Question

creat a Matlab simulation the solves the 2D heat equation.

Answer 1

The solution has been give below :-

%--- Heat Equation in two dimensions-------------------------------------

%--- Solves Ut=alpha*(Uxx+Uyy)-------------------------------------------

clc;

close all;

clear all;

%--dimensions...........................................................

N = 51;

DX=0.1; % step size

DY=0.1;

Nx=5;

Ny=5;

X=0:DX:Nx;

Y=0:DY:Ny;

alpha=7; % random thermal diffusivity

%--boundary conditions----------------------------------------------------

U(1:N,1:N) = 0 ;

U(1,1:N) = 100;

U(N,1:N) = 0;

U(1:N,1) = 0;

U(1:N,N) = 0;

%--initial condition------------------------------------------------------

U(23:29,23:29)=1000; % a heated patch at the center

Umax=max(max(U));

%-------------------------------------------------------------------------

DT = DX^2/(2*alpha); % time step

M=2000; % maximum number of allowed iteration

%---finite difference scheme----------------------------------------------

fram=0;

Ncount=0;

loop=1;

while loop==1;

ERR=0;

U_old = U;

for i = 2:N-1

for j = 2:N-1

Residue=(DT*((U_old(i+1,j)-2*U_old(i,j)+U_old(i-1,j))/DX^2 ...

+ (U_old(i,j+1)-2*U_old(i,j)+U_old(i,j-1))/DY^2) ...

+ U_old(i,j))-U(i,j);

ERR=ERR+abs(Residue);

U(i,j)=U(i,j)+Residue;

end

end

if(ERR>=0.01*Umax) % allowed error limit is 1% of maximum temperature

Ncount=Ncount+1;

if (mod(Ncount,50)==0) % displays movie frame every 50 time steps

fram=fram+1;

surf(U);

axis([1 N 1 N ])

h=gca;

get(h,'FontSize')

set(h,'FontSize',12)

colorbar('location','eastoutside','fontsize',12);

xlabel('X','fontSize',12);

ylabel('Y','fontSize',12);

title('Heat Diffusion','fontsize',12);

fh = figure(1);

set(fh, 'color', 'white');

F=getframe;

end

%--if solution do not converge in 2000 time steps------------------------

if(Ncount>M)

loop=0;

disp(['solution do not reach steady state in ',num2str(M),...

'time steps'])

end

%--if solution converges within 2000 time steps..........................

else

loop=0;

disp(['solution reaches steady state in ',num2str(Ncount) ,'time steps'])

end

end

%------------------------------------------------------------------------

%--display a movie of heat diffusion------------------------------------

movie(F,fram,1)

%------END---------------------------------------------------------------