Non Linear Homogenization of Laminate Magnetic Material by
Computing Equivalent Magnetic Reluctivity
Ghania Yousfi
a
and Hassane Mohellebi
b
Electrotechnics Department, Faculty of Electrical Engineering and Computer Science,
Mouloud Mammeri University, Tizi-Ouzou, Algeria
Keywords: Laminate Material, Magnetic Reluctivity, Homogenization Technique, Finite Element Method, Non Linear
Material, Inverse Problem, Optimization.
Abstract: In the present study we present a numerical modeling of a laminate magnetic material using an
homogenization technique which exploit an inverse problem resolution. The laminate magnetic material is
consisting of an alternate of magnetic and insulates layers. The equivalent magnetic reluctivity is computed
for the homogenized domain by considering a non linear behavior of the magnetic layers. A finite element
method is used to solve the 2D non-linear electromagnetic partial differential equation. An optimization
problem is constructed and solved with the association of the 2D finite element resolution and a conjugate bi-
gradient algorithm. The computation of the equivalent magnetic reluctivity is then performed for different
excitation field value according to B-H curve. The comparison of the obtained B-H curves of the laminate
and the homogenized domains to the theoretical B-H curve (experimental data) show a good agreement of
laminate results.
1 INTRODUCTION
Generally used material in electrical, mechanical,
structures are: metallic, polymers, ceramic and
composites. The physical characteristics of a
composite material primarily of the laminates are the
result of a combination of the properties of matrix,
reinforcement and additives. The materials nature are
strongly heterogeneous and anisotropic (Trichet ,
2000). The quality of simulation results is directly
related to the precise knowledge of the physical
properties of composite material.
The goal of homogenization of heterogeneous
material is to reduce the complexity involved by the
wall geometry of the solving domain and the non
linearity of the physical properties with anisotropy.
It's then more suitable and convenient, as proposed by
several researchers the use of methods of The
algorithm of optimization based on the method
homogenisation (Meunier, 2010
; Bensaid, 2006;
Charmoille, 2008; Waki, 2005). To determine the
equivalency of electromagnetic characteristics of an
homogeneous material replacing the heterogeneous
a
https://orcid.org/0009-0009-2674-8299
b
https://orcid.org/0000-0003-3661-1690
one (Charmoille, 2008; Waki, 2005; Ren, X, 2016;
Achkar, 2021).
The current study is focused on the computation
of the anhysteretic curve and the equivalent magnetic
reluctivity of homogeneous material equivalent to
laminate one. For this goal , the method of inverse
problem which is coupled with finite elements is
exploited (Szeliga, 2004; Martin, 2015; Gavazzoni,
2022). The algorithm of optimization is based on the
method of the gradient which uses a function cost
defined as a difference between the stratified and
homogeneous magnetic fields. The homogenization
of electromagnetic characteristics of heterogeneous
material are then computed and compared to
theoretical results (Feliachi, 1991).
2 ELECTROMAGNETIC
EQUATION
The modeling of the electromagnetic problems is
based on the Maxwell's equations. The
348
Yousfi, G. and Mohellebi, H.
Non Linear Homogenization of Laminate Magnetic Material by Computing Equivalent Magnetic Reluctivity.
DOI: 10.5220/0012795200003758
Paper published under CC license (CC BY-NC-ND 4.0)
In Proceedings of the 14th International Conference on Simulation and Modeling Methodologies, Technologies and Applications (SIMULTECH 2024), pages 348-353
ISBN: 978-989-758-708-5; ISSN: 2184-2841
Proceedings Copyright © 2024 by SCITEPRESS – Science and Technology Publications, Lda.

electromagnetic equation, by considering magnetic
vector potential unknown
A

, is then deduced from
Maxwell equations as following:
()
s
JAjAB





⋅=+∧∇∧∇ μωσυ )(
(1)
() B
ν
is the magnetic reluctivity which depends on
magnetic flux density [H/m]
-1
,
σ
is the electric
conductivity [S/m], ω is the angular frequency [rd/s]
and
s
J

is the current density [A/m
2
].
In case of (x,y) Cartesian coordinates, equation
(1) could be written according to each solving sub-
domain ''i'' as fellow:
() ()
siiii
JAj
y
A
B
yx
A
B
x
−=−








