# Reference: Michael M. Kostreva,
# "Elasto-Hydrodynamic Lubrication: a Non-linear Complementarity
# Problem", Int. Journal for Num. Methods in Fluids (4), 377-397 (1984).
# The lubricant film gap and the pressure between two lubricated
# elastic cylinders in line contact are calculated. When the
# pressure is positive, Reynolds's equation must be satisfied;
# when the pressure is 0, the surfaces must diverge.
# The load (in pounds) is represented by alpha. The speed
# (in rpm) of the cylinders is represented by lambda. A more
# elaborate version of this model would involve a table of
# (alpha, lambda) pairs.
# In this variant, h is a "defined variable", which AMPL automatically
# substitutes out: the solver does not see it.
param N integer >= 1, default 100; # grid 0..N;
# half-points go 1 .. N
param pi := 4 * atan(1);
param xa := -3;
param xf > xa, := 2;
param dx := (xf - xa) / N;
param alpha := 2.832;
param lambda := 6.057;
param w {i in 0..N} := if i in {0,N} then 0.5 else 1;
var k := 1.6;
var p {i in 1..N} >= 0, := max(0, 1 - abs((xa +1 + i*dx)/2));
var h {j in 0..N} = ( xa + (j+.5)*dx)^2 + k + 1
+ 1/pi * (sum {i in 0..N} w[i] * (i-j-0.5)*dx
* log(abs(i-j-.5)*dx) *
((if i < N then p[i+1])
- (if i > 1 then p[i-1]) ) ) ;
s.t. psum: ( 1 - dx*2/pi * sum {i in 1..N} w[i]*p[i] ) == 0;
s.t. reynolds {i in 1..N}: p[i] >= 0 complements
( (lambda / dx) * (h[i]-h[i-1]) - (1/dx^2) * (
h[i]^3 * ((if i < N then p[i+1])-p[i]) /
exp(alpha*((if i < N then p[i+1])+p[i])*.5)
- h[i-1]^3 * (p[i]-(if i > 1 then p[i-1])) /
exp(alpha*(p[i]+(if i > 1 then p[i-1]))*.5) )
) >= 0;