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  1. Home/
  2. Nikith M/
  3. Steady Vs Unsteady (Mid Term)

Steady Vs Unsteady (Mid Term)

Objective: To solve steady and unsteady 2D heat conduction equation using different iterative techniques for implicit and explicit methods in MATLAB.   Introduction:   Assumptions: - The domain is a square domain - Number of points along the 'x-direction' is equal to the number of points along the 'y-direction'…

  • MATLAB

  • Nikith M

    updated on 28 Jan 2020

Objective: To solve steady and unsteady 2D heat conduction equation using different iterative techniques for implicit and explicit methods in MATLAB.

 

Introduction:

 

Assumptions:

- The domain is a square domain

- Number of points along the 'x-direction' is equal to the number of points along the 'y-direction' (nx=nynx=ny)

- Boundary conditions :  * Left side:      400K

                                    * Right side:    800K

                                    * Top side:       600K

                                    * Bottom side:  900K

- Absolute error is taken as 1e-4

 

2D Heat Conduction Equation:

Steady State: ∂2T∂x2+∂2T∂y2=0∂2T∂x2+∂2T∂y2=0

UnSteady State: ∂2T∂x2+∂2T∂y2=(1α)∂T∂t∂2T∂x2+∂2T∂y2=(1α)∂T∂t

 

Steady State:

 

 

Steady State 

clear all
close all
clc

% Given Conditions 
nx = 10;
ny = 10;

% Domain is unit square along x and y
 x = linspace(0,1,nx);
 y = linspace(0,1,ny);
 dx = 1/(nx-1);
 dy = 1/(ny-1);
 % let the initial temp be 100k
 T = 100*ones(nx,ny);
 
 % BC
 T(:,1)  =   400; % Left
 T(:,nx) =   800; % Right
 T(1,:)  =   600; % Top
 T(ny,:) =   900; % Bottom
 
 % Tolerance and eeror
 error = 9e9;
 absolute_error = 1e-4;
 
 % STEADY STATE ANALYSIS
 % EXPLICIT METHOD
 T_old = T;
 T_initial = T;
  solver = input('number =');
 
 alpha = 1.25;
 K = 2*(dx^2+dy^2)/((dx^2)*(dy^2));
 
 % JACOBI
 
 tic;
 if solver == 1
     jacobi_iter=1;
     while (error>absolute_error)
         for i = 2:nx-1
             for j = 2:ny-1
                 T(i,j) = (K^-1)*((T_old(i-1,j)+T_old(i+1,j))/(dx^2)+(T_old(i,j+1)+T_old(i,j-1))/(dy^2));
             end
         end
         error = max(max(abs(T-T_old)));
         T_old = T;
         jacobi_iter = jacobi_iter+1;
         toc
         simulation_steady_jacobi = toc;
     end
     figure(1)
     [X,Y] = contourf(x,y,T);
     clabel(X,Y)
     colormap(jet)
     colorbar
     xlabel('X-Axis')
     ylabel('Y-Axis')
     title_text =sprintf('Steady state Jacobi Iteration =%d, simulation time =%s', jacobi_iter, simulation_steady_jacobi);
     title(title_text)
     pause(0.0005)
 end
 
 % Gauss Siedel Method

 tic;
 if solver ==2 
     gauss_iter = 1;
     while (error>absolute_error)
         for i = 2:nx-1
             for j = 2:ny-1
                 T(i,j) = (K^-1)*((T(i-1,j)+T_old(i+1,j))/(dx^2)+(T_old(i,j+1)+T(i,j-1))/(dy^2));
             end
         end
         error = max(max(abs(T-T_old)));
         T_old = T;
         gauss_iter = gauss_iter+1;
         toc
         simulation_steady_gauss = toc;
     end
     figure(2)
     [X,Y] = contourf(x,y,T);
     clabel(X,Y)
     colormap(jet)
     colorbar
     xlabel('X-Axis')
     ylabel('Y-Axis')
     title_text =sprintf('Steady State Gauss Iteration =%d, Simulation time =%s', gauss_iter, simulation_steady_gauss);
     title(title_text)
     pause(0.0005)
 end
 
 % Successive over relaxation
 
 tic;
 if solver == 3;
     sor_iter=1;
     while (error>absolute_error)
         for i = 2:nx-1
             for j = 2:ny-1
                 k1 = (T(i-1,j)+T_old(i+1,j))/(dx^2);
                 k2 = (T_old(i,j+1)+T(i,j-1))/(dy^2);
                 T_gauss(i,j) = (k1+k2)/K;
                 T(i,j) = T_old(i,j)*(1-alpha)+alpha*(T_gauss(i,j));
             end
         end
         error = max(max(abs(T-T_old)));
         T_old = T;
         sor_iter = sor_iter+1;
         toc
         simulation_steady_sor=toc;
     end
     
     figure(3)
     [X,Y] = contourf(x,y,T);
     clabel(X,Y)
     colormap(jet)
     colorbar
     xlabel('X-Axis')
     ylabel('Y-Axis')
     title_text =sprintf('Steady State SOR Iteration =%d, Simulation time =%s', sor_iter, simulation_steady_sor);
     title(title_text)
     pause(0.0005)
 end

