Can MATLAB Handle Finite Difference Modeling for Spherical Heat Transfer?

[cayusbonus]

Hello,

I'm triying to solve the unidimensional heat transfer equation in transient scheme for an sphere with Crank Nicholson discretization. Because I must to obtain the Heisler Charts, I'm triying to solve the adimensional equation, that is this equation:

Where 4.1 is the equation, 4.2 the initial condition and 4.3 and 4.4 the boundary conditions (Neumann). In this equation, the Fo is an adimensional time, R is an adimensional radius and theta, an adimensional temperature. Then, the objective is to obtain Theta in all the grid points. The grid is the discretization of R in x-axe and Fo (time) in y-axe.

I've used the fictitious point system for implement the boundary conditions, and after a hard work, I've made this code:

M=8;
N=15;
DR=1/M;
DFo=0.2/N;
a=DFo/(2*DR);
hold on
for j=0.01:0.05:0.8
Bi=j;
c=-2*DR*Bi;
for m=1:M+1
    dA0(m)=-1-2*a;
    if m>1
        dA1(m-1)=a*(1+1/(m-1));
        dA_1(m-1)=a*(1-1/m);
    end
end
A=diag(dA0,0)+diag(dA1,1)+diag(dA_1,-1);
A(1,1)=-1-6*a;
A(1,2)=6*a;
a1=a*(1+1/M);
a2=-1-2*a;
a3=a*(1-1/M);
A(M+1,M)=a1+a3;
A(M+1,M+1)=a1*c+a2;
for m=1:M+1
    dB0(m)=1-2*a;
    if m>1
        dB1(m-1)=a*(1+1/(m-1));
        dB_1(m-1)=a*(1-1/m);
    end
end
B=diag(dB0,0)+diag(dB1,1)+diag(dB_1,-1);
B(1,1)=1-6*a;
B(1,2)=6*a;
b1=a1;
b2=1-2*a;
b3=a3;
B(M+1,M)=b1+b3;
B(M+1,M+1)=b1*c+b2;
Th=ones(M+1,N+1);
C=-inv(A)*B;
for k=1:N
    Th(:,k+1)=C*Th(:,k);
end
Tita=Th(1,:);
Fo=(0:DFo:0.2);
plot(Fo,Tita)
end

With this code, I'm triying to plot the center sphere temperature evolution, this mean to plot the Theta in R=0 for all the times (Fo). But the problem is that it's not ploting correct, and my question is if you can see anything incorrrect in the code (matrix implementations, index evolutions, discretizations...). An extrange issue is that when the number of points is increased (N and M), the integration goes more and more away from the correct expected result.

Thanks.

 

[mmwave]

Hello,

Thank you for sharing your code. After reviewing it, I noticed a few potential issues that may be causing the incorrect results you are experiencing. First, it seems that your implementation of the boundary conditions may not be correct. In your code, you are using the fictitious point method, but it looks like you are only applying it to the first and last points in your grid. However, for a sphere, you will need to apply the boundary conditions at all points on the surface. This may be why your results are not matching the expected outcome.

Additionally, I noticed that your code only accounts for the Neumann boundary conditions, but it does not include the initial condition (4.2) or the Dirichlet boundary condition (4.3). These conditions are also important in accurately solving the heat transfer equation for a sphere.

Finally, I would suggest checking your matrix implementations and making sure they are correct. It's possible that there may be an error in your code that is causing the incorrect results.

I hope this helps and good luck with your research. If you have any further questions or concerns, please feel free to reach out.