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

AAbhijith K Menon20 Nov 2021

The heat Equation is a PDE that describes how the distribution of quantities like heat changes with time in a solid medium as it flows from higher to lower places.

Heat equation:

α = diffusivity of medium

Types of heat schemes:

  1. Steady-state condition
  2. Transient state condition

Equations used for the codes:

  • Steady-state equation

  • unsteady-state equation

  • unsteady-state Explicit equation:

JACOBI ITERATION:

close all
clear all
clc

%jacobi method

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;

%boundry condn
T_L = 400;
T_T = 600;
T_R = 800;
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;

%avg temp

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;

%calculating jacobi

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

%plotting

figure(1);
contourf(x,y,T)
clabel(contourf(x,y,T))
colorbar
colormap(jet)
set(gca,'ydir','reverse')

xlabel('Xaxis')
ylabel('yaxis')
title(sprintf('no:of jacobi=%d',jacobi_iteration));
pause(0.03);

GAUSS SIDEL IMPLICIT METHOD:

close all
clear all
clc

%gauss siedel method

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;

%boundry condn
T_L = 400;
T_T = 600;
T_R = 800;
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;

%avg temp

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;

%calculating gauss sidel method

iterative_solver = 2;

if iterative_solver ==2
    Gaussseidel_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;
         Gaussseidel_iteration= Gaussseidel_iteration+1;
        
    end
end

%plotting

figure(1);
contourf(x,y,T)
clabel(contourf(x,y,T))
colorbar
colormap(jet)
set(gca,'ydir','reverse')

xlabel('Xaxis')
ylabel('yaxis')
title(sprintf('no:of  Gaussseidel=%d', Gaussseidel_iteration));
pause(0.03);

succesive_relaxation implicit:

close all
clear all
clc

%succesive_relaxation  method

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;
omega=1.2;

%boundry condn
T_L = 400;
T_T = 600;
T_R = 800;
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;

%avg temp

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;

%calculating succesive_relaxation method

iterative_solver = 3;

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

%plotting

figure(1);
contourf(x,y,T)
clabel(contourf(x,y,T))
colorbar
colormap(jet)
set(gca,'ydir','reverse')

xlabel('Xaxis')
ylabel('yaxis')
title(sprintf('no:of  succesive_relaxation=%d', succesive_relaxation_iteration));
pause(0.03);

UNSTEADY:

Jacobi method:

close all
clear all
clc

%jacobi method

nx=10;
ny=nx;
nt=1000;

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;

%boundry condn
T_L = 400;
T_T = 600;
T_R = 800;
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;

%avg temp

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_intial=T;
T_old=T;

%calculating jacobi
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
                
                Term_1= 1/(1+(2*k1)+(2*k2));
                Term_2= k1*Term_1;
                Term_3= k2*Term_1;
                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_intial(i,j)*Term_1)+(H*Term_2)+(V*Term_3);
                
            end
        end
        
        error = max(max(abs(T_old -T)));
        T_old=T;
        jacobi_iteration=jacobi_iteration+1;
        
    end
    T_intial=T;
                

%plotting

figure(1);
contourf(x,y,T)
clabel(contourf(x,y,T))
colorbar
colormap(jet)
set(gca,'ydir','reverse')

xlabel('Xaxis')
ylabel('yaxis')
title(sprintf('no:of unsteady jacobi=%d',jacobi_iteration));
pause(0.03);

end

gauss seidel method:

close all
clear all
clc

%gausssidel method

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;

%boundry condn
T_L = 400;
T_T = 600;
T_R = 800;
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;

%avg temp

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_intial=T;
T_old=T;

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

gausssidel_iteration=1;

for k=1:nt
    error=9e9;
    
    while(error > tolerance)
        for i=2:nx-1
            for j=2:ny-1
                
                Term_1= 1/(1+(2*k1)+(2*k2));
                Term_2= k1*Term_1;
                Term_3= k2*Term_1;
                H= (T(i-1,j))+(T_old(i+1,j));
                V= (T(i,j+1))+(T_old(i,j-1));
                
