2D steady state heat conduction equation using Jacobi iteration

2D steady heat conduction equation on the unit square subject to the following

Dirichlet boundary conditions:

T(x,0)=100x T(0,y)=200y

T(x,1) =200+100sin(pi*x) T(1,y)=100(1+y)

T(x,y) =0 (initial condition)

Use uniform grid in x and y of 21 points in each direction. Iterate until the maximum change is less than 0.1.

I set up the problem using a Jacobi iteration method;

and I have an issue setting up the boundary conditions correctly. Whenever I run the code it show me this (Unable to perform assignment because the size of the left side is 21-by-1 and the size of the right side is 21-by-21.) if any one could solve the issue for me? Thanks in advance !

Here's my code

for I=1:3
    tic
    % problem setup 
    Lx = 1;  %length of x grid 
    Ly = Lx; % length of y grid
    nx = 21; % number of nodes in x- direction
    ny = nx; %number of nodes in y- direction 
    dx = Lx/(nx-1); % step size for spatial x-cord
    dy = Ly/(ny-1); % step size for spatial y-cord
    tol = 0.1;  %error criteria 
    error = 9e9; %initialising error 
    k=0;  %initialising iteration
   % creating 1D and 2D grid 
   x = linspace(0,Lx,nx);   %1d x grid 
   y = linspace(0,Ly,ny);   %1d y grid
   [x y] = meshgrid(x,y);   %2d x-y grid
  % initialising temperature array and BC 
  T = zeros(nx,ny);   %initial condition array 
  % BC
  T(1:ny,ny)= 100*(1+y) ;        
  T(1:ny,1)= 200*y ;   
  T(nx,1:nx)= 200+100*sin(pi*x) ;        
  T(1,1:nx)= 100*x ;       
  %creating a copy of temperature array after BC
  Told = T; 
  while error > tol   %while loop runs untill error > tolerance value 
      for i = 2:nx-1  %for loop runs for x values from 2 to last node -1 
          for j = 2:ny-1 %for loop runs for y value form 2 to last node -1 
              %iterative solver 
              if I== 1 %jacobi 
                  figure(1)
                  T(i,j) = 0.25*(Told(i-1,j) + Told(i+1,j) + Told(i,j-1) + Told(i,j+1));
              end 
          end 
      end
      error_T = max(abs(Told - T));  %calculating error to decide wether while loop should stop or not 
  error = max(error_T);    %calculating maximum error between absolute value of old arry T and new arry of T 
  Told = T;   %updating temperature array for next iteration 
  k = k+1;    %incrementing iterations by 1 
  % contour map 
  [C, h] = contourf(x,y,T);           %contourf function used 
  clabel(C, h,'FontSize',8,'FontWeight','bold','color','k') 
  xlabel('spatial X -coordinate','FontSize',12)
  ylabel('spatial Y -coordinate','FontSize',12)
  title (sprintf('Temperature distribution on a %d x %d gridn using Jacobi method at iteration no %d', Lx, Ly, k),'FontSize',10)
  pause(0.003)
  end

[John D'Errico]

A picture may be worth a thousand words, but a picture of code is worthless, if you really want help. In fact, it is easier by far to paste in the code itself as simple text. That way someone can edit the code, and look far more easily at the code. Worse yet, you don't even show enough for someone to look at your specific problem.

[Zahraa]

thank you for your comment, I just edit it !

Answer 1

[esat gulhan]

Try this instead of BC;

T(1:ny,ny)= 200+100*(sin(pi*y(:,1))) ;        
T(1:ny,1)= 200+100*y(:,1)   
T(nx,1:nx)= 200*x(1,:) ;        
T(1,1:nx)= 100*(1+x(1,:)) ; 

[Zahraa]

yes it works, thanks !