Mathematical Sciences: Computational Techniques from Geometry and Statistical Physics

数学科学：几何和统计物理的计算技术

• 批准号: 9414631
• 负责人: Alexandre Chorin
• 金额: $ 5万
• 依托单位: University of California-Berkeley
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 1994
• 资助国家: 美国
• 起止时间: 1994-06-15 至 1996-05-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9414631&HistoricalAwards=false
• 关键词: Mathematical; Sciences; Computational; Techniques; Geometry

Over the past few years, the investigator and his colleagues have developed numerical techniques with esoteric and even speculative features. The common theme in these techniques is the use of mathematical machinery and algorithms from one area in a seemingly different arena. In particular, the investigators have used methods from statistical physics and geometry to attack seemingly unrelated problems in computational physics. The problems under investigation in this proposal are both technologically interesting and among the most difficult in computing. The work encompasses: (1) applications of statistical mechanics to fluid mechanics and turbulence theory; (2) the analysis of stochastic systems with filamentary subsystems, in particular polymers and quantum vortices; (3) the exploitation of variable temperature and annealing techniques to untangle and recognize knots and possibly fold proteins; (4) the analysis of kinking in moving filaments and singularities in moving surfaces by interface algorithms based on hyperbolic conservation law theory and differential geometry; and (5) coupling these moving interface algorithms with mean curvature flow analysis to design algorithms for grid generation. Through this work, a considerable body of state-of-the-art, powerful mathematical and numerical techniques will be transformed into practical tools of importance in manufacturing, energy processes, and material science. Sophisticated advancing interface algorithms will be used to compute detailed processes in the manufacture of microelectronic components in computer chips. Research on statistical mechanics and fluid turbulence will be used as part of computer simulations of fluid mixing in combustion processes. Finally, surface advancement techniques for grid generation are aimed at automating a key component of computational fluid mechanics simulations present in modeling and designing flows past aircraft, in engines, and in hydrodynamic devices.

在过去的几年里，研究者和他的同事开发了具有深奥甚至推测性特征的数值技术。 这些技术的共同主题是在看似不同的领域使用来自一个领域的数学机制和算法。 特别是，研究人员使用统计物理学和几何学的方法来解决计算物理学中看似不相关的问题。 该提案中所研究的问题既在技术上很有趣，又是计算中最困难的问题之一。 工作内容包括：（1）统计力学在流体力学和湍流理论中的应用； (2) 具有丝状子系统的随机系统的分析，特别是聚合物和量子涡旋； (3) 利用可变温度和退火技术来解开和识别结以及可能折叠的蛋白质； (4)基于双曲守恒定律理论和微分几何的接口算法分析运动细丝的扭结和运动表面的奇点； (5)将这些移动界面算法与平均曲率流分析结合起来设计网格生成算法。 通过这项工作，大量最先进、强大的数学和数值技术将转化为制造、能源过程和材料科学领域重要的实用工具。 先进的先进接口算法将用于计算计算机芯片中微电子元件制造的详细过程。 统计力学和流体湍流的研究将被用作燃烧过程中流体混合的计算机模拟的一部分。 最后，网格生成的表面推进技术旨在实现计算流体力学模拟的关键组成部分的自动化，该模拟存在于飞机、发动机和流体动力装置的建模和设计流中。