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ROCKET.SIF
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***************************
* SET UP THE INITIAL DATA *
***************************
NAME ROCKET
* Problem :
* *********
* Maximize the final altitude of a vertically-lauched rocket, using
* the thrust as a control and given the initial mass, the fuel mass
* and the drag characteristics of the rocket.
* This is problem 10 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 in mesh
*IE NH 50 $-PARAMETER
*IE NH 100 $-PARAMETER
*IE NH 200 $-PARAMETER
IE NH 400 $-PARAMETER
* Initial velocity
RE V0 0.0
* Gravity at the surface; normalization of the equations allows g_0 = 1
RE G0 1.0
* Initial height; normalization of the equations allows h_0 = 1
RE H0 1.0
* Initial mass; normalization of the equations allows m_0 = 1
RE M0 1.0
* Parameters for the model
RE TC 3.5
RE HC 500.0
RE VC 620.0
RE MC 0.6
* Derived parameters
R* G0H0 G0 H0
R* G0H02 G0H0 H0
R( C SQRT G0H0
RM C C 0.5
R* MF MC M0
R/ DC M0 G0
R* DC DC VC
RM DC DC 0.5
R* TMAX M0 G0
R* TMAX TMAX TC
* Other useful values
IE 0 0
IE 1 1
RE ONE 1.0
RM TMAX/2 TMAX 0.5
RI RNH NH
RD 1/NH RNH 1.0
R- MF-M0 MF M0
R/ HC/H0 HC H0
RD -1/2C C -0.5
VARIABLES
X STEP
DO I 0 NH
X H(I) $ Height of rocket
X V(I) $ Velocity of rocket
X M(I) $ Mass of rocket
X T(I) $ Thrust of rocket
X D(I) $ Drag
X G(I) $ Gravity
ND
GROUPS
* Objective: maximize final_velocity:
XN OBJ H(NH) -1.0
DO I 0 NH
* linear part of drag constraint {i in 0..nh}:
* -D[i] + DC *(v[i]^2)*exp(-HC *(h[i]-H0 )/H0 ) = 0
XE D(I) D(I) -1.0
* linear part of gravity constraint {i in 0..nh}:
* -G[i] + G0*(H0 /h[i])^2 = 0
XE G(I) G(I) -1.0
ND
DO J 1 NH
IA J-1 J -1
* linear part of h_eqn {j in 1..nh}:
* - h[j] + h[j-1] + .5*step*(v[j] + v[j-1]) = 0
XE H(J) H(J) -1.0 H(J-1) 1.0
* linear part of v_eqn {j in 1..nh}:
* - v[j] + v[j-1] + .5*step*((T[j] - D[j] - m[j]*g[j])/m[j]
* + (T[j-1] - D[j-1] - m[j-1]*g[j-1])/m[j-1]) = 0
XE V(J) V(J) -1.0 V(J-1) 1.0
* linear part of m_eqn {j in 1..nh}:
* - m[j] + m[j-1] - .5*step*(T[j] + T[j-1])/c = 0
XE M(J) M(J) -1.0 M(J-1) 1.0
ND
CONSTANTS
BOUNDS
XR ROCKET 'DEFAULT'
XL ROCKET STEP 0.0
DO I 0 NH
XL ROCKET V(I) 0.0
ZL ROCKET H(I) H0
XL ROCKET T(I) 0.0
ZU ROCKET T(I) TMAX
ZL ROCKET M(I) MF
ZU ROCKET M(I) M0
ND
ZX ROCKET H(0) H0
ZX ROCKET V(0) V0
ZX ROCKET M(0) M0
ZX ROCKET M(NH) MF
START POINT
Z ROCKET STEP 1/NH
DO I 0 NH
RI RI I
R= HI ONE
Z ROCKET H(I) HI