Plots for the Steady State Analysis

 

 


                              Figure 1: Jacobi

                        Figure 2: Gauss Siedel

                             Figure 3: SOR

 

Transient Explicit 

% TRANSIENT EXPLICIT METHOD

close all
clear all
clc

% Given conditions 
nx = 10;
ny = 10;

% Domain is unit square along x and y
 x = linspace(0,1,nx);
 y = linspace(0,1,ny);
 dx = 1/(nx-1);
 dy = 1/(ny-1);

 % Time step dt 
 dt = 1e-2;
 nt = 2/dt;
 
% Initial temperature be 100k
T = 100*ones(nx,ny);

% BC
 % BC
 T(:,1)  =   400; % Left
 T(:,nx) =   800; % Right
 T(1,:)  =   600; % Top
 T(ny,:) =   900; % Bottom
 
% Tolerance and error
error = 9e9;
absolute_error = 1e-4;
beta = 0.25;
alpha = 1.25;

% Transient state analysis
m1 = beta*dt/(dx^2);
m2 = beta*dt/(dy^2);
m = 1-(2*m1)-(2*m2);

% Storing the initial values of T
T_old = T;
T_initial = T;

solver = input('number = ');

%       EXPLICIT METHOD

% JACOBI

tic
jacobi_iter = 1;
if solver == 1
    for s = 1:nt
        while (error>absolute_error)
            for i = 2:nx-1
                for j = 2:ny-1
                    T(i,j) = T_old(i,j)*m+m1*(T_old(i+1,j)+T_old(i-1,j))+m2*(T_old(i,j+1)+T_old(i,j-1));
                end
            end
            error = max(max(abs(T-T_old)));
            jacobi_iter = jacobi_iter+1
            T_old = T;
        end
    end
    toc
    transient_explicit_jacobi = toc;
    
    % Plot
    figure(4)
    [X,Y] = contourf(x,y,T);
    clabel(X,Y)
    colormap(jet)
    colorbar
    xlabel('X-Axis')
    ylabel('Y-Axis')
    title_text =sprintf('Transient Explicit Jacobi Iteration =%d, simulation time =%s', jacobi_iter, transient_explicit_jacobi);
     title(title_text)
     pause(0.0005)
end

% Gauss Siedel 

tic
if solver == 2
    gauss_iter = 1;
    for s = 1:nt
        while (error>absolute_error)
            for i = 2:nx-1
                for j = 2:ny-1
                    T(i,j) = T_old(i,j)*m+m1*(T_old(i+1,j)+T(i-1,j))+m2*(T_old(i,j+1)+T(i,j-1));
                end
            end
            error = max(max(abs(T-T_old)));
            gauss_iter = gauss_iter+1
            T_old = T;
        end
    end
    toc
    transient_explicit_gauss = toc;
    
    % Plot
    figure(5)
    [X,Y] = contourf(x,y,T);
    clabel(X,Y)
    colormap(jet)
    colorbar
    xlabel('X-Axis')
    ylabel('Y-Axis')
    title_text =sprintf('Transient Explicit Gauss Iteration =%d, simulation time =%s', gauss_iter, transient_explicit_gauss);
     title(title_text)
     pause(0.0005)
end

% SOR

tic
if solver == 3
    sor_iter = 1;
    for s = 1:nt
        while (error>absolute_error)
            for i = 2:nx-1
                for j = 2:ny-1
                    T(i,j) = T_old(i,j)*(1-alpha)+alpha*(T_old(i,j)*m+m1*(T_old(i+1,j)+T(i-1,j))+m2*(T_old(i,j+1)+T(i,j-1)));
                end
            end
            error = max(max(abs(T-T_old)));
            sor_iter = sor_iter+1
            T_old = T;
        end
    end
    toc
    transient_explicit_sor = toc;
    
    % Plot
    figure(6)
    [X,Y] = contourf(x,y,T);
    clabel(X,Y)
    colormap(jet)
    colorbar
    xlabel('X-Axis')
    ylabel('Y-Axis')
    title_text =sprintf('Transient Explicit SOR Iteration =%d, simulation time =%s', sor_iter, transient_explicit_sor);
     title(title_text)
     pause(0.0005)
end

 

Plots for Transient Explicit Method

                              Figure 4: Jacobi

                        Figure 5: Gauss Siedel

                              Figure 3: SOR

 

Transient Implicit 

% TRANSIENT IMPLICIT METHOD

close all
clear all
clc

% Given conditions 
nx = 10;
ny = 10;

% Domain is unit square along x and y
 x = linspace(0,1,nx);
 y = linspace(0,1,ny);
 dx = 1/(nx-1);
 dy = 1/(ny-1);

