My research deals with the Front Tracking method in which surfaces of discontinuity are given explicit computational degrees of freedom, supplementing the continuous solution values at regular grid points to provide high quality and high resolution numerical solutions to physical problems. We have developed the concept of Front Tracking into a robust simulation code, parallelized, tested on multi-physics, and used for production simulations in three dimensions.
Two tracking methods, called grid-free and grid-based tracking, have been used to describe the three dimensional interface propagation and its topological bifurcation. The former is a pure Lagrangian method in which the interface propagation and redistribution are independent of the underlying Eulerian grid. This method is more accurate in the propagation of the interface position, but it is not robust in resolving the interface geometry and the topological bifurcations. The latter is just the opposite. That is, it is robust in resolving the interface topology, but poses a larger error in interface propagation. Its handling of the interface topology is through the reconstruction of the interface on Eulerian mesh blocks similar to that by Lorensen and Cline. Since topological changes are frequent in the computation of fluid interface instabilities, the grid-based tracking method has been used for most simulations, after an initial time interval. In a new development, we have combined these two methods to form the locally grid based tracking method, or LGB. In this method, we use Lagrangian algorithm for propagation and confine the topological bifurcation in small boxes to do Eulerian bifurcation. The use of Eulerian method is reduced to minimum and only used when and where the topological bifurcation is needed.
In addition, we have recently introduced a conservative front tracking algorithm. It preserves conservation properties of the system by enforcing conservation for all grid cells, including the ones cut by the front. We have extended the grid-based tracking into an interface separating multiple components. The most useful interface, after the case of two components in a block, is an interface separating three material components in a block. For such an interface in three dimensions, after rotation and commutation, the block interface can be attributed to 57 isomorphically distinct cases. We have built 57 subroutines for the front tracking code to handle all these cases.
Automatic mesh refinement (AMR) is another powerful tool to concentrate computational power in regions of computational difficulty. Block-structured adaptive Cartesian mesh refinement was proposed and developed by Berger and Colella. Our Front Tracking code has adopted the AMR by inter-operating with the Overture code developed by the Lawrence Livermore National Laboratory.
The Front Tracking method has been applied to the research and computation of various physical and scientific problems including the study of acceleration driven fluid interface instabilities, the computation of supernova formation, the simulation of spray in diesel fuel-injection jet and migration of white blood cells.