2D Implicit Transient Heat conduction Problem

Question

Sunag R A 2018-6-3

0
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https://ww2.mathworks.cn/matlabcentral/answers/403795-2d-implicit-transient-heat-conduction-problem

回答： darova 2020-6-11

Dear all,

I am trying to solve a 2D transient implicit Heat conduction problem using Iterative methods like Jacobi, Gauss Siedel and SOR method.

I have written a code for it. But I have a little problem in looping the inner nodes. I have properly assigned Boundary conditions, also given the inner loop

Iteration. But I am not able to figure out the reason for the proper profile. The solution is stopping after only 3 Iterations.

I have attached the code below. Please let me know if anyone get it. Thanks !!

 % Space Initialization 
Lx = 1; Ly = 1;
nx = 11;
ny = 11;
dx = Lx/(nx-1); dy = Ly/(ny-1);
 % Time Initialization
time = 0.2;
k = 1.65e-4;             % Thermal Diffusivity
dt = 1e-4;
nt = time/dt;
k1 = k*dt/(dx^2);
k2 = k1;
 % Mesh
x = linspace(0,Lx,nx);
y = linspace(0,Ly,ny);
[X, Y] = meshgrid(x,y);
 % Limits
tol = 1e-4;
error = 9e9;
 % Temp Initialization
T = ones(nx,ny);
 % Boundary conditions
T(:,1) = 400; % Left
T(:,end) = 800; % Right
T(1,2:end-1) = 600; % Top
T(end,2:end-1) = 900; % Bottom
 % Copying T
Told = T;
 % Implicit method
iter = 1;
T_prev_dt = Told;
 % Loop
 for k = 1:nt
    while(error > tol)
        for i = 2:nx-1
            for j = 2:ny-1
                term1 = (1+(2*k1)+(2*k2))^-1;
                term2 = k1*term1;
                term3 = k2*term1;
                H = (Told(i-1,j)+Told(i+1,j));
                V = (Told(i,j-1)+Told(i,j+1));
                T(i,j) = (T_prev_dt(i,j)*term1)+(H*term2)+(V*term3);
            end
        end          
            error = max(max(abs(Told - T)));
            Told = T;
            iter = iter+1;
    end
    T_prev_dt = T;
end
 %%Plotting
C = contourf(X,Y,T,'edgecolor','none');
set(gca,'YDIR','reverse');
colormap(jet);
clabel(C,'FontSize',12);