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Finite Elements in Analysis and Design

journal homepage: www.elsevier.com/locate/finel

Electromagnetic computations with Preisach hysteresis model

Alfredo Bermúdeza, Luc Dupréb, Dolores Gómeza,⁎, Pablo Venegasc

a Departamento de Matemática Aplicada, Universidade de Santiago de Compostela, 15706 Santiago de Compostela, Spainb Department of Electrical Energy, Systems and Automation, Ghent University, Belgiumc GIMNAP, Departamento de Matemática, Universidad del Bío-Bío, Chile

A R T I C L E I N F O

Keywords:Transient eddy currentElectromagnetic lossesNonlinear partial differential equationsFinite element methodHysteresis

A B S T R A C T

This paper presents a novel algorithm for the computation of transient electromagnetic fields in nonlinearmagnetic media with hysteresis. We deal with an axisymmetric transient eddy current problem where theconstitutive relation between H and B is given by a hysteresis operator, i.e., the values of the magnetic inductionB depend not only on the present values of the magnetic field H but also on its past history. First, we introducethe mathematical model of the problem and, by applying some abstract result, we show the well posedness of aweak formulation written in terms of the magnetic field. For the numerical solution, we consider the Preisachmodel as hysteresis operator, a finite element discretization by piecewise linear functions, and the backwardEuler time discretization. By taking into account the monotonicity property of the Preisach model, we propose afixed point algorithm to deal with hysteresis effects which is numerically validated: we report a numerical test inorder to assess the order of convergence and we compare the results with experimental data. For the later, weconsider a physical application: the numerical computation of eddy current and hysteresis losses in laminatedmedia as those used in transformers or electric machines.

1. Introduction

It is widely known that the performance of electric machines ismainly defined by the power losses. These losses include iron lossesthat are due to the fact that the magnetic field variations in theferromagnetic materials composing the core of the machine produceenergy dissipation. The efficiency, the thermal behavior and thecompactness are some of the design constraints which are stronglyinfluenced by the losses. Consequently, it is very important to predictthem accurately for an optimum design of the device.

The losses can be divided into three main components: eddycurrent (or classical) losses, hysteresis losses and excess losses (see,for instance, [1]).

In this work we are only interested in the computation of eddycurrent and hysteresis losses. Eddy current losses are caused by thecurrents induced in the magnetic material by the time varying magneticinduction. These currents are dissipated as heat due to the Joule effect.Hysteresis losses are related to the magnetic properties of the materialscomposing the core. Ferromagnetic materials spontaneously divideinto magnetic domains, each with a uniform magnetization. Whenexposing the core to a magnetic field, its magnetic particles tend to lineup with the magnetic field. Then, if the applied magnetic field variesalong the time, the continuous movement of the magnetic particles,

which are trying to align themselves with the magnetic field, producesmolecular friction, which, in its turn, produces heat. These heat lossesare referred to as magnetic hysteresis losses.

In the literature there are numerous publications devoted to obtainanalytical simplified expressions to approximate the different compo-nents of these losses (see, for instance, [1–3]), which are only validunder certain assumptions that do not hold in many practical situa-tions. Numerical simulation is an interesting alternative in order toovercome these limitations and thus, in the last years, we can findseveral works focusing on this approach (see [4–8] and referencestherein).

The first step is the numerical solution of the underlying quasi-static Maxwell's model, which is the aim of this work. In the frameworkof parabolic equations with hysteresis, there are several publicationsdevoted to the mathematical analysis of the problem (see for instance[9,10]). In particular, [10] deals with an abstract parabolic equationmotivated by a two-dimensional (2D) eddy current model with hyster-esis but the numerical analysis and computer implementation of theproblem are not included. Numerical approximation of parabolicproblems with hysteresis is studied, for instance, in [11] where anonlinear parabolic problem with the classical Preisach hysteresismodel is considered. At the macroscopic level, this model [12], whichis based on some hypotheses concerning the physical mechanisms of

http://dx.doi.org/10.1016/j.finel.2016.11.005Received 11 August 2016; Received in revised form 2 November 2016; Accepted 18 November 2016

⁎ Corresponding author.E-mail addresses: alfredo.bermudez@usc.es (A. Bermúdez), luc.dupre@ugent.be (L. Dupré), mdolores.gomez@usc.es (D. Gómez), pvenegas@ubiobio.cl (P. Venegas).

Finite Elements in Analysis and Design 126 (2017) 65–74

Available online 05 January 20170168-874X/ © 2016 Elsevier B.V. All rights reserved.

MARK

magnetization, is the most popular to represent the magnetic hysteresisphenomenon. In [11], the existence of solution is established and anumerical approximation is proposed, but numerical examples orvalidations are not presented.

In the context of computational methods for a 2D eddy currentmodel with hysteresis we mention [8,13]. We can also mention [14]where a fixed-point iteration is proposed to deal with Preisach-typemodels of hysteresis in a magnetostatic problem (see also [15] for a 2Dmagnetostatic problem).

The aim of this work is to consider the mathematical analysis andthe numerical implementation of an axisymmetric parabolic problemwith hysteresis arising from a real world application. The former isbased on [9] whereas, for the latter, a new iterative algorithm isconsidered in order to handle the hysteresis operator. Unlike [14],where the Preisach model is implemented in the reversed fashionunder periodic input (magnetostatic problem), the algorithm wepropose can be used in the above-mentioned parabolic problem witha smooth (not necessarily periodic) source term. Moreover, we considerthe numerical implementation of the scheme and its validation withexperimental data. In view of applications we consider that the sourceinput is the magnetic field on the boundary of the domain (Dirichletboundary condition). This source term is physically realistic in thesense that there exist industrial applications where it can be readilyobtained from measurable quantities (see [16,17]). The results ob-tained in this work complement those in [18,19], where the mathema-tical and numerical analysis of a 2D nonlinear axisymmetric eddycurrent model was performed without considering hysteresis effects.