∂
∂
∂
∂
+






∂
∂
∂
∂
ωσνν
In case of ferromagnetic layers 0=
si
J and for
insulator layers
0=
si
J and 0=
i
σ
.
3 FINITE ELEMENTS
FORMULATION
The finite element formulation of the electromagnetic
problem defined by non linear equation (2) consists
on the substitution of the partial differential equation
by an integral formulation of the problem. According
to that, we choice the projective formulation based on
Galerkin method which has the advantages, with
some considerations, of providing a system whose
mass matrix become symmetrical. Thus, finite
element formulation of Equ. (2) is:

Ω
=








−+








∂
∂
∂
∂
+
∂
∂
∂
∂
0dxdyJAj
y
A
yx
A
x
szizi
zizi
ασωα
αα
υ
The final matrix system to be solve is given by:
() [] [][]
[
]
[] SAMjK =+ ων
imr
jAAA +=
A
is a complex unknown which represents the
magnetic vector potential of real (
r
A ) and imaginary
(
im
A ) components respectively.
() .

Ω








∂
∂
∂
∂
+
∂
∂
∂
∂
= dxdy
yyxx
BK
j
i
j
i
ij
α
α
α
α
υ

Ω
= dxdyM
jiij
αασ

Ω
= dxdyJS
szii
α
ij
K
are coefficients of non linear mass matrix.
ij
M
are coefficients of harmonic matrix.
i
S are coefficients of source vector.
i
α
and
j
α
are projection and shape function at nodes
''i'' and j'' respectively.
4 MAGNETIC LAYERS
MODELING
The magnetic layers used in the electromagnetic
arrangement are a ferromagnetic type by considering
nonlinear hypothesis. A non linear model of the
magnetic reluctivity which governs the magnetic
behavior of ferromagnetic layers, according to
magnetic flux density variation, is given by
expression below(Feliachi,1984; 1991)
() (
)








+
−+=
τ
ννννυ
η
η
2
2
0
B
B
fi
B
i
(9)
B
is the magnetic flux density modulus,
0
ν
is the
magnetic reluctivity of vacuum,
i
ν
and
f
ν
are initial
and final magnetic reluctivity respectively. When
η
and
τ
are parameters of magnetic reluctivity model
which could be deduced from experimental curve
(data). The values of the different parameters of
magnetic reluctivity model are summarized in table 1.
The theoretical curve representing anhysteretic
curve obtained when using non linear magnetic
reluctivity model with the associated parameters
presented in Table 1 is plotted and shown in Fig.1.
Table 1: Parameters of magnetic reluctivity model.
Parameters
i
ν
f
ν
η
τ
(USI)
Values 0.005 1 5.419 1339
(8)
(7)
(6)
(4)
(3)
(2)
(5)
Non Linear Homogenization of Laminate Magnetic Material by Computing Equivalent Magnetic Reluctivity
349

Figure 1: Anhysteretic curve B-H.
5 OPTIMIZATION PROBLEM
The method of inverse problem is applied in the
current study of the homogenization problem (fig. 3).
The goal consists to search for the behavior of
electromagnetic characteristics of the homogenized
solving domain with the determination of the optimal
value of the magnetic reluctivity by taking account of
non linearity. This is performed with the use of the
proposed objectiv
e function J above:
()
2
hom
2
1
HHJ
stra
−=
stra
H is a magnetic field of laminated material,
hom
H
is a magnetic field of homogenized material.
The flowchart of the optimization procedure in shown
in Fig.2.
The treatment of the inverse problem is carried out
while being based on the method of the gradient with
the algorithm described by (
Szeliga, 2004):
()
kkkk
uJuu ∇−=
+

α
1
α
k
: optimal step; u
k
: search point at k iteration;
()
k
uJ∇

: gradient of objective function.
The computation of the gradient of the objective
function is obtained using the proposed formula:
()
ν ∂
∂
∂
∂
=∇
k
k
k
u
u
J
uJ .

kk
kkk
uuu
νν
ν
−
−
=
∂
∂
+
+
1
1
The optimal step was calculated with the method
of Quasi-Newton, whose algorithm is
written as
follows:
(
)
kkkkk
xfsxx ∇−=
+

