Surface Evolver catenoid example

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# Example: Catenoid.

The catenoid is the minimal surface formed between two rings not too far apart. In cylindrical coordinates, its equation is `r = (1/a)cosh(az)`. In cat.fe, both the upper and lower rings are given as one-parameter boundary wires. The separation and radius are parameters, so you can change them during a run with assignment statements or the A command. The initial radius given is the minimum for which a catenoid can exist for the given separation of the rings. To get a stable catenoid, you will have to increase this value. However, if you do run with the original value, you can watch the neck pinch out. The initial surface consists of six rectangles forming a cylinder between the two circles. The vertices on the boundaries are fixed, elsewise they would slide along the boundary to short-cut the curvature; two diameters is shorter than one circumference. The boundary edges are fixed so that vertices arising from subdividing the edges are likewise fixed.

| The initial catenoid skeleton, with vertices and edges numbered.

Here is the catenoid datafile:

    // cat.fe
    // Evolver data for catenoid.

    PARAMETER  RMAX = 1.5088795   // minimum radius for height
    PARAMETER  ZMAX = 1.0

    boundary 1 parameters 1     //  upper ring
    x1:  RMAX * cos(p1)
    x2:  RMAX * sin(p1)
    x3:  ZMAX

    boundary 2 parameters 1    //   lower ring
    x1:  RMAX * cos(p1)
    x2:  RMAX * sin(p1)
    x3:  -ZMAX

    vertices   // given in terms of boundary parameter
    1    0.00  boundary 1   fixed
    2    pi/3  boundary 1   fixed
    3  2*pi/3  boundary 1   fixed
    4    pi    boundary 1   fixed
    5  4*pi/3  boundary 1   fixed
    6  5*pi/3  boundary 1   fixed
    7    0.00  boundary 2   fixed
    8    pi/3  boundary 2   fixed
    9  2*pi/3  boundary 2   fixed
    10   pi    boundary 2   fixed
    11 4*pi/3  boundary 2   fixed
    12 5*pi/3  boundary 2   fixed

    edges
    1    1  2  boundary 1   fixed
    2    2  3  boundary 1   fixed
    3    3  4  boundary 1   fixed
    4    4  5  boundary 1   fixed
    5    5  6  boundary 1   fixed
    6    6  1  boundary 1   fixed
    7    7  8  boundary 2   fixed
    8    8  9  boundary 2   fixed
    9    9  10 boundary 2   fixed
    10   10 11 boundary 2   fixed
    11   11 12 boundary 2   fixed
    12   12 7  boundary 2   fixed
    13   1  7
    14   2  8
    15   3  9
    16   4  10
    17   5  11
    18   6  12

    faces
    1   1 14 -7 -13
    2   2 15 -8 -14
    3   3 16 -9 -15
    4   4 17 -10 -16
    5   5 18 -11 -17
    6   6 13 -12 -18

The parameter in a boundary definition is always `p1` (and `p2` in a two-parameter boundary). The Evolver can handle periodic parameterizations, as is done in this example. Try this sequence of commands (displaying at your convenience):

        r       (refine to get a crude, but workable, triangulation)
        u       (equiangulation makes much better triangulation)
        g 120   (takes this many iterations for neck to collapse)
        t 0.05  (collapse neck to single vertex by eliminating all
                          edges shorter than 0.05)
        o       (split neck vertex to separate top and bottom surfaces)
        g       (spikes collapse)

The catenoid shows some of the subtleties of evolution. Suppose the initial radius is set to `RMAX = 1.0` and the initial height to `ZMAX = 0.55` (these are pre-set in `catman.fe`). Fifty iterations with optimizing scale factor result in an area of 6.458483. At this point, each iteration is reducing the area by only .0000001, the triangles are all nearly equilateral, everything looks nice, and the innocent user might conclude the surface is very near its minimum. But this is really a saddle point of energy. Further iteration shows that the area change per iteration bottoms out about iteration 70, and by iteration 300 the area is down to 6.4336. The triangulation really wants to twist around so that there are edges following the lines of curvature, which are vertical meridians and horizontal circles. Hence the optimum triangulation appears to be rectangles with diagonals.

The evolution can be speeded up by turning on the conjugate gradient method with the U command. With `catman.fe`, try the script "`r; u; U; g 70`". For conjugate gradient cognoscenti, the saddle point demonstrates the difference between the Fletcher-Reeves and Polak-Ribiere versions of conjugate gradient. The saddle point seems to confuse the Fletcher-Reeves version (which used to be the default). However, the Polak-Ribiere version (the current default) has little problem. The U command toggles conjugate gradient on and off, and ribiere toggles the Polak-Ribiere version. With Fletcher-Reeves conjugate gradient in effect, the saddle point is reached at iteration 17 and area starts decreasing again until iteration 30, when it reaches 6.4486. But then iteration stalls out, and the conjugate gradient mode has to be turned off and on to erase the history vector. Once restarted, another 20 iterations will get the area down to 6.4334. In Polak-Ribiere mode, no restart is necessary.

Exercise for the reader: Get the Surface Evolver to display an unstable catenoid by declaring the catenoid facets to be the boundary of a body, and adjusting the body volume with the b command to get zero pressure. See the sample datafile `catbody.fe`.

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