The outline of this work is as follows. In Section 2 we introduce thetransient eddy current model with hysteresis to be analyzed, and theaxisymmetric case is considered. By using classical weighted two-dimensional Sobolev spaces a weak formulation in terms of themagnetic field is obtained and an existence result is recalled. Section3 is devoted to the numerical implementation of the fully discreteproblem. For the numerical solution, a finite element discretization bypiecewise linear functions on triangular meshes, and the backwardEuler time discretization are used. In order to handle the nonlinearityat the discrete level, we propose a fixed point algorithm which is basedon some properties of the Yosida regularization of maximal monotoneoperators. This algorithm, introduced by Bermúdez and Moreno [20],has been extensively used for a wide range of applications with goodnumerical results. We discuss in detail how it can be adapted to thePreisach model and report some results allowing us to assess itsperformance. In particular, the algorithm reveals to be a suitable tool todeal with the hysteresis nonlinearity because it takes advantage of thespatial independence of the hysteresis operator and allows handling its

lack of differentiability. Finally, in Section 4, we report two numericaltests with different objectives: to assess the order of convergence of theabove numerical method and to validate the numerical scheme withexperimental results.

2. Statement of the problem

Eddy currents are usually modeled by the so-called low-frequencyMaxwell's equations,

H J B E Bt

curl curl 0= , ∂∂

+ = , div = 0,(2.1)

where we have used the standard notation in electromagnetism: E isthe electric field, B the magnetic induction, H the magnetic field and Jthe current density.

In order to obtain a closed system we add Ohm's law in conductors,

J Eσ= ,

where σ is the electrical conductivity, and the magnetic constitutiveequation

B H Mμ= ( + ),0

where M is the magnetization and μ0 the magnetic permeability invacuum. For ferromagnetic materials, where hysteresis phenomenamay occur, the relationship between M and H exhibits a history-dependent behavior and must be represented by a suitable constitutivelaw M H= ( ), where is a vector operator accounting for hysteresis(see [21,22] and, more recently, [23] for a vector model well-suited forfinite element implementation). From the above equations we caneasily obtain the following vector partial differential equation inconductors:

B Ht σ

curl curl 0∂∂

+ 1 = ,⎛⎝⎜

⎞⎠⎟ (2.2)

which has to be solved together with the constitutive equation

B H Hμ= ( + ( ))0 (2.3)

in a conducting domain Ω ⊂ 3 .

2.1. Axisymmetric eddy current model

In many applications, the computational domain Ω has cylindricalsymmetry and all fields are independent of the angular variable θ. Insuch cases, in order to reduce the computational effort, it is convenientto consider a cylindrical coordinate system r θ z( , , ). Let us denote by er,

Fig. 1. Cylindrical coordinate system (left) and sketch of the domain (right).

A. Bermúdez et al. Finite Elements in Analysis and Design 126 (2017) 65–74

66

eθ and ez the corresponding unit vectors of the local orthonormal basisas sketched in Fig. 1 (left). We will assume that the magnetic field hasonly azimuthal component, i.e., it is of the form

H r z t H r z t e( , , ) = ( , , ) .θ (2.4)

If the materials composing the domain have magnetic isotropicbehavior, then B has only azimuthal component too:

B r z t B r z t e( , , ) = ( , , ) .θ (2.5)

We notice that any field of the form (2.5) is divergence-free. Next,according to (2.4),

H r z tz

H r z tr r

rH r z tcurl e e( , , ) = − ∂∂

( , , ) + 1 ∂∂

( )( , , )r z (2.6)

and then Eq. (2.2) reads

Bt r σr

rHr z σ

Hz

∂∂

− ∂∂

1 ∂( )∂

− ∂∂

1 ∂∂

= 0.⎛⎝⎜

⎞⎠⎟

⎛⎝⎜

⎞⎠⎟

This equation holds in a meridian section Ω of Ω , for all time t T∈ [0, ](see Fig. 1 (right)). In order to write a well-posed problem we must addan initial condition B r z B r z( , , 0) = ( , )0 in Ω and appropriate sourceterms. In view of applications, we consider a non-homogeneousDirichlet boundary condition:

H r z t g r z t Γ( , , ) = ( , , ) on ,

where g is a given function and Γ Ω≔∂ . For applications of this model,we refer for instance to [16,24], where a Dirichlet problem arises in thesimulation of metallurgical electrodes, or the computation of currentlosses in a toroidal laminated core [17,25]. As we will see in Section4.2, in this last case the boundary condition can be readily obtainedfrom the source current intensity.

Finally, taking into account that the fields involved are scalar, therelation (2.3) can be represented by a suitable scalar constitutive lawaccounting for hysteresis:

B r z t μ H r z t H ξ r z t( , , ) = ( ( , , ) + [ ( , )]( , , )),0 (2.7)

where the given function ξ r z( , ) belongs to a suitable metric space Y andcontains all the information about the initial state needed to compute

(including eventually its history). Here, we are interested only inrate-independent processes, namely we will not take into accountmicrostructural dependent dynamic effects (movement of domainwalls) which produces the so-called excess losses. Thus, we havechosen the well-known classical Preisach model (see [21,12,26] fordifferent Preisach-relays-based models), in which the space-dependenthysteresis operator is defined by

∬H ξ r z t h H r z ξ r z t p ρ dρ r z Ω[ ( , )]( , , ) = [ ( ( , , ·), ( , ))]( ) ( ) , a. e. ( , ) ∈ρ

(2.8)

where ξ: → { − 1, 1}, for ρ > 0,0 is the so-called Preisach triangleρ ρ ρ ρ ρ ρ ρ≔{ = ( , ) ∈ : − ≤ ≤ ≤ }1 2

20 1 2 0 and h = ± 1ρ is the scalar

relay operator with “switch-up” and “switch-down” values at ρ2 and ρ1,respectively. Function p p∈ L( ), > 01 , is known as the Preisachdensity function which can be analytically approximated by, e.g.,Gaussian or Lorentzian distributions (see [1]) or, otherwise, discretelyobtained through the so-called Everett function (see, for instance,[21]). Let us emphasize that hρ depends on the values of H r z s( , , ) fors t≤ . Thus, the operator is local in x but is non-local in t. Furtherdetails can be found, for instance, in [21,27].

