## Preprint 2014-01

## Alfredo Bermúdez, Dolores Gómez, Rodolfo Rodríguez, Pablo Venegas:

### Mathematical analysis and numerical solution of axisymmetric eddy-current problems with Preisach hysteresis model

### Abstract:

This work deals with the mathematical analysis and the numerical computation of transient electromagnetic fields in nonlinear magnetic media with hysteresis. Many ferromagnetic materials present hysteretic behavior, which essentially means that the magnetic induction at each particular point depends, not only on the present magnetic field, but also on the past magnetic history. The present work can be seen as a complement of previous results of the authors, where the mathematical and numerical analysis of a two-dimensional nonlinear axisymmetric eddy current model was performed under fairly general assumptions on the H-B curve, but without considering hysteresis effects. Now, the constitutive relation between H and B is given by a rather general scalar hysteresis operator. Axisymmetry of the fields is assumed and in view of applications the source inputs are either current intensities or voltage drops. With this in mind, two source terms are considered: either the magnetic field on the boundary (Dirichlet condition) or the magnetic flux across a meridian section of the device (magnetic flux condition). These source terms are physically realistic in the sense that there are many real applications where they can be readily obtained from measurable quantities. Moreover, a time and space dependent electrical conductivity is considered, an important issue because this quantity is typically a function of temperature which, in turn, is a time dependent field. For both problems, an existence result is achieved under suitable assumptions. For the numerical solution, the classical Preisach model is used as hysteresis operator, together with a finite element discretization by piecewise linear functions on triangular meshes and a backward Euler scheme. A detailed description of the Preisach model is included and, finally, a numerical test is reported.

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

**Alfredo BERMúDEZ, Dolores GóMEZ, Rodolfo RODRíGUEZ, Pablo VENEGAS: ***Mathematical analysis and numerical solution of axisymmetric eddy-current problems with Preisach hysteresis model*. Rendiconti del Seminario Matematico. Universita e Politecnico Torino, vol. 72, 1-2, pp. 73 - 117, (2014).