α
1
S
k
: symmetric matrix; α
k
: optimal step given by linear
minimization
Figure 2: Flowchart of the optimization problem.
6 APPLICATIONS AND RESULTS
The electromagnetic device considered in the current
work is consisting of an alternates arrangement of
ferromagnetic and insulator layers. The complex
algebraic system (4) is solved using finite element
code developed under Matlab PDETOOL package.
6.1 Study of the Laminate Material
6.1.1 Geometric and Physical Characteristics
The laminated armature is made of two kinds of
materials: ferromagnetic and insulator. Ten (10)
layers are ferromagnetic ones with 0.25 mm width
and nine (09) layers are insulators with 0.1 mm width,
the height of the stratified material is about 8 mm
(Fig. 3).
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5
x 10
5
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2
Magnetic Field H(A/m)
Magnetic Flux Density B(T)
(10)
(13)
Electromagnetic problem resolution
Field computation
Objective function evaluation
Computation of the gradient
End
Initialization: , H
0
,
Mesh domain
Geometr
y
and
p
arameters descri
p
tion
No
Optimal value of reluctivity
Start
Yes
(12)
(11)
(14)
SIMULTECH 2024 - 14th International Conference on Simulation and Modeling Methodologies, Technologies and Applications
350

Figure 3: Laminate solving domain.
The physical properties of the studied device with
stratified material are given in the following Table 2:
Table 2: Physical parameters.
Electric Conductivity Magnetic
Reluctivity
Magnetic
material
4 10
6
].[ mS
)(B ν ]/[ Hm
Insulator 0
5
0
10 95.7= ν
6.1.2 Mesh Domain
The mesh of the resolution domain with stratified
material is shown in Fig. 4.
Figure 4: Meshed solving domain.
The mesh domain consists of 10694 nodes and 21314
triangles.
6.1.3 Results and Discussion
After resolution of the non linear problem, the results
obtained are shown in figures 5, 6, 7 and 8. In Figure
5 and 6 are given the behaviors of the magnetic vector
potential distribution and the magnetic anhysteretic
curve compared to experimental one. It shows a good
agreement between results. In Fig. 7 is given the
magnetic reluctivity behavior with magnetic flux
density of laminate material.
Figure 5: Magnetic vector potential distribution (Laminate
domain).
Figure 6: Anhysteretic magnetic curves.
Figure 7: Magnetic reluctivity variation of laminate.
6.2 Study of the Homogenized Structure
The homogenized structure is shown in Fig. 8 where
the load region is consisting of a single domain
0 0.005 0.01 0.015 0.0 2 0.025
0
0.005
0.01
0.015
0.02
0.025
0.03
0.035
0 0.005 0.01 0.015 0.02 0.025
0
0.005
0.01
0.015
0.02
0.025
0.03
0.035
x(m)
y(m)
0 0.005 0.01 0.015 0.02 0.025
0
0.005
0.01
0.015
0.02
0.025
0.03
0.035
-1
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
1
x 10
-3
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5
x 10
5
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2
Magnetic Field H(A/m)
Magnetic flux Desnsity (T)
Theoritical Results
Simulation Results
0 0.05 0.1 0.15 0.2 0.25 0.3 0.35
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
Magnetic Reluctivity (H/m)-1
Magnetic Flux Density (T)
Magnetic vector potential A
Laminate Material
Inductor coil
x(m)
y(m)
Non Linear Homogenization of Laminate Magnetic Material by Computing Equivalent Magnetic Reluctivity
351

replacing the laminate one as shown in the previous
Fig.3.
Figure 8: Homogenized solving domain.
6.2.1 Mesh Domain
The mesh of homogenized domain is presented in
Fig.9.
Figure 9: Mesh of homogenized domain.
6.2.2 Results and Discussion
The complex algebraic system is solved using finite
element code developed under Matlab PDETOOL
package and coupled to the optimization conjugate
gradient algorithm as illustrated in flowchart (Fig.2).
After resolution of the optimization problem, the
results obtained are shown in figures 10, 11 and 12.
The distribution of the magnetic vector potential
shown in Fig. 10 is reproduced correctly in the
homogenized material and seems to be identical to the
distribution obtained in the case of laminate material.
Figure 10: Magnetic vector potential distribution
(Homogenized domain).
Figure 11: Anhysteretic magnetic curves.
In Fig. 11, the anhysteretic curve obtained in the
homogenized material is compared to theoretical
anhysteretic curve (experimental data) where the
results seem to be comparable with some difference
for high excitation field.
Figure 12: Magnetic reluctivity behavior curves.
The results are acceptable and permit to conclude to
the validity of the optimization process used. Thee
magnetic reluctivity behavior with iterations shown
for both laminate and homogeneous material in Fig.
12 have a similar variation with a difference which
appears when the excitation magnetic field becomes
0 0.005 0.01 0.015 0.02 0.025
0
0.005
0.01
0.015
0.02
0.025
0.03
0.035
x(m)
y(m)
Hogenized Domain Resolution
0 0.005 0.01 0.015 0.02 0.025
0
0.005
0.01
0.015
0.02
0.025
0.03
0.035
x(m)
y(m)
0 0.005 0.01 0.015 0.02 0.025
0
0.005
0.01
0.015
0.02
0.025
0.03
0.035
x(m)
y(m)
Magnetic vector potential A
-1
-0.5
0
0.5
1
x 10
-3
0 2 4 6 8 10 12 14
x 10
4
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
Magnetic Field (A/m)
Magnetic Fl ux Densi ty (T)
Theori tic al Res ults
Simulation Res ults
0 2 4 6 8 10 12 14 16 18
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
reluctivité magnétique
Itérations
Réluc t i v i t é magnét i que ( H/m) -1
Solution stratifiée
Solution homogène
Inductor coil
Homogenized Material
♦Laminate results
+ Homogenized results
Ma
g
netic reluctivit
y
ν [H/m]
-1
SIMULTECH 2024 - 14th International Conference on Simulation and Modeling Methodologies, Technologies and Applications
352

