## 1 Introduction  
Ferromagnetic materials, both soft and hard, are widely used as magnetic cores or permanent magnets in electrical devices, such as inductors, transformers, generators and motors, due to their high permeabilities or magnetization properties [1]. When designing such devices, the knowledge and characterization of their hysteretic behavior are therefore of paramount importance, and having a good mathematical model capable of reproducing this behavior is important as well [2; 3]. However, to achieve an efficient design procedure, the used models have to be accurate and fast. Satisfying these two criteria simultaneously is not easy; therefore, a compromise has to be made. During the past century, a large number of hysteresis models have been proposed and applied in computational electromagnetics [4; 5; 6; 7; 8; 9; 10]. Based on their application level, they can be classified into _scalar_ and _vector_ models, depending on whether or not the field remains collinear at all times, as well as between _static_ and _dynamic_ models, depending on whether or not the constitutive relationships are indifferent to rates. Another classification is based on their accuracy and distinguishes between _mathematical_ or _physical_ models. The former are generally faster but less accurate [4; 5], as they ignore the underlying physics of the material behavior, whereas the latter can describe the complex magnetization process more faithfully and are thus more applicable to engineering problems. The physical models can be further classified into _physics-based_ and _phenomenological_ models, depending on whether they are based or not on the actual physical processes occurring in the matter subjected to magnetic fields. Among the former are the Stoner-Wohlfarth (S-W) [6] and Jiles-Atherton (J-A) [7] models, while the Preisach [8], Play [9] and neural network-based [10] models are usually considered as phenomenological ones. The implementation of the S-W model requires the knowledge of the distribution of the particles in the ferromagnetic material. However, the statistical distribution function is difficult to determine. In order to achieve sufficient accuracy, a complex identification process is needed based on a vast number of experimental measurements, which limits its practical application [11]. The J-A model is therefore the only physics-based hysteretic model that can be easily implemented into the finite element method (FEM) and hence finds a wide use in the design of electromagnetic devices. The original J-A model was a static scalar model, but an extension to dynamic and vector hysteresis models has been proposed in [12; 13], respectively. Phenomenological models are widely used and can be incorporated into FEM simulations as well. The classical Preisach model (CPM) was born as a static scalar model; however, many extensions and modification attempts have been made over the years [14; 15; 16; 17]. For instance, [PERSON] included the dynamic effect and proposed a moving Preisach model (MPM) [14], while [PERSON] provided an extension to the vector Preisach model (VPM) [15]. Attempts to reduce the computational efforts for either CPM and VPM have been made in [16; 17], respectively. Even the Play model and neural network-based models have been improved to account for the vectorization or numerical speed-up of the hysteresis models [18; 19; 20; 21; 22; 23; 24; 25]. In this study, only the CPM and original J-A models have been considered, as they are the most popular ones. A numerical comparison between these two models can be found in [26; 27]. The static J-A model is considered to be more efficient in terms of computational time and memory requirements. On the other hand, the CPM model is not only more accurate but can represent a wider class of materials because of the arbitrariness of the material function [27]. Besides these, a novel phenomenological model has been proposed by the authors in [28], referred to as the D'Aloia-Di Francesco-De Santis (D-D-D) model.