// bga-6.fe
// Simple ball grid array joint.
// Circular, parallel, coaxial wetted pads. With gravity.
// Same as bga-5.fe, but with upper pad load and optimizing pad height.
// Both pads represented with constraints.
// Liquid entirely bounded by facets.
evolver_version "2.11c" // minimum Evolver version needed
// physical constants, in cgs units
parameter S_TENSION = 480 // liquid solder surface tension, erg/cm^2
parameter SOLDER_DENSITY = 9 // grams/cm^3
gravity_constant 980 // cm/sec^2
// configuration parameters
optimizing_parameter height = 0.12 delta=1e-5 // height of upper pad, cm
parameter radius = 0.1 // radius of pads, cm
parameter pad_weight = 500 // weight of upper pad, dynes
// upper pad energy
quantity pad_energy energy method vertex_scalar_integral
scalar_integrand: pad_weight*z
// lower pad
constraint 1
formula: z = 0
// upper pad
constraint 2
formula: z = height
// rim of pads
constraint 3
formula: x^2 + y^2 = radius^2
// lower barrier (substrate off pad, nonwetting)
constraint 4 nonnegative
formula: z
// upper barrier (upper chip off pad, nonwetting)
constraint 5 nonpositive
formula: z - height
// lower substrate off pad, exact constraint
constraint 6
formula: z = 0
// upper chip off pad, exact constraint
constraint 7
formula: z = height
vertices
// lower pad
1 radius*cos(0*pi/3) radius*sin(0*pi/3) 0 constraints 1,3 fixed
2 radius*cos(1*pi/3) radius*sin(1*pi/3) 0 constraints 1,3 fixed
3 radius*cos(2*pi/3) radius*sin(2*pi/3) 0 constraints 1,3 fixed
4 radius*cos(3*pi/3) radius*sin(3*pi/3) 0 constraints 1,3 fixed
5 radius*cos(4*pi/3) radius*sin(4*pi/3) 0 constraints 1,3 fixed
6 radius*cos(5*pi/3) radius*sin(5*pi/3) 0 constraints 1,3 fixed
// upper pad
7 radius*cos(0*pi/3) radius*sin(0*pi/3) height constraints 2,3 fixed
8 radius*cos(1*pi/3) radius*sin(1*pi/3) height constraints 2,3 fixed
9 radius*cos(2*pi/3) radius*sin(2*pi/3) height constraints 2,3 fixed
10 radius*cos(3*pi/3) radius*sin(3*pi/3) height constraints 2,3 fixed
11 radius*cos(4*pi/3) radius*sin(4*pi/3) height constraints 2,3 fixed
12 radius*cos(5*pi/3) radius*sin(5*pi/3) height constraints 2,3 fixed
// dummy vertex for upper pad load
13 0 0 height constraint 2 fixed bare pad_energy
edges // defined by endpoints
// lower pad edges
1 1 2 constraints 1,3 fixed
2 2 3 constraints 1,3 fixed
3 3 4 constraints 1,3 fixed
4 4 5 constraints 1,3 fixed
5 5 6 constraints 1,3 fixed
6 6 1 constraints 1,3 fixed
// upper pad edges
7 7 8 constraints 2,3 fixed
8 8 9 constraints 2,3 fixed
9 9 10 constraints 2,3 fixed
10 10 11 constraints 2,3 fixed
11 11 12 constraints 2,3 fixed
12 12 7 constraints 2,3 fixed
// vertical edges
13 1 7 constraints 4,5
14 2 8 constraints 4,5
15 3 9 constraints 4,5
16 4 10 constraints 4,5
17 5 11 constraints 4,5
18 6 12 constraints 4,5
faces // defined by oriented edge loops to have outward normal
// lateral faces
1 1 14 -7 -13 tension S_TENSION constraints 4,5
2 2 15 -8 -14 tension S_TENSION constraints 4,5
3 3 16 -9 -15 tension S_TENSION constraints 4,5
4 4 17 -10 -16 tension S_TENSION constraints 4,5
5 5 18 -11 -17 tension S_TENSION constraints 4,5
6 6 13 -12 -18 tension S_TENSION constraints 4,5
// lower pad
7 -6 -5 -4 -3 -2 -1 fixed no_refine color red tension 0
// upper pad
8 7 8 9 10 11 12 fixed no_refine color green tension 0 constraint 2
bodies // defined by oriented face list
1 1 2 3 4 5 6 7 8 volume 1.3*pi*radius^2*height density SOLDER_DENSITY
read
hessian_normal
// command to transfer vertices that hit one-sided constraints to exact constraints
nail := { nailcount := 0;
foreach vertex vv where hit_constraint 4 do
{ unset vv constraint 4; set vv constraint 6; nailcount := nailcount + 1 };
foreach vertex vv where hit_constraint 5 do
{ unset vv constraint 5; set vv constraint 7; nailcount := nailcount + 1 };
printf "Nailed vertices: %g\n",nailcount;
}
// command to loosen up vertices that maybe should not be nailed.
unnail := { unnailcount := 0;
foreach vertex vv where on_constraint 6 do
{ unset vv constraint 6; set vv constraint 4; unnailcount := unnailcount + 1 };
foreach vertex vv where on_constraint 7 do
{ unset vv constraint 7; set vv constraint 5; unnailcount := unnailcount + 1 };
printf "Unnailed vertices: %g\n",unnailcount;
}
// Typical evolution
gogo := { u; g 3; r; g 5; r; g 5; hessian; hessian }