Volume preserving mean curvature flow in three space dimensions

Uwe F. Mayer

System:486DX4-100, Linux 1.2.13
Computation time:About 6 or 7 hours
Scale:Edge length = 1, box = 1 x 1 x 15, time step = 0.01
Vertices:Initially 498, finally 112
Facets:Initially 992, finally 220
Facet threshold:Eliminated facet if area is less than 1/5 of average of all facets
Comment:The facets all vanished at the ends of the tube.

The initial configuration.
after 0 steps

After 100 time steps.
after 100 steps

After 200 time steps.
after 200 steps

After 300 time steps.
after 300 steps

After 400 time steps.
after 400 steps

After 500 time steps.
after 500 steps

After 600 time steps.
after 600 steps

After 700 time steps.
after 700 steps

After 800 time steps.
after 800 steps

After 900 time steps.
after 900 steps

After 1000 time steps.
after 1000 steps

After 2000 time steps.

The flow has for all practical purposes reached its stationary limit: a sphere. It does not look perfectly round because the facets are of very different interior angles and sizes.

after 2000 steps

After refining the triangulation of the previous object and going another 200 steps it looks like this (442 vertices and 880 facets):

after 2200 steps

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mayer@math.utah.edu
Last updated: 01/29/97