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Week 5.1 - Mid term project - Solving the steady and unsteady 2D heat conduction problem

AArunkumar Awati3 Nov 2021

Steady state analysis & Transient State Analysis

Solve the 2D heat conduction equation by using the point iterative techniques that were taught in the class. The Boundary conditions for the problem are as follows;

Top Boundary = 600 K

Bottom Boundary = 900 K

Left Boundary = 400 K

Right Boundary = 800 K

1. Jacobi

clear all
close all
clc
nx=10;
ny=10;
T=ones(nx,ny);
TL=400;
TR=800;
TT=600;
TB=900;
T(:,1)=TL;
T(:,ny)=TR;
T(1,:)=TT;
T(nx,:)=TB;
T(1,1)=0.5*(T(2,1)+T(1,2));
T(nx,1)=0.5*(T(nx-1,1)+T(nx,2));
T(nx,ny)=0.5*(T(nx-1,ny)+T(nx,ny-1));
T(1,ny)=0.5*(T(2,ny)+T(1,ny-1));
Told=T;
error=9e9;
tol=1e-5;
jac_iter=0
while(error>tol)
     for i=2:nx-1
        for j=2:ny-1

       % T(i,j)=0.25*(Told(i+1,j)+Told(i-1,j)+Told(i,j-1)+Told(i,j+1));
       T(i,j)=0.25*(Told(i+1,j)+Told(i-1,j)+Told(i,j-1)+Told(i,j+1));
    end
end
error=max(max(abs(T-Told)));
Told=T;
jac_iter=jac_iter+1
contourf(T,'showtext','on')
title(sprintf('steday state jacobi inerations=%d',jac_iter));
pause(0.005)


colorbar
end

2. Gauss-seidel

clear all
close all
clc
nx=10;
ny=10;
T=ones(nx,ny);
TL=400;
TR=800;
TT=600;
TB=900;
T(:,1)=TL;
T(:,ny)=TR;
T(1,:)=TT;
T(nx,:)=TB;
T(1,1)=0.5*(T(2,1)+T(1,2));
T(nx,1)=0.5*(T(nx-1,1)+T(nx,2));
T(nx,ny)=0.5*(T(nx-1,ny)+T(nx,ny-1));
T(1,ny)=0.5*(T(2,ny)+T(1,ny-1));
Told=T;
error=9e9;
tol=1e-5;
gs_iter=0
while(error>tol)
     for i=2:nx-1
        for j=2:ny-1

       % T(i,j)=0.25*(Told(i+1,j)+Told(i-1,j)+Told(i,j-1)+Told(i,j+1));
       T(i,j)=0.25*(Told(i+1,j)+T(i-1,j)+T(i,j-1)+Told(i,j+1));
    end
end
error=max(max(abs(T-Told)));
Told=T;
gs_iter=gs_iter+1
contourf(T,'showtext','on')
title(sprintf('steday stategauss inerations=%d',gs_iter));
pause(0.005)
colorbar

end

3. Successive over-relaxation

clear all
close all
clc
nx=10;
ny=10;
T=ones(nx,ny);
TL=400;
TR=800;
TT=600;
TB=900;
T(:,1)=TL;
T(:,ny)=TR;
T(1,:)=TT;
T(nx,:)=TB;
T(1,1)=0.5*(T(2,1)+T(1,2));
T(nx,1)=0.5*(T(nx-1,1)+T(nx,2));
T(nx,ny)=0.5*(T(nx-1,ny)+T(nx,ny-1));
T(1,ny)=0.5*(T(2,ny)+T(1,ny-1));
Told=T;
error=9e9;
tol=1e-5;
sor_iter=0
omega=1.6;
while(error>tol)
     for i=2:nx-1
        for j=2:ny-1

