Vector Preisach model and Maxwell's equations

Physica B 306 (2001) 21–25

Vector Preisach model and Maxwell’s equations A. Visintin* Dipartimento di Matematica dell’Universita" di Trento, via Sommarive 14, 38050 Povo (Trento), Italy

Abstract We represent processes in highly anisotropic, nonhomogeneous, either ferromagnetic or ferrimagnetic materials, by coupling the Maxwell equations with a vector extension of the standard rectangular hysteresis loop. We then derive a two-length-scale vector model via a homogenization procedure. Here, the fine-scale magnetization M depends on the fine-scale magnetic field H; at variance with the corresponding scalar problem. This amends the usual formulation of the vector Preisach model, which is not consistent with the law r  B ¼ 0: r 2001 Elsevier Science B.V. All rights reserved. PACS: 75.60.Ej Keywords: Hysteresis; Vector Preisach model; Homogenization; Two-length-scale model

1. Introduction The scalar Preisach model of hysteresis is based on integration of elementary rectangular hysteresis loops, often named relays [1]. Along the lines of Refs. [2–4], we extend these relays to the vector setting. We then reformulate them in such a way that their coupling with the Maxwell equations is mathematically consistent; i.e., associated initialand boundary-value problems have a solution. In order to represent processes in highly anisotropic, composite either ferromagnetic or ferrimagnetic materials, we derive a two-lengthscale vector model, by means of a homogenization procedure. At variance with the scalar counterpart, here the fine-scale magnetization M depends on the fine-scale magnetic field H: This amends the usual formulation of the vector Preisach model, which is not consistent with the law r  B ¼ 0 on the fine scale.

This note announces results of Ref. [5], and is part of a research on homogenization of hysteresis models.

2. Maxwell’s equations Let a bounded region O of the three-dimensional space, R3 ; be occupied by an either ferromagnetic or ferrimagnetic material, surrounded by air. We assume that the dielectric permeability, e; is a scalar constant, denote by Je a prescribed exterior electric current, and use Gauss units. We distinguish three cases. (i) For low-frequency processes in a ferromagnetic metal we can neglect displacement currents. Let Ea be a given applied electromotive force. By the Maxwell equations and by Ohm’s law, J ¼ sðE þ Ea Þ; we get 4ps

*Tel.: +39-461-881-635; fax: +39-461-881-624. E-mail address: [email protected] (A. Visintin).

qB þ c2 r  r  H ¼ 4pcsr  Ea  G qt

0921-4526/01/$ - see front matter r 2001 Elsevier Science B.V. All rights reserved. PII: S 0 9 2 1 - 4 5 2 6 ( 0 1 ) 0 0 9 5 7 - 7

in O; ð2:1Þ

22

e

A. Visintin / Physica B 306 (2001) 21–25

q2 B þ c2 r  r  H ¼ 4pcr  Je qt2

outside O: ð2:2Þ

High-frequency processes in ferromagnetic metals are represented by Eq. (2.2) in the whole space, with the extra term 4psqB=qt in O: (ii) In a ferrimagnetic insulator we account for displacement currents, and get Eq. (2.2) in the whole space. (iii) For quasi-stationary processes in either ferromagnetic or ferrimagnetic materials, we deal with the magnetostatic equations rB¼0 cr  H ¼ 4pJ

in R3 :

ð2:3Þ

Initial conditions must be provided in cases (i) and (ii). The constitutive relation between M  ð1=4pÞðB HÞ and H is the main issue of this note. Under severe restrictions on the geometry and on the symmetry of the fields, the above problems can be reduced to an either one- or two-dimensional problem for scalar variables. The scalar fields B and H can then be related by a scalar hysteresis operator, e.g., the Preisach model. Several mathematical results are known for corresponding initial- and boundary-value problems, see Refs. [6,7]. Here, we deal with the fully three-dimensional setting.

Fig. 1. (a) Relay operator; M ¼ 71: (b) Completed relay operator; 1pMp1:

mount to 3. Scalar Preisach model

ðM 1ÞðH r2 ÞX0 ðM þ 1ÞðH r1 ÞX0

for any t:

ð3:1Þ

3.1. Scalar relay

jMjp1

Any pair r  ðr1 ; r2 Þ (r1 or2 ) corresponds to the elementary (delayed) relay operator hr : H/M outlined in Fig. 1(a). On account of its discontinuity, this operator is not closed in function spaces which naturally arise in partial differential equations. Along the lines of Ref. [6], we then replace hr by the (multivalued) completed relay operator, kr : H/M (cf. Fig. 1(b)). The operator kr : H/M is characterized by the following conditions: (i) Confinement condition: The point ðH; MÞ is confined to the rectangle ½r1 ; r2   ½ 1; 1 joint with two half-lines. By Refs. [7,8], this is tanta-

