Internal boundary conditions in subdomains

N=1;
modello=createpde(1);
thicness_e=0.05; %%mm
thicness_d=1.95; %%mm
thicness_s=10; %%mm
L=10; %mm
H1=thicness_s;
H2=thicness_s+thicness_d;
H3=thicness_s+thicness_d+thicness_e;

R1=[3 4 0 L L 0 0 0 H1 H1];
R2=[3 4 0 L L 0 H1 H1 H2 H2];
R3=[3 4 0 L L 0 H2 H2 H3 H3];
gdm=[R1; R2; R3]';
g=decsg(gdm,'R1+R2+R3',['R1';'R2';'R3']');
geometryFromEdges(modello,g);

figure (1)
pdegplot(modello,'EdgeLabels','on','FaceLabels','on')
xlabel('x, millimeters')
ylabel('y, millimeters')

How do i include in the Edge E3 the following internal boundary conditions :

where and are the temperature of Face 1 and Face 2 ?

Answer 1

[Sameer]

Hi @Carmelo Filippo

If you want to apply an internal boundary condition on edge E3 (the interface between Face 1 and Face 2) in MATLAB’s "PDE Toolbox", specifically a condition that involves the temperature on both sides of the edge (let’s say ( T_1 ) on Face 1 and ( T_2 ) on Face 2)—here’s what you need to know:

1. For most internal boundary conditions (like thermal contact or interface conditions), you need to model each region with its own PDE.

So, set N=2 when you create your model:

N = 2;
modello = createpde(N);

2. After defining your geometry and identifying the edge number of E3 (use pdegplot to check), you can apply a mixed (Robin-type) boundary condition that couples the two regions.

For example, if you want the flux across E3 to be proportional to the temperature difference (( q = h(T_1 - T_2) )), use:

h = 100; % Example heat transfer coefficient

applyBoundaryCondition(modello, ...
    'mixed', ...
    'Edge', E3, ...
    'EquationIndex', [1 2], ...
    'q', [h -h; -h h], ...
    'g', [0; 0]);

This enforces the relationship between the temperatures on both sides of the edge.

3. If you only have a single PDE (N=1), you can only set standard boundary conditions (Dirichlet, Neumann, Robin) on edges, not interface conditions involving both sides.

In summary:

• For interface conditions between regions, use a system with N=2 and a 'mixed' boundary condition with 'EquationIndex'.
• For simple flux or temperature conditions, use 'neumann' or 'dirichlet' as usual.

For more information, Please refer the following MathWorks documentation link:

https://www.mathworks.com/help/pde/ug/pde.pdemodel.applyboundarycondition.html

Hope this helps!

[marc]

Hello Sameer

Do you have a full working code for this example? It seems that Matlab will ignore the boundary conditions at internal boundaries except if you specify InternaBC=true. But then how do you assign a PDE to a specific region? Or else: how do you specify the coefficients?

Thank you for your input.

[Torsten]

If you only want to prescribe the condition from the question: this condition is automatically fulfilled. The finite element method satisfies continuity of temperature and heat flux at internal interfaces.