elec.gms : Distribution of electrons on a sphere COPS 2.0 #2

**Description**

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

**References**

- 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. Physical Review B 53, 4-15 (1996), R1740-R1743.
- Saff, E B, and Kuijlaars, A, Distributing Many Points on a Sphere. The Mathematical Intelligencer 19, 1 (1997), 5-11.

**Large Model of Type :** NLP

**Category :** GAMS Model library

**Main file :** elec.gms

```
$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
$offText
$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;
```