All together, the resulting axisymmetric problem read as follows:

Problem 1. Find H r z t( , , ) and B r z t( , , ) such that

Bt r σr

rHr z σ

Hz

f Ω T∂∂

− ∂∂

1 ∂( )∂

− ∂∂

1 ∂∂

= in × (0, ),⎛⎝⎜

⎞⎠⎟

⎛⎝⎜

⎞⎠⎟ (2.9)

B μ H H ξ Ω T= ( + ( , )) in × (0, ),0 (2.10)

H g Γ T= on × (0, ), (2.11)

B B Ω| = in ,t=0 0 (2.12)

where σ r z t f r z t g r z t ξ r z( , , ), ( , , ), ( , , ), ( , ) and B r z( , )0 are givenfunctions.

Remark 1. For the sake of completeness, in (2.9) we have considereda general right-hand side f. Moreover, we consider a space and timedependent electrical conductivity because, in practical applications, σ isa function of temperature which, in turn, is a space and time dependentfield.

Remark 2. Notice that in order to compute the hysteresis operatorH ξ[ ( , )] we need to provide an appropriate “initial state” ξ. From the

practical point of view, a typical initial condition (cf. (2.12)) is the so-called demagnetized or virginal state of the material, namelyB H( , )| = (0, 0)t=0 . The demagnetized state can be achieved, forinstance, by heating the material above its Curie temperature.Another method that returns the material to a nearly demagnetizedstate is to apply a magnetic field with a direction that changes back andforth, while at the same time its amplitude reduces to zero.

2.2. Weak formulation

To derive a suitable variational form of Problem 1, we introduce thefunction space Ω v v Ω v ΩH ( )≔{ : ∈ L ( ) and ∇ ∈ L ( )} r r r

11/2 2 , where ΩL ( )r

2

and ΩL ( )r1/2 denote the weighted Lebesgue spaces of all measurable

functions u defined in Ω for which

∫ ∫u u r drdz u ur

drdz∥ ∥ ≔ | | < ∞ and ∥ ∥ ≔ | | < ∞,Ω Ω Ω ΩL ( )

2 2L ( )2

2

r r2

1/2

respectively. We recall from [28, Section 3] that functions in ΩH ( ) r1

have

traces on Γ . Here, we denote the space of these traces by ΓH ( ) r1/2

.Moreover, in order to deal with the above boundary condition wedefine G Ω G Ω≔{ ∈ H ( ): | = 0} ⊂ H ( ) r Γ r

1 1.

Then, a weak formulation of Problem 1 reads

Problem 2. Given g T Γ∈ H (0, ; H ( )) r2 1/2

, f T∈ H (0, ; ′)1 , ξ Ω Y: →and B Ω∈ L ( )r0

2 , find H T Ω T Ω∈ L (0, ; H ( )) ∩ L (0, ; L ( )) r r2 1 ∞ 2 and

B T Ω∈ L (0, ; L ( ))r2 2 with T∈ L (0, ; ′)B

t∂∂

2 , such that

∫Bt

Gσr

rHr

rGr

rHz

rGz

drdz f G

GT B μ H H ξ Ω T

H g Γ T B B Ω

∂∂

, + 1 ∂( )∂

∂( )∂

+ ∂( )∂

∂( )∂

= ⟨ , ⟩ ∀

∈ ,a. e. in [0, ], = ( + ( , )) in × (0, ),

= on × (0, ), | = in .

Ω

t

, ′, ′

0

=0 0

⎛⎝⎜

⎞⎠⎟

In the expressions above, we have used the classical notation⟨·, ·⟩ , ′ for the duality product between and its dual space ′.The following existence result can be deduced by using the classicalRothe's method.

Theorem 2.1. Let us assume the following hypotheses hold:

1. Function σ T Ω: (0, ) × → belongs to T ΩW (0, ; L ( ))1,∞ ∞ and thereexist non-negative constants σ* and σ* such that

σ σ r z t σ t T Ω* ≤ ( , , ) ≤ * ∀ ∈ [0, ], a. e. in .

2. There exist H M Ω Ω( , ) ∈ H ( ) × L ( ) r r0 01 2 , such that B μ H M= ( + )0 0 0 0

and M H ξ= [ ( , ))](0)0 0 in Ω.Then, there exists (H, B) solution toProblem 2.

Proof. The proof is carried out by using the properties of the Preisachoperator through three different steps: time discretization, a prioriestimates and passage to the limit by using compactness. It can befound in [27, Theorem 3].

A. Bermúdez et al. Finite Elements in Analysis and Design 126 (2017) 65–74

67

3. Numerical approximation

In this section a numerical method to solve a full discretization ofProblem 2 is proposed. From now on, we will assume that Ω is apolygon. We associate a family of partitions { }h h>0 of Ω into triangles,where h denotes the mesh size. Let h be the space of continuouspiecewise linear finite elements vanishing on the symmetry axis (r=0).Then Ω⊂ H ( )h r

1. We also consider the finite-dimensional space

≔ ∩h h and denote by Γ( )h the space of traces on Γ of functionsin h.

By using the above finite element space for space discretization andthe backward Euler scheme for time discretization, we are led to thefollowing Galerkin approximation of Problem 2:

Problem 3. Given B μ H M= ( + )h0

0 0 0 , find H ∈hn

h and B Ω∈ L ( )hn

r2 ,

n m= 1, …, , such that

∫ ∫

∫t

B G r drdzrσ

rHr

rGr

rHz

rGz

drdz

f Gt

B G r drdz G

1Δ

+ 1 ∂( )∂

∂( )∂

+ ∂( )∂

∂( )∂

= ⟨ , ⟩ + 1Δ

∀ ∈ ,

Ωhn

hΩ n

hn

h hn

h

nh

Ωhn

h h h, ′−1

⎛⎝⎜

⎞⎠⎟

(3.1)

B r z H r z Ω( , ) = ( )( , ) in ,hn n

hn (3.2)

H g Γ= on ,hn

hn

(3.3)

where g Γ∈ ( )hn

h is a convenient approximations of g t( )n , n m= 1, …, .To completely describe the previous problem, it is necessary to definehow the nonlinear operators Ω Ω: L ( ) → L ( )n

r r2 2 , n m= 1, …, , act.