       % T(i,j)=0.25*(Told(i+1,j)+Told(i-1,j)+Told(i,j-1)+Told(i,j+1));
       TGS(i,j)=0.25*(Told(i+1,j)+T(i-1,j)+T(i,j-1)+Told(i,j+1));
       T(i,j)=(1-omega)*Told(i,j)+omega*TGS(i,j);
    end
end
error=max(max(abs(T-Told)));
Told=T;
sor_iter=sor_iter+1
contourf(T,'showtext','on')
title(sprintf('steday state sor inerations=%d',sor_iter));
pause(0.005)


end

unsteady

Jacobi

clear all
close all
clc
nx=10;
ny=10;
T=ones(nx,ny);
TL=400;
TR=800;
TT=600;
TB=900;
T(:,1)=TL;
T(:,ny)=TR;
T(1,:)=TT;
T(nx,:)=TB;
T(1,1)=0.5*(T(2,1)+T(1,2));
T(nx,1)=0.5*(T(nx-1,1)+T(nx,2));
T(nx,ny)=0.5*(T(nx-1,ny)+T(nx,ny-1));
T(1,ny)=0.5*(T(2,ny)+T(1,ny-1));
Told=T;
Tprev=T;
dt=1e-2;
dx=1e-2;
dy=1e-2;
%thermaldiffusivity
alpha=1.5;
k1=alpha *dt/dx^2;
k2=alpha *dt/dy^2
error=9e9;
conv=9e9;
tol=1e-5;
jac_iter=0
jac_time_iter=0;
while(conv>tol)
while(error>tol)
     for i=2:nx-1
        for j=2:ny-1

       % T(i,j)=0.25*(Told(i+1,j)+Told(i-1,j)+Told(i,j-1)+Told(i,j+1));
      % T(i,j)=0.25*(Told(i+1,j)+Told(i-1,j)+Told(i,j-1)+Told(i,j+1));
      T(i,j)=(1+2*k1+2*k2)^-1*(Tprev(i,j)+k1*(Told(i-1,j)+Told(i+1,j))+k2*(Told(i,j-1)+Told(i,j+1)));

    end
end
error=max(max(abs(T-Told)));
Told=T;
jac_iter=jac_iter+1
contourf(T,'showtext','on')
title(sprintf('trasient implicit jocobi inerations=%d\nTime iterations=%d',jac_iter,jac_time_iter));
pause(0.005)


colorbar
end
conv=max(max(abs(T-Tprev)));
Tprev=T;
jac_time_iter=jac_time_iter+1
error=9e9;

end

guss

clear all
close all
clc
nx=10;
ny=10;
T=ones(nx,ny);
TL=400;
TR=800;
TT=600;
TB=900;
T(:,1)=TL;
T(:,ny)=TR;
T(1,:)=TT;
T(nx,:)=TB;
T(1,1)=0.5*(T(2,1)+T(1,2));
T(nx,1)=0.5*(T(nx-1,1)+T(nx,2));
T(nx,ny)=0.5*(T(nx-1,ny)+T(nx,ny-1));
T(1,ny)=0.5*(T(2,ny)+T(1,ny-1));
Told=T;
Tprev=T;
dt=1e-2;
dx=1e-2;
dy=1e-2;
%thermaldiffusivity
alpha=1.5;
k1=alpha *dt/dx^2;
k2=alpha *dt/dy^2
error=9e9;
conv=9e9;
tol=1e-5;
gs_iter=0
gs_time_iter=0;
while(conv>tol)
while(error>tol)
     for i=2:nx-1
        for j=2:ny-1

       % T(i,j)=0.25*(Told(i+1,j)+Told(i-1,j)+Told(i,j-1)+Told(i,j+1));
      % T(i,j)=0.25*(Told(i+1,j)+Told(i-1,j)+Told(i,j-1)+Told(i,j+1));
      T(i,j)=(1+2*k1+2*k2)^-1*(Tprev(i,j)+k1*(T(i-1,j)+Told(i+1,j))+k2*(T(i,j-1)+Told(i,j+1)));

    end
end
error=max(max(abs(T-Told)));
Told=T;
gs_iter=gs_iter+1
contourf(T,'showtext','on')
title(sprintf('trasient implicit guss inerations=%d\nTime iterations=%d',gs_iter,gs_time_iter));
pause(0.005)


colorbar
end
conv=max(max(abs(T-Tprev)));
Tprev=T;
gs_time_iter=gs_time_iter+1
error=9e9;

