Побудова RBF-метамоделей структур збудження рухомого концентричного вихрострумового перетворювача
Розроблено програмне забезпечення для розрахунку розподілу густини вихрових струмів в зоні контролю накладного вихрострумового перетворювача із врахуванням ефекту швидкості за «точними» електродинамічними математичними моделями. Розроблено програмне забезпечення для формування точок плану експеримен...
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| Published in: | Електротехніка і електромеханіка |
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| Date: | 2019 |
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| Language: | Ukrainian |
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Інститут технічних проблем магнетизму НАН України
2019
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| Online Access: | https://nasplib.isofts.kiev.ua/handle/123456789/159059 |
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| Cite this: | Побудова RBF-метамоделей структур збудження рухомого концентричного вихрострумового перетворювача / В.Я. Гальченко, Р.В. Трембовецька, В.В. Тичков // Електротехніка і електромеханіка. — 2019. — № 2. — С. 28-38. — Бібліогр.: 22 назв. — укр., англ. |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine| _version_ | 1860008465832869888 |
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| author | Гальченко, В.Я. Трембовецька, Р.В. Тичков, В.В. |
| author_facet | Гальченко, В.Я. Трембовецька, Р.В. Тичков, В.В. |
| citation_txt | Побудова RBF-метамоделей структур збудження рухомого концентричного вихрострумового перетворювача / В.Я. Гальченко, Р.В. Трембовецька, В.В. Тичков // Електротехніка і електромеханіка. — 2019. — № 2. — С. 28-38. — Бібліогр.: 22 назв. — укр., англ. |
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| container_title | Електротехніка і електромеханіка |
| description | Розроблено програмне забезпечення для розрахунку розподілу густини вихрових струмів в зоні контролю накладного вихрострумового перетворювача із врахуванням ефекту швидкості за «точними» електродинамічними математичними моделями. Розроблено програмне забезпечення для формування точок плану експерименту із використанням ЛПτ–послідовностей, що дозволило здійснювати відбір планів з рівномірним заповненням точками гіперпростору пошуку. Для нерухомого та рухомого накладних вихрострумових перетворювачів створено нейромережеві метамоделі на радіально-базисній функції Гауса. Оцінено адекватність та інформативність отриманих метамоделей накладних вихрострумових перетворювачів. Результати дослідження можуть бути використані при синтезі рухомих накладних вихрострумових перетворювачів із апріорі заданим розподілом густини вихрових струмів в зоні контролю.
Goal. Creation of surface circular concentric eddy current probe RBF-metamodels, which can be used to calculate eddy currents density distribution in the control zone and suitable for use in optimal synthesis problems. Method. To develop an approximation model, a mathematical apparatus for artificial neural networks, namely, RBF–networks, has been used, whose accuracy has been increased with the help of the neural networks committee. Correction of errors in the committee was reduced by applying the bagging procedure. During the network training the regularization technique is used, which avoids re-learning the neural network. The computer experiment plan was performed using the Sobol LPt–sequences. The obtained multivariable regression model quality evaluation was performed by checking the response surface reproducibility correctness in the entire region of variables variation. Results. The modelling of eddy currents density distribution calculations on exact electrodynamic mathematical models in the experimental plan points are carried out.
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Theoretical Electrical Engineering and Electrophysics
28 ISSN 2074-272X. Electrical Engineering & Electromechanics. 2019. no.2
© V.Ya. Halchenko, R.V. Trembovetska, V.V. Tychkov
UDC 620.179.147:519.853.6 doi: 10.20998/2074-272X.2019.2.05
V.Ya. Halchenko, R.V. Trembovetska, V.V. Tychkov
DEVELOPMENT OF EXCITATION STRUCTURE RBF-METAMODELS OF MOVING
CONCENTRIC EDDY CURRENT PROBE
Introduction. The work is devoted to metamodels creation of surface circular concentric eddy current probe. Formulation of
the problem. In the problem of surface circular concentric eddy current probe synthesis in the general formulation, apriori
given desired eddy currents density distribution in the control zone was used. The realization of the optimal synthesis problem
involves a multiple solution to the analysis problem for each current structure of numerical calculations excitation, which are
very costly in terms of computational and time costs, which makes it impossible to solve the synthesis problem in the classical
formulation. By solving the critical resource intensiveness problem, there is the surrogate optimization technology using of
that uses the surface circular concentric eddy current probe metamodel, which is much simpler in realization and is an
approximation of the exact electrodynamic model. Goal. Creation of surface circular concentric eddy current probe RBF-
metamodels, which can be used to calculate eddy currents density distribution in the control zone and suitable for use in
optimal synthesis problems. Method. To develop an approximation model, a mathematical apparatus for artificial neural
networks, namely, RBF–networks, has been used, whose accuracy has been increased with the help of the neural networks
committee. Correction of errors in the committee was reduced by applying the bagging procedure. During the network training
the regularization technique is used, which avoids re-learning the neural network. The computer experiment plan was
performed using the Sobol LP–sequences. The obtained multivariable regression model quality evaluation was performed by
checking the response surface reproducibility correctness in the entire region of variables variation. Results. The modelling of
eddy currents density distribution calculations on exact electrodynamic mathematical models in the experimental plan points
are carried out. For the immovable and moving surface circular concentric eddy current probe, RBF–metamodels were
constructed with varying spatial coordinates and radius. Scientific novelty. Software was developed for eddy currents density
distribution calculation in the surface circular concentric eddy current probe control zone taking into account the speed effect
on exact electrodynamic mathematical models and for forming experiment plan points using the Sobol LP–sequences. The
geometric surface circular concentric eddy current probe excitation structures models with homogeneous sensitivity for their
optimal synthesis taking into account the speed effect are proposed. Improved computing technology for constructing
metamodels. The RBF-metamodels of the surface circular concentric eddy current probe are built and based on the speed
effect. Practical significance. The work results can be used in the surface circular concentric eddy current probe synthesis
with an apriori given eddy currents density distribution in the control zone. References 22, tables 6, figures 8.
