Application of Evolutionary Strategies for Optimisation of
Parameters during the Modelling of the Magnetic Hysteresis Loop of
the Construction Steel
D. Jackiewicz and R. Szewczyk
Institute of Metrology and Biomedical Engineering, Warsaw University of Technology,
sw. A. Boboli 8, 02-525 Warsaw, Poland
Keywords: Evolutionary Strategies, Magnetic Characteristics Modelling, Non-destructive Testing.
Abstract: This paper concerns the possibility of use evolutionary strategies for optimisation of magnetic
characteristics model's parameters. The Jiles-Atherton extended model was used for modelling the magnetic
hysteresis loop of construction steel St10. In this model k parameters change their value during the
magnetisation process. However, determination of model’s parameters by gradient optimisation was not
succesfull. Only use of evolutionary strategies for optimisation enables achievement of very good
agreement with results of experimental measurements. This agreement was confirmed by high values of the
R
2
determination coefficient.
1 INTRODUCTION
Magnetic method is frequently used in non-
destructive testing of ferromagnetic components
(Blitz, 2007). The most important disadvantage of
this method is its limitation only to the comparative
measurements. For this reason, it is important to
develop the material’s magnetisation model, which
will enable a generalised description of
the characteristics of magnetisation changes and
only accordant to the mechanical state of that
material. This model should be based on physical
principles, and to include the influence of
mechanical stress or fatigue failures.
Among many models of magnetisation of
ferromagnetic materials (Andrei et al., 1998), used
in non-destructive testing of construction steel, the
Jiles-Atherton model, seems to be the most useful
(Jiles and Atherton, 1986). This model not only
mathematically reproduces the magnetic hysteresis
loops, but also takes into account the physical aspect
of the material magnetisation process. For this
reason, it is used in stress assessment of
ferromagnetic construction materials and widely
documented in literature (Chwastek and
Szczyglowski, 2006).
However, Jiles-Atherton model has significant
limitations. For one set of calculated model
parameters, results of the modelling are in
accordance with results of experimental
measurements, but only for one of the amplitudes of
the magnetising field.
Therefore, in these studies the Jiles-Atherton
extended model is used, which allows to avoid this
disadvantage. In all models based on Jiles-Atherton
approach, model parameters are determined on the
base of minimisation of the sum of squared
differences, between the hysteresis loop obtained by
modelling, and the hysteresis loop resulting from
experimental measurements. However, for extended
Jiles-Atherton model of magnetic characteristics, the
determination of model’s parameters on the base of
commonly used gradient optimisation is not
successful, due to the presence of local minima
(Szewczyk R. 1, 2007). In this case, the evolutionary
strategies were proposed.
The paper presents a novel method of
determination of Jiles-Atherton model parameters.
The method can enable technological breakthrough
in non-destructive testing of construction steels, such
as St10.
2 JILES-ATHERTON EXTENDED
MODEL
The Jiles-Atherton model is based on an analysis of
the thermodynamic potentials (Sablik et al., 1988).
297
Jackiewicz D. and Szewczyk R..
Application of Evolutionary Strategies for Optimisation of Parameters during the Modelling of the Magnetic Hysteresis Loop of the Construction Steel.
DOI: 10.5220/0004155602970301
In Proceedings of the 4th International Joint Conference on Computational Intelligence (ECTA-2012), pages 297-301
ISBN: 978-989-8565-33-4
Copyright
c
2012 SCITEPRESS (Science and Technology Publications, Lda.)
From a physical point of view these potentials
characterise the thermodynamic transformations and
are described by the following relations:
MHGA
0
(1)
2
3
STUG
(2)
2
0
2
1
MU
(3)
where: A - Helmholtz free energy, G - Gibbs free
energy, U - materials internal energy, S - materials
free entropy, M - magnetisation, H - magnetising
field, μ
0
=4π10
7
H/m is vacuum permeability, T -
materials temperature, σ - stress mechanics in
material, λ - magnetostrictive strain, a - coefficient
describing the coupling between the domain
(according to the Bloch model) (Liorzou et al.,
2000).
The original Jiles-Atherton model of
magnetisation process utilizes seven parameters: a -
quantifies domain walls density, k - quantifies
average energy required to break pining site, c –
coupling coefficient, α - is interdomain coupling, K
an
– anisotropy energy density, t - participation of
anisotropic phase, M
s
- saturation magnetisation.
The Jiles-Atherton model should include the
anisotropy of the material (Szewczyk 2, 2007).
