Mathematical Sciences: Computational Techniques from Geometry and Statistical Physics

在过去的几年里，这位研究者和他的同事们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一个看似不同的领域中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和几何的方法来解决计算物理中看似无关的问题。本提案中所研究的问题在技术上是有趣的，在计算中也是最困难的。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)具有细丝子系统的随机系统的分析，特别是聚合物和量子涡旋；(3)利用变温和退火技术来解开和识别结和可能折叠的蛋白质；(4)基于双曲守恒理论和微分几何的界面算法分析运动细丝的扭结和运动表面的奇异性；(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，大量先进的、强大的数学和数值技术将转化为制造业、能源过程和材料科学中重要的实用工具。复杂先进的接口算法将用于计算微电子元件在计算机芯片中制造的详细过程。统计力学和流体湍流的研究将作为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的关键组成部分，这些模拟和设计存在于飞机、发动机和流体动力装置中的流动。

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.