Key words: surface eddy current probe, eddy currents density distribution, excitation structure, mathematical model, optimal
synthesis, computer experiment plan, LPτ–sequence, RBF–metamodel, neural networks committee.
Розроблено програмне забезпечення для розрахунку розподілу густини вихрових струмів в зоні контролю накладного
вихрострумового перетворювача із врахуванням ефекту швидкості за «точними» електродинамічними
математичними моделями. Розроблено програмне забезпечення для формування точок плану експерименту із
використанням ЛПτ–послідовностей, що дозволило здійснювати відбір планів з рівномірним заповненням точками
гіперпростору пошуку. Для нерухомого та рухомого накладних вихрострумових перетворювачів створено
нейромережеві метамоделі на радіально-базисній функції Гауса. Оцінено адекватність та інформативність
отриманих метамоделей накладних вихрострумових перетворювачів. Результати дослідження можуть бути
використані при синтезі рухомих накладних вихрострумових перетворювачів із апріорі заданим розподілом густини
вихрових струмів в зоні контролю. Бібл. 22, табл. 6, рис. 8.
Ключові слова: накладний вихрострумовий перетворювач, розподіл густини вихрових струмів, структура збудження,
математична модель, оптимальний синтез, комп’ютерний план експерименту, ЛПτ–послідовність, RBF–метамодель,
комітет нейронних мереж.
Разработано программное обеспечение для расчета распределения плотности вихревых токов в зоне контроля
накладного вихретокового преобразователя с учетом эффекта скорости по «точным» электродинамическим
математическим моделям. Разработано программное обеспечение для формирования точек плана эксперимента с
использованием ЛПτ-последовательностей, что позволило осуществлять отбор планов с равномерным заполнением
точками гиперпространства поиска. Для неподвижного и движущегося накладных вихретоковых преобразователей
созданы нейросетевые метамодели на радиально-базисной функции Гаусса. Оценены адекватность и
информативность полученных метамоделей накладных вихретоковых преобразователей. Результаты исследования
могут быть использованы при синтезе движущихся накладных вихретоковых преобразователей с априори заданным
распределением плотности вихревых токов в зоне контроля. Библ. 22, табл. 6, рис. 8.
Ключевые слова: накладной вихретоковый преобразователь, распределение плотности вихревых токов, структура
возбуждения, математическая модель, оптимальный синтез, компьютерный план эксперимента, ЛПτ–
последовательность, RBF–метамодель, комитет нейронных сетей.
Introduction. The eddy current control method and
the devices on its basis are widely used to determine the
parameters of various objects of control (OC): imperfect
material defects, control of the dimensions of the OC and
vibration parameters, quality control of thermal and
chemical-thermal processing of parts, the state of surface
layers after machining, the presence of residual
mechanical stresses, the reconstruction of the distribution
of electrical conductivity and the magnetic permeability
within the objects, and others.
ISSN 2074-272X. Electrical Engineering & Electromechanics. 2019. no.2 29
Along with the significant advantages of the eddy
current control method, there are some disadvantages, for
example, the ability to control only the conductive
objects, the relatively small depth of the eddy currents
penetration, the heterogeneous sensitivity of the probes of
classical design.
The typical surf ace eddy current probes (SECP) are
characterized by a characteristic distribution of the eddy
currents density (ECD) in the OC, which depends on the
geometrical, electromagnetic parameters and the relative
position of its exciting coil relative to the controlled
surface. In SECP, the ECD is maximal in the surface layer
of the conductive object and decreases at the removal
from the excitation coil windings along the surface (Fig.