 % Time step dt 
 dt = 1e-2;
 nt = 2/dt;
 
% Initial temperature be 100k
T = 100*ones(nx,ny);

 % BC
 T(:,1)  =   400; % Left
 T(:,nx) =   800; % Right
 T(1,:)  =   600; % Top
 T(ny,:) =   900; % Bottom
 
% Tolerance and error
error = 9e9;
absolute_error = 1e-4;
beta = 0.25;
alpha = 1.25;

% Storing the initial values of T
T_old = T;
T_initial = T;

solver = input('number = ');

% IMPLICIT METHOD
k1 = (beta*dt/(dx^2));
k2 = (beta*dt/(dy^2));
k = 1+(2*k1)+(2*k2);

% JACOBI

tic
jacobi_iter = 1;
if solver == 1
    for s = 1:nt
        while(error>absolute_error)
            for i = 2:nx-1
                for j = 2:ny-1
                   T_left = T_old(i-1,j);
                   T_right = T_old(i+1,j);
                   T_top = T_old(i,j+1);
                   T_bottom = T_old(i,j-1);
                   T(i,j) = ((T_initial(i,j)+k1*(T_left+T_right)+k2*(T_top+T_bottom))/k);                   
                end
            end
            error = max(max(abs(T-T_old)));
            T_old = T;
            jacobi_iter = jacobi_iter+1
            
        end
    end
    toc
    transient_implicit_jacobi = toc;
    
    % Plot
    figure(1)
    [X,Y] = contourf(x,y,T);
    clabel(X,Y)
    colormap(jet)
    colorbar
    xlabel('X-Axis')
    ylabel('Y-Axis')
    title_text =sprintf('Transient Explicit Jacobi Iteration =%d, simulation time =%s', jacobi_iter, transient_implicit_jacobi);
    title(title_text)
    pause(0.0005)
end

% GAUSS SIEDEL

tic
gauss_iter = 1;
if solver == 2
    for s = 1:nt
        while(error>absolute_error)
            for i = 2:nx-1
                for j = 2:ny-1
                   T_left = T(i-1,j);
                   T_right = T(i+1,j);
                   T_top = T(i,j+1);
                   T_bottom = T(i,j-1);
                   T(i,j) = ((T_initial(i,j)+k1*(T_left+T_right)+k2*(T_top+T_bottom))/k);                   
                end
            end
            error = max(max(abs(T-T_old)));
            T_old = T;
            gauss_iter = gauss_iter+1
            
        end
    end
    toc
    transient_implicit_gauss = toc;
    
    % Plot
    figure(1)
    [X,Y] = contourf(x,y,T);
    clabel(X,Y)
    colormap(jet)
    colorbar
    xlabel('X-Axis')
    ylabel('Y-Axis')
    title_text =sprintf('Transient Explicit Gauss Iteration =%d, simulation time =%s', gauss_iter, transient_implicit_gauss);
    title(title_text)
    pause(0.0005)
end
    
% SOR

tic
sor_iter = 1;
if solver == 3
    for s = 1:nt
        while(error>absolute_error)
            for i = 2:nx-1
                for j = 2:ny-1
                   T_left = T(i-1,j);
                   T_right = T(i+1,j);
                   T_top = T(i,j+1);
                   T_bottom = T(i,j-1);
                   T(i,j) = (1-alpha)*(T_old(i,j))+alpha*((T_initial(i,j)+k1*(T_left+T_right)+k2*(T_top+T_bottom))/k);                   
                end
            end
            error = max(max(abs(T-T_old)));
            T_old = T;
            sor_iter = sor_iter+1
            
        end
    end
    toc
    transient_implicit_sor = toc;
    
    % Plot
    figure(1)
    [X,Y] = contourf(x,y,T);
    clabel(X,Y)
    colormap(jet)
    colorbar
    xlabel('X-Axis')
    ylabel('Y-Axis')
    title_text =sprintf('Transient Explicit SOR Iteration =%d, simulation time =%s', sor_iter, transient_implicit_sor);
    title(title_text)
    pause(0.0005)
end

 

Plots for Transient Implicit Method

                             Figure 7: Jacobi

 

                        Figure 8: Gauss Siedel

                                  Figure 9: SOR

 

Conclusion

  • The transient state requires more computation time than steady-state equations. In the steady-state scheme, the time-derivative term is zero. Hence it requires lesser computational time.
  •  The implicit method is unconditionally stable. If the CFL number is greater than 0.25 then the implicit method is preferred.
  • It can be seen that the Gauss-Seidel iterative technique is preferred over Jacobi as it takes lesser iterations for convergence.
  • SOR is preferred over Gauss-Seidel as it takes less time and iterations for convergence.
  • Iterations for Implicit > Explicit.
  • Iterations for Jacobi > Gauss-Seidel > SOR
  • Iterations for Transient State > Steady State
  • For the explicit case, there was no need to apply

 

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