[Getdp] Enforcing magnetic scalar potential jump in H-Phi model

[DILASSER Guillaume]

Dear GetDP developpers,

I am writing looking for some advice with a model that I am presently implementing.

I would like to model the quench of superconducting coil. This is the transition of an originally superconducting coil to the normal, resistive, state. In the initial state, the current is able to flow inside the windings without causing heat dissipation. As a part of the coil "quench", it becomes resistive and starts generating heat. In turn, that heat tends to propagate the quench to the neighbouring regions, which can lead to a domino effect making the whole coil transition.

This problem is coupled between electrodynamics and thermics but for now I am solely focused on the electromagnetic aspect of it and I have opted for a H-Phi formulation. The idea I would like to try is to avoid modelling the coil down to the conductor level and instead use an homogenized approach. I my mind, I would like to represent the properties of the superconducting winding using a non-linear, local, orthotropic matrix. This matrix would locally define a plane in which the material properties are those of a superconductor while there is a normal, ohmic resistivity along the orthogonal axis. For now, I am not using using non-linear superconductor properties though and I simply try to make a model with a linear matrix that has a low resistivity in a plane and a large resistivity in the transverse axis. I hope that homogenizing the model with such a matrix, I would be able to use a coarser mesh inside the winding region and hopefully run some large simulations.

I have enclosed an example of what I am trying to achieve. This is an attempt at modelling a single-pancake (spiral) coil, the red torus, with current leads and a bridge to close the electrical circuit (yellow parts), all enclosed in a large air box. My main issue comes with the H-Phi formulation and the use of cuts to enforce jumps in the magnetic field scalar potential Phi in the air region. In the system I am trying to model, there are two cohomology cuts needed. On one cut, the one shaped like a "football goal" running below the yellow current leads, I know that I want to enforce a constrained potential jump, corresponding to the power supply current . The issue is with the second cut, the one in the hole of the coil.

I know that if the coil turns are perfectly insulated, then the current is forced to make the full number of coil turns and the value of the jump is the number of ampere-turns of the coil. Simple. However, I would also like to be able to model non-insulated coils, meaning coils in which the current is allowed to jump from one turn to the next without making a full turn around the coil. In this case, the value of the Phi jump in the cut is not necessarily the number of ampere-turns. It can be lower in the case there is some radial bypassing current.

I am stuck here because I do not understand if I can make that model work within GetDP:

*         Do I need to leave unconstrained the DoF corresponding to the potential jump across the coil bore cut?

*         On the contrary, do I need to enforce the value of the jump?

*         If so, how can I determine the value of the jump. If have a feeling that I might have to write an additional 0-D equation using the circulations of the electrical field along parts of the homology cycles of the conducting domain. I am not sure if this is the correct way to proceed.

On a side note, I also have the issue that the current density tends to be very messy at the junction between the current leads and the coil. I understand that this is because the current initially starts flowing in both direction when it enters the coil but that then the current going in the wrong direction makes a U-turn within the low-resistivity plane. That problem can be "fixed", using a resistivity of Rotate[ TensorDiag[ 1.e9, 1., 1.e9 ], 0., 0., alpha[] ] ; at line 53 of the .pro file. This is an ugly fix though and I wonder if there are better ways to proceed.

I any case, I would like to thank you for reading this very long email. Should you have any idea on how to solve this issue, I would be very interested to hear your comments.

Sincerely Yours,

Guillaume Dilasser
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