Design and development of sensors, measurement systems, and
measurement methods in the NDE 4.0 framework.
More info about this article: https://www.ndt.net/?id=29302
Alessandro Sardellitti1
1 Department of Electrical and Information Engineering, University of Cassino and South-
ern Lazio, Cassino, Italy
E-mail: alessandro.sardellitti@unicas.it
Abstract
The concept of Industry 4.0 has revolutionized the manufacturing sector by integrating
advanced technologies, real-time monitoring and data analysis to improve production processes. This thesis highlights the critical role of quality control in Industry 4.0, employing
IoT sensors, artificial intelligence and data analytics for early defect detection. The focus
is on continuous and comprehensive inspection using Non-Destructive Testing and Evaluation (NDT&E) methods. This thesis addresses the challenge of developing methodologies
aligned to NDE 4.0, employing low-cost sensors for on-line and real-time measurements.
In particular, the study explores eddy current methods for thickness estimation overcoming
limitations through optimization techniques. Furthermore, the application of dimensional
analysis is introduced to reduce the complexity of NDT&E problems. The research contributes valuable insights and methodologies for efficient quality control in the context of
Industry 4.0.
Keywords: NDT&E, NDE 4.0, Eddy Current Techniques, Dimensional analysis, Methods
Development.
1
Introduction
In recent years, production processes have developed increasingly stringent quality standards
and there is a growing need to carry out not just spot inspections, but continuous and comprehensive checks on the produced samples. Quality control methods ensure that the products
meets the specified parameters, minimizing variations and defects. The real strength lies in the
early detection of defects; indeed, using IoT sensors, artificial intelligence and data analysis,
deviations from design or project parameters can be quickly identified, preventing problems
from arising and reducing waste [1]. Sensors and data analysis work together to predict maintenance and production needs, avoiding unplanned downtime that could interrupt production and
compromise quality [2].
In the framework of quality control, Non-Destructive Testing and Evaluation (NDT&E)
methods play a key role. The importance of NDT&E lies in its ability to detect defects, imperfections or irregularities that could compromise the performance of products or systems without
altering the physical and chemical characteristics of the product being analyzed [3, 4]. Through
the application of these methods, the integrity, safety and reliability of materials, components
and structures is ensured.
© 2024 The Authors. Published by NDT.net under License CC-BY-4.0 https://creativecommons.org/licenses/by/4.0/
https://doi.org/10.58286/29302
Many parameters can be analyzed using these methods. For example, geometrical parameters of the sample (e.g. thickness parameter) are indicative of how the mechanical forming
process is performing (such as pressing processes, rolling etc.) and determines the structural
characteristics of the produced sample [5,6]. Another application is the analysis of the chemical
characteristics of the sample. For instance, the electrical conductivity determines the quality of
the part in the molding (in the case of metallic materials) and curing (in the case of composite
materials) processes [7, 8]. In addition, parameters that indicate the degradation of the sample
are crucial; an example is the loss of material that occurs in the case of corrosion processes [9].
All these parameters determine the quality and remaining lifetime of the sample under analysis.
These issues have been studied within the NDE community for a long time and there are
now many technologies and methodologies that provide good accuracy. The challenge that this
research project aims to solve is to identify methodologies that are in line with the context of
NDE 4.0. This context is made of low-cost sensors, easily integrated into automated measurement systems and optimized measurement time and processing [10, 11]. The challenge of the
research activity has been to develop sensors, measurement systems, and methodologies that
have the purpose of applicability in the NDE 4.0 context.
In Section 2 the study of the geometric parameters of greatest interest in the industrial and
research area has been realized. In this context, the thickness of metallic samples was observed
to be an important parameter in the context of industrial quality control. Two different methods
based on Eddy Current Techniques (ECT) have been developed in order to apply this techniques
in industrial environment with low measurement time and good accuracies.
