Abstract

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.