PDE Neumann boundary condition depends on u; qmatrix(u)

Hi, I want to solve following parabolic equation (heat transfer)

u = parabolic(u0, tlist, b,p,e,t,c,a,f,d,rtol,atol);

The Geometry is a simple rectangle, and to creat the mesh I use

[p, e, t] = initmesh(rectangle, 'Hmax', 1e-3);

Both sides of the rectangle are adiabatic. On the bottom a constant heat flux is given. On top I want to set a heat flux coefficient which depends on the temperature (h = 1.5*u-300; u = temp.)

The boundary conditions are defined as follows:

function [q_matrix,g_matrix,h_matrix,r_matrix] = boundary_conditions(p,e,u,time);
q_matrix = zeros(1,ne);
g_matrix = q_matrix;
h_matrix = zeros(1,2*ne);
r_matrix = h_matrix;

for k = 1:ne
       switch e(5,k)
        case {1,3} %boundarys edge 1 and 3 adiabatic
            q_matrix(k) = 0; 
            g_matrix(k) = 0;

          case {2} %boundary edge 2 heat transfer coefficient
               q_matrix(k) = 1.5 * (u(e(1,k))+u(e(2,k)))/2 -300; 
               g_matrix(k) = (1.5 * (u(e(1,k))+u(e(2,k)))/2 -300) * 273; 
                                                        %T_ambient = 273K 
          otherwise %boundary edge 4 heat flux
              q_matrix(k) = 0;
              g_matrix(k) = 2000;

      end
  end

My Problem now is, that this is not working when I use a function of u in q_matrix! I get following error:

??? Attempted to access u(2); index out of bounds because numel(u)=0.

Error in ==> boundary_conditions at 53
             q_matrix(k) = 1.5 * (u(e(1,k))+u(e(2,k)))/2 -300;

I hope somebody can help me,

Thanks Felix

Answer 1

[Bill Greene]

Hi,

Are you by any chance using a version of MATLAB older than R2012b?

The capability to have coefficients and boundary conditions that are functions of the dependent variables in parabolic (and hyperbolic) was an enhancement for R2012b.

Other than that, the code snippets you show above basically look OK to me.

Regards.

Bill

Answer 2

[Felix]

Hi Bill,

thanks for your answer! Unfortunately I don't have the change to use MATLAB R2012b. I use MATLAB R2012a.

Is an upgrade the only possibility to solve my problem? I thought that the described problem is more or less a standart problem by using MATLAB PDE.

Regards Felix

Answer 3

[Bill Greene]

Yes, you have boundary conditions that are a function of u.

That was not supported in PDE Toolbox for transient PDE (parabolic and hyperbolic) before R2012b. Older versions of PDE Toolbox do let you solve a stationary problem with coefficients and boundary conditions that are a function of u using the the pdenonlin function.

Bill