CAREER: Numerical Methods For Liquid Crystals And Their Optimal Design

液晶在现代技术设备中很常见，最著名的是其光学特性在电子显示器（例如lcd）中的应用。它们之所以如此有用，是因为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，液晶使物质的精细操作成为可能。本研究项目的目标是创造新的数学方法/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。该研究将对当地社区、教育和液晶科学家产生影响：(i)中学科学博览会项目。PI将在少数民族中学指导有关液晶的科学展览项目（液晶的视觉吸引力使其成为项目主题的理想选择）。PI将有博士后和研究生参与指导。（ii）公共图书馆业务。PI将通过当地图书馆与中学生互动，创建一个“与科学家坐在一起”的项目。每节课都在图书馆分馆举行，包括一个简短的介绍，然后是实践活动，让学生积极学习液晶的物理和数学。PI将包括博士后和研究生，让他们在会议期间协助。(3)教育。PI将继续通过reu和高级项目指导本科生，并在此基础上开设研究生课程“液晶计算方法”。（iv）开放软件。PI将创建该研究的软件实现、教程和演示。该研究将创造新的数学算法来正确有效地模拟分子尺度以上的液晶现象。目前的算法做不到这一点。它们要么对底层模型进行临时更改以方便数学计算，要么计算成本很高（或两者兼而有之）。效率对于促进将这些方法“插入”到高级设计和优化程序（例如材料设计）中至关重要。lc的分子模拟对于迭代设计工作来说过于昂贵。q -张量模型更好，但仍然昂贵且难以求解。本研究主要做了以下工作：(1)利用离散极大值原理（DMP）为q -张量模型创造了求解成本更低、更忠实于物理模型的新方法；（ii）通过“切割”有限元法（cutFEM）扩展我们的方法来处理任意几何形状；（iii）扩展我们的cutFEM来进行LC系统的最佳形状设计，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，利用精细的数学工具，如伽马收敛和DMP。这与我们的核心专业知识非常吻合，例如之前在lc， 3d网格生成/实现，多物理场几何流和形状优化方面的工作。咨询LC建模专家将有助于验证我们的结果。该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，并将形状优化与切削fem相结合。

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.