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LIPPERT1.SIF
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***************************
* SET UP THE INITIAL DATA *
***************************
NAME LIPPERT1
* Problem :
* *********
* A discrete approximation to a continuum optimal flow problem
* in the unit square. The continuum problem requires that the
* divergence of a given flow should be given everywhere in the
* region of interest, with the restriction that the capacity of
* the flow is bounded. The aim is then to maximize the given flow.
* The discrete problem (primal formulation 1) in the unit square is to
* maximize t
* subject to dx( u_ij - ui-1j ) + dx( v_ij - vij-1 ) = t s_ij
* u_ij^2 + v_ij^2 <= 1
* u_i-1j^2 + v_ij^2 <= 1
* u_ij^2 + v_ij-1^2 <= 1
* u_i-1j^2 + v_ij-1^2 <= 1
* where 1 <= i <= nx, 1 <= j <= ny
* and t >= 0
* Source: R. A. Lippert
* "Discrete approximations to continuum optimal flow problems"
* Tech. Report, Dept of Maths, M.I.T., 2006
* following a suggestion by Gil Strang
* SIF input: Nick Gould, September 2006
* classification LQR2-MN-V-V
* Number of nodes in x direction
*IE NX 2 $-PARAMETER
*IE NX 3 $-PARAMETER
*IE NX 10 $-PARAMETER
*IE NX 40 $-PARAMETER
IE NX 100 $-PARAMETER
* Number of nodes in y direction
*IE NY 2 $-PARAMETER
*IE NY 3 $-PARAMETER
*IE NY 10 $-PARAMETER
*IE NY 40 $-PARAMETER
IE NY 100 $-PARAMETER
* Other useful parameters
IA X+ NX 1
IA X- NX -1
IA Y+ NY 1
IA Y- NY -1
IE 1 1
IE 0 0
RE ONE 1.0
RE -ONE -1.0
* Source value
RE S 1.0
R* -S S -ONE
* Discretization intervals
RI RX NX
R/ DX ONE RX
R/ -DX -ONE RX
RI RY NY
R/ DY ONE RY
R/ -DY -ONE RY
VARIABLES
T
DO I 0 NX
DO J 1 NY
X U(I,J)
ND
DO I 1 NX
DO J 0 NY
X V(I,J)
ND
GROUPS
* objective function (maximize)
N OBJ T -1.0
* conservation constraints
DO I 1 NX
IA I-1 I -1
DO J 1 NY
IA J-1 J -1
ZE O(I,J) 'SCALE' DX
ZE O(I,J) U(I,J) DX
ZE O(I,J) U(I-1,J) -DX
ZE O(I,J) V(I,J) DY
ZE O(I,J) V(I,J-1) -DY
ZE O(I,J) T -S
ND
* capacity constraints
DO I 1 NX
DO J 1 NY
XL A(I,J)
XL B(I,J)
XL C(I,J)
XL D(I,J)
ND
RHS
DO I 1 NX
DO J 1 NY
X LIPPERT1 A(I,J) 1.0
X LIPPERT1 B(I,J) 1.0
X LIPPERT1 C(I,J) 1.0
X LIPPERT1 D(I,J) 1.0
ND
BOUNDS
FR LIPPERT1 'DEFAULT'
LO LIPPERT1 T 0.01
START POINT
* LIPPERT1 'DEFAULT' 1.0
ELEMENT TYPE
EV SQR ALPHA
ELEMENT USES
T 'DEFAULT' SQR
DO I 0 NX
DO J 1 NY
ZV P(I,J) ALPHA U(I,J)
ND
DO I 1 NX
DO J 0 NY
ZV Q(I,J) ALPHA V(I,J)
ND
GROUP USES
DO I 1 NX
IA I-1 I -1
DO J 1 NY
IA J-1 J -1
XE A(I,J) P(I,J)
XE A(I,J) Q(I,J)
XE B(I,J) P(I-1,J)
XE B(I,J) Q(I,J)
XE C(I,J) P(I,J)
XE C(I,J) Q(I,J-1)
XE D(I,J) P(I-1,J)
XE D(I,J) Q(I,J-1)
ND
OBJECT BOUND
* Solution
*LO SOLTN -3.77245385
ENDATA
***********************
* SET UP THE FUNCTION *
* AND RANGE ROUTINES *
***********************
ELEMENTS LIPPERT1
INDIVIDUALS
* square of x
T SQR
F ALPHA * ALPHA
G ALPHA ALPHA + ALPHA
H ALPHA ALPHA 2.0
ENDATA