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

CChinmay Vithal Chavan25 Aug 2021

Jacobi Method for steay flow

clear all
close all
clc

%solving the steady state 2D equation (jacobi method)

%Input grids
nx=30;
ny=nx;

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

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

%error criteria
error= 9e9;
tol = 1e-4;

%defining the boundary condition 
T_l= 400;
T_t= 600;
T_r= 800;
T_b= 900;

T= 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(nx,2:ny-1) = T_b;
 
 %calculating the temperature at the corners
 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); 
 
 
 %assigning the T values
 T_intial = T;
 T_old = T;
 
 
  %jacobi method 
  iterative_solver =1;
  
  if iterative_solver ==1
     jacobi_iter = 1;
     while (error > tol)
         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_iter = jacobi_iter + 1;
     end
     
     
     % plotting
    figure(1)
    contourf(x,y,T)
    clabel(contourf(x,y,T))
    colorbar
    colormap(jet)
        
    xlabel('X_axis')
    ylabel('Y_axis')
    title_text=sprintf('Jacobi steady state No of iterations(implicit)=%d',jacobi_iter);
    title(title_text)
    pause(0.03);
    
  end

steady state Gauss Seidal method

clear all
close all
clc

%solving the steady state 2D equation (gs methid)

%Input grids
nx=30;
ny=nx;

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

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

%error criteria
error= 9e9;
tol = 1e-4;

%defining the boundary condition 
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;
 
 %calculating the temperature at the corners
 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); 
 
 
 %assigning the T values
 T_intial = T;
 T_old = T;
 
 
  %gauss scidal method 
  iterative_solver =2;
  if iterative_solver ==2
     gs_iter = 1;
     while (error > tol)
         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;
     gs_iter = gs_iter + 1;
     end
     % plotting
    figure(1)
    contourf(x,y,T)
    clabel(contourf(x,y,T))
    colorbar
    colormap(jet)
        
    xlabel('X_axis')
    ylabel('Y_axis')
    title_text=sprintf('Gauss seidal steady state(implicit) No of iterations=%d',gs_iter);
    title(title_text)
    pause(0.03);
    
  end

Steady state with SOR method

clear all
close all
clc

%solving the steady state 2D equation (Sor methid)

%Input grids
nx=30;
ny=nx;

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

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

%error criteria
error= 9e9;
tol = 1e-4;
omega = 1.2;
%defining the boundary condition 
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;
 
 %calculating the temperature at the corners
 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); 
 
 
 %assigning the T values
 T_intial = T;
 T_old = T;
 
 
  %gauss scidal method 
  iterative_solver =3;
  if iterative_solver ==3
     sor_iter = 1;
     while (error > tol)
         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;
     sor_iter = sor_iter + 1;
     end
     % plotting
    figure(1)
    contourf(x,y,T)
    clabel(contourf(x,y,T))
    colorbar
    colormap(jet)
        
    xlabel('X_axis')
    ylabel('Y_axis')
    title_text=sprintf('SOR steady state (Implicit)No of iterations=%d',sor_iter);
    title(title_text)
    pause(0.03);
    
  end

Jacobi steady state No of iterations(impilcit)

Gauss Seidal steady state (implicit)No of iterations

SOR Steady state (Implicit) No of iterations

Transient state for Jacobi Method 

clear all
close all
clc

%solving the  Unsteady state 2D equation (jacobi methid)

%Input grids
nx=30;
ny=nx;

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

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


%error criteria
error= 9e9;
tol = 1e-4;
dt=2000;
alpha=1.88e-5;

%defining the boundary condition 
T_l= 400;
T_r= 800;
T_t= 600;
T_b= 900;

T= 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;
 
 %calculating the temperature at the corners
 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); 
 
 
 %assigning the T values
 T_intial = T;
 T_old = T;
 
 %calculating the 2D heat conduction equation by explicit 
 k1=(alpha*dt)/(dx^2);
 k2=(alpha*dt)/(dy^2);
 T_prev_dt= T_old;
 
  %jacobi method 
  term1= (1+2*k1+2*k2)^(-1);
  term2=k1*term1;
  term3=k2*term1;
  iterative_solver = 1;
  if iterative_solver ==1
      jacobi_iteration = 1;
     while (error > tol)
         for i = 2:nx-1
              for j = 2:ny-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_prev_dt(i,j)*term1)+(H*term2+V*term3);
              end
         end
    
     error=max(max(abs(T_old-T)));
     T_old=T;
     jacobi_iteration = jacobi_iteration + 1;
    
    % plotting
    contourf(x,y,T)
    clabel(contourf(x,y,T))
    colorbar
    colormap(jet)
    xlabel('X_axis')
    ylabel('Y_axis')
    title_text=sprintf('Jacobi Unsteady iterations(implicit)=%d',jacobi_iteration);
    title(title_text)
    pause(0.03);
     end
  end

Transient State for Gauss Seidal 

clear all
close all
clc

%solving the unsteady state 2D equation (gs method)

%Input grids
L=1;
nx=30;
ny=nx;

