Laplace Eqn
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Assignment 3 Rajarshi Guha, Vishal Dalvi and Rahul Kumar M.Tech. 2nd year, 07302009, 07302005 and 07302006 Modeling & Simulation, CL676 Monte Carlo Simulation of 2 dimensional Laplace Equation: 10 x 10 Temperature Grid: 400 300 300 300 300 300 300 300 300 400
500 330 310 310 330 440 420 410 360 500
500 340 300 450 360 350 340 400 350 500
500 420 480 310 400 360 460 480 380 500
500 440 300 310 470 350 410 340 400 500
500 500 400 300 410 420 300 370 410 500
500 420 500 380 380 320 310 320 440 500
500 440 420 410 360 320 310 440 440 500
500 500 300 340 430 300 470 450 380 500
400 300 300 300 300 300 300 300 300 400
500 480 460 440 420 400 380 360 340 10 320 300 10
8 6 9
8
4 7
6
5
4
2 3
2
1
0
Fig.1. 3-dimensional plot of grid point temperatures using MC simulation in MATLAB. Walls are at temperatures 300 & 500 K with the corner points having 400K.
Finite Difference Simulation of 2 dimensional Laplace Equation: 10 x 10 Temperature Grid:
500 480 460 440 420 400 380 360 340 320 300 10
9
8
7
6
5
4
3
2
1
1
2
3
4
5
6
7
8
9
10
Fig.2. 3-dimensional plot of grid point temperatures using FD simulation in MATLAB. Walls are at temperatures 300 & 500 K with the corner points having 400K.
Monte Carlo Simulation of 2 dimensional Laplace Equation: 10 x 10 Temperature Grid: 500 500 500 500 500 500 500 500 500 500
500 500 500 500 500 500 500 500 500 500
500 500 500 500 500 500 500 500 500 500
500 500 500 500 500 500 500 500 500 500
500 500 500 500 500 500 500 500 500 500
500 500 500 500 500 500 500 500 500 500
500 500 500 500 500 500 500 500 500 500
500 500 500 500 500 500 500 500 500 500
500 500 500 500 500 500 500 500 500 500
500 500 500 500 500 500 500 500 500 500
501 5 0 0 .8 5 0 0 .6 5 0 0 .4 5 0 0 .2 500 4 9 9 .8 4 9 9 .6 4 9 9 .4 10 4 9 9 .2 8 499 10
9
6 8
7
4 6
5
4
2 3
2
1
0
Fig.3. 3-dimensional plot of grid point temperatures using MC simulation in MATLAB. Wall temperatures are at 500K throughout.
Finite Difference Simulation of 2 dimensional Laplace Equation: 10 x 10 Temperature Grid:
6 00 5 50 5 00 4 50 4 00 35 0 30 0 10 9 8 7 6 5
10 9 4
8 6
3
7
5 4
2
3 1
2 1
Fig.4. 3-dimensional plot of grid point temperatures using MC simulation in MATLAB. Wall temperatures are at 500K throughout.
Simulation with MC and FD with different Boundary Temperatures: The below figures are for Boundary Temperatures of 500 & 800 K with corners are at 500, 600, 650 & 700K.
800
750
700
650
600 10
550 8 500 10
6 9
8
4
7
6
5
2
4
3
2
1
0
800
750
700
650
600
550
500 10
9
8
7
6
5
4
3
2
1
10
9
8
7
6
5
4
3
2
1
Appendix: Coding: 1. Monte Carlo Simulation: % Defining the boundary points for i=1:10 for j=1:10 if(i==1||i==10||j==1||j==10) C(1,:)=500; C(10,:)=500; C(:,1)=300; C(:,10)=300; C(1,1)=400; C(1,10)=400; C(10,1)=400; C(10,10)=400; else C(i,j)=0; end end end %C; % Taking intermediate grid points and iterating for i=2:9 for j=2:9 x=i; y=j; avgT=0; for m=1:10000 conv=1; while(conv>0) % 2-dimensional Random Walk r = rand(1,1); if(r < 0.25) x=x-1;
elseif (r < 0.50) x=x+1; elseif (r < 0.75) y=y+1; else y=y-1; end if(x<1 || x>10 || y<1 || y>10) if(x<=1) x=x+1; end if(y<=1) y=y+1; end if(x>=10) x=x-1; end if(y>=1) y=y-1; end continue; end % Whether boundary is reached if(C(x,y)==300 || C(x,y)==400|| C(x,y)==500) d(m)=C(x,y); break; else continue; end end avgT=avgT+d(m); if(m==10) C(i,j)=avgT/m; end end end end
% Plotting the Mesh for i=1:10 for j=1:10 Px(i)=i; Py(j)=j; Temp(i,j)=C(i,j); end end Temp mesh(Px,Py,Temp)
2. Finite Difference Simulation: a. Function File: function f = laplace_eqn(Tn) v=Tn; for i=2:9 for j=2:9 v(i,j)=0.25*(v(i-1,j)+v(i+1,j)+v(i,j+1)+v(i,j-1)); end end f=v; b. Main file: for i=1:10 for j=1:10 if(i==1||i==10||j==1||j==10) C(1,:)=500; C(10,:)=500; C(:,1)=300; C(:,10)=300; C(1,1)=400; C(1,10)=400; C(10,1)=400; C(10,10)=400; else
C(i,j)=450; end end end % Running the iterations until consecutive matrix difference norm becomes less than tolerance for i=1:50000 Co=C; soln=feval('laplace_eqn',C); Cn=soln; C=soln; if(norm(Cn-Co)<.01) break; end end norm(Cn-Co) Temp=C; for i=1:10 for j=1:10 Px(i)=i; Py(j)=j; Temp(i,j)=C(i,j); end end Temp mesh(Px,Py,Temp)
500 330 310 310 330 440 420 410 360 500
500 340 300 450 360 350 340 400 350 500
500 420 480 310 400 360 460 480 380 500
500 440 300 310 470 350 410 340 400 500
500 500 400 300 410 420 300 370 410 500
500 420 500 380 380 320 310 320 440 500
500 440 420 410 360 320 310 440 440 500
500 500 300 340 430 300 470 450 380 500
400 300 300 300 300 300 300 300 300 400
500 480 460 440 420 400 380 360 340 10 320 300 10
8 6 9
8
4 7
6
5
4
2 3
2
1
0
Fig.1. 3-dimensional plot of grid point temperatures using MC simulation in MATLAB. Walls are at temperatures 300 & 500 K with the corner points having 400K.
