**1. Introduction**

Hysteresis is a very complex nonlinear behavior affecting many physical phenomena. Systems affected by this behavior are characterized by the fact that the way they evolve in response to a stimulus depends not only on the cause of the stimulus, but also on the preceding states of the system. Thus, the same instantaneous values of the input can give different outputs depending on the history of the input applied, which gives rise to a relationship that is not only nonlinear but also multivalued, making it very difficult to model and control. This memory-based property is found in various areas of science and engineering such as mechanics, biology, economics and multiphase flow in porous media or magnetism, among others. It was precisely in the latter that the term was initially coined since most ferromagnetic materials exhibit hysteresis [1]. This means that when the material is subjected to the application of a magnetic field, the magnetic induction reached at each point of the material depends not only on the intensity of the applied field at a given instant of time, but also on its previous magnetic history. Despite the difficulties involved in its study, having a good hysteresis model is essential in a multitude of applications in the field of electrical engineering; an example of this is the estimation of energy losses in electrical machines.

The variety of systems with hysteresis is very broad and there is a large amount of bibliography devoted to hysteresis models in different communities of physicists,

engineers, and mathematicians, where great efforts have been devoted to their development and analysis. As a result, nowadays there are several hysteresis models, ranging from simple to complex, each of them valid in some specific situation; see, e.g., the monographs and reviews [2–5].

The mathematical approach tries to deal with this phenomenon under a common mathematical framework, but this is not always possible. We highlight the monograph by Krasnosel'ski˘i and Pokrovski˘i [2], who have conceptually introduced the notion of hysteresis operator and carried out a systematic analysis of its properties. More recently, research on hysteresis models as well as their coupling with partial differential equations has been progressing; see, among others, the works by Visintin [5, 6] and Brokate and Sprekels [7]; from the physical point of view, Mayergoyz [8] and Bertotti [9] are the classical references.

Hysteresis models can be classified into two main classes according to whether or not the system response depends on the input velocity. In *static* or *rate-independent* models, output values are not affected by the velocity of the input but only by the values in the input range and by the order in which these values have been attained. Consequently, the model cannot reflect the frequency or waveform dependence of the field. This is a characteristic of the classical hysteresis phenomenon and in this sense some authors [6] consider hysteresis as a rate-independent memory only. On the opposite side are the so-called *dynamic* or *rate-dependent* models, which account for the effect of the velocity of changes in the applied input.

The so-called classical Preisach operator [10] is a rate-independent model originally designed to model hysteresis of ferromagnetic materials which is based on physical assumptions derived from the concept of magnetic domains [8, 9, 11, 12]. It is suitable for modeling scalar hysteresis and for this purpose, the magnetic field strength H is used as the input variable while the magnetization M (or the magnetic flux density strength, B) acts as the output variable. Its main advantage is the ability to describe not only the major hysteresis loop, but also inner loops and other complex characteristics of the magnetization processes. There are several extensions of this model that consider the rate of change of the stimulus to be able to consider phenomena where this factor influences the magnetization. These are generically referred to as *dynamic Preisach models* (see, for instance [13–17]).

Currently, the Preisach-type operators [18] are used not only in the area of magnetism, but also to describe hysteresis phenomena in other fields as diverse as fluid flow in porous media [19], elastoplasticity [20], solid phase transitions [7], shape memory alloys [21], or biology [22], to name a few.

The mathematical and numerical treatment of PDEs with hysteresis operators is a challenging issue because, despite the importance of the topic, still few results are available. Due to the strong interdisciplinary character of hysteresis phenomena, the interest for their study is continuously increasing in a great variety of applications. During the last few years, the authors of this chapter have been working on the mathematical modeling, numerical analysis and computation of PDE hysteresisrelated problems motivated by real applications, with special emphasis on the hysteresis of ferromagnetic materials. This work is a survey of previous co-authored studies [17, 23–25], where we have intended to provide original steps into the mathematical and numerical treatment of parabolic problems with hysteresis. In all of them, the Preisach model was considered as hysteresis operator. The problems were addressed in different aspects: mathematical analysis, numerical methods, until convergence of approximate solutions and computational results to confirm the theory and to compare with experimental data. The rigorous mathematical framework or the numerical

analysis are not included here, and we will be concerned with the results obtained for different industrial problems.

The chapter is organized as follows. In Section 2, we start by recalling the basic properties of the hysteresis operators. The well-known Preisach hysteresis model, both in the rate-independent and rate-dependent cases, is introduced in Section 3. Finally, in Section 4 we show some examples of the applications of the Preisach model in electrical engineering. These examples include the computation of hysteresis losses, the numerical simulation of magnetization and demagnetization processes in ferromagnetic workpieces and the simulation of batteries for electric vehicles.
