## Preprint 2018-12

## Alfredo Bermúdez, Bibiana López-Rodríguez, Rodolfo Rodríguez, Pilar Salgado:

### Numerical solution of a transient three-dimensional eddy current model with moving conductors

### Abstract:

The aim of this paper is to propose and analyze a numerical method to solve a time-dependent eddy current problem in a domain containing moving non magnetic conductors. To this end, we choose a formulation in terms of the magnetic field, what leads to a parabolic problem for which we prove an existence result. For space discretization, we propose a finite element method based on N´ed´elec edge elements on a mesh that remains fixed over the time. The curl-free constraint in the dielectric domain is relaxed by means of a penalty strategy that can be easily implemented, without the need that the mesh fits the moving conducting and dielectric domains. For time discretization, we use a backward Euler scheme. We report some numerical results. First, we solve a test problem with a known analytical solution, which allows us to assess the convergence of the method as the penalization and discretization parameters go to zero. Finally, we solve a problem with cylindrical symmetry, which allows us to compare the results with those obtained with an axisymmetric code.

This preprint gave rise to the following definitive publication(s):

**Alfredo BERMúDEZ, Bibiana LóPEZ-RODRíGUEZ, Rodolfo RODRíGUEZ, Pilar SALGADO: ***Numerical solution of a transient three-dimensional eddy current model with moving conductors*. International Journal of Numerical Analysis and Modeling, vol. 16, 5, pp. 695-717, (2019).