Mathematical Problems in Engineering

Volume 2015 (2015), Article ID 715018, 13 pages

http://dx.doi.org/10.1155/2015/715018

## On the Generalization Capabilities of the Ten-Parameter Jiles-Atherton Model

Department of Engineering, Roma Tre University, Via Vito Volterra 62, 00146 Rome, Italy

Received 5 October 2015; Accepted 24 November 2015

Academic Editor: Xiao-Qiao He

Copyright © 2015 Gabriele Maria Lozito et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

This work proposes an analysis on the generalization capabilities for the modified version of the classic Jiles-Atherton model for magnetic hysteresis. The modified model takes into account the use of dynamic parameterization, as opposed to the classic model where the parameters are constant. Two different dynamic parameterizations are taken into account: a dependence on the excitation and a dependence on the response. The identification process is performed by using a novel nonlinear optimization technique called Continuous Flock-of-Starling Optimization Cube (CFSO^{3}), an algorithm belonging to the class of swarm intelligence. The algorithm exploits parallel architecture and uses a supervised strategy to alternate between exploration and exploitation capabilities. Comparisons between the obtained results are presented at the end of the paper.

#### 1. Introduction

Hysteretic behavior of magnetic materials is a common topic found in literature [1–7] especially in the numerical modeling field [8–19]. Presently, multiple phenomenological models exist to represent this behavior. The lack of a physical model suggested the use of advanced numerical techniques to represent it, like, for instance, Neural Networks [20–30]. Among the phenomenological models, one that is based on physical considerations and that is very common model in literature is the one proposed by Jiles-Atherton in 1983 [31–34]. After being proposed, the model was later refined through the physical explanation on some details concerning the reversible component and the main equation to give a physically sensible result for minor loops. The model is parametric and is defined by five coefficients that describe different physical aspects. The identification process of the model is numerical and is approached through solving a least squares (LSQ) problem against some reference data that can be either experimental or simulated [35, 36]. The use of this model for practical applications has been favored by the development of new optimization techniques, which allowed a finer identification of the model parameters [16, 37–49]. On the other hand, the core of the model was widely revised in the years. An approach that is commonly found in literature imposes some dynamic dependence for the parameters on the model excitation [50, 51]. These modified models feature a higher number of degrees of freedom and thus are able to represent complex dynamics. Indeed, this comes with the drawback that the identification procedure is more difficult. The identification problem for the classic model is nonconvex [52] and different solutions exist for the same material. The different solutions provide different generalization capabilities. Once the model has been identified for a particular set of excitations/responses, the goal is to use it to represent the material for any generic excitation. However, it has been seen that complex waveforms that feature minor loops are poorly reproduced unless particular care has been taken in the choice of the identification waveform. Nevertheless, using a complex waveform for identification makes the identification problem more difficult to solve. For this reason, a powerful optimization algorithm is needed, to solve both the problem of a difficult convergence and the risk of being trapped into the local minima of a nonconvex problem. For such a task, in this work, we propose the use of the Continuous Flock-of-Starling Optimization Cube (CFSO^{3}) algorithm, described in [53–55]. This algorithm belongs to the swarm intelligence class of metaheuristics algorithms and is an analytic evolution of the Flock-of-Starlings (FSO) algorithm found in literature and already applied with success on different problems [56, 57]. The main difference between the FSO and the CFSO^{3} lies in the criterion for position update of the particles: the FSO updates the positions according to a discrete law; the CFSO computes analytical trajectories for each particle. Since the trajectories are computed analytically, it is possible to supervise the optimization behavior: it is possible to modify the algorithm parameters to exhibit convergence, divergence, or local oscillations of the particles. The proposed algorithm runs natively on master-slave parallel architecture. The main idea is to have a master process performing the exploration of the solution space and to refine the solutions into the candidate areas through local exploitations performed by slave processes. The purpose of this work is to investigate the generalization capabilities of a modified Jiles-Atherton model, featuring five additional parameters. This new ten-parameter model will be shortly addressed as J10P. The additional parameters are not constants (like the classic ones) but are rather scaled according to the model state. In particular, two different dependences are going to be investigated: a dependence on the excitation and a dependence on the response. In both cases, the model will be identified on a wide set of reference materials, and the results will be validated against random waveforms on the same materials. In the first part of this paper, the derivation of the J10P model from the classic one will be shortly reviewed. In the second part, the CFSO^{3} algorithm will be analyzed, remarking its parallel capabilities. In the third part, the identification and validation benchmark will be described, along with the results obtained in terms of convergence and generalization. Conclusions and final considerations will follow.

