Mathematical Sciences: Computational Techniques from Geometry and Statistical Physics Applied to Fluid Mechanics and Interface Problems

Sethian和他的同事研究了微分方程解中奇点的运动，特别是用Level-Set方法描述并应用于界面运动问题的界面运动，以及用涡旋方法和其他拉格朗日技术描述的三维空间中的涡旋运动。界面技术将哈密顿-雅可比方法与迎风格式相结合用于双曲型求解器；新的工作集中于开发快速方法，多界面方法，以及在表面轮廓运动中结合可见性、表面扩散和反射/再发射的影响。通过拉格朗日方法的涡旋运动发展了从重整化方法到聚合物分析的机械，从应用于流体现象的统计物理到与谢诺伊-威廉姆斯方法和Kosterlitz-Thouless公式的联系。研究人员开发了跟踪界面和漩涡运动的方法。例如，这样的问题出现在物流中，每当物体或材料相遇或结束的地方随着时间的变化而变化。这些情况在整个科学和工程领域都很常见。这项工作适用于许多领域。在微电子元件制造中的蚀刻、沉积和光刻中，这项工作被用来分析导致更高产量和电子设备效率的工艺步骤。在应用于医学成像的图像增强、分割和识别中，这项工作导致了用于识别CT和MRI扫描中的心脏数据的自动软件。在网格生成中，这些技术被用来开发新的技术，以建立复杂物体周围的网格，以满足飞机和发动机设计的需要。拉格朗日粒子和涡旋粒子的工作集中在尾迹中大湍流结构的产生，聚合物在超流体中的运动，以及超导的基本原理。

Sethian The investigator and his colleague study the motion of singularities in solutions of differential equations, in particular, the motion of interfaces as described by level-set methods and applied to problems in interface motion, and the motion of vortices in three space dimensions, as described by vortex methods and other Lagrangian techniques. The interface techniques exploit Hamilton-Jacobi methodology coupled to upwind schemes for hyperbolic solvers; new work focuses on developing fast methods, methods for multiple interfaces, and the incorporation of effects of visibility, surface diffusion, and reflection/re-emission in surface profile motion. The vortex motion through Lagrangian methods develops machinery from renormalization methods and polymer analysis from statistical physics applied to fluid phenomena through linkage with Shenoy-Williams methodology, and Kosterlitz-Thouless formulations. The investigators develop methods to follow the motion of interfaces and vortices. Such problems arise in flows of materials, for instance, and whenever the places where objects or materials meet or end change with time. These are common situations throughout science and engineering. The work is applied to a number of areas. In etching, deposition, and lithography in the manufacture of microelectronic components, the work is used to analyze process steps leading to higher yield and efficiency of electronic devices. In image enhancement, segmentation, and recognition applied to medical imaging, the work leads towards automatic software for recognizing cardiac data in CT and MRI scans. In grid generation, the techniques are used to develop new techniques for building grids around complex bodies, as required in aircraft and engine design. The Lagrangian and vortex particle work focuses on generation of large turbulent structures in wakes, motions of polymers in superfluids, and fundamentals of superconductivity.