Biomedical Engineering Reference
In-Depth Information
The second fundamental idea was the adaptation of numerical methods de-
veloped for hyperbolic conservation laws. The field of numerical hyperbolic
conservation laws is a mature field with a substantial body of research devoted
to both the theory and practice of these methods. Much of this field is concerned
with the construction of numerical flux functions, which approximate the phys-
ical flux function in a way which respects the propagating characteristics of
the problem. The resulting numerical methods more accurately compute the
speeds for propagating shocks, and find the unique entropy condition satisfy-
ing rarefaction fans. In [102], Sethian observed that the theory of hyperbolic
conservation laws could be applied to the problem of propagating interfaces.
This naturally led to [85], where the equation for propagating the interface
using the implicit representation was formulated as the integral of a hyper-
bolic conservation law. In the context of moving interfaces, the shocks became
corners in the interface, and the rarefaction fans became regions of interface
expansion.
The coupling of the numerical hyperbolic conservation laws with the implicit
representation led to the first level set method, which was demonstrated to be
a powerful, robust method for solving the flame propagation problem.
Though the level set method, in its original form, was successful for the
original application, it was soon observed by Chopp [19] that a fundamental
problem in the method still existed. At that point, nearly all of the applications of
the level set method involved interface speed functions which depended solely
upon mean curvature. This class of problems is very special, as indicated in
[39-42], because the embedding function maintained bounded gradients almost
everywhere, giving the method additional stability properties. This property
does not hold for a general interface speed function, and so for the level set
method to be generalizable, one important modification was required in order
to maintain a stable level set method.
The key modification to the level set method, proposed in [19], was to observe
that forcing the embedding function to maintain bounded gradients was possible,
without changing the underlying motion of the interface. This process was called
reinitialization, and it essentially forced the embedding function to be the signed
distance function, even if the level set evolution equation would not do it on its
own. Once this piece was added to the level set method toolbox, the level set
method exploded in popularity, being used in a wide array of interface motion
applications.
