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

SSwasthika D.S20 Oct 2021

Aim: To solve the 2D heat conduction equation by using the point iterative techniques.

The given boundary condetion is as follows

Top Boundary=600K

Bottom Boundary=900k

Left Boundary=400K

Right Boundary=800K

Abosolute error criteria=1e-4

The method implemented to solve the following equation are

1)Jacobi

2)Gauss-seidel

3)Successive-over relaxation

Plots

Explanation

  • In Each plot, the domain is considered as square of side lengh 1m
  • In plot the title name, method type, and nymber of iteration is displyed
  • In plot, each color indicates specific temperature in kelvin

Conclusion

  • 2D heat conduction equation is solved by using jacobi,gauss-seidel,and SOR for both explicit and implicit scheme
  • The temperature distribution is plotted for each method 
    clear all
close all
clc

%input
nx=10;
ny=10;
T=300*ones(nx,ny);

% Boundary Condetion
T(1,:)=600;
T(end,:)=900;
T(:,1)=400;
T(:,end)=800;

% Temperture at edges
T(1,1)=(600+400)/2;
T(1,end)=(600+800)/2;
T(end,1)=(900+400)/2;
T(end,end)=(900+800)/2;

x=linspace(0,1,nx);
dx=x(2)-x(1);
y=linspace(0,1,ny);
dy=y(2)-y(1);
dt=1e-3;

k=2*((1/dx^2)+(1/dy^2));
alpha=1.4;
k1=alpha*(dt/(dx^2));
k2=alpha*(dt/(dy^2));
cfl=k1+k2;

for method=linspace(1,7,7)
    if method==1
        error_1=steady_jacobi(x,y,nx,ny,dx,dy,T,k);
        fprintf('Error using jacobimethod(steady state)=%d',error_1)
        
    elseif method==2
        error_2=steady_gs(x,y,nx,ny,dx,dy,T,k);
        fprintf('n Error using gauss seidel method(steady state)=%d',error_2)
        
    elseif method==3
        error_3=steady_sor(x,y,nx,ny,dx,dy,T,k);
        fprintf('n Error using SOR method(steady state)=%d',error_3)
        
    elseif method==4
        error_4=transient_explicit_jacobi(x,y,nx,ny,dx,dy,T,k1,k2);
        fprintf('n error using jacobi method(explicit transient state)=%d',error_4)
        
    elseif method==5
        error_5=transient_implicit_jacobi(x,y,nx,ny,dx,dy,T,k1,k2);
     fprintf('n error using jacobi method(implicit transient state)=%d',error_5)
    
     
      elseif method==6
        error_6=transient_implicit_gs(x,y,nx,ny,dx,dy,T,k1,k2);
     fprintf('n error using gauss siedal method(implicit transient state)=%d',error_6)
     
     elseif method==7
        error_7=transient_implicit_sor(x,y,nx,ny,T,k1,k2);
     fprintf('n error using SOR method(implicit transient state)=%d',error_7)
     
    end
    
end

     
        

%steady state gs
function [error]=steady_gs(x,y,nx,ny,dx,dy,T,k)

error=9e9;
tol=1e-4;
Told=T;
gs_iter=1;

while (error>tol)
    for i=2:nx-1
        for j=2:ny-1
        term1=(Told(i-1,j)+Told(i+1,j))/(k*(dx^2));
        term2=(Told(i,j-1)+Told(i,j+1))/(k*(dy^2));
        T(i,j)=term1+term2;
    end
end

error=max(max(abs(Told-T)));
Told=T;
gs_iter=gs_iter+1;
end

figure(2)
meshgrid(x,y);
[a, b]=contourf(x,y,T);
clabel(a,b)
colorbar
colormap(jet)
set(gca,'ydir','reverse')
xlabel('X Domain Length')
ylabel('Y Domain Length')
title_text=sprintf('No of iterations(steady state gauss seidal)=%d',gs_iter);
title(title_text)
end

%steady state jacobi
function [error]=steady_jacobi(x,y,nx,ny,dx,dy,T,k)

error=9e9;
tol=1e-4;
Told=T;
jacobi_iter=1;

while(error>tol)
    for i=2:nx-1
        for j=2:ny-1
        term1=(Told(i-1,j)+Told(i+1,j))/(k*(dx^2));
        term2=(Told(i,j-1)+Told(i,j+1))/(k*(dy^2));
        T(i,j)=term1+term2;
    end
end

error=max(max(abs(Told-T)));
Told=T;
jacobi_iter=jacobi_iter+1;
end

figure(1)
meshgrid(x,y);
[a, b]=contourf(x,y,T);
clabel(a,b)
colorbar
colormap(jet)
set(gca,'ydir','reverse')
xlabel('X Domain Length')
ylabel('Y Domain Length')
title_text=sprintf('No of iterations(steady state jacobi)=%d',jacobi_iter);
title(title_text)
end