1,a) and in deeper layers according to the exponential
law. That is, in such a heterogeneous distribution of the
ECD (Fig. 1,a), the relative position of the SECP with
respect to the OC significantly influences the sensitivity
of the method. In the defectoscopy, for example, in the
case where a surface fracture of a finite length is located
under the geometric center of the excitation coil, the
sensitivity will be close to zero (Fig. 1,c), the minimum
sensitivity is observed for the case of the location of the
surface crack in parallel to the vortex currents (Fig. 1,d)
and the maximum one – if the crack is perpendicular to
the direction of eddy currents (Fig. 1,e).
a b
c d e
Fig. 1. Features of the SECP: distribution of ECD, inherent in
classical designs of probes (a); uniform distribution of ECD (b);
sensitivity close to zero (c); minimum sensitivity (d); maximum
sensitivity (e)
In order to reduce the effect of the dependence of the
sensitivity of the probes to the defect, regardless of its
location in the control zone, it is desirable to have the
distribution of the ECD in it homogeneous (Fig. 1,b). The
problem arises of the creation of the SECP with
homogeneous sensitivity, and, consequently, a
homogeneous distribution of the ECD in the control zone
of the object. This problem can be solved within the
framework of optimal synthesis as a result of determining
the rational structure of the excitation system of the SECP
with the corresponding parameters that provide the
necessary distribution of the ECD. It is also important to
achieve homogeneous sensitivity of the SECP which are
not only stationary relative to the OC or move at a low
speed, when the effect of the transfer currents can be
neglected, but also for mobile probes.
Literature review. In [1], the problems of linear
synthesis of a fixed SECP are considered, where the
dependence of the output signal on the gap or the specific
electrical conductivity of the investigated object is taken
as the initial data. In order to solve an incorrect synthesis
problem, the method of regularization is applied, i.e.
certain restrictions were introduced on the desired
functions. In [2] the questions of linear synthesis of a
fixed SECP are considered. The plane of the control zone
is parallel to the working face of the probe, where a given
magnetic field structure was created. In [3] an algorithm
for nonlinear synthesis of magnetic fields of excitation of
a fixed SECP with a predefined configuration is
presented. The solution of the problem is obtained by
minimizing the average stepped approximation of the
minimax functional, which provides the minimum
deviation of the desired distribution of the
electromagnetic field from the given one. In the paper [4]
a structural-parametric synthesis of the excitation system
of a fixed SECP is performed. The searched parameters
are the number of sections, their radii and coordinates.
The search for an optimal solution is performed using a
genetic algorithm. The optimum values of the parameters
of the coil sections, as well as the most constructively
simple excitation system that provide the given
distribution of the probe field in space, are obtained.
Significant improvement of the quality of the generated
field of the synthesized magnetic system, the essential
simplification of the structure by the number of sections
and the reduction of the length of the system, as well as
reduced number of turns in sections at the same values of
currents have been achieved. In [5] a methodology for
optimizing the design of a coil of an eddy current probe
(ECP) is proposed, which allows maximum
approximation to the ideal excitation field in the multi-
purpose statement of the problem. The research presents a
method for optimizing the design of the excitation system
to obtain a tangential and uniform distribution of
multilinear eddy currents. In [6], a method for optimizing
the parameters of an excitation coil was developed by
solving a multi-parameter multi-purpose optimization
problem. The imitation modeling of the behavior of an
infinite coil with a tangential uniform field on the surface
of the OC is carried out. As a result, a non-uniform
multilayer design of the ECP coil has been obtained,
which provides a uniform field of excitation. In [7], a
genetic algorithm for solving the optimization problem of
selecting the parameters of the ECP excitation field was
used. For the excitation coil of the probe, the optimal
values of frequency and size are obtained.
Thus, previously published studies devoted to the
questions of synthesis of ECP [1-7] with a given
configuration of the probe field in the control zone,
considered the fixed OC and did not take into account the
reaction of the conductive medium. It was enough to
create a system of excitation of the SECP with a uniform
distribution of the electromagnetic field, which was
guaranteed to ensure a uniform distribution of the ECD in
the OC. The account of the speed effect involves the
synthesis of a homogeneous distribution of the ECD in
the environment of the OC, which is a fundamental
difference from the results of previous studies and can not
be carried out by the means previously proposed.
The goal and objectives of the study. The object of
the study is the processes of eddy current control of the
30 ISSN 2074-272X. Electrical Engineering & Electromechanics. 2019. no.2
quality of objects. The subject of research is a mobile
circular SECP with a homogeneous distribution of the
ECD in the control zone. The goal of the work is to create
a RBF-metamodel for a mobile concentric circular SECP
that can be used to calculate the distribution of the ECD
in the control area and be suitable for use in optimal
synthesis problems.
Mathematical model of the mobile SECP. As the
initial input data for designing in the problem of synthesis
of the SECP in the general formulation a priori given the
desired distribution of the ECD Jreference in the control
zone is used. For the purpose of some simplification of
the problem, we restrict ourselves first by obtaining this
distribution of the ECD on the surface of the OC,
specifying certain values of the ECD in the set of N
control points Q.
The structure of the excitation of the SECP consists
of a system of M coils of varying height of position z0k,
k = 1…M of the corresponding coil relative to the OC and
radii rk. The circuit of their connection is counter or
coherent, and the supply current I can be the same or
different for each of the coils. As an option of the
structure of excitation, Fig. 2,a shows a system of
concentric coils with different radii which is located at the
same height z0 over OC. Fig. 2,b shows the excitation
system of coils of different radii, which are located at the
same height, with the centers of the coils shifted, that is,
the coils are not concentric. Fig. 2,c shows a system of
coils with different radii, which are located at different
heights and with the displacement of the centers of one
relative to the other.
