Simulation of Voltage-Driven Coils

[paul-hilscher]

Dear MFEM team,

is there a way to model a voltage driven (electromagnetic) stranded coil in mfem ?

The weak-forms in the coil in the domain $\Omega$ are given by:

1. The field problem in the a-formulation

$$\left( \nu \, \mathrm{curl} \times \mathbf{a}, \mathrm{curl} \times \mathbf{a}' \right) _\Omega + I \left( \mathbf{\tau}, \mathbf{a}' \right) _\Omega = 0 \quad,$$

where $\mathbf{a}$ is the magnetic vector potential, $I$ the (unknown) current flowing through each strand of the coil, and $\tau$ is the electric current density in the coil when a current of 1A is applied.

2. The electric circuit problem (which introduces a single additional dof)

$$\int_\Omega \partial_t \mathbf{a} \cdot \mathbf{\tau} d\Omega + R I = V \quad,$$

where $R$ is the specified resistance and $V$ is the specified voltage.

Thanks a lot for any reply and many thanks for an amazing library.

[mlstowell]

Hi, @paul-hilscher,

This is not a class of problems that I am familiar with but I think MFEM could be used to solve them. A quick google search turns up an article which you may be familiar with: Voltage-driven coils in finite-element formulations using a current vector and a magnetic scalar potential. Their section 2.2 seems to describe the same type of problem that you outline above.

From looking at your description and the paper I think this could be implemented as a block system of size (e+1)x(e+1) where e is the number of edges in the mesh of your coil (I'm not sure how to handle multiple strands). One e x e sized block would be a curl-curl operator (as in MFEM's example 3) with the material coefficient $\nu$. The 1 x e and e x 1 blocks would need to be custom-built operators based on LinearForm objects (we've done this sort of thing before but we may not have any examples in the code yet). For a similar type of operator see issue #2818 and the prototype implementation in PR #3435.

If this sounds like a feasible approach I'd be happy to offer further advice if needed.

Best wishes,
Mark

[paul-hilscher]

Hi @mlstowell,

thanks a lot for your reply and pointers on how to move forward. I will give this a try and report back in a couple of days.