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

Sethian 研究员和他的同事研究微分方程解中奇点的运动，特别是水平集方法描述的界面运动，并应用于界面运动问题，以及三维空间中涡的运动，如涡方法和其他拉格朗日技术所述。 接口技术利用汉密尔顿-雅可比方法耦合到迎风计划的双曲型求解器;新的工作重点是开发快速的方法，多个接口的方法，并将能见度，表面扩散和反射/再发射的影响，在表面轮廓运动。 通过拉格朗日方法的涡运动开发机械重整化方法和聚合物分析从统计物理应用到流体现象，通过链接与Shenoy-Williams方法，和Kosterlitz无尾公式。 研究人员开发了跟踪界面和涡流运动的方法。 例如，在物质流动中，当物体或物质相遇或结束的地方随时间而变化时，就会出现这样的问题。 这些都是科学和工程中常见的情况。 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，该工作用于分析导致电子器件更高产量和效率的工艺步骤。 在应用于医学成像的图像增强、分割和识别方面，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，这些技术用于开发围绕复杂物体构建网格的新技术，如飞机和发动机设计所需。 拉格朗日和涡旋粒子的工作集中在尾迹中大湍流结构的产生，超流体中聚合物的运动和超导性的基本原理。

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.