// bga-4.fe // Simple ball grid array joint. // Circular, parallel, coaxial wetted pads. // Energy is surface tension of free surface, with. // Both pads represented with constraints. // Liquid entirely bounded by facets. // Same as bga-3.fe, but with variational integrals for z force. evolver_version "2.11d" // needed for command files // 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 parameter height = 0.12 // height of upper pad, cm parameter radius = 0.10 // radius of pads, cm // 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 // variational integrals for z force // first, area change method_instance area_change method facet_general_integral scalar_integrand: S_TENSION*(x4^2+x5^2)/sqrt(x4^2+x5^2+x6^2)/height // second, gravitational energy change method_instance grav_change method facet_vector_integral vector_integrand: q1: 0 q2: 0 q3: G*SOLDER_DENSITY*z^2/height // third, correction for volume change method_instance vol_fix method facet_vector_integral vector_integrand: q1: 0 q2: 0 q3: -body[1].pressure*z/height // put them together, modulus -1 since force is negative of gradient quantity zforce7 info_only modulus -1 global_method area_change global_method grav_change global_method vol_fix 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 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 14 2 8 15 3 9 16 4 10 17 5 11 18 6 12 faces // defined by oriented edge loops to have outward normal // lateral faces 1 1 14 -7 -13 tension S_TENSION 2 2 15 -8 -14 tension S_TENSION 3 3 16 -9 -15 tension S_TENSION 4 4 17 -10 -16 tension S_TENSION 5 5 18 -11 -17 tension S_TENSION 6 6 13 -12 -18 tension S_TENSION // 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 read "zforce.cmd" // contains a bunch of force calculations // To set volume to spherical segment, for zforce = 0 test (requires no grav) setvol := { body[1].target := pi*(radius^2+height^2/6)*height } // Typical evolution gogo := { u; g 3; r; g 5; r; g 5; hessian; hessian } // command to calculate and print all the ways to do zforce forces := { zforce1; zforce2; zforce3; zforce4; zforce5; zforce6; printf "zforce: %18.15f (variational integrands)\n", zforce7.value; } // A script for automatically generating force data for a range // of volumes and heights. read "runforce.cmd" // Command used to create data in chart example. // To be run right after loading. chartdata := { u; height := .10; recalc; V; V; V; body[1].target := .002; gogo; r; g 5; hessian; hessian; runforce; }