(ii) Dissipation condition: The dynamics outlined by arrows in Fig. 1(b) yields Z t Z t H dMX ½r2 ðdMÞþ r1 ðdMÞ  0 0 Z r2 r1 t r þ r1 ½MðtÞ Mð0Þ ð3:2Þ jdMj þ 2 ¼ 2 2 0 for any t: Here, we omit information about the initial condition. This formulation can be extended to space-structured (i.e., x-dependent) systems, just requiring Eqs. (3.1) and (3.2) to hold pointwise in O:

23

A. Visintin / Physica B 306 (2001) 21–25

3.2. Scalar Preisach integral Let m be a finite positive Borel measure over the half-plane of admissible thresholds, P  frAR2 : r1 or2 g: We thenR define the (completed) Preisach operator Fm  P kr dmðrÞ: More precisely, for any continuous function H; we set MAFm ðHÞ if and only if for any rAPR there exists a function Mr Akr ðHÞ such that M ¼ P Mr dmðrÞ: (We still omit information about the initial condition.)

4. Vector Preisach model 4.1. Vector relay Let us denote by y  ðy1 ; y2 ÞAS 2 the angular coordinates, by ey the corresponding unit vector, and set P3  P  S2 : For any ðr; yÞAP3 ; we introduce the (completed) vector relay operator kðr;yÞ ðHÞ  kr ðH  ey Þey for any continuous function t/HðtÞ: This operator is multivalued and inherits several properties from kr : 4.2. Vector Preisach integral Let us fix a finite Borel measure m over P3 ; and introduce the corresponding vector Preisach operator [8] Z Fm  kðr;yÞ dmðr; yÞ: P3

We amend this model, for it may not be consistent with the law r  B ¼ 0; see Section 8.

5. A two-scale model The Preisach integral can be interpreted as an average of fine length-scale elements. This suggests us to regard the medium as a composite material, and to apply a homogenization procedure. Besides the (macroscopic) coarse length scale, which is represented by the parameter xAO; we

introduce a (mesoscopic) fine length scale, and represent it by yAY  ½0; 13 : To any yAY we associate a pair ðrðyÞ; yðyÞÞAP3 and the corresponding vector relay kðrðyÞ;yðyÞÞ : The measure mðr; yÞ then represents the fine-scale relay distribution Z Z jðr; yÞ dmðr; yÞ ¼ jðrðyÞ; yðyÞÞ dy Y

P3

for any smooth function jðr; yÞ: The pair of functions ðr; yÞ : Y-P3 represents the fine-scale structure of the material, and contains more information than the distribution m: We denote the characteristic fine length scale by e; and relate the two scales by y ¼ x=e: Any function Y-R3 is then extended by periodicity.

6. Homogenization of the magnetostatic problem Let the electric current density field J  ðc=4pÞr  U be prescribed. We represent slow evolution in a time interval 0; T½ by coupling the vector-relay constitutive law with the magnetostatic equations. Problem 1e . To find He ðx; tÞ and Me ðx; tÞ such that; setting Be  He þ 4pMe for 0otoT r  Be ¼ 0 cr  He ¼ 4pJ

in R3 ;

Me ðx; tÞA½krðx=eÞ ðHe ðx; Þ  eyðx=eÞ ÞðtÞeyðx=eÞ in O; and Me  0 outside O: (The dot in the argument indicates dependence on the whole function t/He ðx; tÞ:) This problem has a generalized solution. One can show that there exist fields Hðx; y; tÞ and Mðx; y; tÞ such that, possibly extracting subsequences, He -H and Me -M in the sense of two-scale convergence (cf. Ref. [9]). That is, for any smooth y-periodic function cðx; yÞ; as e-0; Z He ðx; tÞcðx; x=eÞ dx R3 ZZ Hðx; y; tÞcðx; yÞ dx dy R3 Y

24

A. Visintin / Physica B 306 (2001) 21–25

and similarly for M: Moreover, Mðx; y; tÞA½krðyÞ ðHðx; y; Þ  eyðyÞ ÞðtÞ eyðyÞ in O; R r  Y Bðx; y; tÞ dy ¼ 0 ry  Bðx; y; tÞ ¼ 0

exists a y-periodic field Kðx; y; tÞ such that ð6:1Þ

in R3  Y;

qB ðx; y; tÞ þ c2 rx  rx  Hðx; y; tÞ qt ¼ Gðx; tÞ ry  Kðx; y; tÞ;

4ps

ry  Hðx; y; tÞ ¼ 0;

there exist two scalar functions uðx; tÞ and u1 ðx; y; tÞ; the latter being Y-periodic, such that Hðx; y; tÞ ¼ ruðx; tÞ þ ry u1 ðx; y; tÞ þ Uðx; tÞ in R3  Y: Finally, ry  rx  H ¼ 0 in R3  Y: The fields B and H do not depend on y outside O; that is, the mesostructure is confined to O: By rx (ry ; resp.) we denote the gradient vector w.r.t. the variable x (y; resp.). The coarse-scale fields Z Bðx; tÞ  Bðx; y; tÞ dy; Y Z Hðx; tÞ  Hðx; y; tÞ dy ¼ ruðx; tÞ þ Uðx; tÞ Y

then solve the coarse-scale magnetostatic equations (2.3).

