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CHAIN.SIF
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***************************
* SET UP THE INITIAL DATA *
***************************
NAME CHAIN
* Problem :
* *********
* Find the chain (of uniform density) of length L suspended between
* two points with minimal potential energy
* This is problem 3 in the COPS (Version 2) collection of
* E. Dolan and J. More'
* see "Benchmarking Optimization Software with COPS"
* Argonne National Labs Technical Report ANL/MCS-246 (2000)
* SIF input: Nick Gould, November 2000
* classification OOR2-AN-V-V
* The number of subintervals
*IE NH 50 $-PARAMETER
*IE NH 100 $-PARAMETER
*IE NH 200 $-PARAMETER
IE NH 400 $-PARAMETER
* The length of the suspended chain
RE L 4.0
* The height of the chain at t=0 (left)
RE A 1.0
* The height of the chain at t=1 (right)
RE B 3.0
* The ODE is defined in [0,TF]
RE TF 1.0
* The uniform interval length
RI RNH NH
R/ H TF RNH
* If B > A then TMIN = 0.25 else TMIN = 0.75
RE TMIN 0.25
* Other useful values
IE 0 0
IE 1 1
IA NH-1 NH -1
RM H/2 H 0.5
R- B-A B A
R( ABSB-A ABS B-A
RM 4ABSB-A ABSB-A 4.0
VARIABLES
DO I 0 NH
X X(I) $ Height of the chain
X U(I) $ Derivative of X
ND
GROUPS
* Objective function: potential energy
N PE
* U is derivative of X:
* X[I] - X[I+1] + 0.5*H*(U[I] + U[I+1]) = 0
DO I 0 NH-1
IA I+1 I 1
XE D(I) X(I) 1.0 X(I+1) -1.0
ZE D(I) U(I) H/2
ZE D(I) U(I+1) H/2
ND
* Length is L
E LENGTH
CONSTANTS
Z CHAIN LENGTH L
BOUNDS
XR CHAIN 'DEFAULT'
ZX CHAIN X(0) A
ZX CHAIN X(NH) B
START POINT
DO K 0 NH
RI RK K
R/ K/NH RK RNH
RM 0.5K/NH K/NH 0.5
* X[K] := 4 abs(B-A) * (K/NH) * (0.5*(K/NH) - TMIN) + A
R- PAREN 0.5K/NH TMIN
R* XK 4ABSB-A PAREN
R* XK XK K/NH
R+ XK XK A
ZV CHAIN X(K) XK
* U[K] := 4 abs(B-A) * ((K/NH) - TMIN)
R- PAREN K/NH TMIN
R* UK 4ABSB-A PAREN
ZV CHAIN U(K) UK
ND
ELEMENT TYPE
EV XSU X U
EV SU U
ELEMENT USES
DO I 0 NH
XT XSU(I) XSU
ZV XSU(I) X X(I)
ZV XSU(I) U U(I)
ND
DO I 0 NH
XT SU(I) SU
ZV SU(I) U U(I)
ND
GROUP USES
* Objective function:
* 0.5 * H * SUM (X[I] * SQRT(1 + U[I]^2) + X[I+1] * SQRT(1 + U[I+1]^2))
ZE PE XSU(0) H/2
DO I 1 NH-1
ZE PE XSU(I) H
ND
ZE PE XSU(NH) H/2
* Length constraint:
* 0.5 * H * SUM (SQRT(1 + U[I]^2) + SQRT(1 + U[I+1]^2)) - L = 0
ZE LENGTH SU(0) H/2
DO I 1 NH-1
ZE LENGTH SU(I) H
ND
ZE LENGTH SU(NH) H/2
OBJECT BOUND
* Solution
*LO SOLUTION 5.07226D+00 $ (NH=50)
*LO SOLUTION 5.06987D+00 $ (NH=100)
*LO SOLUTION 5.06891D+00 $ (NH=200)
*LO SOLUTION 5.06862D+00 $ (NH=400)
ENDATA
***********************
* SET UP THE FUNCTION *
* AND RANGE ROUTINES *
***********************
ELEMENTS CHAIN
TEMPORARIES
R ONEPU2
R S1PU2
M SQRT
INDIVIDUALS
T XSU
A ONEPU2 1.0 + U * U
A S1PU2 SQRT( ONEPU2 )
F X * S1PU2
G X S1PU2
G U X * U / S1PU2
H U X U / S1PU2
H U U X / S1PU2 - X * U * U / S1PU2 ** 3
T SU
A ONEPU2 1.0 + U * U
A S1PU2 SQRT( ONEPU2 )
F S1PU2
G U U / S1PU2
H U U 1.0 / S1PU2 - U * U / S1PU2 ** 3
ENDATA