2D induction cold crucible with impedance boundary conditions

[Roland]

Hi,
I would like to model a 2D axisymetric harmonic high frequency induction cold crucible by using the impedance boundary condition (say IBC) BC on the crucible surface. This makes that the inner part of the copper crucible can be removed from the model, as the induced current densities become surface current densities just on the crucible surface.
The geometry is here attached, showing the surrounding inductor (the outer right rectangle) which carries the imposed inductor current density, and the cold crucible which is the inner left empty rectangle since its surface should be treated in IBC.
Here attached is also the zipped GUI project in its 1rst state without IBC, meaning that in the Cold crucible surface BC (boundary condition 2) the corresponding IBC relations (!Layer Electric Conductivity = Real 6e7
!Layer Relative Permeability = Real 1) are just comments since they start with the "!" character).
Unfortunately when activating these IBC relations (by removing the "!" character) the solver gives a SIGSEGV error message.
I used these IBC relations in a 3D cold crucible model with the WithneyAVHarmonic solver and it works very well.
So is it possible to use these IBC relations in 2D for this kind of 2D cold crucible?
Thanks in advance for any help about this exciting issue!
Roland

[raback]

Hi Roland, Unfortunately this feature is only available in 3D (where it is often essential). -Peter

[Roland]

Hi Peter,
Yes I can understand that and it works very well in 3D.
But as we would like to couple the former crystal_growth models with a cold crucible configuration and that this will couple many physics, we would like to do this in 2D in a first state (also in order to get a model with not too many meshes...).
This is why I would like to set up a cold crucible model in 2D which uses the IBC boundary condition, which is E = Zs*(n^H), and which could perhaps be directly written in MATC language in the cold crucible surface BC.
Do you think that this could be possible?
Roland

[raback]

Hi

Sure it could be done. Just not in this form since the only field we are solving for in 2D is A_phi. In 3D we are solving for both A and V and the BC looks probably quite different.

-Peter

[Roland]

Yes I will try to think about it.
It must be possible since, as it is axi symetric and there is the only one perpendicular A_phi vector potential component and as:
H = (1/µ)*curl(A) and E = -i*omega*A in harmonic (no grad(V) electric potential gradient in axi symetric).
So H and E can be expressed in function of A, meaning that the IBC relation can perhaps be completely expressed with A and the surface impedance Zs. The problem could come from the former H = (1/µ)*curlA expression which makes appear the A derivatives with r and z coordinates...
Is there a way to express these derivatives in MATC language?
As I am not very clever with the MATC language could you please give some advices about how this could be done?
Thanks in advance for your precious help!
Roland

[raback]

Hi Roland,

I guess it will be a form of Robin boundary condition. I think MATC is not sufficient since there will be some implicit terms too. But I'm also sure that somehow it can be expressed in terms of A_phi and involve adding a few lines of code.

BR, Peter

[Roland]

Hi Peter,
Yes this is very exciting and interesting!
I think you can understand that this IBC boundary condition in 2D could be very interesting since it is always more careful to setup the many physics coupled models in 2D before wanting to make them in 3D (which consumes much more meshes and hence dofs than in 2D!).
So do you think that you could spend a few moments to give me a start for doing this?
Thanks in advance!
Roland

[Roland]

Hi,
After thinking about this IBC matter and looking at my old calculations, I think that this IBC is written in 2D axi(in harmonic high frequency) as following:
n^H = E/Zs = -i*omega*A/Zs
with:
n^H : Js surface current density (in A/m)
E : tangential electric field (= -dA/dt) (only 1 phi component)
A: vector potential (only 1 phi component)
omega: angular frequency
Zs: surface impedance (classicaly expressed with µ,omega, epsilon, sigma)
n: normal vector

This relation can be written as a generalized Neuman BC (of type n*(c*nabla(u)+alpha*u)+q*u = g, u beeing the variable, here the vector potential A) which is written in weak form as a weak contribution as following (after showing that alpha = g = 0, and that c*nabla(u) = n^H and that q = i*omega/Zs):
-i*omega*(Ar*Ar_test+Aphi*Aphi_test+Az*Az_test)/Zs = -i*omega*Aphi*Aphi_test/Zs (since Ar = Az = 0)
Ar_test,Aphi_test,Az_test are the test functions of respectively Ar,Aphi,Az (notice that here, on the boundary, Ar,Aphi,Az are vector elements functions)
So my question is:
How is it possible to write this IBC boundary condition in weak form ? More generally is it possible to write a relation in weak form in Elmer (we did this in Comsol) .
Or is there perhaps another way to express this 2D IBC relation on the cold crucible surface? Perhaps, Peter, could you give your opinion about that?
I would appreciate any help about this exciting problem!
Thanks in advance!
Roland

[raback]

Hi Roland,

I copy-pasted the logic from 3D, from other 2D BCs and combined with your formula:
https://github.com/ElmerCSC/elmerfem/co ... 8e05dbcaef
This is not tested but may work, or might not...

-Peter

[Roland]

Hi Peter,
Thank you very much for your efforts!
I will take a look at this within today and keep you informed.
Roland