2d Unsteady state heat conduction equation using Gauss-seidel method

I Wrote the code for 2d unsteady state using jacobi method. Please check it is correct or not. close all

clear all

clc

%Solving the Unsteady state 2D heat conduction by Gauss Seidel Method(IMPLICIT SCHEME)

%Input Parameters

%Number of grid points

nx = 10;

ny = nx;

nt = 1400;

x = linspace(0,1,nx);

y = linspace(0,1,ny);

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

dy = dx;

%Absolute error criteria > tolerance

error = 9e9;

tolerance = 1e-4;

dt = 1e-3;

%Defining Boundary conditions

T_L = 400;

T_T = 600;

T_R = 800;

T_B = 900;

T = 300*ones(nx,ny);

T(2:ny - 1, 1) = T_L;

T(2:ny - 1, nx) = T_R;

T(1, 2:nx - 1) = T_T;

T(ny, 2:nx - 1) = T_B;

%Calculating Average temperature at corners

T(1,1) = (T_T + T_L)/2;

T(nx,ny) = (T_R + T_B)/2;

T(1,ny) = (T_T + T_R)/2;

T(nx,1) = (T_L + T_B)/2;

%Assigning orginal values to T

T_old = T;

T_intial = T;

%Calculation of 2D steady heat conduction EQUATION by Gauss-seidel method

A= 1.1*(dt/(dx^2));

Gauss_Seidel_iteration = 1;

for k = 1:nt

    error = 9e9;

    while(error > tolerance)

        for i = 2:nx - 1

            for j = 2:ny - 1

                T(i,j)= T_intial(i,j).*(1-4*A) +  A *(T(i-1,j)+ T_old(i+1,j)+T(i,j+1)+T_old(i,j-1));

            end

        end

        error = max(max(abs(T_old - T)));

        T_old = T;

        Gauss_Seidel_iteration = Gauss_Seidel_iteration + 1;

    end

    T_intial = T;

    %Plotting

    figure(1)

    contourf(x,y,T)

    clabel(contourf(x,y,T))

    colorbar

    colormap(jet)

    set(gca, 'ydir', 'reverse')

    xlabel('X-Axis')

    ylabel('Y-Axis')

    title(sprintf('No. of Unsteady Gauss-Seidel Iterations(IMPLICIT) = %d', Gauss_Seidel_iteration));

    pause(0.03)

end

[Ahmed Fawky]

why do you take average temperatures in corners?

[Torsten]

Do you have a better idea if the temperature where the edges meet is not identical on both edges ?

If fixed temperatures are set on all 4 boundary edges, the temperatures set in the corners will not influence the resulting temperature distribution over time, at least for the discretization used above.

Answer 1

[Wan Ji]

Hi, Rajat Powade

I have checked your code carefully and I know you have used the 5-point Gauss-Seidel difference method to obtain the solution of 2d unsteady state in heat transfer modelling. I don't think there is anything wrong with your code, and the final result looks beautiful. Only one thing that should be modfied from your code is to make it run faster by avoiding loops:

Therefore, you need to replace the following code lines

for i = 2:nx - 1
    for j = 2:ny - 1
        T(i,j)= T_intial(i,j).*(1-4*A) +  A *(T(i-1,j)+ T_old(i+1,j)+T(i,j+1)+T_old(i,j-1));
    end
end

with these three lines below

i = 2:nx - 1;
j = 2:ny - 1;
T(i,j)= T_intial(i,j).*(1-4*A) +  A *(T(i-1,j)+ T_old(i+1,j)+T(i,j+1)+T_old(i,j-1));

Wish you all the best!

Wan Ji