Example Calculations

There are numerous examples of applications for the level set and fast marching methods. A comprehensive list of the most recent applications will be given in Section 4.4. Two simple examples which illustrate the basic level set method are presented here.

Minimal Surfaces with Voids

As noted in the introduction, the original paper on the level set function involved a speed function which depended on mean curvature. Flow by mean curvature was also used to compute examples of minimal surfaces [19]. In the present example, flow by mean curvature is again used, but this time there are void regions where the surface area contained in the void is not counted in the total surface area.

It is shown in [123] that the minimal surface in this case will meet the voids orthogonally. The orthogonality boundary condition can be rewritten in a way that is familiar. Suppose $ is the evolving surface moving by mean curvature, and assume f is a level set function representing the voids with the surface of the voids identified by f = 0. The orthogonality boundary condition is equivalent to the surface normals being orthogonal; in other words, we must have

This equation is reminiscent of the equation for velocity extensions, Eq. 4.32. In fact, the velocity extension algorithm is used to determine $ inside the voids.

In Fig. 4.9, a surface which passes through five spherical voids is illustrated. Initially, the surface passes over the central void. As the surface relaxes, it strikes the center sphere and finally reaches equilibrium on the lower side of the sphere. The voids are semitransparent so that the results of applying the velocity extension code to $ can be seen. Also, the shading on the surface indicates the magnitude of the mean curvature.

Curvature Flow in Hyperbolic Space

In [25], mean curvature flow in hyperbolic space mapped onto the upper halfspace is investigated. In particular, foliations of the space are computed using a

Figure 4.9: Example of curvature flow with voids.

Figure 4.10: Change in topology of prescribed level for a notched annulus. Time steps are (a) t = 0, (b) t = 0.05, (c) t = 0.1, (d) t = 0.15, (e) t = 0.2, and (f) t = 0.5. Reprinted with permission from [25].

Figure 4.10: Change in topology of prescribed level for a notched annulus. Time steps are (a) t = 0, (b) t = 0.05, (c) t = 0.1, (d) t = 0.15, (e) t = 0.2, and (f) t = 0.5. Reprinted with permission from [25].

flow of the form dp

where V is a closed curve in the 2 = 0 plane.

In Fig. 4.10, a sample evolution of one of the level curves is shown. One of the questions addressed in [25] is whether all disks in a foliation are topologically disks. In Fig. 4.10, the resolving of the topology for a particular leaf in a foliation is illustrated as it evolves in time. The numerical experiments conducted in [25] suggest that the answer is that the foliation is of disks, even for very complicated boundary curves.

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