1996, 1999, 2006
J.A. Sethian

Coupling Level Set Methods to Physical Problems
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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.

In order to develop a general level set implementation of such problems, we need to make an important observation. The Narrow Band level set methods contains two embeddings. First, the propagating interface is embedded as the zero level set of the higher dimensional level set function. This allows one to track topological changes in the interface, and calculate geometric quantities from the smooth level set function.

Equally important, the speed F of the interface in its normal direction is embedded in this higher dimensional view; an extension speed function F_ext must be defined must be defined in a neighborhood of the zero level set. In other words, we must extend the speed F off of the interface to the neighboring level sets.

The choice of which extension to use is crucial. While tempting to use those given by the underlying physics (such as, for example, the fluid velocity away from the front), this leads to poor results, including the loss of mass and inaccuracy. Remedies to repair these errors through so-called "re-initialization techniques" (that is, stopping the calculation every time step and building a wholly new level set function) are fraught with their own problems; they can change the position of the interface.

Adding Material Transport

Narrow Band level set methods track the trace of the propagating front: all information about the original parameterization is lost. This is the reason it handles topological changes so effortlessly. However, sometimes additional quantities are carried by transport along the front, and these are required in order to correctly solve the associated physics.

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.

Annotated References:

In Ref. 1 below, a general technique was presented to build extension velocities, based only on the value of the speed function F on the front. This choice of extension velocity F_ext moves the interface under the correct speed, and can be shown to keep the smoothness of the interface as it evolves. The equation that gives this extension velocity may be quickly solved using the Fast Marching Method . This provides a general way to couple level set methods to physical problems.

In Ref. 2 below, the idea of adding an additional embedded function to carry material transport is discussed.

New Book and Resource on Level Set and Fast Marching Methods


  1. The Fast Construction of Extension Velocities in Level Set Methods : Adalsteinsson, D., and Sethian, J.A., 148, pp. 2-22, 1999.

    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.

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  2. Transport and Diffusion of Material Quantities on Propagating Interfaces via Level Set Methods : ,
          Adalsteinsson, D., and Sethian, J.A., J. Comp. Phys, 185, 1, pp. 271-288, 2002

    We develop theory and numerical algorithms to apply level set methods to problems involving the transport and diffusion of material quantities in a level set framework. Level set methods are computational techniques for tracking moving interfaces; they work by embedding the propagating interface as the zero level set of a higher dimensional function, and then approximate the solution of the resulting initial value partial differential equation using upwind finite difference schemes. The traditional level set method works in the trace space of the evolving interface, and hence disregards any parameterization in the interface description. Consequently, material quantities on the interface which themselves are transported under the interface motion are not easily handled in this framework. We develop model equations and algorithmic techniques to extend the level set method to include these problems. We demonstrate the accuracy of our approach through a series of test examples and convergence studies.

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