In [8-13] a mathematical model of a single
excitation coil of the SECP was developed, which allows
to determine the distribution of the ECD in the OC, which
we agree to call «accurate». For this, the following
assumptions were made: the medium is linear,
homogeneous, isotropic; the OC is mobile, conductive, of
infinite width and length and has a finite thickness d; the
coil is excited by alternating current I of frequency ; the
conductor of the coil is represented as infinitely thin; the
electrical conductivity , the relative magnetic
permeability r and the speed of the probe 0,, yx
are constant. In accordance with this mathematical model,
three calculated areas are considered in which the
complex values of magnetic flux density are determined:
in the area 0 < z < z0
,0rot,0
,
d
4
,rot
,
0
1
rr
liii
ri
BB
R
lJ
AAB
BBB
(1)
where iB
describes the own magnetic field of a turn of
length l and current density J
, and rB
is the magnetic
field of eddy currents induced in the medium of the OC;
in the area –d < z < 0
;0div
,0
2
20
22
02
B
Bj
y
B
x
B
B yx
(2)
in the area z < –d
.0rot,0 33 BB
(3)
GCS
LCS
GCS
LCSk
a b
GCS
LCSk
GCS
LCSk
c d
Fig. 2. Geometric models of SECP excitation structures:
a system of concentric coils, where the coils are located at the
same height z0 (a); a system of coils where the coils are located
at one height z0, the centers of the coils shifted (b); a system of
coils, where the coils are located at different heights, the centers
of the coils shifted (c); general location of global and local
coordinate systems of coils (d)
The solution of the system of equations (1)-(3) in
conjunction with the conditions of continuity of the
tangential component of the magnetic field strength and
the normal component of magnetic flux density on the
boundaries of the media interfaces z = 0 and z = –d,
allows to obtain the distribution of components of the
magnetic flux density in the medium of the OC:
;,
1
1
18
22
0
22
22
00
0
2
0
22
0
2
ddeSe
ee
eee
e
I
B
yxjz
z
d
z
d
d
d
r
x
(4)
;,
1
1
1
1
8
22
0
22
22
00
0
2
0
22
0
2
ddeSe
ee
eee
e
I
B
yxjz
z
d
z
d
d
d
r
y
(5)
ISSN 2074-272X. Electrical Engineering & Electromechanics. 2019. no.2 31
,,
1
1
18
22
0
22
22
00
0
2
0
2
22
2
0
2
ddeSe
ee
eee
e
I
jB
yxjz
z
d
z
d
d
d
r
z
(6)
where B2x, B2y, B2z are the components of the magnetic
flux density by spatial coordinates; S(,) is the function
of the shape of the coil,
22
1
22
2
,
rJ
r
jS ;
r
yxr
j
j
0
0
22
;
d
rr
d
r
e
e
2
2
22
2
22
22222
0
1
;
d
rr
d
r
e
e
2
2
22
2
22
22
0
22
4 ,
where x, y are the components of the velocity of the
circular SECP relative to the OC; d is the OC thickness;
, are the variables of integration.
These expressions are adequate in the local
coordinate system (LCS), where the origin of the
coordinates coincides with the center of the turn. Multiple
non-proper integrals of the first kind, which they contain,
are calculated numerically by the truncation method.
Expressions (4) – (6) allow to obtain an «exact»
mathematical model of the distribution of the ECD in the
OC for the circular SECP. The components of the ECD
by spatial coordinates x, y, z are respectively determined
by the formulas:
.
1
;
1
;
1
22
0
22
0
22
0
y
B
x
B
J
x
B
z
B
J
z
B
y
B
J
xy
r
z
zx
r
y
yz
r
x
(7)
The coordinates of the control points Qi, i = 1…N
are specified in the global coordinate system (GCS), then
they are recalculated to the k-th LCS. In the LCS, the
ECD calculation is performed at each control point, and
then the resulting values are obtained as a superposition at
each point i = 1…N from all M coils (Fig. 2,d).
In the general case, the objective function for the
problem of optimal synthesis in the classical formulation
has the form:
min
2
1 1
N
i
M
k
referenceiktarget JJF , (8)
where Jreference is the desired value of the eddy current
currents at the control point; Jik is the density of the eddy
current in the control point of the OC with the number i,
created by the k-th coil of the excitation system of the
SECP; N is the number of control points in the area; M is
the number of coils in the system of excitation of the
circular SECP. As a result of the synthesis, the spatial
configuration and geometric parameters of the structure of
excitation of the SECP are obtained, which collectively
provide the implementation of the required
characteristics. The realization of the problem of optimal
synthesis involves a multiple solution to the problem of
analysis for each current structure of excitation by
numerical calculations. In [14, 15] it is established that
calculations on these expressions are very costly in terms
of computational and time costs, which makes it
impossible to solve the problem of synthesis.
One of the solutions of the problem of critical
resource intensity is the use of surrogate optimization
technologies [16, 17] and stochastic meta-heuristic
optimization [18, 19]. That is, for the purpose of
formulating the goal function within the framework of the
optimal synthesis problem, a SECP metamodel can be
used, which is much simpler in implementation and less
resource-intensive [14, 15] and is an approximation of the
«exact» electrodynamic model.
