the fine-scale manipulation of matter. The goal of this research project is to create new mathematical methods/algorithms for simulating liquid crystal phenomena, and for designing new materials that utilize liquid crystals. In other words, the research will provide the groundwork for developing functionalized and switchable materials that are "driven" by liquid crystal physics. The research will impact local communities, education, and liquid crystal scientists: (i) Middle School Science Fair Projects. The PI will mentor science fair projects on liquid crystals at minority serving middle schools (the visual appeal of liquid crystals makes them ideal for project topics). PI will involve post-docs and graduate students in the mentoring. (ii) Public Library Engagements. PI will interact with middle/high school students through local libraries by creating a "sit-with-a-scientist" program. Each session takes place at a library branch location and includes a short, introductory presentation followed by hands-on activities to allow the students to actively learn about the physics and mathematics of liquid crystals. PI will involve post-docs and graduate students by having them assist during the sessions. (iii) Education. PI will continue to mentor undergraduates through REUs and senior projects, and create a graduate course "Computational Methods For Liquid Crystals" based on this research. (iv) Open Software. PI will create software implementations, tutorials, and demos of the research.The research will create new mathematical algorithms to correctly and efficiently simulate liquid crystal (LC) phenomena above the scale of molecules. Current algorithms do not do this. They either make ad-hoc changes to the underlying model for mathematical convenience, or they are expensive to compute (or both). Efficiency is crucial to facilitate "plugging" these methods into high level design and optimization procedures for, say, material design. Molecular simulations of LCs are too expensive for iterative design work. The Q-tensor model is better, but can still be expensive and hard to solve. The research does the following: (i) creates new methods for the Q-tensor model that are cheaper to solve and more faithful to the physical model by taking advantage of the discrete maximum principle (DMP); (ii) extend our method to handle arbitrary geometry through a "cut" finite element method (cutFEM); (iii) extend our cutFEM to do optimal shape design of LC systems, including self-assembly of colloidal inclusions. The research will create new methods to simulate and optimize liquid crystal systems that capitalize on delicate mathematical tools, such as Gamma-convergence and the DMP. This fits well within our core expertise, such as prior work in LCs, 3-D mesh generation/implementation, multi-physics geometric flows, and shape optimization. Consulting with LC modeling experts will help validate our results. An added benefit of the research is that it will further the development of finite element methods for non-linear degenerate partial differential equations, and combine shape optimization with cutFEMs.

液晶在现代技术设备中是常见的，并且最着名的是用于电子显示器（例如LCD）中的光学特性。 使它们如此有用的是，它们很容易通过施加电压或磁场来操纵和控制。 此外，液晶可以与精细尺度的“物质粒子”相互作用。“因此，液晶使物质的精细尺度操纵成为可能。 该研究项目的目标是创建新的数学方法/算法来模拟液晶现象，并设计利用液晶的新材料。 换句话说，这项研究将为开发由液晶物理“驱动”的功能化和可切换材料提供基础。这项研究将影响当地社区、教育和液晶科学家：（i）中学科学博览会项目。 PI将指导少数民族服务中学的液晶科学展览项目（液晶的视觉吸引力使其成为项目主题的理想选择）。 PI将涉及博士后和研究生的指导。(ii)公共图书馆参与。 PI将通过当地图书馆与初中/高中学生互动，创建一个“与科学家坐在一起”的项目。 每次会议都在图书馆的分支位置举行，包括简短的介绍性演讲，然后进行动手活动，让学生积极学习液晶的物理和数学。 PI将通过让博士后和研究生在会议期间提供协助来参与。(iii)教育PI将继续通过雷乌斯和高级项目指导本科生，并根据本研究创建研究生课程“液晶的计算方法”。(iv)开放软件。PI将创建研究的软件实现、教程和演示。该研究将创建新的数学算法，以正确有效地模拟分子尺度以上的液晶（LC）现象。目前的算法不这样做。 它们要么为了数学上的方便而对底层模型进行特别的修改，要么计算起来很昂贵（或者两者兼而有之）。效率是至关重要的，以促进“插入”这些方法到高层次的设计和优化程序，比如说，材料设计。LC的分子模拟对于迭代设计工作来说太昂贵了。 Q张量模型更好，但仍然很昂贵，很难解决。 本文的研究工作包括：（i）利用离散极大值原理（DFM），为Q张量模型建立了新的求解方法，该方法不仅求解成本低，而且更接近物理模型;（ii）通过“切割”有限元方法（cutFEM），将我们的方法扩展到处理任意几何形状的问题;（iii）将我们的cutFEM扩展到LC系统的优化形状设计，包括胶体包裹体的自组装。 这项研究将创造新的方法来模拟和优化液晶系统，利用微妙的数学工具，如伽玛收敛和反射。 这非常符合我们的核心专业知识，例如LC，3D网格生成/实现，多物理几何流和形状优化的先前工作。 咨询LC建模专家将有助于验证我们的结果。 研究的另一个好处是，它将进一步发展非线性退化偏微分方程的有限元方法，并将联合收割机形状优化与切割有限元相结合。