                T(i,j)= (T_intial(i,j)*Term_1)+(H*Term_2)+(V*Term_3);
                
            end
        end
        
        error = max(max(abs(T_old -T)));
        T_old=T;
        gausssidel_iteration=gausssidel_iteration+1;
        
    end
    T_intial=T;
                

%plotting

figure(1);
contourf(x,y,T)
clabel(contourf(x,y,T))
colorbar
colormap(jet)
set(gca,'ydir','reverse')

xlabel('Xaxis')
ylabel('yaxis')
title(sprintf('no:of unsteady gausssidel=%d',gausssidel_iteration));
pause(0.03);

end

succesive_relaxation(SOR):

close all
clear all
clc

%succesive_relaxation method

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;
omega=1.1;

%boundry condn
T_L = 400;
T_T = 600;
T_R = 800;
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;

%avg temp

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_intial=T;
T_old=T;
T_end=T;

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

succesive_relaxation_iteration=1;

for k=1:nt
    error=9e9;
    
    while(error > tolerance)
        for i=2:nx-1
            for j=2:ny-1
                
                Term_1= 1/(1+(2*k1)+(2*k2));
                Term_2= k1*Term_1;
                Term_3= k2*Term_1;
                H= (T(i-1,j))+(T_old(i+1,j));
                V= (T(i,j+1))+(T_old(i,j-1));
                
                T(i,j)= (T_intial(i,j)*Term_1)+(H*Term_2)+(V*Term_3);
                T_end(i,j)= ((1-omega)*T_old(i,j))+(T(i,j)*omega);
                
            end
        end
        
        error = max(max(abs(T_old -T)));
        T_old=T_end;
        succesive_relaxation_iteration=succesive_relaxation_iteration+1;
        
    end
    T_intial=T_end;
                

%plotting

figure(1);
contourf(x,y,T)
clabel(contourf(x,y,T))
colorbar
colormap(jet)
set(gca,'ydir','reverse')

xlabel('Xaxis')
ylabel('yaxis')
title(sprintf('no:of unsteady succesive_relaxation=%d',succesive_relaxation_iteration));
pause(0.03);

end

EXPLICIT method - unsteady/Transient condition:

close all
clear all
clc

%succesive_relaxation method(EXPLICIT)

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;
omega=1.1;

%boundry condn
T_L = 400;
T_T = 600;
T_R = 800;
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;

%avg temp

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_intial=T;
T_old=T;


%calculating jacobi
iterative_solver=1;
k1= 1.1*(dt/(dx^2));
k2= 1.1*(dt/(dy^2));

if iterative_solver==1
    steady_explicit=1;
    
    for k=1:nt
        error=1;
        
        while(error>tolerance)
            
            for i=2:nx-1
                for j=2:ny-1
                    Term_1=T_old(i,j);
                    Term_2=k1*(T_old(i+1,j)-2*T_old(i,j)+(T_old(i-1,j)));
                    Term_3=k2*(T_old(i,j+1)-2*T_old(i,j)+T_old(i,j-1));
                    T(i,j)=Term_1+Term_2+Term_3;
                    
                end
            end
            
            error=max(max(abs(T_old-T)));
            T_old=T;
            steady_explicit= steady_explicit+1;
            
        end
    end
end

                    


%plotting

figure(1);
contourf(x,y,T)
clabel(contourf(x,y,T))
colorbar
colormap(jet)
set(gca,'ydir','reverse')

xlabel('Xaxis')
ylabel('yaxis')
title(sprintf('no:of unsteady iteration(EXPLICIT)=%d', steady_explicit));
pause(0.03);

RESULTS:

TYPE OF SCHEME METHODS ITERATIONS
Implicit steady-state analysis Jacobi 208
Implicit steady-state analysis

Gauss seidel

 112
Implicit steady-state analysis SOR 46
Implicit Transient-state analysis Jacobi 3880
Implicit Transient-state analysis SOR 3290
Implicit Transient-state analysis Gauss seidel 3612
  • Jacobi > Gauss sidel>SOR according to number of iterations
  • Steady-state takes less time to compute and fewer iterations
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