To achieve this goal, the following tasks were
solved: creation of software for calculating the
distribution of the ECD in the control zone of the SECP
taking into account the effect of speed by «exact»
electrodynamic mathematical models; creation of
software for forming points of an experiment plan using
the Sobol LP–sequences to select the most perfect
experimental plans individually for the approximated
surfaces of the response; to create geometrical models of
excitation structures of circular SECPs with homogeneous
sensitivity for their optimal synthesis taking into account
the effect of speed; to improve the computational method
of constructing metamodels of objects that are
characterized by considerable computational resource
intensities in the simulation of physical processes; to
create RBF-metamodels of the concentric circular SECP
(fixed one and taking into account the effect of speed).
To calculate the «exact» electrodynamic
mathematical models (4) – (7), software was developed in
the MathCAD 15 package.
The calculation of the distribution of the ECD for
the turn of the excitation coil of the circular shape in order
32 ISSN 2074-272X. Electrical Engineering & Electromechanics. 2019. no.2
to visualize it was performed for the case of variation of
the two parameters J = f(x, y) (Fig. 2,a) and the other
fixed ones by the formulas (4) – (7) of the «exact»
mathematical models with the following input data: for
the case of a fixed SECP –x = 0…30 mm, y = 0…30 mm,
r = 5 mm; for the case of a moving SECP – = (40; 0; 0) m/s;
x = –30…30 mm, y = 0…30 mm, r = 5, 10, 15 mm;
thickness of the conductive material d = 10 mm; height of
the placement turn of the coil over OC z0 = 3 mm;
frequency f = 100 Hz; electrophysical parameters of the
material = 3.745107 S/m, r = 1, current I = 1 A.
Fig. 3,a-h show the 3D distribution of the ECD and
the level lines for some excitation coil turns radiuses.
For example, Fig. 3,a,b show the simulation results for a
fixed SECP, and Fig. 3,c-h present the results of
calculating the distribution of the ECD taking into
account the effect of speed.
The computational complexity of a one-time
calculation of the ECD distribution by an «exact»
mathematical model with variations of only two spatial
coordinates J = f(x, y) at r = const is sufficiently large and
ranges from 5 to 8 hours.
Main points and development of metamodels. In
[14, 17], the authors proposed a general computational
method for constructing metamodels using modern
achievements in the field of artificial intelligence and the
theory of experiment planning. A number of examples have
proved the effectiveness of its use. Neural networks have
been used to construct a substitute model, which provide
the ability to quickly and easily calculate the output of the
network, even with a sufficiently large number of neurons
in hidden layers. In [15, 16] some features of the
application of this technology in relation to the problems of
the synthesis of the SECP are considered. Below, attention
is focused on the details of the construction of metamodels
of the circular SECP with certain structures of the
excitation system, namely, the variant illustrated in Fig.
2,a, that is, approximation ).,,(=ˆ ryxfJ
In contrast to the previous studies of authors,
increasing the accuracy of the neural network solution of
approximation problems was achieved with the help of
the neural networks committee [20]. The committee
makes the final decision using separate solutions of
several neural networks, that is, the method of bagging.
Thus, the bagging committee is used to reduce the
correlation of neural network errors. This methodology
involves training neural networks on bootstrap-samples,
which represent a set of elements with repetitions from a
previous training set of data. Bagging provides the most
effectiveness in the case of a fairly large number of
input training data. Thus, for the construction of an
approximation model, the mathematical apparatus of
artificial neural networks was used namely the bagging
committee of RBF-networks with the Gaussian
activation core function.
The creation of a metamodel involves the
construction of a computer experiment plan, at which
points the distribution of the ECD is calculated by the
«exact» mathematical model, the construction of an
approximation model and validation of the model.
The experiment plan is implemented with the help of
uniform computer filling by the points of the 3D search
space, namely, using the Sobol LP–sequences [21]. The
points of the experiment plan are generated using LP–
sequences (1, 2, 4) and their total number is: for the
case of a fixed SECP – N = 2048 and N = 3315 – for a
moving SECP. For each section of the surface by the
radius there is approximately Ncut = 146 and Ncut = 255
points, respectively.
The obtained ECD values at the points of the plan
are used as the initial data for the implementation of the
next stage – construction of the metamodel. The number
of points of calculation essentially depends on the
symmetry of the distribution of the ECD relative to the
coordinate axes (Fig. 3), so for the fixed SECP the points
of the plan are given in the I quadrant, and for the moving
one – in the I and II quadrants.
Fig. 4 is presented in order to provide a visual
representation of the experiment plan. Fig. 4,a shows the
location of the points of LP–sequences for their small
number N = 250 in the 3D space k = 3, and Fig. 4,b shows
the location of the indicated points in subspaces of
smaller dimension k = 2 for the combined factors (1, 2,
4). Fig. 4,c-f present a 3D distribution of points for fixed
radii 1, 5, 10 and 15 mm when generating them according
to this plan.
For realization of the second stage the heuristic
method of constructing metamodels with the help of
neural networks is used. The construction of the RBF-
metamodels is accomplished with the help of an
automatic strategy and multiple sub-samples.
In the automatic mode, the samples are formed by
random division in the ratio: 70 % – training, 15 % –
control, 15 % – test, where the test population was used
for cross-checking.