end

3. Successive over-relaxation

clear all
close all
clc
nx=10;
ny=10;
T=ones(nx,ny);
TL=400;
TR=800;
TT=600;
TB=900;
T(:,1)=TL;
T(:,ny)=TR;
T(1,:)=TT;
T(nx,:)=TB;
T(1,1)=0.5*(T(2,1)+T(1,2));
T(nx,1)=0.5*(T(nx-1,1)+T(nx,2));
T(nx,ny)=0.5*(T(nx-1,ny)+T(nx,ny-1));
T(1,ny)=0.5*(T(2,ny)+T(1,ny-1));
Told=T;
omega=1.005;
Tprev=T;
dt=1e-2;
dx=1e-2;
dy=1e-2;
%thermaldiffusivity
alpha=1.5;
k1=alpha *dt/dx^2;
k2=alpha *dt/dy^2
error=9e9;
conv=9e9;
tol=1e-5;
sor_iter=0
sor_time_iter=0;
while(conv>tol)
while(error>tol)
     for i=2:nx-1
        for j=2:ny-1

       % T(i,j)=0.25*(Told(i+1,j)+Told(i-1,j)+Told(i,j-1)+Told(i,j+1));
      % T(i,j)=0.25*(Told(i+1,j)+Told(i-1,j)+Told(i,j-1)+Told(i,j+1));
      TGS(i,j)=(1+2*k1+2*k2)^-1*(Tprev(i,j)+k1*(Told(i-1,j)+Told(i+1,j))+k2*(Told(i,j-1)+Told(i,j+1)));
       T(i,j)=(1-omega)*Told(i,j)+omega*(TGS(i,j));
    end
end
error=max(max(abs(T-Told)));
Told=T;
sor_iter=sor_iter+1
contourf(T,'showtext','on')
title(sprintf('trasient implicit gosinerations=%d\nTime iterations=%d',sor_iter,sor_time_iter));
pause(0.005)


colorbar
end
conv=max(max(abs(T-Tprev)));
Tprev=T;
sor_time_iter=sor_time_iter+1
error=9e9;

end

Explicit Jacobi

clear all
close all
clc
nx=10;
ny=10;
T=ones(nx,ny);
TL=400;
TR=800;
TT=600;
TB=900;
T(:,1)=TL;
T(:,ny)=TR;
T(1,:)=TT;
T(nx,:)=TB;
T(1,1)=0.5*(T(2,1)+T(1,2));
T(nx,1)=0.5*(T(nx-1,1)+T(nx,2));
T(nx,ny)=0.5*(T(nx-1,ny)+T(nx,ny-1));
T(1,ny)=0.5*(T(2,ny)+T(1,ny-1));
Told=T;
Tprev=T;
dt=1e-2;
dx=7e-2;
dy=7e-2;
%thermaldiffusivity
alpha=0.05
k1=alpha *dt/dx^2;
k2=alpha *dt/dy^2
cfl=k1+k2;
conv=9e9;
tol=1e-5;

time_iter=0;
while(conv>tol)

     for i=2:nx-1
        for j=2:ny-1

       % T(i,j)=0.25*(Told(i+1,j)+Told(i-1,j)+Told(i,j-1)+Told(i,j+1));
      % T(i,j)=0.25*(Told(i+1,j)+Told(i-1,j)+Told(i,j-1)+Told(i,j+1));
      T(i,j)=(1-2*k1-2*k2)*Tprev(i,j)+k1*(Tprev(i-1,j)+Tprev(i+1,j))+k2*(Tprev(i,j-1)+Tprev(i,j+1));

    end
end
conv=max(max(abs(T-Tprev)));
Tprev=T;
time_iter=time_iter+1
contourf(T,'showtext','on')
title(sprintf('trasient explicit iteration=%d ', time_iter));
pause(0.005)


colorbar
end

Types methods iterations
Implicit steady jacobi 253
gauss sledel 134
successive over relaxation 41
Implicit transient jacobi 672
gauss sledel 370
successive over relaxation 668
explicit jacobi 593

*Jocobi>Gauss seidel>Successive over-relaxation according to the num of iterations both steady and unsteady

*SOR iteration method is the most effective iteration method as it took least No iteration for converging in both explicit and implicit compare to other two

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