Finite Difference Simulation of 2 dimensional Laplace Equation: 10 x 10 Temperature Grid:
500 480 460 440 420 400 380 360 340 320 300 10
9
8
7
6
5
4
3
2
1
1
2
3
4
5
6
7
8
9
10
Fig.2. 3-dimensional plot of grid point temperatures using FD simulation in MATLAB. Walls are at temperatures 300 & 500 K with the corner points having 400K.
Monte Carlo Simulation of 2 dimensional Laplace Equation: 10 x 10 Temperature Grid: 500 500 500 500 500 500 500 500 500 500
500 500 500 500 500 500 500 500 500 500
500 500 500 500 500 500 500 500 500 500
500 500 500 500 500 500 500 500 500 500
500 500 500 500 500 500 500 500 500 500
500 500 500 500 500 500 500 500 500 500
500 500 500 500 500 500 500 500 500 500
500 500 500 500 500 500 500 500 500 500
500 500 500 500 500 500 500 500 500 500
500 500 500 500 500 500 500 500 500 500
501 5 0 0 .8 5 0 0 .6 5 0 0 .4 5 0 0 .2 500 4 9 9 .8 4 9 9 .6 4 9 9 .4 10 4 9 9 .2 8 499 10
9
6 8
7
4 6
5
4
2 3
2
1
0
Fig.3. 3-dimensional plot of grid point temperatures using MC simulation in MATLAB. Wall temperatures are at 500K throughout.
Finite Difference Simulation of 2 dimensional Laplace Equation: 10 x 10 Temperature Grid:
6 00 5 50 5 00 4 50 4 00 35 0 30 0 10 9 8 7 6 5
10 9 4
8 6
3
7
5 4
2
3 1
2 1
Fig.4. 3-dimensional plot of grid point temperatures using MC simulation in MATLAB. Wall temperatures are at 500K throughout.
Simulation with MC and FD with different Boundary Temperatures: The below figures are for Boundary Temperatures of 500 & 800 K with corners are at 500, 600, 650 & 700K.
800
750
700
650
600 10
550 8 500 10
6 9
8
4
7
6
5
2
4
3
2
1
0
800
750
700
650
600
550
500 10
9
8
7
6
5
4
3
2
1
10
9
8
7
6
5
4
3
2
1
Appendix: Coding: 1. Monte Carlo Simulation: % Defining the boundary points for i=1:10 for j=1:10 if(i==1||i==10||j==1||j==10) C(1,:)=500; C(10,:)=500; C(:,1)=300; C(:,10)=300; C(1,1)=400; C(1,10)=400; C(10,1)=400; C(10,10)=400; else C(i,j)=0; end end end %C; % Taking intermediate grid points and iterating for i=2:9 for j=2:9 x=i; y=j; avgT=0; for m=1:10000 conv=1; while(conv>0) % 2-dimensional Random Walk r = rand(1,1); if(r < 0.25) x=x-1;
elseif (r < 0.50) x=x+1; elseif (r < 0.75) y=y+1; else y=y-1; end if(x<1 || x>10 || y<1 || y>10) if(x<=1) x=x+1; end if(y<=1) y=y+1; end if(x>=10) x=x-1; end if(y>=1) y=y-1; end continue; end % Whether boundary is reached if(C(x,y)==300 || C(x,y)==400|| C(x,y)==500) d(m)=C(x,y); break; else continue; end end avgT=avgT+d(m); if(m==10) C(i,j)=avgT/m; end end end end
% Plotting the Mesh for i=1:10 for j=1:10 Px(i)=i; Py(j)=j; Temp(i,j)=C(i,j); end end Temp mesh(Px,Py,Temp)
2. Finite Difference Simulation: a. Function File: function f = laplace_eqn(Tn) v=Tn; for i=2:9 for j=2:9 v(i,j)=0.25*(v(i-1,j)+v(i+1,j)+v(i,j+1)+v(i,j-1)); end end f=v; b. Main file: for i=1:10 for j=1:10 if(i==1||i==10||j==1||j==10) C(1,:)=500; C(10,:)=500; C(:,1)=300; C(:,10)=300; C(1,1)=400; C(1,10)=400; C(10,1)=400; C(10,10)=400; else
C(i,j)=450; end end end % Running the iterations until consecutive matrix difference norm becomes less than tolerance for i=1:50000 Co=C; soln=feval('laplace_eqn',C); Cn=soln; C=soln; if(norm(Cn-Co)<.01) break; end end norm(Cn-Co) Temp=C; for i=1:10 for j=1:10 Px(i)=i; Py(j)=j; Temp(i,j)=C(i,j); end end Temp mesh(Px,Py,Temp)