Given an initial state ξ and the known values H r z( , )0 , H r z( , )hl ,

l n= 1, …, − 1, n is defined, for any u Ω∈ L ( )r2 , as

u r z μ u r z U ξ r z t Ω( )( , ) ≔ ( ( , ) + [ ( , )]( , , )) in ,n n0 (3.4)

with U being the continuous piecewise linear in time function such thatU r z t H r z( , , ) = ( , )0

0 , U r z t H r z( , , ) = ( , )lhl , l n= 1, …, − 1, and

U r z t u r z( , , ) = ( , )n a.e. in Ω. Notice that the operator n is a naturaldiscrete version of the continuous operator defined by (2.7) and, sincethis operator is nonlinear, in general Bh

n does not belong to thediscrete space h. From [27, Lemma 4] it follows that n m, = 1, …,n ,is continuous and strongly monotone, i.e.,

∫ u v u v r drdz μ u v( ( ) − ( ))( − ) ≥ ∥ − ∥ .Ω

n nΩ0 L ( )

2

r2

Different algorithms have been proposed to approximate nonlinearequations with hysteresis (see, for instance, [14,8,13,11]). In general,the nonlinearity is handled by means of fixed-point techniques thatallow dealing with non-smooth curves and ensure convergence. Forexample, in [14] a vector potential formulation of a magnetostaticproblem is solved. The nonlinear term reads H ζ B= ( ) and thehysteretic term ζ is decomposed as ζ B νB R( ) = + where ν dependson ζ and R is evaluated iteratively. In particular, the Preisach model ofhysteresis is implemented in a reversed fashion which is possible forthe static Preisach model under periodic time evolution. Then, the timeevolution of H is reconstructed from a given B in a waveform. Adifferent approach is presented in [13] where the nonlinear term isgiven by the differential permeability μ B H= ∂ /∂d instead of the B(H)(or H(B)) curve.

In this paper, by using the monotonicity of n, we propose a noveliterative algorithm which has been introduced in [20] in a differentsetting. This algorithm, based on the properties of maximal monotoneoperators and their Yosida regularization, has been extensively used fora wide range of applications with good numerical performance (see, forinstance, [29]).

Before introducing this algorithm, and for the sake of clarity, let usconsider the following definitions: for a positive number β, let

G G βG G Ω( ) ≔ ( ) − ∀ ∈ L ( ).n β nr

, 2 (3.5)

Then we recall that the Yosida regularization of n β, is defined by

GG J G

λ( ) ≔

− ( ),λ

n β λβ

,(3.6)

where Jλβ is the resolvent operator of n β, , i.e.,

J I λ≔ ( + ) ,λβ n β, −1 (3.7)

which is defined for any positive λ such that λβ ≤ 1.A simple way to deal with the nonlinearities is to use the following

lemma, which is the basis for the algorithm given below (see [20]). Weinclude the proof for the sake of completeness.

Lemma 3.1. The following statements are equivalent:

(i) G G= ( )n β1

,2 ,

(ii) G G λG= ( + )λn β

1,

2 1 , G G Ω, ∈ L ( )r1 22 .

Proof. The proof follows by straightforward calculations:

G G λG G

G J G λGI λ G G λG

G G

= ( + ) ⟺ =

⟺ = ( + )⟺ ( + )( ) = +⟺ ( ) = . ▫

λn β G λG J G λG

λ

λβ

n β

n β

1,

2 1 1+ − ( + )

2 2 1,

2 2 1,

2 1

λβ

2 1 2 1

At time-step n, we introduce the new function q B βH= −hn

hn

hn.

According to (3.2)

q r z H r z βH r z Ω( , ) = ( )( , ) − ( , ) in ,hn n

hn

hn

and hence from (3.5) and Lemma 3.1 we deduce, for λβ0 < ≤ 1

q r z H λq r z Ω( , ) = ( + )( , ) in .hn

λn β

hn

hn,

Now, in order to solve (3.1)–(3.3), the idea is to replace Bhn by q βH+h

nhn

in (3.1), leading to the following problem which is equivalent toProblem 3:

Given B μ H M= ( + )h0

0 0 0 in Ω, find H ∈hn

h and q Ω∈ L ( )hn

r2 ,

n m= 1, …, , such that

∫ ∫

∫ ∫t

βH G r drdzσ r

rHr

rGr

rHz

rGz

drdz

tq G r drdz f G

tB G r drdz

G q H λq in Ω H

g on Γ B βH q in Ω n

1Δ

+ 1 ∂( )∂

∂( )∂

+ ∂( )∂

∂( )∂

+ 1Δ

= ⟨ , ⟩ + 1Δ

∀

∈ , = ( + ) ,

= , ≔ + , > 1.

Ωhn

hΩ n

hn

h hn

h

Ω hn

hn

hΩ

hn

h

h h hn

λn β

hn

hn

hn

hn

hn

hn

hn

, ′−1

,

⎛⎝⎜

⎞⎠⎟

Then, the algorithm consists of a fixed-point iteration using theprevious formulation of the problem. At time-step n, field Hh

n iscomputed as the limit of sequence H{ }h s

ns,[ ] ∈

which is obtained asfollows:

• At the beginning, function qhn,[0] is given arbitrarily in h.

• Iteration s: qh sn,[ −1] is known.

1. Hh sn,[ ] is computed as the unique solution of the following linear

problem:

∫ ∫

∫ ∫

βH G r drdz tσ r

rHr

rGr

rHz

rGz

t f G B G r drdz q G r drdz

G

+ Δ ∂( )∂

∂( )∂

+∂( )

∂∂( )

∂

= Δ ⟨ , ⟩ + − ∀

∈ ,

Ωh sn

hΩ n

h sn

h h sn

h

nh

Ωhn

hΩ h s

nh

h h

,[ ],[ ] ,[ ]

, ′−1

,[ −1]

⎛⎝⎜

⎞⎠⎟

(3.8)

H g Γ= on ,h sn

hn

,[ ] (3.9)

2. qh sn,[ −1] is updated at the quadrature nodes P Ω∈ (in our case the

vertices of the mesh), by formula

q P H P λq P( ) = ( ( ) + ( )).h sn

λn β

h sn

h sn

,[ ],

,[ ] ,[ −1] (3.10)

The convergence of this algorithm is proved in [20] in an abstractgeneral setting that includes our case, provided λβ ≤ 1/2.

