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

HHarish T24 Nov 2021

Solving 2D heat conduction under steady, usteady,transit conditions:

Steady state jacobi iterations:

close all
clear all
clc

nx=10;
ny=nx;

x=linspace(0,1,nx);
y=linspace(0,1,ny);

dx=x(2)-x(1);
dy=dx;

error=9e9;
tolerance=1e-4;

T_L=400;
T_R=800;
T_T=600;
T_B=900;

T=300*ones(nx,ny);
T(2:ny-1,1)=T_L;
T(2:ny-1,nx)=T_R;
T(1,2:nx-1)=T_T;
T(ny,2:nx-1)=T_B;

T(1,1)=(T_T+T_L)/2;
T(nx,ny)=(T_R+T_B)/2;
T(1,ny)=(T_T+T_R)/2;
T(nx,1)=(T_L+T_B)/2;

T_initial=T;
T_old=T;

iterative_solver=1;

if iterative_solver ==1
    jacobi_iteration=1;
    while (error>tolerance)
        for i=2:nx-1
            for j=2:ny-1
                T(i,j)=0.25*(T_old(i-1,j)+T_old(i+1,j)+T_old(i,j-1)+T_old(i,j+1));
            end
        end
        error=max(max(abs(T_old-T)));
        T_old=T;
        jacobi_iteration=jacobi_iteration+1;
    end
end

figure(1)
contourf(x,y,T)
clabel(contourf(x,y,T))
colorbar
colormap(jet)
set (gca,'ydir', 'reverse')
xlabel('X-Axis')
ylabel('Y-Axis')
title (sprintf('No. of jacobi iterations (Implicit)= %d' , jacobi_iteration));
pause(0.03);

Steady state gauss siedel iterations:

close all
clear all
clc

nx=10;
ny=nx;

x=linspace(0,1,nx);
y=linspace(0,1,ny);

dx=x(2)-x(1);
dy=dx;

error=9e9;
tolerance=1e-4;

T_L=400;
T_R=800;
T_T=600;
T_B=900;

T=300*ones(nx,ny);
T(2:ny-1,1)=T_L;
T(2:ny-1,nx)=T_R;
T(1,2:nx-1)=T_T;
T(ny,2:nx-1)=T_B;

T(1,1)=(T_T+T_L)/2;
T(nx,ny)=(T_R+T_B)/2;
T(1,ny)=(T_T+T_R)/2;
T(nx,1)=(T_L+T_B)/2;

T_initial=T;
T_old=T;

iterative_solver=2;

if iterative_solver ==2
    gauss_siedel_iteration=1;
    while (error>tolerance)
        for i=2:nx-1
            for j=2:ny-1
                T(i,j)=0.25*(T(i-1,j)+T_old(i+1,j)+T(i,j-1)+T_old(i,j+1));
            end
        end
        error=max(max(abs(T_old-T)));
        T_old=T;
        gauss_siedel_iteration=gauss_siedel_iteration+1;
    end
end

figure(1)
contourf(x,y,T)
clabel(contourf(x,y,T))
colorbar
colormap(jet)
set (gca,'ydir', 'reverse')
xlabel('X-Axis')
ylabel('Y-Axis')
title (sprintf('No. of gauss siedel iterations (Implicit)= %d' , gauss_siedel_iteration));
pause(0.03);

Steady state successive over relaxation iterations:

close all
clear all
clc

nx=10;
ny=nx;

x=linspace(0,1,nx);
y=linspace(0,1,ny);

dx=x(2)-x(1);
dy=dx;

error=9e9;
tolerance=1e-4;
alpha=1.2;

T_L=400;
T_R=800;
T_T=600;
T_B=900;

T=300*ones(nx,ny);
T(2:ny-1,1)=T_L;
T(2:ny-1,nx)=T_R;
T(1,2:nx-1)=T_T;
T(ny,2:nx-1)=T_B;

T(1,1)=(T_T+T_L)/2;
T(nx,ny)=(T_R+T_B)/2;
T(1,ny)=(T_T+T_R)/2;
T(nx,1)=(T_L+T_B)/2;