Z ROCKET T(I) TMAX/2
R/ I/NH RI RNH
R- 1-I/NH ONE I/NH
R* VI I/NH 1-I/NH
Z ROCKET V(I) VI
R* MI MF-M0 I/NH
R+ MI MI M0
Z ROCKET M(I) MI
R- DI H0 HI
R/ DI DI H0
R* DI DI HC
R( DI EXP DI
R* DI DI VI
R* DI DI VI
R* DI DI DC
Z ROCKET D(I) DI
A/ GI H0 HI
R* GI GI GI
R* GI GI G0
Z ROCKET G(I) GI
ND
ELEMENT TYPE
EV EXPFUN V H
EP EXPFUN C D
EV RECIP2 H
EV HTYPE V1 V2
EV HTYPE STEP
IV HTYPE V STEP
EV VTYPE STEP
EV VTYPE T D
EV VTYPE M G
IV VTYPE STEP TD
IV VTYPE M G
ELEMENT USES
DO I 0 NH
XT E(I) EXPFUN
ZV E(I) V V(I)
ZV E(I) H H(I)
ZP E(I) C HC/H0
ZP E(I) D HC
XT R(I) RECIP2
ZV R(I) H H(I)
XT V(I) VTYPE
ZV V(I) STEP STEP
ZV V(I) T T(I)
ZV V(I) D D(I)
ZV V(I) M M(I)
ZV V(I) G G(I)
ND
DO J 1 NH
IA J-1 J -1
XT H(J) HTYPE
ZV H(J) V1 V(J)
ZV H(J) V2 V(J-1)
ZV H(J) STEP STEP
XT M(J) HTYPE
ZV M(J) V1 T(J)
ZV M(J) V2 T(J-1)
ZV M(J) STEP STEP
ND
GROUP USES
DO I 0 NH
* nonlinear part of drag constraint {i in 0..nh}:
* -D[i] + DC *(v[i]^2)*exp(-HC *(h[i]-H0 )/H0 ) = 0
ZE D(I) E(I) DC
* nonlinear part of gravity constraint {i in 0..nh}:
* -G[i] + G0*(H0 /h[i])^2 = 0
ZE G(I) R(I) G0H02
ND
DO J 1 NH
IA J-1 J -1
* nonlinear part of h_eqn {j in 1..nh}:
* - h[j] + h[j-1] + .5*step*(v[j] + v[j-1]) = 0
XE H(J) H(J) 0.5
* nonlinear part of v_eqn {j in 1..nh}:
* - v[j] + v[j-1] + .5*step*((T[j] - D[j] - m[j]*g[j])/m[j]
* + (T[j-1] - D[j-1] - m[j-1]*g[j-1])/m[j-1]) = 0
XE V(J) V(J) 0.5 V(J-1) 0.5
* nonlinear part of m_eqn {j in 1..nh}:
* - m[j] + m[j-1] - .5*step*(T[j] + T[j-1])/c = 0
ZE M(J) M(J) -1/2C
ND
OBJECT BOUND
* Solution
*LO SOLUTION -1.0128D+00 $ (NH=50)
*LO SOLUTION -1.0128D+00 $ (NH=100)
*LO SOLUTION -1.0128D+00 $ (NH=200)
*LO SOLUTION -1.0128D+00 $ (NH=400)
ENDATA
***********************
* SET UP THE FUNCTION *
* AND RANGE ROUTINES *
***********************
ELEMENTS ROCKET
TEMPORARIES
R EXPARG
M EXP
INDIVIDUALS
T EXPFUN
A EXPARG EXP( D - C * H )
F V ** 2 * EXPARG
G H - C * V ** 2 * EXPARG
G V 2.0 * V * EXPARG
H H H ( C * V ) ** 2 * EXPARG
H V V 2.0 * EXPARG
H H V - 2.0 * C * V * EXPARG
T RECIP2
F 1.0 / H ** 2
G H - 2.0 / H ** 3
H H H 6.0 / H ** 4
T HTYPE
R V V1 1.0 V2 1.0
R STEP STEP 1.0
F STEP * V
G V STEP
G STEP V
H V STEP 1.0
T VTYPE
R TD T 1.0 D -1.0
R M M 1.0
R G G 1.0
R STEP STEP 1.0
F STEP * ( ( TD / M ) - G )
G STEP ( ( TD / M ) - G )
G TD STEP / M
G M - STEP * TD / M ** 2
G G - STEP
H TD STEP 1.0 / M
H M STEP - TD / M ** 2
H G STEP - 1.0
H M TD - STEP / M ** 2
H M M 2.0 * STEP * TD / M ** 3
ENDATA