Find the surface with minimal area, given boundary conditions, and above an obstacle.
Plateau's problem is to determine the surface of minimal area with a given
closed curve in
as boundary. We assume that
the surface can be represented in nonparametric form
,
and we add the requirement that for some obstacle .
The solution of this obstacle problem [13]
minimizes the function
,
A finite element approximation to the minimal surface problem is obtained by triangulating and minimizing over the space of piecewise linear functions with values at the vertices of the triangulation. We set and use a triangulation with, respectively, and internal grid points in the coordinate directions. Data for this problem appears in Table 16.1.
We provide results for the AMPL formulation in Table 17.2.
For these results we fix and vary .
The starting guess is the function
evaluated at the grid nodes. We used
boundary data
Solver | ||||
LANCELOT | 2.77 s | 5.9 s | 10.34 s | 16.33 s |
2.51948e+00 | 2.51488e+00 | 2.50568e+00 | 2.50694e+00 | |
violation | 0.00000e+00 | 0.00000e+00 | 0.00000e+00 | 0.00000e+00 |
iterations | 8 | 9 | 10 | 13 |
LOQO | 2.98 s | 9.76 s | 23.32 s | |
2.51948e+00 | 2.51488e+00 | 2.50568e+00 | ||
violation | 2.4e-15 | 3.8e-15 | 3.4e-15 | |
iterations | 20 | 28 | 46 | |
MINOS | 103.76 s | 984.81 s | ||
2.51948e+00 | 2.51488e+00 | |||
violation | 0.0e+00 | 0.0e+00 | ||
iterations | 1 | 1 | ||
SNOPT | 137.88 s | |||
2.51948e+00 | ||||
violation | 0.0e+00 | |||
iterations | 171 | |||
Errors or warnings. Timed out. |