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

点击翻译按钮获取中文摘要

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.