HISTORY OF THE METHODS/FLOW CHART
ABOUT THE AUTHOR/CV
1996, 1999, 2006
Coupling Level Set Methods to Physical Problems
Coupling to General Physics
In order to couple level set methods to most physical applications,
one must link the position and motion of the interface to physical
effects on one or both sides of the interface. This linking often
requires solving relevant partial differential equations on either side
of the front, and employs jump boundary conditions at the interface
itself. Thus, the position of the front influences the physics, and the
physics then prescribes the motion of the interface.
To handle this problem, a new technique is used to carry an additional embedded function, defined in all of space. The time-evolution partial differential equation for this function is solved as part of the simultaneous update along with the level set advancement.
Level set techniques are numerical techniques for tracking the evolution of interfaces. They rely on two central embeddings; first the embedding of the interface as the zero level set of a higher dimensional function, and second, the embedding (or extension) of the interface's velocity to this higher dimensional level set function. This paper applies Sethian's Fast Marching Method, which is a very fast technique for solving the Eikonal and related equations, to the problem of building fast and appropriate extension velocities for the neighboring level sets. Our choice and construction of extension velocities serves several purposes. First, it provides a way of building velocities for neighboring level sets in the cases where the velocity is defined only on the front itself. Second, it provides a sub-grid resolution in some cases not present in the standard level set approach. Third, it provides a way to update an interface according to a given velocity field prescribed on the front in such a way that the signed distance function is maintained, and the front is never re-initialized; this is valuable in many complex simulations. In this paper, we describe the details of such implementations, together with speed and convergence tests, and applications to problems in visibility relevant to semi--conductor manufacturing and thin film physics.