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CAMSHAPE.SIF
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***************************
* SET UP THE INITIAL DATA *
***************************
NAME CAMSHAPE
* Problem :
* *********
* Maximize the area of the valve opening for one rotation of a convex cam
* with constraints on the curvature and on the radius of the cam
* This is problem 4 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 LOR2-AN-V-V
* The number of discretization points
*IE N 100 $-PARAMETER
*IE N 200 $-PARAMETER
*IE N 400 $-PARAMETER
IE N 800 $-PARAMETER
* design parameter related to the valve shape
RE RV 1.0
* maximum allowed radius of the cam
RE RMAX 2.0
* minimum allowed radius of the cam
RE RMIN 1.0
* average radius
R+ RAV RMIN RMAX
RM RAV RAV 0.5
* approximation of pi
RF PI/4 ARCTAN 1.0
RM PI PI/4 4.0
* curvature limit parameter
RE ALPHA 1.5
* angle between discretization points
IA N+1 N 1
IM 5(N+1) N+1 5
RI 5(N+1) 5(N+1)
RM DTHETA PI 2.0
R/ DTHETA DTHETA 5(N+1)
* Other useful values
IE 0 0
IE 1 1
IE 2 2
IA N-1 N -1
RI RN N
R* PIRV PI RV
R/ PIRV/N PIRV RN
RM -PIRV/N PIRV/N -1.0
R( CDTHETA COS DTHETA
RM 2CDTHETA CDTHETA 2.0
R* ADTHETA ALPHA DTHETA
RM -ADTHETA ADTHETA -1.0
RM 2ADTHETA ADTHETA 2.0
RM -RMIN RMIN -1.0
RM -RMAX RMAX -1.0
RM -2RMAX RMAX -2.0
R* RMIN2 RMIN RMIN
R* RMIN2CD RMIN 2CDTHETA
R* RMAX2CD RMAX 2CDTHETA
VARIABLES
DO I 1 N
X R(I) $ radius of the cam at discretization points
ND
GROUPS
* Objective function: maximize valve_area: ((PI*RV)/n)*sum {i in 1..n} r[i]
* = - minimize valve_area: - ((PI*RV)/n)*sum {i in 1..n} r[i]
DO I 1 N
ZN AREA R(I) -PIRV/N
ND
* convexity {i in 2..n-1}:
* - r[i-1]*r[i] - r[i]*r[i+1] + r[i-1]*r[i+1]*2CDTHETA <= 0;
DO I 2 N-1
XL CO(I)
ND
* convex_edge1:
* - RMIN*r[1] - r[1]*r[2] + RMIN*r[2]*2CDTHETA <= 0;
ZL E1 R(1) -RMIN
ZL E1 R(2) RMIN2CD
* convex_edge2:
* - RMIN^2 - RMIN*r[1] + RMIN*r[1]*2CDTHETA <= 0;
R- R RMIN2CD RMIN
ZL E2 R(1) R
* convex_edge3:
* - r[n-1]*r[n] - r[n]*RMAX + r[n-1]*RMAX*2CDTHETA <= 0;
ZL E3 R(N) -RMAX
ZL E3 R(N-1) RMAX2CD
* convex_edge4:
* - 2*RMAX*r[n] + r[n]^2*2CDTHETA <= 0;
ZL E4 R(N) -2RMAX
* curvature_edge1:
* -ALPHA*DTHETA <= (r[1] - RMIN)^2 <= ALPHA*DTHETA;
XG CU(0) R(1) 1.0
* curvature {i in 1..n-1}:
* -ALPHA*DTHETA <= (r[i+1] - r[i]) <= ALPHA*DTHETA;
DO I 1 N-1
IA I+1 I 1
XG CU(I) R(I+1) 1.0 R(I) -1.0
ND
* curvature_edge n:
* -ALPHA*DTHETA <= (RMAX - r[n]) <= ALPHA*DTHETA;
XG CU(N) R(N) -1.0
CONSTANTS
Z CAMSHAPE E2 RMIN2
R+ R -ADTHETA RMIN
Z CAMSHAPE CU(0) R
DO I 1 N-1
Z CAMSHAPE CU(I) -ADTHETA
ND
R- R -ADTHETA RMAX
Z CAMSHAPE CU(N) R
RANGES
DO I 0 N
Z CAMSHAPE CU(I) 2ADTHETA
ND
BOUNDS
DO I 1 N
ZL CAMSHAPE R(I) RMIN
ZU CAMSHAPE R(I) RMAX
ND
START POINT
DO I 1 N
ZV CAMSHAPE R(I) RAV
ND
ELEMENT TYPE
EV SQR X
EV PROD X Y
ELEMENT USES
DO I 2 N-1
IA I-1 I -1
IA I+1 I 1
XT RA(I) PROD
ZV RA(I) X R(I)
ZV RA(I) Y R(I-1)
XT RB(I) PROD
ZV RB(I) X R(I+1)
ZV RB(I) Y R(I-1)
ND
XT RA(N) PROD
ZV RA(N) X R(N)
ZV RA(N) Y R(N-1)
XT R2 SQR
ZV R2 X R(N)
GROUP USES
* convexity {i in 2..n-1}:
* - r[i-1]*r[i] - r[i]*r[i+1] + r[i-1]*r[i+1]*2CDTHETA <= 0;
DO I 2 N-1
IA I+1 I 1
XE CO(I) RA(I) -1.0 RA(I+1) -1.0
ZE CO(I) RB(I) 2CDTHETA
ND
* convex_edge1:
* - RMIN*r[1] - r[1]*r[2] + RMIN*r[2]*2CDTHETA <= 0;
XE E1 RA(2) -1.0
* convex_edge3:
* - r[n-1]*r[n] - r[n]*RMAX + r[n-1]*RMAX*2CDTHETA <= 0;
XE E3 RA(N) -1.0
* convex_edge4:
* - 2*RMAX*r[n] + r[n]^2*2CDTHETA <= 0;
ZE E4 R2 2CDTHETA
OBJECT BOUND
* Solution
*LO SOLUTION -4.2841D+00 $ (NH=100)
*LO SOLUTION -4.2785D+00 $ (NH=200)
*LO SOLUTION -4.2757D+00 $ (NH=400)
*LO SOLUTION -4.2743D+00 $ (NH=800)
ENDATA
***********************
* SET UP THE FUNCTION *
* AND RANGE ROUTINES *
***********************
ELEMENTS CAMSHAPE
TEMPORARIES
R ONEPU2
R S1PU2
M SQRT
INDIVIDUALS
T PROD
F X * Y
G X Y
G Y X
H X Y 1.0
T SQR
F X * X
G X X + X
H X X 2.0
ENDATA