T_initial=T;
T_old=T;

iterative_solver=3;

if iterative_solver ==3
    successive_over_relaxation_iteration=1;
    while (error>tolerance)
        for i=2:nx-1
            for j=2:ny-1
                T(i,j)=alpha*(0.25*(T(i-1,j)+T_old(i+1,j)+T(i,j-1)+T_old(i,j+1)))+(1-alpha)*T_old(i,j);
            end
        end
        error=max(max(abs(T_old-T)));
        T_old=T;
        successive_over_relaxation_iteration=successive_over_relaxation_iteration+1;
    end
end

figure(1)
contourf(x,y,T)
clabel(contourf(x,y,T))
colorbar
colormap(jet)
set (gca,'ydir', 'reverse')
xlabel('X-Axis')
ylabel('Y-Axis')
title (sprintf('No. of SOR iterations (Implicit)= %d' , successive_over_relaxation_iteration));
pause(0.03);

Unsteady state jacobi iterations:

close all
clear all
clc

nx=10;
ny=nx;
nt=1400;

x=linspace(0,1,nx);
y=linspace(0,1,ny);

dx=x(2)-x(1);
dy=dx;

error=9e9;
tolerance=1e-4;
dt=1e-3;

T_L=400;
T_R=800;
T_T=600;
T_B=900;

T=300*ones(nx,ny);
T(2:ny-1,1)=T_L;
T(2:ny-1,nx)=T_R;
T(1,2:nx-1)=T_T;
T(ny,2:nx-1)=T_B;

T(1,1)=(T_T+T_L)/2;
T(nx,ny)=(T_R+T_B)/2;
T(1,ny)=(T_T+T_R)/2;
T(nx,1)=(T_L+T_B)/2;

T_old=T;
T_initial=T;

k1=1.1*(dt/(dx^2));
k2=1.1*(dt/(dy^2));

jacobi_iteration=1;
for k=1:nt
    error=9e9;
    while (error>tolerance)
        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_old(i-1,j))+(T_old(i+1,j));
                V=(T_old(i,j+1))+(T_old(i,j-1));
                T(i,j)=(T_initial(i,j)*term1)+(H*term2)+(V*term3);
            end
        end

        error=max(max(abs(T_old-T)));
        T_old=T;
        jacobi_iteration=jacobi_iteration+1;
    end
    T_initial=T;
end 

figure(1)
contourf(x,y,T)
clabel(contourf(x,y,T))
colorbar
colormap(jet)
set (gca,'ydir', 'reverse')
xlabel('X-Axis')
ylabel('Y-Axis')
title (sprintf('No. of unsteady jacobi iterations (Implicit)= %d' , jacobi_iteration));
pause(0.03);

Unsteady state gauss siedel iterations:

close all
clear all
clc

nx=10;
ny=nx;
nt=1400;

x=linspace(0,1,nx);
y=linspace(0,1,ny);

dx=x(2)-x(1);
dy=dx;

error=9e9;
tolerance=1e-4;
dt=1e-3;

T_L=400;
T_R=800;
T_T=600;
T_B=900;

T=300*ones(nx,ny);
T(2:ny-1,1)=T_L;
T(2:ny-1,nx)=T_R;
T(1,2:nx-1)=T_T;
T(ny,2:nx-1)=T_B;

T(1,1)=(T_T+T_L)/2;
T(nx,ny)=(T_R+T_B)/2;
T(1,ny)=(T_T+T_R)/2;
T(nx,1)=(T_L+T_B)/2;

T_old=T;
T_initial=T;

k1=1.1*(dt/(dx^2));
k2=1.1*(dt/(dy^2));

gauss_siedel_iteration=1;
for k=1:nt
    error=9e9;
    while (error>tolerance)
        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))+(T_old(i+1,j));
                V=(T(i,j+1))+(T_old(i,j-1));
                T(i,j)=(T_initial(i,j)*term1)+(H*term2)+(V*term3);
            end
        end

        error=max(max(abs(T_old-T)));
        T_old=T;
        gauss_siedel_iteration=gauss_siedel_iteration+1;
    end
    T_initial=T;
end 

figure(1)
contourf(x,y,T)
clabel(contourf(x,y,T))
colorbar
colormap(jet)
set (gca,'ydir', 'reverse')
xlabel('X-Axis')
ylabel('Y-Axis')
title (sprintf('No. of unsteady gauss siedel iterations (Implicit)= %d' , gauss_siedel_iteration));
pause(0.03);