A. Bermúdez et al. Finite Elements in Analysis and Design 126 (2017) 65–74

68

Remark 3. Notice that qh sn,[ ] is used only to compute the last integral in

(3.8). Since λn β, is a nonlinear mapping, this integral needs to be

computed, which is done by means of a quadrature rule. Therefore,only the values of qh s

n,[ ] at the integration points P of the rule has to be

computed by means of (3.10). Moreover, from (3.5) to (3.7) it followsthat the nonlinear function λ

n β, depends on n, which in turn dependson the history of the fully discrete scheme, namely H H H{ , , , …}h h0

1 2 .

Remark 4. In order to compute qh sn,[ ] at the particular quadrature node

P Ω∈ by using (3.10), it is necessary to solve a scalar nonlinearequation to evaluate Jλ

β (cf. (3.7)). However, an interesting feature ofthe algorithm is that instead of solving a nonlinear system of coupledequations (cf. Problem 3), we have to solve a linear system at eachiteration step (cf. (3.8)) and one scalar nonlinear equation perquadrature node (cf. (3.10)). The latter can be done because thealgorithm takes advantage of the spatial independence of thehysteresis operator (cf. (2.8)). Another feature is that, in cases whereσ is time independent, the matrix associated to the linear problem (3.8)is independent of n and s, and then it can be assembled (and eventuallyfactorized) only once before the time-step loop.

Notice that the main difficulty of the proposed algorithm is thecomputation of qh s

n,[ ] by using (3.10). In the following section we give

some details about this procedure for a particular choice of Preisachfunction.

3.1. Updating procedure to compute qh sn,[ ]

From (3.10) and definitions (3.5)–(3.7), it is easy to check that if wedenote by G P H P λq P( )≔ ( ) + ( )h s

nh sn

,[ ] ,[ −1] at any quadrature node P Ω∈ ,then

q P G P Zλ

( ) = ( ) − ,h sn,[ ]

where Z is the unique root of the scalar equation

Z λ Z βZ G P+ ( ( ) − ) = ( ).n

To solve this equation we need to evaluate x( )n for different values ofx ∈ . The definition of n is given in (3.4) where (r,z) are thecoordinates of point P. This expression involves U ξ P t[ ( , )]( , )n .

Now, let us compute such a function at a specific point P r z Ω≔( , ) ∈and n m∈ {1, …, }. At time step n, the values H P( )h

i , i n= 0, …, − 1,have been previously computed. They represent the history of the fullydiscrete problem at point P. Here, H ∈h h

0 is a convenient approxima-

tion of H Ω∈ H ( ) r01

. We assume that the B–H relation (cf. (2.10)) isgiven by a Preisach operator with Preisach function p and “initial state”ξ. Because of the latter, we may define : → (cf. (3.4)) by

x U ξ t( ) ≔ [ ( , )]( ),xn

with Ux being the continuous piecewise linear in time function suchthat U t H P( ) = ( )x

lhl , l n= 0, …, − 1, and U t x( ) =x

n .As an example, we can compute x( ) in the interval ρ ρ[ − , ]0 0 by

using the Preisach function p defined by a Factorized-Lorentziandistribution (see [1]):

p ρ ρ Nρ ω

γωρ ω

γω( , ) ≔ 1 +

−1 +

+1 2

22 −1

12 −1⎛

⎝⎜⎜

⎛⎝⎜

⎞⎠⎟

⎞⎠⎟⎟

⎛⎝⎜⎜

⎛⎝⎜

⎞⎠⎟

⎞⎠⎟⎟

where ω and γ are the identification parameters, and N is a normalizedconstant depending on the value of magnetization at saturation state.Here, the parameters are set to be: N=1, ω = 0.8 and γ = 0.6. In thiscase, the Preisach triangle is characterized by ρ = 50 . Functionis computed for three different sequences H P{ ( )}h

ll∈, more speci

fically, S ≔{5.0, − 0.3}1 , S ≔{5.0, − 2.0, 2.0, − 1.5, − 0.3}2 andS ≔{ − 5.0, 1.0, − 0.3}3 which correspond to different histories to arriveat the same state ( − 0.3). The piecewise linear interpolation of thesevalues has been depicted in Fig. 2 (left-top). The correspondingcurves for each of these sequences are shown in Fig. 2. Notice that the

sequences start with values satisfying either H P ρ( ) ≤ −h0

0 or H P ρ( ) ≥h0

0so that we do not need any initial state ξ to compute because allrelays are “switched-down”.

Notice that the algorithm reveals itself as a useful tool because, ingeneral, the convergence of Newton's method is not guaranteed asthere is a lack of differentiability at some points (see Fig. 2 right-topand bottom).

4. Numerical example

The aim of this section is twofold: firstly, to analyze convergenceproperties of the numerical scheme proposed in the previous sectionsin order to approximate the solution to Problem 2. For this purpose, itis applied to a test problem where several successively refined meshesand time-steps have been considered. Secondly, to validate thenumerical scheme with experimental results. With this aim, weconsider a physical application: the numerical computation of eddycurrent and hysteresis losses in a toroidal laminated media as thoseused in transformers or electric machines.

For both examples we have used an in-house Fortran code. Inparticular, the characterization of the Preisach operator has been madeby the procedure proposed by Mayergoyz [21] (further details can beseen, for instance, in [27]).

4.1. Convergence analysis

In order to numerically estimate the order of convergence of thealgorithm, we have solved the eddy current Problem 2 in a thinrectangular domain Ω R R d= [ , ] × [0, ]1 2 along the time interval[0, 0.02]. For simplicity, we have considered a time dependentDirichlet boundary condition g r z t πft( , , ) = 200sin(2 ) however, theanalysis presented above applies to more general boundary terms (cf.Problem 2) as, for instance, the one used in [17]. The geometrical andphysical data are summarized in Table 1.