#### 2. The Dynamic Parameterization of the Jiles-Atherton Model

Several models are available in literature to represent the magnetic hysteresis phenomenon [1]. Models span from representing the phenomenon at microscopic level [2] through domain analysis to less physical and more application oriented approaches [7, 13]. Among the different models, the most commonly used in literature are the one from Preisach and the one from Jiles-Atherton. The former is widely used in both scalar [5, 8–11, 15] and vector [3, 19] forms. A recent work extended the hysteresis representation of the Preisach vector model in 3D domain [12]. The original model proposed by Jiles-Atherton [31–34] for the hysteretic response of ferromagnetic material is a differential model that expresses the derivative of the magnetization with respect to the magnetic field, through a nonlinear relation. The model is composed of two equations (1) and (2), which express, respectively, the differential relation and the nonlinear component of the anhysteretic magnetization :There are five parameters involved: , the magnetization saturation; , the shape parameter; , the mean field parameter; , the domain wall pinning constant; and , the domain flexing constant. The term takes into account whether the integration is taking place in the ascending or the descending part of the loop. It is called* directional factor*, and its value is either +1 or −1, respectively. The strong advantage presented by the classic Jiles-Atherton parameter lies in the reduced number of parameters and the physical interpretation that was attributed to each parameter. The identification process for these parameters can be solved in several ways as proposed by the same author of the model [35, 36] and other notable examples that can be found in literature [16, 43]. Still, the most common approach is to identify it by solving a least squares problem. Different algorithm can be used to solve the inverse problem: classic Particle Swarm Optimization [14, 45], Simulated Annealing [48], Differential Evolution [51], and hybrid approaches based on Genetic Algorithms [47, 52]. The goal of the identification procedure is to identify the set of five parameters that describes the material under study. In the classic model, the five parameters are constant. The model used in this work deviates from the physical nature of the original J-A model, introducing a phenomenological approach where each generic parameter is expressed bywhere can be either the current excitation (i.e., the magnetic field ) or the last computed value for the response of the model (i.e., the flux density ). Introducing a dependency on the excitation/response of the model is a strategy already used in other modeling approaches where the hysteretic phenomenon is modeled through Neural Networks [20, 22, 24, 27, 30]. Representation of the ferromagnetic material under distorted excitation [23, 25] of a multifrequency domain [28] is a difficult generalization problem for an ANN that must be solved by introducing some sort of memory in the neural estimator. This approach is not mandatory in this model, since its differential nature naturally takes into account the memory effect. However, the representation of minor loops can be enhanced by applying this paradigm to model parameterization. The dynamic part of the parameter is distributed according to a Gaussian distribution, with the width of the curve proportional to . The value of is related to the saturation values for either the magnetic field or the flux density, according to the model used. It is defined during the identification procedure and is equal to 0.2–0.8 times the or the value. Indeed, the dynamic expansion for all the five parameters may be redundant for some simple problems. An incremental analysis from the J5P model to the J10P model can be found in Appendix B. The degrees of freedom introduced by the J10P model, in terms of flexibility in representing a hysteresis phenomenon, can be seen by operating alternatively on the two components of the same parameter. For example, the parameter is basically a vertical scaling factor for the whole hysteresis loop: larger will give a taller loop, whereas smaller will give a shorter one. By modifying alternatively the component and the components, it is apparent how the two changes affect the hysteresis curve. In Figure 1, two sets of hysteresis curves are shown. On Figure 1(a), the parameter assumes the values of , and the parameter is kept at 0. In Figure 1(b), the parameter is set to , and the parameter assumes the values of .