Mathematical Sciences: Computational Techniques from Geometry and Statistical Physics

在过去的几年里，这位研究人员和他的同事们已经开发出具有深奥甚至推测特征的数值技术。这些技术的共同主题是在一个看似不同的领域使用数学机器和算法。特别是，研究人员使用统计物理和几何的方法来攻击计算物理中看似无关的问题。这项提案中正在研究的问题既是技术上有趣的，也是计算中最困难的问题之一。这项工作包括：(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)利用变温和退火法来解开和识别节点以及可能折叠的蛋白质；(4)利用基于双曲守恒律理论和微分几何的界面算法来分析运动细丝中的扭结和运动表面中的奇点；以及(5)将这些移动界面算法与平均曲率流分析相结合，以设计网格生成算法。通过这项工作，相当多的最先进的、强大的数学和数值技术将转化为在制造、能源工艺和材料科学中具有重要意义的实用工具。复杂的先进接口算法将用于计算计算机芯片中微电子部件制造中的详细工艺。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自动化计算流体力学模拟中的一个关键组件，该模拟存在于对绕过飞机、发动机和流体动力装置的流动进行建模和设计中。

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.