The identification of the Preisach model is based on the experi-mental evaluation of return branches called first order curves (see [6]).Nevertheless, it is not the Preisach function the one used in thecomputations, but the associated Everett function. In practice, themeasurable data is the B–H curve associated to different peak values ofthe magnetic induction. We have computed the Everett function (seeFig. 3, left) from experimental data and we have assessed the validity ofthis approximation. Fig. 3 (right) shows a comparison between themeasured B–H curves and the ones computed from the Everettfunction.

Since there is no analytical solution to this problem, we assess theperformance of the method by comparing the computed results withthose obtained with a very fine uniform mesh of size h /1280 and time-step tΔ /640 . The solution to this problem is taken as the “exact” solutionH.

The method has been used on several successively refined meshesand time-steps, both chosen in a convenient way in order to analyze theconvergence with respect to these parameters. Fig. 4 shows a piece ofthe coarser used mesh. We denote by h0 the corresponding mesh sizeand take as coarser time-step tΔ = 0.0040 . The rest of the meshes areuniform refinements of this one. The numerical approximations arecompared with the “exact” solution by computing the percentage errorfor H in a discrete T ΩL (0, ; L ( ))r

2 2 -norm as follows:

E Ht H t H

t H t( ) ≔ 100

∑ Δ ∥ ( ) − ∥

∑ Δ ∥ ( )∥.h

tim i

hi

Ω

im i

Ω

Δ=0−1 +1 +1

L ( )2

1/2

=0−1 +1

L ( )2

1/2

r

r

2

2

⎛⎝⎜

⎞⎠⎟

⎛⎝⎜

⎞⎠⎟

Table 2 shows the percentage errors for the magnetic field, E H( )htΔ , at

different levels of discretization. Taking a small enough time-step tΔ ,one can observe the behavior of the error with respect to the space

A. Bermúdez et al. Finite Elements in Analysis and Design 126 (2017) 65–74

69

discretization (see, for instance, the last row of the table). On the otherhand, by considering a small enough mesh-size h, one can inspect theorder of convergence with respect to tΔ (see, for instance, the last

column). In this example, we observe a linear order of convergence intime and a higher order than linear in space (see Fig. 5) but notquadratic as expected when hysteresis is not considered (see [19]).

We have also computed the percentage error JE ( )htΔ for the current

density J Hcurl= (cf. (2.6)) in the analogous discreteT ΩL (0, ; L ( ) )r

2 2 2 -norm:

JH H

HE

t t

t t

curl curl

curl( ) ≔ 100

∑ Δ ∥ ( ) − ∥

∑ Δ ∥ ( )∥,h

tim i

hi

Ω

im i

Ω

Δ=0−1 +1 +1

L ( )2

1/2

=0−1 +1

L ( )2

1/2

r

r

2 2

2 2

⎛⎝⎜

⎞⎠⎟

⎛⎝⎜

⎞⎠⎟

Fig. 2. Left-top: Piecewise linear interpolation of history data Si, denoted by I i, = 1, 2, 3Si , respectively. The dotted dashed, dashed and dotted lines represent the history. The solid and

small dotted lines in figures (a), (b) and (c) correspond to the curve x( ) and the major hysteresis loop, respectively. The dot corresponds to the last value H P( ) = − 0.3hn .

Table 1Geometrical and physical data for Test 1.

Internal radius of the core, R1 1 mExternal radius of the core, R2 1.03 mThickness of the laminate, d 0.0005 mElectrical conductivity, σ 4,064,777 (Ω m)−1

Frequency, f 50 Hz

Fig. 3. Everett function (left) and comparison between the computed and measured B–H curves (right).

A. Bermúdez et al. Finite Elements in Analysis and Design 126 (2017) 65–74

70

with H H e≔hi

hi

θ+1 +1 . The percentage errors are summarized in Table 3. As

in Table 2, we can observe the behavior of the error with respect tospace and time discretization by taking the time-step and mesh-sizesmall enough. In this case we deduce an order of convergence

h t( + Δ ).

4.2. Computation of electromagnetic losses

In this section, we compare the eddy current and the hysteresislosses obtained from the numerical computations with experimentaland analytical approximate values. With this goal in mind, we solve theeddy current problem in a toroidal laminated core which is assumed tobe surrounded by an infinitely thin coil. Firstly, we show that thisproblem fits within the axisymmetric setting described in the previoussections so we can apply the numerical method proposed above. Inparticular, we will show that the computational domain can be reducedto a domain with one single sheet for which appropriate boundaryconditions are deduced. Then, in order to obtain an expression for thelosses, we write an energy conservation principle for the whole domainΩ and deduce the corresponding electromagnetic losses in theaxisymmetric setting.

4.3. Boundary condition in a toroidal laminated core

Let us consider a toroidal laminated core consisting of N sheets ofrectangular section and thickness d surrounded by a coil (see Fig. 6).Let R1 and R2 be the internal and external radii of the core andD R R≔ −2 1, respectively. Let ne be the number of turns of the coil andI(t) the current intensity flowing through the coil at time t. We willassume that the coil is infinitely thin in such a way that it can bemodeled as a surface current of surface density (A/m) given by

J J J

J

R z t n I tπR

r L t n I tπr

R z t

n I tπR

r t n I tπr

e e

e e

( , , ) = ( )2

, ( , , ) = ( )2

, ( , , )

= − ( )2

, ( , 0, ) = − ( )2

,

Se

z Se

r S

ez S

er

11

2

2

on the inner, upper, outer and lower surfaces of the core, respectively.In order to reduce the eddy current losses, the sheets are isolated by avarnish layer. We neglect the thickness of this dielectric between eachtwo sheets so that the thickness L of the meridian section is L=Nd.Moreover, we use cylindrical coordinates in order to exploit thecylindrical symmetry of the problem. In particular, the magnetic fieldhas only azimuthal component, namely,

H r z t H r z t e( , , ) = ( , , ) ,θ

and the current density in the sheets J Hcurl= has the form given in(2.6).

By using Ampère's law and the axisymmetry of the problem, it iseasy to see that the magnetic field is null outside the core. Hence, sincethe jump through the boundary of its tangential component is equal tothe surface current density, we easily obtain the following Dirichletboundary condition for the magnetic intensity:

H r z t n I tπr

( , , ) = ( )2

.e(4.1)

In particular, H is independent of the z-coordinate on the boundary ofthe core. Moreover, on the internal surfaces between sheets, the normalcomponent of the current density has to be null because they areisolated. Then, according to (2.6),

rH r z t C t( , , ) = ( )

where C(t) varies, in principle, with the internal surface. However,since for r R= 2

R H R z t n I tπ

( , , ) = ( )2

,e2 2

Fig. 4. Detail of the coarser mesh.

