Christopher J. Kimmer, Ph.D.
IU Southeast Informatics
iSci
the informatics of scientific computing
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Implementation
Evolving a triangulated surface is more an art than a science as triangles must be interactively added or deleted, and the triangulation needs to be kept sufficiently equiangular. Only very close to a minimum (say after the surface area has converged to 8 places) or at very high levels of refinement (10000s of faces) may the evolution be left to run on its own.
Consequently, systematic methods for evolution cannot be precisely defined, but guidelines can be stated. If necessary, an initial surface patch is first modified using the Surface Evolver's capabilities. Specifically, small triangles are eliminated so that all the triangles' areas are the same within an order of magnitude, and edge lengths are also kept within an order of magnitude of each other. Triangle and edge deletion is accomplished by merging one endpoint of an edge with the other endpoint, deleting two triangles in the process. Then, the triangulation is ``vertex averaged'' so that a vertex is moved based on a weighting function designed to equalize the areas of the triangles. The Evolver offers an equiangulation command that transforms the triangulation to one that is more equiangular by utilizing an update move that examines two triangles sharing a common edge. If the angles opposite the edge add to more than 180 o , the triangulation can locally be made more equiangular by deleting the common edge and replacing it with the line connecting the two vertices opposite the original edge. During a surface's evolution, equiangulations are done every 100 to 1000 iterations until the area is well-converged or the triangulation has stabilized and is sufficiently equiangular. After a minimum has been reached, the vertices are given small random displacements to search for lower minima, and if it appears that a global minimum has been reached, the surface is refined and equiangulated before the process is repeated for the newly-obtained triangulation. Details for these triangulation-modifying algorithms are given by Brakke[ 52 , 51 ].
Frequently, the instabilities in evolution using discrete mean curvature mean that several triangles quickly become three or more orders of magnitude smaller in area than the average area over the triangles. These small triangles are elongating as they attempt to degenerate into an edge. The appearance of these triangles can be discerned by watching the step-size taken by the Surface Evolver along its minimization direction. A rapid decrease in the order of magnitude of this quantity when the energy is not at a minimum signifies that these very small triangles should be deleted before the evolution proceeds.
Chris Kimmer 2011-06-01