higher. The difference shown could be investigated
by performing an experimental setup.
7 CONCLUSION
This works presents the characterization of laminated
material in terms of physical properties. The finite
elements method is used and associated to inverse
algorithm problem based on gradient search method
of optimum value. The magnetic reluctivity is
calculated considering the objective function
depending on the square difference between magnetic
fields of laminated and homogenized materials.
Results seems interesting to be able to apply the
model to the identification of the electric conductivity
of laminated material used in electrical machinery
which are subjected to eddy current and saturation
effects.
The difference recorded on the magnetic
anhysteretic curves and the magnetic reluctivity
behavior could be investigated by performing an
experimental setup. This brings us to conclude that
the used algorithm supplies reproducible results were
from his robustness.
REFERENCES
Trichet, D. (2000). Contributuin à la modélisation, à la
conception et au développement du chauffage par
induction des matériaux composites conducteurs, Thèse
de Doctorat, Université de Nantes, France, 2000.
Meunier, G., Charmoille, M. V., Guérin, C., Labie, P.,
Maréchal, Y. (2010). Homogenization for periodical
Electromagnetic Structure: Which Formulation?, IEEE
Trans. On Magn., 46(8), pp.3409-3412.
Bensaid, S. (2006). Contribution à la Modélisation
Electromagnétique et Thermique des Matériaux
Composites Anisotropes, Thèse de Doctorat, Université
de Nantes, France.
Charmoille, M. V., Meunier, G., Guérin, C., Labie, P.
(2008).Modélisation des Pertes Dans les Bobines
Conductrices par l'utilisation d’une Technique
d’homogénéisation Faisant Intervenir la Perméabilité
magnétique complexe, Numelec’08, Liège, 8-10
Décembre, France.
Waki, H., Igarashi, H., Honna, T. (2005). Estimation of
Effective Permeability of Magnetic Composite
Materials, IEEE Transactions on Magnetics, 41( 5)..
Ren, X., Corcolle, R., Daniel, R. (2016). Losses
Approximation for Soft Composites Based on a
homogenized Equivalent Conductivity, Adv. Electtro.,
5(2),
Achkar G., Pichon, L., Bensetti, M., Daniel, L.(2021)
Homogenization of Metal Grid Reinforced Composites
for Near-Field Low Frequency Magnetic Shielding,
Progress In Electromagnetics Research M, 99, 153–
163.
Gavazzoni, M., Ferro, N., Perotto, S., Foletti, S. (2022).
Multi-physique inverse homogenization for the design
of innovative cellular material: Application to
thermoelastic problems, Math. Comput. 27(1), 15.
Szeliga, D., Gawad, J., Pietrzyk, M. (2004). Parameters
Identification of Material Models Based on the Inverse
Analysis Int. J. Appl. Math. Comput. Sci., 14 (4), 549–
556
Martin, F., Belahcene, Anouar.,Lehikoinen, A., and Rasilo,
Paavo, (2015). Homogenization Techniquefor Axially
Laminated Rotorsof Synchronous Reluctance
Machines, IEEE Trans. On Magn., 51(12),
Feliachi, M. (1991). Modelisation du chauffage par
induction de pieces ferromagnetiques en mouvement.
Rapport, Contrat EDF-IUT-DT2I, IUT-Saint Nazaire,
LRTI, France.
Feliachi, M. (1984). Modelisation numeriquedes
phenomenes electromagnetiques lies à l'enregistrement
magnetique, These de Doctorat d'etat, Grenoble,
France.
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