报告题目：MARS: An Analytic and Computational Framework for Incompressible Flows with Moving Boundaries
报告摘要：Current methods such as VOF methods, level-set methods, and phase-field methods avoid geometry and topology by converting them into problems of numerical PDEs. In comparison, we try to tackle geometric and topological problems with tools in geometry and topology.
As the first part of our MARS framework, we propose a topological space called the Yin space as a mathematical model for physically meaningful material regions. Each element in the Yin space is a regular open semianalytic set with bounded boundaries. We further equip the Yin space with Boolean algebra so that the topology info (such as the Betti numbers of a material region) can be extracted in constant time. In particular, non-manifold points on the fluid boundary, a key problem in studying topological changes, are handled naturally. The second part of MARS is the donating region theory in the context of hyperbolic conservation laws. For a fixed simple curve in a nonautonomous flow, the fluxing index of a passively advected Lagrangian particle is the total number of times it goes across the curve within a given time interval. Such indices naturally induce donating regions, equivalence classes of the particles at the initial time. Under the MARS framework, many explicit methods such as VOF methods and fronting tracking methods can be unified and proved to be second-order accurate. MARS also leads to new methods of fourth- and higher-order accuracy for interface tracking and curvature estimation.
The MARS framework can be further expanded with a fourth-order projection method called GePUP for numerically solving the incompressible Navier-Stokes equations (INSE). We have augmented GePUP to irregular domains and are currently working on coupling GePUP with our new interface tracking methods to form a fourth-order solver for INSE with moving boundaries.
报告人介绍:张庆海，1998年获清华大学学士学位，2001年获清华大学工学硕士学位，2008年获美国Cornell大学博士学位。之后在美国劳伦斯伯克利国家实验室从事计算数学方面的研究。2012年起任美国犹他大学数学系科研助理教授。2015年入选第十一批国家“千人计划”青年人才重点支持。2016年初开始任浙江大学数学科学学院教授。主要研究领域是与多相流相关的计算数学，重点是高精度自适应并行有限体积法、边界追踪问题、计算拓扑、流固耦合以及算法的数学分析。同时用数值模拟的手段研究一些具有重大意义的实际问题，如海啸模拟和甲壳类动物游泳等等。在国际高影响力期刊上发表SCI论文数十篇，包括SIAM Review, PNAS, CMAME, SIAM Journal on Numerical Analysis, SIAM Journal on Scientific Computing，Journal of Computational Physics等国际知名期刊。