In the second series of constructing metamodels,
using the method of multiple sub-samples, a bagging
algorithm was used in which 20 repetitive samples were
generated based on the training set and training was
performed based on these bootstrap-samples of the 20
neural networks. Elements not included in the next
sample are used as a test set for the corresponding neural
network. For neural networks, the problem of «retraining»
is inherent, which is associated with the number of
neurons in the hidden layer. During the training of the
network the technique of regularization is used, which
avoids retraining the neural network. Unsuccessful
versions of networks with productivity less than 90 %
were filtered off. All other networks were evaluated by
subjective analysis of histograms of residues, scattering
diagrams and numerical values of indicators:
determination coefficient R2 (performance), the ratio of
standard deviations of forecast error and training data
S.D.ratio, the average relative value of the model error
MAPE, the residual median square МSR.
ISSN 2074-272X. Electrical Engineering & Electromechanics. 2019. no.2 33
a b
c d
e f
g h
Fig. 3. Exact function of the ECD distribution on the surface of the OC in the control zone 3030 mm: fixed SECP, excitation coil
r = 5 mm (a, b); movable SECP, excitation coil r = 5 mm (c, d); movable SECP, excitation coil r = 10 mm (e, f); movable SECP,
excitation coil r = 15 mm (g, h)
34 ISSN 2074-272X. Electrical Engineering & Electromechanics. 2019. no.2
a b
c d e f
Fig. 4. Locations of the points of LPτ-sequences (1, 2, 4) in the 3D factor space: for r = 1 ... 15 mm,
number of points N = 250 (a); matrix representation of sequences (1, 2, 4) in 2D projections (b);
for a radius of an excitation coil of 1 mm (c); for a radius of 5 mm (d); for a radius of 10 mm (e); for a radius of 15 mm (f)
To construct a metamodel of a fixed SECP with
variation of three parameters within x = 0 ... 30 mm;
y = 0 ... 30 mm; r = 1 ... 15 mm almost 320 RBF-neuron
networks are created for the plan N = 2048 with the
number of hidden neurons from 280 to 350, of which the
best (Table 1) are selected for the indicated indicators.
Networks with a productivity of more than 0.9 were used
together, organizing a networks committee.
To improve accuracy, as a rule to make a decision
the average value of the networks included in the
committee is used. For the neural networks committee, in
Fig. 5,b,d,f lines of the level of the surface of the response
are presented in the previously determined ranges of
variations of variables reproduced at 2048 points of the
training sample. Each section of the surface in the radius
accounts for about 145 points. Table 2 shows the results
of the approximation of the ECD distribution by the
created committee for radii 5, 10, 15 mm.
To construct the metamodel of the SECP taking into
account of the effect of speed = (40; 0; 0) m/s and
variation of three parameters within x = –30…30 mm;
y = 0…24 mm; r = 2…15 mm, almost 95 RBF-neuron
networks were created for the plan N = 3315 with the
number of hidden neurons from 200 to 700, of which the
best (Table 3, 4) were selected for the indicated
indicators.
For the neural networks committee, Fig. 6,b,d,f show
lines of the level of the surface of the response,
reproduced at 3315 points of the training sample. In each
section of the surface in a radius in this plan is 255 points.
Table 1
The best RBF-metamodels for a fixed SECP
No. Neural network R2 for training, control and test samples S.D.ratio MAPE, % MSR
1 RBF-3-282-1(156) 0.9949; 0.9946; 0.993 0.086 22.6 0.00057
2 RBF-3-293-1(218) 0.993; 0.994; 0.994 0.0904 27.9 0.000614
3 RBF-3-293-1(219) 0.994; 0.992; 0.989 0.0939 28.6 0.000674
4 RBF-3-300-1(254) 0.9949; 0.993; 0.989 0.0891 26.8 0.000631
5 RBF-3-322-1(284) 0.995; 0.992; 0.988 0.09 22.9 0.000613
6 RBF-3-343-1(307) 0.996; 0.993; 0.996 0.0739 22.1 0.000424
ISSN 2074-272X. Electrical Engineering & Electromechanics. 2019. no.2 35
Table 2
Results of approximation of the ECD distribution by the networks committee for a fixed SECP
Radius, mm S.D.ratio MAPE, % MSR
5 0.164 13.08 0.000506
10 0.061 5.89 0.000316
15 0.083 6.43 0.000947
a b c
d e f
Fig. 5. Lines of the level of the surface of the response of a fixed SECP: the plan of experiment N = 145, applied on the lines of the
level of the «exact» model, for the sections of the surface of radii r = 5, 10, 15 mm respectively (a, c, e); the surface of the response,
reproduced at the points of the training sample using the networks committee (b, d, f)
Table 3
The best RBF-metamodels for a movable SECP
No. Neural network R2 for training, control and test samples S.D.ratio MAPE, % MSR
1 RBF-3-610-1(2) 0.944; 0.933; 0.926 0.278 46 0.00458
2 RBF-3-620-1(8) 0.958; 0.942; 0.935 0.263 41.2 0.00355
3 RBF-3-627-1(15) 0.96; 0.941; 0.918 0.272 44.1 0.00367
4 RBF-3-635-1(28) 0.96; 0.947; 0.933 0.265 37.74 0.00345
5 RBF-3-635-1(29) 0.96; 0.949; 0.924 0.261 38.3 0.00349
6 RBF-3-665-1(31) 0.958 0.95; 0.938 0.261 39.2 0.00347
7 RBF-3-665-1(34) 0.96; 0.948; 0.937 0.262 32.9 0.00341
Table 4
Results of approximation of the ECD distribution by the networks committee for a movable SECP
Radius, mm S.D.ratio MAPE, % MSR
5 0.242 31 0.001151
10 0.293 23 0.002382
15 0.381 21.7 0.008434
36 ISSN 2074-272X. Electrical Engineering & Electromechanics. 2019. no.2
a b c
d e f
Fig. 6. Lines of the level of the surface of the response of a movable SECP: the plan of experiment N = 255, applied on the lines of
the level of the «exact» model, for the sections of the surface of radii r = 5, 10, 15 mm respectively (a, c, e); the surface of the
response, reproduced at the points of the training sample by the networks committee (b, d, f)
Validation and verification of SECP metamodels.
