$title Distribution of Electrons on a Sphere COPS 2.0 #2 (ELEC,SEQ=230)

$onText
Given n electrons, find the equilibrium state distribution (of
minimal Coulomb potential) of the electrons positioned on a
conducting sphere.

This model is from the COPS benchmarking suite.
See http://www-unix.mcs.anl.gov/~more/cops/.

The number of electrons can be specified using the command line
parameter --np. COPS performance tests have been reported for np = 25,
50, 100, 200

Dolan, E D, and More, J J, Benchmarking Optimization
Software with COPS. Tech. rep., Mathematics and Computer
Science Division, 2000.

Morris, J R, Deaven, D M, and Ho, K M, Genetic Algorithm
Energy Minimization for Point Charges on a Sphere. Phys.
Rev. B. 53 (1996), R1740-R1743.

Saff, E B, and Kuijlaars, A, Distributing Many Points on
the Sphere. Math. Intelligencer 19 (1997), 5-11.

Keywords: nonlinear programming, Thomson problem, equilibrium state distribution,
          engineering, Coulomb's law
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$if     set n  $set np %n%
$if not set np $set np 25

Set
   i       'electrons' /i1*i%np%/
   ut(i,i) 'upper triangular part';

Alias (i,j);
ut(i,j)$(ord(j) > ord(i)) = yes;

Variable
   x(i)      'x-coordinate of the electron'
   y(i)      'y-coordinate of the electron'
   z(i)      'z-coordinate of the electron'
   potential 'Coulomb potential';

Equation
   obj     'objective'
   ball(i) 'points on unit ball';

obj.. potential =e= sum{ut(i,j), 1.0/sqrt(sqr(x[i] - x[j]) + sqr(y[i] - y[j]) + sqr(z[i] - z[j]))};

ball(i).. sqr(x(i)) + sqr(y(i)) + sqr(z(i)) =e= 1;

* Set the starting point to a quasi-uniform distribution
* of electrons on a unit sphere

Parameter theta(i), phi(i);
theta(i) = 2*pi*uniform(0,1);
phi(i)   =   pi*uniform(0,1);

x.l(i)   = cos(theta(i))*sin(phi(i));
y.l(i)   = sin(theta(i))*sin(phi(i));
z.l(i)   =               cos(phi(i));

Model elec / all /;

elec.workFactor = 5;

solve elec using nlp minimizing potential;