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

• Solve the 2D heat conduction equation by using the point iterative techniques.

• Solve the steady state equation time defendent varaible dx,dy.
• create the codes x direction,y direction.
• Initialize the matrix form gives to ones value for all.
• Give the boundry condition 2d plate top,bottom,right,left.
• Give x,y coordinates value for using linspace
• Iterative solver using compare less than error values.
• error value mention that 9e-9.
• Using numeric formula central differencing equation.
• Equation should be value is less than given value.
• So implementing the implicit method
• Jacobi method
• Gauss seidel method
• Over relaxation method
• Using for loop,elseif
• Create the contour plots.

Contour plots

close all

clear all

clc

%inputs

nx=10;

ny=nx;

time=1;

% initialize

t=ones(nx,ny);

% copy of t old value

told=t;

% boundry condition

t(:,1)=400;

t(1,:)=900;

t(:,end)=800;

t(end,:)=600;

% x,y coordinates

x=linspace(0,1,nx);

y=linspace(0,1,ny);

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

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

alpha=1;

dt=0.001;

% iteration process

iterative_solver=1;

% iterative_solver(gs)=2;

% iterative_solver(sor)=3;

% error=9e9;

nt=time/dt;

%trainsient explicit method

tol=1e-5;

k1=alpha*(dt/dx^2);

k2=alpha*(dt/dy^2);

if iterative_solver==1

iteration=1;

% while(error>tol)

for k=1:nt

for i=2:nx-1

for j=2:ny-1

t(i,j)=told(i,j)+k1*(told(i-1,j)-2*told(i,j)+told(i+1,j))+k2*(told(i,j-1)-2*told(i,j)+told(i,j+1));

end

end

% error=max (max(abs(told-t)));

told=t;

colormap(jet)

contourf(x,y,t)

iteration = iteration + 1;

title_text = sprintf('time step = %d', iteration);

title(title_text)

colorbar

pause(0.001 ...

)

end

end

Transient explicit method procedure

• Solve the steady state equation time defendent varaible dx,dy.
• Create the codes x direction,y direction.
• Initialize the matrix form gives to ones value for all.
• Give the boundry condition 2d plate top,bottom,right,left.
• Give x,y coordinates value for using linspace
• Using number of time steps heat conduction of plate.
• Using numeric formula central differencing equation.
• Equation should be value is less than given value.
• So implementing the implicit method
• Jacobi method
• Gauss seidel method
• Over relaxation method
• Using for loop,elseif
• Create the contour plots.

Contour plots

Transient implicit method

close all

clear all

clc

%inputs

nx=10;

ny=nx;

dt=0.01;

alpha=1;

% initialize

t=ones(nx,ny);

% copy of t old value

told=t;

% boundry condition

t(:,1)=400;

t(1,:)=900;

t(:,end)=800;

t(end,:)=600;

% x,y coordinates

x=linspace(0,1,nx);

y=linspace(0,1,ny);

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

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

% iteration process

iterative_solver=2;

error=9e9;

%jacobi method steady state analysis

tol=1e-5;

k1=alpha*(dt/dx^2);

k2=alpha*(dt/dy^2);

if iterative_solver==1

jacobi_iteration=1;

while(error>tol)

for i=2:nx-1

for j=2:ny-1

term1=1+2*k1+2*k2;

t(i,j)=(told(i,j)+k1*(told(i-1,j)+told(i+1,j))+k2*(told(i,j+1)+told(i,j-1)))/term1;

end

end

error=max (max(abs(told-t)));

told=t;

colormap(jet)

contourf(x,y,t)

jacobi_iteration = jacobi_iteration + 1;

title_text = sprintf('Iteration Number = %d', jacobi_iteration);

title(title_text)

colorbar

pause(0.001 ...

)

end

% Gauss seidel theorem

elseif iterative_solver==2

gauss_seidel_iteration=1;

while(error>tol)

for i=2:nx-1

for j=2:ny-1

term1=1+2*k1+2*k2;

t(i,j)=(t(i,j)+k1*(t(i-1,j)+t(i+1,j))+k2*(t(i,j+1)+t(i,j-1)))/term1;

end

end

error=max (max(abs(told-t)));

told=t;

colormap(jet)

contourf(x,y,t)

gauss_seidel_iteration = gauss_seidel_iteration + 1;

title_text = sprintf('Iteration Number = %d', gauss_seidel_iteration);

title(title_text)

colorbar

pause(0.001 ...)

% over relaxation

end

elseif iterative_solver== 3

omega=2;

sor_iteration=1;

while(error>tol)

for i=2:nx-1

for j=2:ny-1

term1=1+2*k1+2*k2;

t(i,j)=(t(i,j)+k1*(t(i-1,j)+t(i+1,j))+k2*(t(i,j+1)+t(i,j-1)))/term1;

t(i,j)=t(i,j)*(1-omega)+(omega*t(i,j));

end

end

error=max (max(abs(told-t)));

told=t;

colormap(jet)

contourf(x,y,t)

sor_iteration = sor_iteration + 1;

title_text = sprintf('Iteration Number = %d', sor_iteration);

title(title_text)

colorbar

pause(0.001 ...

)

end

end

Contour plots

Jacobi method iteration=330

Gauss-Seidel method iteration=213

Over relaxation method iteration=101

Observered :

Finally compare the 3 equation result iteration values

Transient state implicit analysis for

• Jacobi iteration=330
• Gauss seidel iteration=213
• SOR iteration=101 this method short iteration give solution is connverged.

Transient Explicit method

Solution converging takes time step to 1001.

Steady state anaysis

• jacobi iteration =235
• Gauss seidel iteration=155
• SOR method=35

Finally we observed Successive over realaxation methods takes less iteration value solution is coverged.