Anisotropy can be caused by stress arising in the
material. The total magnetisation M is given as the
sum of reversible magnetisation M
rev
and irreversible
magnetisation M
irr
. Reversible magnetisation M
rev
are be calculated from the equation (Jiles and
Atherton, 1986):
)(
irranrev
MMcM
(4)
where M
an
is the anhysteretic magnetisation, which
should be calculated as a weighted sum of the
anisotropic magnetisation M
aniso
and isotropic
magnetisation M
iso
(Jiles et al., 1997).
isoanisoan
MttMM )1(
(5)
where t - weight coefficient, describing participation
of anisotropic phase in the material.
Isotropic magnetisation M
iso
in material is given
by the equation (Jiles and Atherton, 1986):
eff
eff
siso
H
a
a
H
MM cath
(6)
where H
eff
= H + αM - effective magnetising field,
where α represents interdomain coupling.
Anisotropic magnetisation M
aniso
in material is
given by the equation (Ramesh et al., 1996):
0
21
0
21
dsine
dcossine
)E()E(
)E()E(
saniso
MM
(7)
where E(1) and E(2) are energies and are given by
the equation:
2
0
sincos)1(
aM
K
a
H
E
s
an
eff
(8)
2
0
sincos)2(
aM
K
a
H
E
s
an
eff
(9)
where K
an
- anisotropic energy density, ψ - angle
between the easy axis of the material and the
magnetising field direction.
Equation for the anisotropic magnetisation can
be calculated only by numerical methods, because
the primary functions of the integral functions are
not known.
The original model allows to model hysteresis
loops only for one value of the magnetising field.
Extended model can be used to model the hysteresis
loop for different values of the magnetising field.
This is possible, because the model's parameters are
dependent on change of the value of magnetisation.
In the Jiles-Atherton extended model parameter k
is connected with magnetisation M in material and is
given by the equation (Szewczyk, 2009):
)
01
(
1
1
)/-(1
0
)/(
2
2
gg
g
e
s
MMg
e
g
s
MMk
(10)
where: g
0
- defines the value k in demagnetized
state, g
1
- defines the value k of magnetic saturation,
g
2
- factor of the waveform functions k (|M|/M
s
),
where M
s
is saturation magnetisation.
3 METHODOLOGY OF
MEASUREMENTS
Experimental measurements of magnetic
characteristics of steel were made for the ring-
shaped core made of construction steel St10. This
core has the following dimensions: inner diameter
30,9 mm, outer diameter 44,9 mm, height 9,55 mm.
Experimental setup for measurements of B(H)
characteristics of ring-shaped samples is presented
in figure 1. The measuring setup is controlled by PC.
The hysteresis loop was measured using HBP 2.0
IJCCI2012-InternationalJointConferenceonComputationalIntelligence
298
hysteresismeter. Measurement was carried out for
the speed of gain of the magnetising field H of
150 A/m/s. Hysteresis loops were determined for
increasing amplitude of magnetic field intensity in
the range from 20 to 1142 A/m. Also the initial
magnetisation curves were measured. Between the
measurements of magnetic hysteresis loop the core
was demagnetised with sinusoidal waveform of the
exponentially decreasing amplitude. Frequency of
this waveform was 10 Hz, initial amplitude was
1142 A/m, ratio of successive amplitudes was 1,03.
Computer
Fluxmeter
Magnetizing and
demagnetizing signal
generator
U/I
Magnetizing
winding
Measuring
winding
Sample
Interface
Figure 1: Schematic block diagram of the measuring
setup.
4 METHOD OF OPTIMISATION
The optimisation process bases on the minimisation
of target function, which is given by the equation
(Szewczyk, 2009):
n
i
ipomiSAJ
HBHBF
1
2
))()((
(11)
where: n - number of measurement points, H
i
-
magnetic field,
B
J-A-S
(H
i
) – results of the modelling,
B
pom
(H
i
) – results of the experimental
measurements.
In presented case, the best is to use a two-stage
optimisation. In the first step, the evolutionary
strategies (μ+λ) (Schwefel, 1995), combined with
simulated annealing (Schwefel, 1981), (Wilson et
al., 2001), should be used. In the second step, the
gradient optimisation should be used, for the 20 best
results obtained after the first step.
