Numerical Approximation of Liquid Crystal Flows

9973285技术描述：本项目围绕向列液晶运动模拟算法的构建和分析展开。这样的算法将有助于模拟由液晶构成的设备（如计算机显示器）的制造和操作。表征液晶运动的方程是由Ericksen和Leslie在20世纪60年代提出的；然而，直到最近，很少有关于这些方程的存在性、唯一性和稳定性的一般结果。F. Lin和C. Liu最近关于新能量估计的发现使许多基本问题得以解答，这反过来又促进了这些方程近似算法的开发和分析。作者与C. Liu合作，对Ericksen- Leslie方程的一个具体实例的初步结果非常有希望，本建议概述了一个研究计划，该计划将为这些方程的全面一般性模拟提供鲁棒和可靠的代码。非技术描述：在过去的半个世纪里，计算机已经刺激了美国几乎所有工业产品的设计和制造的重大变化。计算机模拟产品的强度、操作和性能已经取代了任何产品推向市场之前所需的大量昂贵的原型和测试阶段。为了进一步降低设计和开发成本，从先进材料的模拟到复杂系统的操作，制造过程的每一步都需要更好的计算机算法。这个项目计划利用最近的理论突破来预测材料的行为，就像在现代液晶电脑显示器中发现的那样。这些材料表现出许多复杂的现象，其性质的模拟和预测需要现代高性能计算机。

9973285Technical Description:This project centers around the construction and analysis of algorithms for the simulation of the motion of nematic liquid crystals. Such algorithms will facilitate simulation of the manufacture and operation of devices (such as computer displays) built from liquid crystals. Equations characterizing the motion of liquid crystals were proposed by Ericksen and Leslie in the 1960's; however, until recently there were very few general results concerning the existence, uniqueness, and stability of these equations. Recent discoveries by F. Lin and C. Liu of new energy estimates enables many of these fundamental questions to be answered, which, in turn, facilitates the development and analysis of algorithms for the approximation of these equations. Preliminary results by the author, in collaboration with C. Liu, for a specific instance of the Ericksen--Leslie equations have been very promising, and this proposal outlines a program of research that will result in robust and reliable codes for the simulation of these equations in their full generality.Non-Technical Description:Over the past half century computers have stimulated a major change in the design and manufacture of virtually every industrial product produced in the U.S. Computer simulation of product strength, operation, and performance has replaced much of the expensive prototyping and testing phase required before any product is brought to market. In order to further reduce design and development costs better computer algorithms are required for each step of the manufacturing process, ranging from the simulation of advanced materials to the operation of complex systems. This project plans to draw upon recent theoretical breakthroughs to predict the behavior of materials like those found in modern liquid crystal computer displays. These materials exhibit many complex phenomena, and simulation and prediction of their properties requires modern high performance computers.