Unsteady state successive over relaxation iterations:

close all
clear all
clc

nx=10;
ny=nx;
nt=1400;

x=linspace(0,1,nx);
y=linspace(0,1,ny);

dx=x(2)-x(1);
dy=dx;

error=9e9;
tolerance=1e-4;
dt=1e-3;
alpha=1.2;

T_L=400;
T_R=800;
T_T=600;
T_B=900;

T=300*ones(nx,ny);
T(2:ny-1,1)=T_L;
T(2:ny-1,nx)=T_R;
T(1,2:nx-1)=T_T;
T(ny,2:nx-1)=T_B;

T(1,1)=(T_T+T_L)/2;
T(nx,ny)=(T_R+T_B)/2;
T(1,ny)=(T_T+T_R)/2;
T(nx,1)=(T_L+T_B)/2;

T_old=T;
T_initial=T;
T_end=T;

k1=1.1*(dt/(dx^2));
k2=1.1*(dt/(dy^2));

successive_over_relaxation_iteration=1;
for k=1:nt
    error=9e9;
    while (error>tolerance)
        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))+(T_old(i+1,j));
                V=(T(i,j+1))+(T_old(i,j-1));
                T(i,j)=(T_initial(i,j)*term1)+(H*term2)+(V*term3);
                T_end(i,j)=((1-alpha)*T_old(i,j))+(T(i,j)*alpha);
            end
        end

        error=max(max(abs(T_old-T)));
        T_old=T_end;
        successive_over_relaxation_iteration=successive_over_relaxation_iteration+1;
    end
    T_initial=T_end;
end 

figure(1)
contourf(x,y,T)
clabel(contourf(x,y,T))
colorbar
colormap(jet)
set (gca,'ydir', 'reverse')
xlabel('X-Axis')
ylabel('Y-Axis')
title (sprintf('No. of unsteady successive over relaxation iterations (Implicit)= %d' , successive_over_relaxation_iteration));
pause(0.03);

Steady state explicit iterations:

close all
clear all
clc

nx=10;
ny=nx;
nt=1400;

x=linspace(0,1,nx);
y=linspace(0,1,ny);

dx=x(2)-x(1);
dy=dx;

error=9e9;
tolerance=1e-4;
dt=1e-3;
alpha=1.2;

T_L=400;
T_R=800;
T_T=600;
T_B=900;

T=300*ones(nx,ny);
T(2:ny-1,1)=T_L;
T(2:ny-1,nx)=T_R;
T(1,2:nx-1)=T_T;
T(ny,2:nx-1)=T_B;

T(1,1)=(T_T+T_L)/2;
T(nx,ny)=(T_R+T_B)/2;
T(1,ny)=(T_T+T_R)/2;
T(nx,1)=(T_L+T_B)/2;

T_old=T;
T_initial=T;

k1=1.1*(dt/(dx^2));
k2=1.1*(dt/(dy^2));


iterative_solver=1;

if iterative_solver==1
    steady_explicit_iteration=1;
    for k=1:nt
        error=1;

        while (error>tolerance)
            for i=2:nx-1
                for j=2:ny-1

                    term1=T_old(i,j);
                    term2=k1*(T_old(i+1,j)-2*T_old(i,j)+(T_old(i-1,j)));
                    term3=k2*(T_old(i,j+1)-2*T_old(i,j)+(T_old(i,j-1)));
                    T(i,j)=term1+term2+term3;

                end
            end

            error=max(max(abs(T_old-T)));
            T_old=T;
            steady_explicit_iteration=steady_explicit_iteration+1;
        end
    end 
end

figure(1)
contourf(x,y,T)
clabel(contourf(x,y,T))
colorbar
colormap(jet)
set (gca,'ydir', 'reverse')
xlabel('X-Axis')
ylabel('Y-Axis')
title (sprintf('No. of steady explicit iterations = %d' , steady_explicit_iteration));
pause(0.03);

Final result:

Scheme Method Iterations
Steady implicit Jacobi 208
Gauss Siedel 112
Successive over relaxation 76
Unsteady implicit Jacobi 3880
Gauss siedel 3612
Successive over relaxation 3124
explicit 1941
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