The evolutionary strategies (μ+λ) are the
heuristic optimisation methods, based on adaptation
and evolution. In evolutionary strategies,
the population of vectors, which contain parameters
of the Jiles-Atherton extended model, is subjected to
the three operators. First - mutation operator, which
randomly changes the value of the parameter of the
model. Second - crossover operator, which
exchanges values between the two vectors. And
third - selection operator, which to select the best
value of the target function F.
From the population of
individuals (parents),
population of
individuals (descendants) is created.
During this process, copies of randomly selected μ
individuals are parents. Then, on the base of μ
parents, the population of
descendants is created
randomly, using the operators of mutation and
crossover. Population of descendants is combined
with the parents population, creating a population of
μ+λ individuals. The best
individuals from the
μ+λ population gives the new population for the
next iteration.
During the optimisation process, physical limits
of the Jiles-Atherton model have to be strictly
observed. If physical limits are exceeded (e.g. value
of anisotropic energy density K
an
is lower than zero)
the value of the target function F is significantly
increased. As a result, the optimisation process is
carried out within physical limits.
In the minimisation process a population of 900
vectors was used. The crossover operator of a group
of
= 3 vectors (parents) generated
= 12 vectors
(descendants). Then the descendants vectors were
subjected to the mutation. The distribution of value
changes of the parameters during the process of
mutation was a normal distribution, of which
standard deviation was equal to 3% of the initial
value of the modified parameter. In every iteration,
in accordance with the simulated annealing, the
standard deviation decreased by 5%.
5 RESULTS
The target function F was calculated for 3 hysteresis
loops (measured for different magnetising fields) at
the same time.
The figure 1 below shows the changes in the
value of the target function F during the
optimisation process by using the evolutionary
strategies.
Because the functions F
min
(for best vector of the
population, calculated during the the optimisation
process) and F
max
(for worst vector of the
population, calculated during the the optimisation
process) decreases monotonically in the next
iterations, the optimisation process can be regarded
as convergent.
The next figure 2 shows the results of
experimental measurement of hysteresis loop B(H)
(marked with •) and modelling results (marked with
ApplicationofEvolutionaryStrategiesforOptimisationofParametersduringtheModellingoftheMagneticHysteresis
LoopoftheConstructionSteel
299
—). The obtained parameters of the Jiles-Atherton
model for steel St10 are shown in table 1.
The figure 3 shows the changes in the
permeability for experimental measurement (marked
with •) and modelling results (marked with —).
Results of the modelling utilising evolutionary
strategies are correspond to results of experimental
measurements. The R
2
determination coefficient
exceeds 99% for each of the magnetic hysteresis
loop B(H) in the amplitude of the magnetising field
below 1000 A/m.
Figure 2: The changes of the target function (F
min
for best
vector, F
max
for worst vector, F
mean
for average) during the
optimisation process.
Figure 3: Results of the experiment (•) and results of
modelling (—) quasistatic magnetic hysteresis loop B(H)
for steel St10.
0
400
800
1200
1600
2000
0 200 400 600 800 1000
µa
Hm (A/m)
Figure 4: Results of the experiment (•) and results of
modelling (—) dependence of amplitudal magnetic
permeability µ
a
on amplitude of magnetic field H for steel
ST10.
The obtained parameters g
0
, g
1
and g
2
indicate
that the value of the parameter k decreases rapidly as
a function of the magnetisation M. The obtained
parameters are consistent with the results of physical
measurements for steel. Particularly, the saturation
magnetisation M
s
amount 1 788 100 A/m is close to
typical values for low carbon steel.
Table 1: Jiles-Atherton model parameters after
optimisation.
Parameter Value
A 819 A/m
g
0
404 A/m
g
1
216 A/m
g
2
13,9
c 0,491
M
s
1 788 100 A/m
α 0,00131
6 CONCLUSIONS
The application of evolutionary strategies to
optimisation process decreased sensitivity to local
minima. This optimisation not only allows to
calculate the model's parameters, but also allows to
obtain very good results of the modelling. These
results are correspond to results of experimental
measurements. This was confirmed by the high
values of the R
2
determination coefficient.
For this reason Jiles-Atherton extended model
may be suitable for determination of stresses in St10
construction steel during non-destructive testing.
This model enables modelling of the magnetisation
characteristics of St10 steel in a wide range of
amplitude of the magnetising field.
The presented application of evolutionary
strategies will be particularly useful in developing
methods for assessing the state of stress in the
material, by measuring the magnetic hysteresis loop.
Calculations were made in the Interdisciplinary
Centre for Mathematical and Computational
Modelling of Warsaw University, grant G36-10.
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LoopoftheConstructionSteel
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