%steady state sor
function [error]=steady_sor(x,y,nx,ny,dx,dy,T,k)

error=9e9;
tol=1e-4;
Told=T;
sor_iter=1;
omega=1.4;

while (error>tol)
    for i=2:nx-1
        for j=2:ny-1
        term1=(T(i-1,j)+Told(i+1,j))/(k*(dx^2));
        term2=(T(i,j-1)+Told(i,j+1))/(k*(dy^2));
        T(i,j)=((Told(i,j))*(1-omega))+(omega*(term1+term2));
        end
    end
error=max(max(abs(Told-T)));
Told=T;
sor_iter=sor_iter+1;
end

figure(3)
meshgrid(x,y);
[a, b]=contourf(x,y,T);
clabel(a,b)
colorbar
colormap(jet)
set(gca,'ydir','reverse')
xlabel('X Domain Length')
ylabel('Y Domain Length')
title_text=sprintf('No of iterations(steady state SOR)=%d',sor_iter);
title(title_text)
end

function [error]=transient_explicit_jacobi(x,y,nx,ny,dx,dy,T,k1,k2)
Told=T;
for nt=1:1400
    for i=2:nx-1
        for j=2:ny-1
            term1=(Told(i-1,j)+Told(i+1,j));
            term2=(Told(i,j-1)+Told(i,j+1));
            T(i,j)=(Told(i,j)*(1-(2*k1)-(2*k2)))+(k1*term1)+(k2*term2);
    end
    end
error=max(max(abs(Told-T)));
Told=T;
end

figure(4)
[a, b]=contourf(x,y,T);
clabel(a,b)
colorbar
colormap(jet)
set(gca,'ydir','reverse')
xlabel('X Domain Length')
ylabel('Y Domain Length')
title('Transient explicit scheme(jacobi)')
end


function [error]=transient_implicit_gs(x,y,nx,ny,dx,dy,T,k1,k2)
Told=T;
Tpre=T;
gs_iter=1;
for nt=1:1400
    error=9e-9
    tol=1e-4;
    while(error>tol)
    for i=2:nx-1
        for j=2:ny-1
            term1=1/(1+(2*k1)+(2*k2));
            term2=k1*term1;
            term3=k2*term1;
            H=(T(i-1,j)+Told(i+1,j));
            V=(T(i,j-1)+Told(i,j+1));
            T(i,j)=(Tpre(i,j)*term1)+(H*term2)+(V*term3);
    end
    end
error=max(max(abs(Told-T)));
Told=T;
gs_iter=gs_iter+1;
    end
    Tpre=T;
end

figure(6)
meshgrid(x,y)
[a, b]=contourf(x,y,T);
clabel(a,b)
colorbar
colormap(jet)
set(gca,'ydir','reverse')
xlabel('X Domain Length')
ylabel('Y Domain Length')
title_text=sprintf('no of iterations(transient implicit gauss seidal)=%d',gs_iter);
title(title_text)
end


function [error]=transient_implicit_jacobi(x,y,nx,ny,dx,dy,T,k1,k2)
Told=T;
Tpre=T;
jacobi_iter=1;
for nt=1:1400
    error=9e9;
    tol=1e-4;
    while(error>tol)
    for i=2:nx-1
        for j=2:ny-1
            term1=1/(1+(2*k1)+(2*k2));
            term2=k1*term1;
            term3=k2*term1;
            H=(Told(i-1,j)+Told(i+1,j));
            V=(Told(i,j-1)+Told(i,j+1));
            T(i,j)=(Tpre(i,j)*term1)+(H*term2)+(V*term3);
    end
    end
error=max(max(abs(Told-T)));
Told=T;
jacobi_iter=jacobi_iter+1;
end
Tpre=T;
end
figure(5)
meshgrid(x,y)
[a, b]=contourf(x,y,T);
clabel(a,b)
colorbar
colormap(jet)
set(gca,'ydir','reverse')
xlabel('X Domain Length')
ylabel('Y Domain Length')
title_text=sprintf('no of iterations(transient implicit scheme(jacobi)=%d',jacobi_iter);
title(title_text)
end


function [error]=transient_implicit_sor(x,y,nx,ny,T,k1,k2)
Told=T;
Tpre=T;
sor_iter=1;
omega=1.2;
for nt=1:1400
    error=9e-9
    tol=1e-4;
    while(error>tol)
    for i=2:nx-1
        for j=2:ny-1
            term1=1/(1+(2*k1)+(2*k2));
            term2=k1*term1;
            term3=k2*term1;
            H=(T(i-1,j)+Told(i+1,j));
            V=(T(i,j-1)+Told(i,j+1));
            T(i,j)=(1-omega)*(Told(i,j))+(omega*((Tpre(i,j)*term1)+(H*term2)+(V*term3)));
    end
    end
error=max(max(abs(Told-T)));
Told=T;
sor_iter=sor_iter+1;
    end
    Tpre=T;
end

figure(7)
meshgrid(x,y)
[a, b]=contourf(x,y,T);
clabel(a,b)
colorbar
colormap(jet)
set(gca,'ydir','reverse')
xlabel('X Domain Length')
ylabel('Y Domain Length')
title_text=sprintf('no of iterations(Transient implicit SOR)=%d',sor_iter);
title(title_text)
end

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