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

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


%error criteria
error= 9e9;
tol = 1e-4;
dt=2000;
alpha=1.88e-5;

%defining the boundary condition 
T_l= 400;
T_t= 600;
T_r= 800;
T_b= 900;

T= 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;
 
 %calculating the temperature at the corners
 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); 
 
 
 %assigning the T values
 T_intial = T;
 T_old = T;
 
 %calculating the 2D heat conduction equation by explicit 
 k1=(alpha*dt)/(dx^2);
 k2=(alpha*dt)/(dy^2);
 T_prev_dt= T_old;
 
  %gs method 
  term1= (1+2*k1+2*k2)^(-1);
  term2=k1*term1;
  term3=k2*term1;
  iterative_solver = 2;
  if iterative_solver ==2
      gs_iteration = 1;
     while (error > tol)
         for i = 2:nx-1
              for j = 2:ny-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_prev_dt(i,j)*term1)+(H*term2+V*term3);
              end
         end
    
     error=max(max(abs(T_old-T)));
     T_old=T;
     gs_iteration = gs_iteration + 1;
    
    % plotting
    contourf(x,y,T)
    clabel(contourf(x,y,T))
    colorbar
    colormap(jet)
    xlabel('X_axis')
    ylabel('Y_axis')
    title_text=sprintf('Gauss Seidal unsteady iterations(implicit)=%d',gs_iteration);
    title(title_text)
    pause(0.03);
     end
  end

Transient state For SOR Method

clear all
close all
clc

%solving the unsteady state 2D equation (SOR method)

%Input grids
L=1;
nx=30;
ny=nx;

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

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


%error criteria
error= 9e9;
tol = 1e-4;
dt=2000;
alpha=1.88e-5;

% For SOR 
alpha1= 1.8;

%defining the boundary condition 
T_l= 400;
T_t= 600;
T_r= 800;
T_b= 900;

T= 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;
 
 %calculating the temperature at the corners
 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); 
 
 
 %assigning the T values
 T_intial = T;
 T_old = T;
 
 %calculating the 2D heat conduction equation by explicit 
 k1=(alpha*dt)/(dx^2);
 k2=(alpha*dt)/(dy^2);
 T_prev_dt= T_old;
 
  %SOR method 
  term1= (1+2*k1+2*k2)^(-1);
  term2=k1*term1;
  term3=k2*term1;
  iterative_solver = 3;
  if iterative_solver == 3
      sor_iteration = 1;
     while (error > tol)
         for i = 2:nx-1
              for j = 2:ny-1
                H=(alpha1*T(i-1,j)+(1-alpha1)*T_old(i-1,j)+T_old(i+1,j));
                V=(alpha1*T(i,j-1)+(1-alpha1)*T_old(i,j-1)+T_old(i,j+1));
                T(i,j)=(T_prev_dt(i,j)*term1)+(H*term2+V*term3);
              end
         end
    
     error=max(max(abs(T_old-T)));
     T_old=T;
     sor_iteration = sor_iteration + 1;
    
    % plotting
    contourf(x,y,T)
    clabel(contourf(x,y,T))
    colorbar
    colormap(jet)
    xlabel('X_axis')
    ylabel('Y_axis')
    title_text=sprintf('Sor unsteady iterations(Implicit)=%d',sor_iteration);
    title(title_text)
    pause(0.03);
     end
  end

Transient (Implicit) for Jacobi Method

Transient Gauss Seidal

Transient SOR method

Transient Explicit

clear all
close all
clc

%solving the Transient state 2D equation for Explicit

%Input grids
nx=30;
ny=nx;
nt=1400;

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

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


%error criteria
error= 9e9;
tol = 1e-4;
dt=0.01;
alpha=1.88e-5;

%defining the boundary condition 
T_l= 400;
T_r= 800;
T_t= 600;
T_b= 900;

T= 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(nx,2:ny-1) = T_b;
 
 %calculating the temperature at the corners
 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); 
 
 
 %assigning the T values
 T_intial = T;
 T_old = T;
 
 %calculating the 2D heat conduction equation by explicit 
 k1=(alpha*dt)/(dx^2);
 k2=(alpha*dt)/(dy^2);
 
    iterative_solver =1;
   if iterative_solver ==1
      explicit_iteration = 1;
     for k=1:nt
         error=1;
     while (error > tol)
         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;
     explicit_iteration = explicit_iteration + 1;
     end
     end
    
    % plotting
    contourf(x,y,T)
    clabel(contourf(x,y,T))
    colorbar
    colormap(jet)
    xlabel('X_axis')
    ylabel('Y_axis')
    title_text=sprintf('Transient state for iterations(Explicit)=%d',explicit_iteration);
    title(title_text)
    pause(0.03);
   end

Transient Explicit

Table :

Types Methods No. of iterations
Implicit- Steady

Jacobi

Gauss Seidal

SOR

1784

956

661
Implicit- Unsteady

Jacobi

Gauss Seidal

SOR

806

433

127
Explicit-Transient 998565
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