One of the criteria for the quality of a multivariate
regression model is to verify the correctness of the
reproducibility of the response surface using the resulting
mathematical model throughout the modeling area. Fig. 7
shows the results of the reproduction of the response
surface for a fixed SECP obtained with the help of the
neural networks committee, executed in the entire range
of variation of variables at a considerably increased
number of points 7154. In this case, the sections of the
surface with radii 5, 10, 15 mm accounted for 511 points.
At the stage of reproduction of the response surface,
the adequacy of the obtained metamodel is evaluated
according to the indicators: the sum of the squares
corresponding to the regression, the remnants, the total;
middle squares; dispersion of reproducibility, adequacy,
general; standard error of reproducibility estimation,
estimation of adequacy, overall; determination factor;
ratio of standard deviations; average relative value of
model error (or average error of approximation) [22]. The
estimation of these indicators is summarized in Table 5.
Fig. 8 shows the result of the reproduction of the
response surface received by the neural networks
committee for a moving SECP, executed throughout the
range of variation of variables at 6643 points. On the
sections of the surface of radii 5, 10, 15 mm in this
example there are 511 points.
For the created neural networks committee, the
indicators characterizing the adequacy and
informativeness of the metamodel are estimated, the
results of which are summarized in Table 6.
Table 5
Verification of the adequacy and informativeness of the metamodel of a fixed SECP
Dispersion components (N = 7154): Sum of squares Middle square Dispersion
Standard
estimation error
regression SSD = 369.265 MSD = 123.088 2
D = 0.051537 SD = 0.227018
remnants SSR = 1.91 MSR = 0.000266 2
R = 0.000266 SR = 0.016325
general SST = 374.088 MST = 0.052221 2
T = 0.052210 ST = 0.228496
Fisher criterion valuetable
;;
alexperiment
RDRD
FF ;νν alexperiment
71503;F = 193.74; valuetable
7150;3;05,0F = 2.6079
determination factor R2 0.9945
r = 5 mm 16.56 %
r = 10 mm 5.92 %
average error of approximation,
MAPE, %
r = 15 mm 5.41 %
ratio of standard deviations S.D.ratio 0.071445
ISSN 2074-272X. Electrical Engineering & Electromechanics. 2019. no.2 37
a b c
Fig. 7. Reproduction of the response surface with the help of the neural networks committee for a fixed SECP.
Level lines reproduced at N = 511 points for sections of the surface with radii 5, 10, 15 mm respectively (a, b, c)
a b c
Fig. 8. Reproduction of the response surface with the help of the neural networks committee for a movable SECP.
Level lines reproduced at N = 511 points for sections of the surface with radii 5, 10, 15 mm respectively (a, b, c)
Table 6
Verification of the adequacy and informativeness of the metamodel of a movable SECP
Dispersion components (N = 6643) Sum of squares Middle square Dispersion
Standard
estimation error
regression SSD = 244.1923 MSD = 81.397 2
D = 0.036111 SD = 0.190030
remnants SSR = 27.5733 MSR = 0.004077 2
R = 0.004077 SR = 0.06385
general SST = 278.9221 MST = 0.041248 2
T = 0.041248 ST = 0.203097
Fisher criterion valuetable
;;
alexperiment
; RDRD
FF alexperiment
66393;F = 8.857; valuetable
6639;3;05,0F = 2.6079
determination factor R2 0.901353
r = 5 mm 40.38 %
r = 10 mm 23.54 %
average error of approximation,
MAPE, %
r = 15 mm 24.79 %
ratio of standard deviations S.D.ratio 0.314381
The results of the study can be used in the synthesis
of mobile SECP with a priori set ECD distribution in the
control zone.
Conclusions.
1. For the first time, RBF-metamodels of a concentric
circular SECP (both fixed one and taking into account the
speed effect) are created.
2. Based on modern computer methods of planning the
experiment, artificial intelligence and data analysis, the
computational technique of constructing metamodels
characterized by a lower computational resource intensity
during simulation is improved.
3. For the first time, geometric models of excitation
structures of circular SECPs with uniformity of sensitivity
for their optimal synthesis taking into account the effect
of speed are proposed.