7. Homogenization of the eddy-current problem For the sake of simplicity, here we uncouple the problem in the conductor O from that outside O; although our analysis can be extended to the coupled problem.

ry  rx  H ¼ 0:

ð7:3Þ

Eq. ð7:3Þ1 entails the coarse-scale eddy-current equation (2.1) and ry  Bðx; y; tÞ ¼ 0; Z r Bðx; y; tÞ dy ¼ 0: Y

The quasi-linear hyperbolic problem (ii) of Section 2 also has a generalized solution, and can be homogenized similarly. An analogous approach can be used for high-frequency processes in ferromagnetic metals.

8. Discussion Moving from a two-scale interpretation of the Preisach model and from a vector generalization of the relay operator, we derived the hysteresis relation in the form Mðx; y; tÞA½kðrðyÞ;yðyÞÞ ðHðx; y; ÞÞðtÞ eyðyÞ : Notice the occurrence of the fine-scale magnetic field Hðx; y; tÞ; instead of its coarse-scale counterpart Z Hðx; tÞ  Hðx; y; tÞ dy; Y

Problem 2e . To find He ðx; tÞ and Me ðx; tÞ such that, setting Be  He þ 4pMe for 0otoT; qBe þ c2 r  r  He ¼ G in R3 ; 4ps ð7:1Þ qt Me ðx; tÞA½krðx=eÞ ðHe ðx; Þ  eyðx=eÞ ÞðtÞ eyðx=eÞ

ð7:2Þ

in O: Moreover; n  r  He on the boundary of O and Be at t ¼ 0 are prescribed. This problem has a generalized solution. As above there exist two fields Hðx; y; tÞ and Mðx; y; tÞ such that, possibly extracting subsequences, as e-0; He -H and Me -M in the sense of two-scale convergence. Moreover, Eq. (6.1) holds, and there

at variance with the vector Preisach model. Indeed H can have large oscillations, for in general r  Hðx; tÞ need not be integrable. Moreover, the example of Fig. 2 shows that the dependence of Mðx; y; tÞ on Hðx; tÞ may violate the law ry  Bðx; y; tÞ ¼ 0: This applies also if the y’s are uniformly distributed on the unit sphere. This behaviour looks as a combined effect of anisotropy, of nonhomogenity, and of the structure of Maxwell equations. According to the present approach, the density of the different relays in a material (which is represented by the measure m) does not provide sufficient information, and the description of the (mesoscopic)

25

A. Visintin / Physica B 306 (2001) 21–25

interpretation of that modelFa point of view which is maintained by distinguished researchers. The above developments can be extended to ferroelectric materials. In this case one can couple a hysteretic D vs. E relation either with the quasilinear hyperbolic equation q2 D þ c2 r  r  E ¼ 0 in R3 qt2 or, for slow processes, with the electrostatic equations r  D ¼ 4pr# cr  E ¼ 0

in R3 :

An analogous approach can be applied to the Prandtl-Ishlinski&ı models of elasto-plasticity, see Ref. [10]. The main features of this discussion also hold for constitutive relations without hysteresis. Fig. 2. (a) Two strongly anisotropic magnetic particles subjected to the same field H; ry  Ba0: (b) A nonuniform H yields ry  B ¼ 0:

fine-scale space distribution of relays is needed. This raises the question of identifying the latter. On the other hand, in the (univariate) scalar setting a mesostructure may occur for M but not for H; because the Maxwell equations entail the integrability of qH=qx: The hysteresis constitutive law (7.2) is then reduced to Mðx; y; tÞA½krðyÞ ðHðx; ÞÞðtÞ; here with Hðx; Þ in place of Hðx; y; Þ; consistently with the scalar Preisach model. The above two-scale approach to the Preisach model is at variance with a purely phenomenologic

References [1] F. Preisach, Z. Phys. 94 (1935) 277. [2] A. Damlamian, A. Visintin, C.R. Acad. Sci. Paris S!er. I 297 (1983) 437. [3] I.D. Mayergoyz, Mathematical Models of Hysteresis, Springer, New York, 1991. [4] I.D. Mayergoyz, G. Friedman, J. Appl. Phys. 61 (1987) 4022. [5] A. Visintin, Maxwell’s equations with vector hysteresis, Preprint, 2001. [6] A. Visintin, Differential Models of Hysteresis, Springer, Berlin, 1994. [7] A. Visintin, C. R. Acad. Sci. Paris S!er. 332 (2001) 315. [8] A. Visintin, J. Mater. Process. Manuf. Sci. 9 (2000) 64. [9] G. Allaire, SIAM J. Math. Anal. 23 (1992) 1482. [10] A. Visintin, Int. J. Nonlinear Mech., submitted for publication.