Table 2Percentage errors of the computed magnetic field: E H( )h

tΔ .

tΔ h0 h /20 h /40 h /80 h /160

tΔ 0 10.3096 10.6189 10.7197 10.7494 10.7574tΔ /20 6.1990 5.7865 5.7013 5.6944 5.6953tΔ /40 4.0384 3.1842 3.0029 2.9851 2.9871tΔ /80 3.6093 2.2140 1.8783 1.8332 1.8328tΔ /160 3.6852 1.7438 1.1019 0.9836 0.9763tΔ /320 3.8209 1.6224 0.7577 0.5184 0.4946tΔ /640 3.9240 1.6174 0.6577 0.3145 0.2616tΔ /1280 3.9903 1.6346 0.6431 0.2533 0.1404

10−6 10−510−3

10−2

10−1

100

101

102

103

104

Perc

enta

ge e

rror

(%)

h

O(h2) convergenceO(h) convergencePercentage error HPercentage error J

Fig. 5. Percentage errors E H( )htΔ and JE ( )h

tΔ versus the mesh-size h for a fixed time-step

tΔ /128 (log–log scale).

Table 3Percentage errors of the computed current density: JE ( )h

tΔ .

tΔ h0 h /20 h /40 h /80 h /160

tΔ 0 75.0323 71.9613 71.0353 70.7283 70.6271tΔ /20 43.9901 36.1454 33.5191 32.7483 32.5284tΔ /40 35.4035 23.7177 19.1064 17.5277 17.1018tΔ /80 34.3057 20.3499 14.0134 11.4529 10.6809tΔ /160 34.7764 19.3202 11.2893 7.4450 6.0035tΔ /320 35.2999 19.1616 10.4616 5.7973 3.6259tΔ /640 35.6374 19.2307 10.3220 5.3516 2.7365tΔ /1280 35.8637 19.3248 10.3305 5.2602 2.4387

A. Bermúdez et al. Finite Elements in Analysis and Design 126 (2017) 65–74

71

then C t n I t π( ) = ( )/2e and therefore the magnetic field is also given by(4.1) on these internal surfaces. Hence, we can reduce the problem toone single sheet which is extremely important from the computationalpoint of view.

Therefore, if the current intensity in the coil I(t) is given, we arriveat a problem which is a particular case of Problem 2 withΩ R R d≔[ , ] × [0, ]1 2 , f ≡ 0 and

g r z t n I tπr

( , , )≔ ( )2

.e

From Section 2.1 and the previous analysis we deduce that the eddycurrent problem on a toroidal laminated media fits within the proposedaxisymmetric setting (see also [25]) and a Dirichlet boundary conditioncan be obtained.

4.4. Energy balance

The energy balance in region Ω reads as follows:

∫ ∫ ∫ ∫ ∫ ∫B H x J E x nt

dt d d dt dS dt∂∂

· + · = ·Ω t

t

t

t

Ω t

t

Ω∂ 1

2

1

2

1

2⎛⎝⎜

⎞⎠⎟

⎛⎝⎜

⎞⎠⎟

⎛⎝⎜

⎞⎠⎟ (4.2)

where vector field E H≔ × is called Poynting vector (see, forinstance, [30]).

The right-hand side in (4.2) represents the rate of ingoing electro-magnetic energy through boundary Ω∂ . The term involving J repre-sents the dissipated energy by the so-called Joule effect, usually callededdy current loss. By using Ohm's law we deduce

∫ ∫ ∫ ∫J E x H xd dtσ

d dtcurl· = | ( )| ≥ 0.t

t

Ω t

t

Ω

2

1

2

1

2⎛⎝⎜

⎞⎠⎟

⎛⎝⎜

⎞⎠⎟

The first term on the left-hand side describes how the electromagneticenergy is stored or dissipated in Ω by hysteresis mechanisms duringthe time interval t t[ , ]1 2 . Indeed, this energy is partially stored throughreversible mechanisms and partially irreversibly transformed in heatbecause of hysteresis effects. The energy dissipation involved in thisterm is called hysteresis loss. Let us assume the axisymmetric setting.Then, (4.2) reads as follows

∫ ∫ ∫ ∫

∫ ∫ E τ

Bt

Hdt r drdz Hσ

r drdz dt

rH dl dt

curl e∂∂

+ | ( )|

= ( ) · ,

Ω t

t

t

t

Ω

θ

t

t

Γ

A

2

H

E

1

2

1

2

1

2

⎛⎝⎜

⎞⎠⎟

⎛⎝⎜

⎞⎠⎟

⎛⎝⎜

⎞⎠⎟ (4.3)

with τ being the tangent vector to Γ. Let us assume that the solution toMaxwell's system is periodic in time with period T t t= −2 1. Then, ateach point r z Ω( , ) ∈ the curve B r z t H r z t( ( , , ), ( , , )) is closed andcontained in the so-called major hysteresis loop. From Stokes' theorem,it can be shown that the term labelled A in (4.3) is equal to the net areaenclosed by this curve (see [30]). Therefore, according to the abovediscussion, this net area is the hysteresis loss density (J/m3) along acycle at point (r,z).

