Tombstone Example tomb-B for the Surface Evolver

This datafile adds a solder blob to the chip-pad model of tomb-A.fe.

Starting geometry.	Labelling of the vertices and edges of the liquid solder.

Evolved shape with 8236 facets.

Notable features:

• The solder fully wets the pad and the bottom face of the chip. This eliminates (for now) the necessity of dealing with contact angles, and lets the solder contact lines all be FIXED.
• The solder is completely enclosed in faces. This simplifies the definition of energy and volume, although we will see it has its inconveniences.
• The liquid solder faces are all given a realistic surface tension, using the parameter TENS. Note that changing TENS at runtime will not change the facet tension; rather one would have to set the facet tensions explicitly, as with "set facet tension 300 where not fixed".
• The datafile parameter scale_limit is set at 1/TENS, since the force is proportional to surface tension, and experience has shown that in minimizing area a normal scale factor is around 0.2, with 1 being a good upper bound on the scale factor. Too large a limit leads to trying too large a motion on the first iteration. Hence the scale_limit should be set to the inverse of the surface tension.
• There is a separate set of level-set constraints for the solder. It is good practice to separate constraints used for display surfaces from constraints used for surfaces used in energy calculations, even if the formulas are exactly the same.
• If the pressure were to turn out negative, there would be a danger of the solder surface going below the pad near the pad corners. It would be necessary to add a one-sided constraint on z to keep the surface above the pad.
• The read section of the datafile starts off with hessian_normal so the hessian command considers only motion perpendicular to the surface, which makes for much easier convergence. It does not converge to the absolutely best vertex positions for the triangulation, so g commands can still decrease energy, since g is not restricted to normal motions. The difference in energy, though, is probably much less than the imprecision due to the finite triangulation.
• The target_tolerance is set to 1e-14. This is the criterion for convergence to the target volume of the solder. The default value of 1e-4 is much too small, since the volume is 2e-8. This is one of the adjustments that has to be made for doing a physically small model.
• There is a command gogo for a typical short evolution. It starts out by refining some longer edges of the solder; the cutoff length of .0048 was chosen to get the desired edges refined. After that, it's just a little evolution at each refinement, just to set up the hessian commands for final convergence.
• The hessian command in gogo causes a conversion from the Evolver's original style of energy calculation to the newer, more systematic, named methods and named quantities style. Many energies have Hessian calculations only in named quantities mode, in this case gravitational energy. This is not the default mode yet, since it is a little slower for regular iteration. (The automatic conversion is new in version 2.11; it may be turned off by starting Evolver with the -a- option.)
• The gofar command evolves to a higher refinement, which was used to make the evolved image on this page.
• The command file "change_simple.cmd" is read in. It holds commands change_x0, change_y0, and change_z0 for smoothly changing the position of the chip. Simply changing the position parameters and relying on the constraints can do some ugly things to vertices in the middle of chip faces. Also, these commands linearly interpolate the motion of the liquid solder surface between the pad and the chip, giving smoother motions over large distances.
• The command file "forces.cmd" is read in. It holds commands calc_xf, calc_yf, and calc_zf to calculate the forces exerted on the chip by the surface tension of the solder. The method is to use the principle of virtual work, moving the chip a bit (using the change position commands) and finding the energy difference. Central differences are used for greater accuracy.
• This is a good file for getting a feeling of the rate of convergence of energies and forces as a function of refinement, since it evolves so easily. There is a script command defined that evolves and prints energies and z forces to a file called "tomb-b.out".

  Refinement	Solder facets	Energy	Z force
  0	20	0.01380870549	-5.331341239
  1	80	0.01285240203	-1.486858656
  2	320	0.01262314047	-1.331459046
  3	1280	0.01256553334	-1.376219965
  4	5120	0.0125521508	-1.387712914
  5	20480	0.01254865829	-1.386935502

  There is a reduction in error of about a factor of 4 for each refinement, which is what is expected for linear facets. If more accuracy is needed, then the quadratic model or an even higher order Lagrange model can be used. But the higher order model should only be invoked after the linear model converges, since evolution goes smoother in the linear model.