Description

Determine the stress potential in an infinitely long cylinder when
torsion is applied.

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

The number of internal grid points can be specified using the command
line parameters --nx and --ny. COPS performance tests have been
reported for nx-1 = 50, ny-1 = 25, 50, 75, 100

Large Model of Type : NLP

Category : GAMS Model library

Main file : torsion.gms

$title Elastic-Plastic Torsion COPS 2.0 #15 (TORSION,SEQ=243)

$onText
Determine the stress potential in an infinitely long cylinder when
torsion is applied.

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

The number of internal grid points can be specified using the command
line parameters --nx and --ny. COPS performance tests have been
reported for nx-1 = 50, ny-1 = 25, 50, 75, 100


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

Glowinski, R, Numerical Methods for Nonlinear Variational
Problems. Springer Verlag, 1984.

Averick, B M, Carter, R G, More, J J, and Xue, G L, The
MINPACK-2 Test Problem Collection. Tech. rep., Mathematics
and Computer Science Division, Argonne National Laboratory,
1992.

Keywords: nonlinear programming, engineering, elastic-plastic torsion problem,
          elastic-plastic analysis
$offText

$if not set nx $set nx 51
$if not set ny $set ny 26

Set
   nx 'grid points in 1st direction' / x0*x%nx% /
   ny 'grid points in 2st direction' / y0*y%ny% /;

Alias (nx,i), (ny,j);

Parameter
   D(nx,ny) 'distance to the boundary'
   hx       'grid spacing for x'
   hy       'grid spacing for y'
   area     'area of triangle'
   c        'some constant' / 5.0 /;

hx   = 1/(card(nx) - 1);
hy   = 1/(card(ny) - 1);
area = 0.5*hx*hy;

D(i,j) = min(min(ord(i)-1,card(nx)-ord(i))*hx, min(ord(j)-1,card(ny)-ord(j))*hy);

Variable
   v(nx,ny) 'defines the finite element approximation'
   stress
   linLower
   linUpper
   quadLower
   quadUpper;

Equation
   defLL
   defLU
   defQL
   defQU
   defstress;

defLL.. linLower   =e= sum((nx(i+1),ny(j+1)), v[i+1,j] + v[i,j] + v[i,j+1]);

defLU.. linUpper   =e= sum((nx(i-1),ny(j-1)), v[i,j] + v[i-1,j] + v[i,j-1]);

defQL.. quadLower  =e= sum((nx(i+1),ny(j+1)), sqr((v[i+1,j] - v[i,j])/hx)
                                            + sqr((v[i,j+1] - v[i,j])/hy));

defQU.. quadUpper  =e= sum((nx(i-1),ny(j-1)), sqr((v[i,j] - v[i-1,j])/hx)
                                            + sqr((v[i,j] - v[i,j-1])/hy));

defstress.. stress =e= area*(   (quadLower + quadUpper)/2
                             -c*(linLower  + linUpper )/3);

v.lo(i,j) = -d(i,j);
v.up(i,j) =  d(i,j);
v.l (i,j) =  d(i,j);

Model torsion / all /;

display d, hx, hy, area;

$if set workSpace torsion.workSpace = %workSpace%

solve torsion minimizing stress using nlp;

$exit

* eliminate intermediate variables
Equation defstressx;

defstressx..
   stress =e= area*((  sum((nx(i+1),ny(j+1)), sqr((v[i+1,j] - v[i,j])/hx) + sqr((v[i,j+1] - v[i,j])/hy))
                     + sum((nx(i-1),ny(j-1)), sqr((v[i,j] - v[i-1,j])/hx) + sqr((v[i,j] - v[i,j-1])/hy)))/2
                   -c*(sum((nx(i+1),ny(j+1)), v[i+1,j] + v[i,j] + v[i,j+1])
                     + sum((nx(i-1),ny(j-1)), v[i,j] + v[i-1,j] + v[i,j-1]))/3);

Model torsionx / defstressx /;

$if set workSpace torsionx.workSpace = %workSpace%

solve torsionx minimizing stress using nlp;