Mathematical Sciences: Computational Techniques from Geometry and Statistical Physics

在过去的几年里，研究人员和他的同事们开发了具有深奥甚至投机特征的数值技术。 这些技术的共同主题是在一个看似不同的竞技场中使用来自一个领域的数学机器和算法。 特别是，研究人员使用统计物理和几何的方法来解决计算物理中看似无关的问题。 在这项提案中调查的问题是技术上有趣的，在计算中最困难的。 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）分析具有非线性子系统的随机系统，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，并可能折叠蛋白质;（4）基于双曲守恒律理论和微分几何的界面算法分析运动细丝的扭结和运动曲面的奇异性;（5）将这些移动界面算法与平均曲率流分析相结合，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中重要的实用工具。 先进的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。 对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的目的是自动化的计算流体力学模拟的一个关键组成部分，目前在建模和设计流动过去的飞机，发动机，和流体动力装置。

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.