4. The task of creating software for calculating the ECD
distribution in the control zone of the SECP taking into
account the effect of speed by «exact» electrodynamic
mathematical models is solved. The task of creating
software for forming points of an experiment plan using the
Sobol LPτ–sequences is solved, which made it possible to
select the most perfect experiment plans individually for
the approximated surfaces of the response.
38 ISSN 2074-272X. Electrical Engineering & Electromechanics. 2019. no.2
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Received 08.10.2018
V.Ya. Halchenko1, Doctor of Technical Science, Professor,
R.V. Trembovetska1, Candidate of Technical Science, Associate
Professor,
V.V. Tychkov1, Candidate of Technical Science, Associate
Professor,
1 Cherkasy State Technological University,
460, Shevchenko Blvd., Cherkasy, 18006, Ukraine,
е-mail: halchvl@gmail.com
How to cite this article:
Halchenko V.Ya., Trembovetska R.V., Tychkov V.V. Development of excitation structure RBF-metamodels of moving
concentric eddy current probe. Electrical engineering & electromechanics, 2019, no.2, pp. 28-38. doi: 10.20998/2074-
272X.2019.2.05.
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| id | nasplib_isofts_kiev_ua-123456789-159059 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 2074-272X |
| language | Ukrainian |
| last_indexed | 2025-12-07T16:40:20Z |
| publishDate | 2019 |
| publisher | Інститут технічних проблем магнетизму НАН України |
| record_format | dspace |
| spelling | Гальченко, В.Я. Трембовецька, Р.В. Тичков, В.В. 2019-09-21T17:54:44Z 2019-09-21T17:54:44Z 2019 Побудова RBF-метамоделей структур збудження рухомого концентричного вихрострумового перетворювача / В.Я. Гальченко, Р.В. Трембовецька, В.В. Тичков // Електротехніка і електромеханіка. — 2019. — № 2. — С. 28-38. — Бібліогр.: 22 назв. — укр., англ. 2074-272X DOI: https://doi.org/10.20998/2074-272X.2019.2.05 https://nasplib.isofts.kiev.ua/handle/123456789/159059 620.179.147:519.853.6 Розроблено програмне забезпечення для розрахунку розподілу густини вихрових струмів в зоні контролю накладного вихрострумового перетворювача із врахуванням ефекту швидкості за «точними» електродинамічними математичними моделями. Розроблено програмне забезпечення для формування точок плану експерименту із використанням ЛПτ–послідовностей, що дозволило здійснювати відбір планів з рівномірним заповненням точками гіперпростору пошуку. Для нерухомого та рухомого накладних вихрострумових перетворювачів створено нейромережеві метамоделі на радіально-базисній функції Гауса. Оцінено адекватність та інформативність отриманих метамоделей накладних вихрострумових перетворювачів. Результати дослідження можуть бути використані при синтезі рухомих накладних вихрострумових перетворювачів із апріорі заданим розподілом густини вихрових струмів в зоні контролю. Goal. Creation of surface circular concentric eddy current probe RBF-metamodels, which can be used to calculate eddy currents density distribution in the control zone and suitable for use in optimal synthesis problems. Method. To develop an approximation model, a mathematical apparatus for artificial neural networks, namely, RBF–networks, has been used, whose accuracy has been increased with the help of the neural networks committee. Correction of errors in the committee was reduced by applying the bagging procedure. During the network training the regularization technique is used, which avoids re-learning the neural network. The computer experiment plan was performed using the Sobol LPt–sequences. The obtained multivariable regression model quality evaluation was performed by checking the response surface reproducibility correctness in the entire region of variables variation. Results. The modelling of eddy currents density distribution calculations on exact electrodynamic mathematical models in the experimental plan points are carried out. uk Інститут технічних проблем магнетизму НАН України Електротехніка і електромеханіка Теоретична електротехніка та електрофізика Побудова RBF-метамоделей структур збудження рухомого концентричного вихрострумового перетворювача Development of excitation structure RBF-metamodels of moving concentric eddy current probe Article published earlier |
| spellingShingle | Побудова RBF-метамоделей структур збудження рухомого концентричного вихрострумового перетворювача Гальченко, В.Я. Трембовецька, Р.В. Тичков, В.В. Теоретична електротехніка та електрофізика |
| title | Побудова RBF-метамоделей структур збудження рухомого концентричного вихрострумового перетворювача |
| title_alt | Development of excitation structure RBF-metamodels of moving concentric eddy current probe |
| title_full | Побудова RBF-метамоделей структур збудження рухомого концентричного вихрострумового перетворювача |
| title_fullStr | Побудова RBF-метамоделей структур збудження рухомого концентричного вихрострумового перетворювача |
| title_full_unstemmed | Побудова RBF-метамоделей структур збудження рухомого концентричного вихрострумового перетворювача |
| title_short | Побудова RBF-метамоделей структур збудження рухомого концентричного вихрострумового перетворювача |
| title_sort | побудова rbf-метамоделей структур збудження рухомого концентричного вихрострумового перетворювача |
| topic | Теоретична електротехніка та електрофізика |
| topic_facet | Теоретична електротехніка та електрофізика |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/159059 |
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