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CLNLBEAM.SIF
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***************************
* SET UP THE INITIAL DATA *
***************************
NAME CLNLBEAM
* Problem :
* *********
* An optimal control version of the CLamped NonLinear BEAM problem.
* The energy of a beam of length 1 compressed by a force P is to be
* minimized. The control variable is the derivative of the deflection angle.
* The problem is discretized using the trapezoidal rule. It is non-convex.
* Source:
* H. Maurer and H.D. Mittelman,
* "The non-linear beam via optimal control with bound state variables",
* Optimal Control Applications and Methods 12, pp. 19-31, 1991.
* SIF input: Ph. Toint, Nov 1993.
* classification OOR2-MN-V-V
* Discretization: specify the number of interior points + 1
*IE NI 10 $-PARAMETER n=33, m=20
*IE NI 50 $-PARAMETER n=153, m=100
*IE NI 100 $-PARAMETER n=303, m=200
*IE NI 500 $-PARAMETER n=1503, m=1000
*IE NI 1000 $-PARAMETER n=3003, m=2000 original value
IE NI 2000 $-PARAMETER n=6003, m=4000
*IE NI 5000 $-PARAMETER n=15003, m=10000
* Set ALPHA to the force divided by the bending stiffness
RE ALPHA 350.0 $-PARAMETER stiffness
* Useful constants
RI RNI NI
IA NI-1 NI -1
RD H RNI 1.0
RM H/4 H 0.25
RM H/2 H 0.5
R* AH ALPHA H
RM AH/2 AH 0.5
RM -H/2 H -0.5
IE 0 0
VARIABLES
* deflection angle
DO I 0 NI
X T(I)
OD I
* displacement
DO I 0 NI
X X(I)
OD I
* derivative of the deflection angle (control variable)
DO I 0 NI
X U(I)
OD I
GROUPS
* objective function: energy
N ENERGY
* state equations
DO I 0 NI-1
IA I+1 I 1
XE EX(I) X(I+1) 1.0 X(I) -1.0
XE ET(I) T(I+1) 1.0 T(I) -1.0
ZE ET(I) U(I+1) -H/2
ZE ET(I) U(I) -H/2
OD I
BOUNDS
* All variables are free
FR CLNLBEAM 'DEFAULT'
* Impose the bound on the displacements
DO I 0 NI
XL CLNLBEAM X(I) -0.05
XU CLNLBEAM X(I) 0.05
OD I
* Impose that the beam does not curve too far backwards
DO I 0 NI
XL CLNLBEAM T(I) -1.0
XU CLNLBEAM T(I) 1.0
OD I
* Boundary conditions
XX CLNLBEAM X(0) 0.0
XX CLNLBEAM X(NI) 0.0
XX CLNLBEAM T(0) 0.0
XX CLNLBEAM T(NI) 0.0
START POINT
* The origin is a saddle point!
* Perturb the origin
DO I 0 NI
RI RI I
R* TT RI H
R( CTT COS TT
RM SCTT CTT 0.05
ZV CLNLBEAM X(I) SCTT
ZV CLNLBEAM T(I) SCTT
OD I
ELEMENT TYPE
EV COS T
EV SIN T
EV SQ U
ELEMENT USES
DO I 0 NI
XT C(I) COS
ZV C(I) T T(I)
XT S(I) SIN
ZV S(I) T T(I)
XT USQ(I) SQ
ZV USQ(I) U U(I)
OD I
GROUP USES
DO I 0 NI-1
IA I+1 I 1
ZE EX(I) S(I+1) -H/2
ZE EX(I) S(I) -H/2
ZE ENERGY USQ(I+1) H/2
ZE ENERGY USQ(I) H/2
ZE ENERGY C(I+1) AH/2
ZE ENERGY C(I) AH/2
OD I
OBJECT BOUND
* Solution
*LO SOLTN(10) 345.0301196587
*LO SOLTN(50) 344.8673691861
*LO SOLTN(100) 344.8801831150
*LO SOLTN(500) 344.8748539754
*LO SOLTN(1000) 344.8788169123
*LO SOLTN(5000)
ENDATA
***********************
* SET UP THE FUNCTION *
* AND RANGE ROUTINES *
***********************
ELEMENTS CLNLBEAM
TEMPORARIES
R CC
R SS
M COS
M SIN
INDIVIDUALS
T SQ
F U * U
G U U + U
H U U 2.0
T COS
A CC COS( T )
A SS SIN( T )
F CC
G T - SS
H T T - CC
T SIN
A CC COS( T )
A SS SIN( T )
F SS
G T CC
H T T - SS
ENDATA