Remark 5. The total losses on the right hand side of (4.3) may berewritten depending on the particular problem we are working on. Forinstance, in order to compute the electromagnetic losses in the toroidallaminated media surrounded by a coil presented in the previoussection, from (4.1), Stokes' theorem and Faraday's law (2.1), itfollows that the total electromagnetic losses are given by

∫ ∫ ∫ ∫E τrH dl dt rHt

B drdz dt( ) · = ( )| ∂∂

.t

t

Γ t

tΓ

Ω

T

1

2

1

2⎛⎝⎜

⎞⎠⎟

⎛⎝⎜

⎛⎝⎜

⎞⎠⎟⎞⎠⎟

We now have an expression for the total ( T ), the eddy current ( E)and the hysteresis ( H) losses. Next, we compute the electromagneticlosses obtained by numerically solving eddy current Problem 2 withsource terms obtained from physical measurements and we comparethe obtained results with experimental and analytically approximatedvalues. The measurements were done on an Epstein frame consideringa material sheet of thickness 0.5 mm and width 30 mm, subject to asinusoidal flux excitation with frequencies f equal to 25 and 150 Hz andinduction peak levels Bm equal to 0.5, 0.9 and 1.4 T. For each of thesefrequencies and peak levels, the physical measurements were themagnetic field on the boundary of the sheet and the total electro-magnetic losses per cycle and per unit volume.

To simulate the experimental setting with our axisymmetric model,we consider the geometrical data summarized in Table 4. In order tocharacterize the magnetic behavior of the ferromagnetic material wehave used the same Everett function as the one described in theprevious section.

Firstly, we compute the numerical total losses T per unit volume:

∫ ∫πd R R

π R H ddt

B drdz dt≔ 1( − )

2 | (J/m ).Th

Th Γ

Ωh

22

12 0

23

⎧⎨⎩⎛⎝⎜

⎞⎠⎟

⎫⎬⎭We report in Table 5 these computed values and the correspondingphysical measurements for the different frequencies f and peak levelsBm. This table reveals that the computed losses are in good agreementwith the values obtained by measurements, particularly forB ∈ {0.9, 1.4}m . One possible explanation of the larger differences forBm=0.5 T may be that our Everett function does not generate accurateB–H cycles for “small” values of B (see Fig. 3, right) and, because ofthat, we do not obtain good approximations of the hysteresis losses.Next we focus on the eddy current losses. Because of the lack ofexperimental data, the goal is to compare our numerical resultswith the eddy current losses computed by the analytical approximation(see [1]):

Fig. 6. Toroidal laminated core (left) and meridian section (right).

Table 4Geometrical and physical data for the test.

Internal radius of the core, R1 100 mExternal radius of the core, R2 100.03 mThickness of the laminate, d 0.0005 mElectrical conductivity, σ 4 064 777 (Ω m)−1

A. Bermúdez et al. Finite Elements in Analysis and Design 126 (2017) 65–74

72

π σd fB≔ (J/m ).an16

2 2 2 3(4.4)

Recall that this equation is valid in cases where there is no skin effectand the geometrical restrictions d D≪ , d R≪ 1 hold. We compare theanalytically approximated eddy current losses per unit volume with thenumerical ones computed from the expression

∫ ∫πd R R

π Hσ

rdrdz dte≔ 1( − )

2 |curl( )| (J/m ).Eh

T

Ω

h θ

22

12 0

23

⎧⎨⎩⎛⎝⎜

⎞⎠⎟

⎫⎬⎭Table 6 shows the two eddy current losses and the difference betweenthem. Although the geometric parameters chosen for the test aresuitable to use formula (4.4), the skin effect may become not negligibledepending on the level of saturation. Indeed, at high frequency and lowinduction values, the skin effect is large and the analytical expressionoverestimates the eddy current losses. This is the case for f = 150 Hzand Bm=0.5 T, where the largest discrepancy is observed. To illustratethis fact, in Fig. 7 we have represented the B waveforms at two differentpoints placed in the middle and at the surface of the sheet. Finally, wehave computed the numerical hysteresis losses H per unit volume

∫ ∫πd R R

π Bt

H dt rdrdz≔ 1( − )

2 ∂∂

, (J/m )Hh

Ω

Th

h22

12 0

3⎧⎨⎩

⎛⎝⎜

⎞⎠⎟

⎫⎬⎭

and also the total electromagnetic losses by summing up both the eddycurrent ( E

h ) and the hysteresis ( Hh ) losses, and we have compared

them with the measured total losses. Table 7 summarizes the results fordifferent frequencies and magnetic induction peak levels. Again, thelargest discrepancy is observed for Bm=0.5 T. However, the valuesobtained for the rest of the assessed data are very good.

Acknowledgments

The work of the authors from Universidade de Santiago deCompostela was supported by Spanish Ministry of Science andInnovation under research project ENE2013-47867-C2-1-R and byXunta de Galicia under research project GRC2013/014. P. Venegas wassupported by Centro de Investigación en Ingenierí a Matemática(CI2MA), Universidad de Concepción, through the BASAL project PFB-03, CMM, Universidad de Chile, CONICYT andMECESUP project UCO0713 (Chile).

Appendix A. Supplementary material

Supplementary data associated with this paper can be found in theonline version at http://dx.doi.org/10.1016/j.finel.2016.11.005.

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25 0.5 121.2594 134.6023 11.00360.9 300.2459 300.3188 0.02401.4 638.9281 676.6006 5.8962

150 0.5 167.5503 186.4886 11.30310.9 459.1568 4570.4152 0.37901.4 1090.1780 1095.6934 0.5059

Table 6Eddy current losses (J/ m3).

f (Hz) Bm (T) an (Analytical) Eh (Numerical) Relative error (%)

25 0.5 10.4473 9.9144 5.10110.9 33.8493 32.7726 3.18101.4 81.9071 91.3716 11.5552

150 0.5 62.6840 48.2460 23.03300.9 203.0961 178.5306 12.09551.4 491.4423 506.3943 3.0425

0 0.002 0.004 0.006 0.008 0.01 0.012 0.014−1

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

B

Time (s)

B(t) surfaceB(t) middle

Fig. 7. B(t) in a point at the surface and in the middle of the domain for f = 150 Hz,Bm=0.5 T.

Table 7Total losses (J/m3).

f (Hz) Bm (T) Eh

Hh

Eh + H

h Total (exp) Relativeerror (%)

25 0.5 9.9144 126.2274 136.1418 121.2594 12.27320.9 32.7726 269.4965 302.2691 300.2454 0.67401.4 91.3716 585.7265 677.0981 638.9281 5.9741

150 0.5 48.2460 146.0968 194.3428 167.5503 15.99070.9 178.5306 282.2810 460.8116 459.1568 0.36031.4 506.3943 588.1854 1094.5797 1090.1780 0.4037

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