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LIPPERT2.SIF
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***************************
* SET UP THE INITIAL DATA *
***************************
NAME LIPPERT2
* 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 (dual formulation 2) in the unit square is to
* minimize r
* subject to dx( u_ij - ui-1j ) + dx( v_ij - vij-1 ) = s_ij
* u_ij^2 + v_ij^2 <= r^2
* u_i-1j^2 + v_ij^2 <= r^2
* u_ij^2 + v_ij-1^2 <= r^2
* u_i-1j^2 + v_ij-1^2 <= r^2
* where 1 <= i <= nx, 1 <= j <= ny
* and r >= 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 HALF 0.5
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
R* DX/2 DX HALF
RI RY NY
R/ DY ONE RY
R/ -DY -ONE RY
R* DY/2 DY HALF
VARIABLES
R
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 R 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
ND
* capacity constraints
DO I 1 NX
DO J 1 NY
XG A(I,J)
XG B(I,J)
XG C(I,J)
XG D(I,J)
ND
RHS
DO I 1 NX
DO J 1 NY
Z LIPPERT2 O(I,J) S
ND
BOUNDS
FR LIPPERT2 'DEFAULT'
LO LIPPERT2 R 0.0
START POINT
XV LIPPERT2 R 1.0
RE ALPHA 0.0
DO I 0 NX
DO J 1 NY
ZV LIPPERT2 U(I,J) ALPHA
OD J
R+ ALPHA ALPHA DX/2
OD I
RE ALPHA 0.0
DO J 0 NX
DO I 1 NY
ZV LIPPERT2 V(I,J) ALPHA
OD I
R+ ALPHA ALPHA DX/2
OD J
ELEMENT TYPE
EV SQR ALPHA
ELEMENT USES
T 'DEFAULT' SQR
V RHO2 ALPHA R
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) -1.0
XE A(I,J) Q(I,J) -1.0
XE A(I,J) RHO2
XE B(I,J) P(I-1,J) -1.0
XE B(I,J) Q(I,J) -1.0
XE B(I,J) RHO2
XE C(I,J) P(I,J) -1.0
XE C(I,J) Q(I,J-1) -1.0
XE C(I,J) RHO2
XE D(I,J) P(I-1,J) -1.0
XE D(I,J) Q(I,J-1) -1.0
XE D(I,J) RHO2
ND
OBJECT BOUND
* Solution
*LO SOLTN 3.77245385
ENDATA
***********************
* SET UP THE FUNCTION *
* AND RANGE ROUTINES *
***********************
ELEMENTS LIPPERT2
INDIVIDUALS
* square of x
T SQR
F ALPHA * ALPHA
G ALPHA ALPHA + ALPHA
H ALPHA ALPHA 2.0
ENDATA