Commit edcb9ab7 authored by Gianluca Frison's avatar Gianluca Frison
Browse files

added lec3_4 & ex03

parent 448e29c3
# Todd Munson
# COPS 3.1 - March 2004
###############################################################################
# Generalized Lane-Emden-Fowler-Henon equation with homogeneous Dirichlet
# boundary conditions:
#
# a*Laplacian - b(x)*u + c(x)*u - d(x) = 0 for x in int Omega
# u(x) = 0 for x in bdry Omega
#
# Equation classes considered:
# Lane-Emden-Fowler: c(x) = 1
# Henon : c(x) = |x|^(2*l)
###############################################################################
###############################################################################
# Description of the discretization
###############################################################################
param NODES; # Number of nodes
param ELEMS; # Number of elements
param DIMEN := 2; # Number of spatial dimensions
set D := {1..DIMEN}; # Set for the dimensions
set N := {1..NODES}; # Set for the nodes
set E := {1..ELEMS}; # Set for the elements
param TRIANG{E, 1..DIMEN+1} in N; # Description of triangular elements
param COORDS{N, D}; # Coordinates for the nodes
param BNDRY{N} binary; # Indicator for the boundary nodes
param EDGE{e in E, d1 in 1..DIMEN+1, d2 in D} := # Edge lengths
COORDS[TRIANG[e,(d1 mod (DIMEN+1)) + 1],d2] - COORDS[TRIANG[e,d1],d2];
param AREA{e in E} := # Area of element
(EDGE[e,1,1]*EDGE[e,2,2] - EDGE[e,1,2]*EDGE[e,2,1]) / 2;
check {e in E}: AREA[e] > 0;
###############################################################################
# Description of the partial differential equation
###############################################################################
param a > 0; # Coefficient for laplacian
param b{N} >= 0; # Coefficient for u
param c{N} >= 0; # Coefficient for u^p
param p{N} > 0; # Power
param d{N}; # Right-hand side
check {n in N: BNDRY[n]}: d[n] = 0;
###############################################################################
# Mountain pass algorithm information
# BREAK -- number of breakpoints for path generated
# US -- starting point
# UE -- ending point
# ALPHA -- constant for a bound on the length of the path generated in
# relation to the distance between the starting and ending point
###############################################################################
param BREAK := 40;
param US{N};
param UE{N};
param ALPHA >= 1;
# Check to make sure the boundary conditions are correctly set
check {n in N: BNDRY[n]}: ((US[n] = 0) and (UE[n] = 0));
# Derived quantities
param H := ALPHA/(BREAK+1)*sqrt(sum {n in N} (US[n]-UE[n])^2);
check: H > 0;
###############################################################################
# Optimization problem specification
###############################################################################
var u{0..BREAK+1, N};
var z;
###############################################################################
# Assume right triangles with right angle at node 1 of the triangle.
# Split into cases depending upon whether the x coordinate of edge 1 is
# positive or the y coordinate of edge 1 is positive.
###############################################################################
var integral{b1 in 0..BREAK+1, e1 in E} =
AREA[e1]*(
1 / (DIMEN+1) *
(sum {c1 in 1..DIMEN+1} (b[TRIANG[e1,c1]]*u[b1,TRIANG[e1,c1]]^2/2-
c[TRIANG[e1,c1]]*u[b1,TRIANG[e1,c1]]^(p[TRIANG[e1,c1]]+1)/(p[TRIANG[e1,c1]]+1)+
d[TRIANG[e1,c1]]*u[b1,TRIANG[e1,c1]])) +
a / (8*AREA[e1]^2)*(
u[b1,TRIANG[e1,1]]^2*(EDGE[e1,2,1]^2 + EDGE[e1,2,2]^2) +
u[b1,TRIANG[e1,2]]^2*(EDGE[e1,3,1]^2 + EDGE[e1,3,2]^2) +
u[b1,TRIANG[e1,3]]^2*(EDGE[e1,1,1]^2 + EDGE[e1,1,2]^2) +
2*u[b1,TRIANG[e1,1]]*u[b1,TRIANG[e1,2]]*(EDGE[e1,2,1]*EDGE[e1,3,1] + EDGE[e1,2,2]*EDGE[e1,3,2]) +
2*u[b1,TRIANG[e1,1]]*u[b1,TRIANG[e1,3]]*(EDGE[e1,2,1]*EDGE[e1,1,1] + EDGE[e1,2,2]*EDGE[e1,1,2]) +
2*u[b1,TRIANG[e1,2]]*u[b1,TRIANG[e1,3]]*(EDGE[e1,1,1]*EDGE[e1,3,1] + EDGE[e1,1,2]*EDGE[e1,3,2])
));
var energy{b1 in 0..BREAK+1} = sum {e1 in E} integral[b1,e1];
minimize cost: z;
subject to max_energy {b1 in 1..BREAK}:
z >= energy[b1];
subject to distance {b1 in 0..BREAK}:
sum {n in N} (u[b1+1,n]-u[b1,n])^2 <= H^2;
subject to boundary {b1 in 1..BREAK, n in N: BNDRY[n]}:
u[b1,n] = 0;
subject to start {n in N}:
u[0,n] = US[n];
subject to end {n in N}:
u[BREAK+1,n] = UE[n];
data;
param NODES := 545;
param COORDS: 1 2 =
1 -1.000000000e+00 -1.000000000e+00
2 1.000000000e+00 -1.000000000e+00
3 1.000000000e+00 1.000000000e+00
4 -1.000000000e+00 1.000000000e+00
5 0.000000000e+00 0.000000000e+00
6 -1.000000000e+00 0.000000000e+00
7 0.000000000e+00 -1.000000000e+00
8 1.000000000e+00 0.000000000e+00
9 0.000000000e+00 1.000000000e+00
10 -5.000000000e-01 -5.000000000e-01
11 -5.000000000e-01 5.000000000e-01
12 5.000000000e-01 -5.000000000e-01
13 5.000000000e-01 5.000000000e-01
14 -5.000000000e-01 0.000000000e+00
15 -1.000000000e+00 -5.000000000e-01
16 -5.000000000e-01 -1.000000000e+00
17 0.000000000e+00 -5.000000000e-01
18 -1.000000000e+00 5.000000000e-01
19 0.000000000e+00 5.000000000e-01
20 -5.000000000e-01 1.000000000e+00
21 5.000000000e-01 -1.000000000e+00
22 1.000000000e+00 -5.000000000e-01
23 5.000000000e-01 0.000000000e+00
24 1.000000000e+00 5.000000000e-01
25 5.000000000e-01 1.000000000e+00
26 -7.500000000e-01 -2.500000000e-01
27 -2.500000000e-01 -2.500000000e-01
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34 -7.500000000e-01 7.500000000e-01
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37 2.500000000e-01 7.500000000e-01
38 7.500000000e-01 -7.500000000e-01
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40 7.500000000e-01 2.500000000e-01
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