Finally, in Section 3 the application of the dimensional analysis on NDT&E framework
has been described. This methodology can systematically reduce the analysis complexity of
the problems by decreasing the number of variables involved. For instance, the reduction of
the number of variables has a major impact when a physical problem is modelled either via a
numerical approach or via a machine learning approach.
2
Eddy Current methods for thickness estimation of metallic plates: proposed optimizations
Thickness measurements have a key role in many industrial applications related to metallic
plates because they provide information about the quality of both the manufacturing process and
the realized product in term of structural strength and elasticity. These aspects are fundamental
in all those application fields, such as in automotive and aerospace, where the integrity of the
metallic plates components has a direct impact on the safety of human beings.
In the industrial practice, some metrological checks are carried out by using touch-trigger
probes, laser-based measurement methods, ultrasound methods [12]. These measurement methods are characterized by high measurement times, high cost of the instrumentation and nonintegrability within automated measurement systems.
Several research groups are investigating the possibility offered by ECT, which typically
assure low costs, simple probes, and easy integration to online, real-time, and automated measurement systems. Inside the broad field of the ECT proposed in the literature, there are several
methods that allow the estimation of thickness by means of a multi-frequency analysis or pulsed
eddy current analysis [5, 13, 14].
In particular, in [5,13], Yin et al. proposed and optimized measurement probes and processing algorithms suitably developed for non-magnetic materials able to give accuracy compatible
with industrial quality standards and a good robustness to lift-off variations. In detail, using
different combinations of coils to perform the generation and detection of the excitation and
reaction magnetic flux respectively, Yin et al. compared measurements performed both in air
and on the sample under test to estimate the thickness of planar samples.
Although the methods proposed [5] appear to be promising, they have some limitations in
terms of applicability in the aforementioned context of real-time, in-line measurements with
good, constant measurement accuracy over the entire range of thicknesses analyzed.
Firstly, the method proposed in [5] relies on the constancy of a certain parameter (α0 ).
Unfortunately, this parameter is highly dependent on the thickness of the sample, and it can
be maintained constantly only asymptotically for thicknesses much smaller than the size of the
probe (see Eq. 1 and Figure 1). This makes the method impractical for thick samples or in the
presence of curved surfaces. For instance, in the latter case, there are two conflicting constraints:
(i) the probe has to be much larger than the thickness of the sample and (ii) the probe has to be
much smaller than the curvature of the sample, so as to approximate the surface of the sample
by means of its tangent plane, in a neighborhood of the probe itself.
α(∆) =
σ µ0 ωmin ∆
.
2
(1)
( ) [1/m]
1000
800
600
400
200
0
0
5
10
[mm] 15
20
25
Figure 1: Complete behaviour of α(∆) for a defined probe (blue line), together with its constant
approximation (α0 = 126.28 m−1 ) valid for small ∆/D (black line) [6].
Secondly, in [5, 13], the key feature for estimating the thickness of plates is the value of the
frequency where a proper quantity achieves its minimum value (∆R/ω). To get proper accuracy
in measuring the thickness of the sample under test, this minimum needs to be located in an
accurate manner. In turn, this requires several measurements at different frequencies with high
frequency resolution which makes the approach time-consuming and not suitable for almost
real-time applications.
In order to solve the first limitation, in [6] was provided a deep, complete, and more general
theoretical framework for the method proposed in [5], allowing it to be extended to a broader
class of thickness measurements. From the technical perspective, it was proved that the critical
constant parameter α0 has to be replaced by a new function α = α(∆) , which depends on
the thickness (∆). With this “minimal ”modification, the method’s range of applicability was
extended while identifying its underlying physical limit. Two algorithms were developed to
make the optimal use of the identified variable α = α(∆). The first algorithm is iterative and is
conceived for applications where the unknown thickness ranges from 0 up to the theoretical limit
of the method (see Figure 2). The second algorithm is based on a polynomial approximation of
α(∆) and provides the thickness as the solution of an algebraic equation of the same degree as
the approximating polynomial (see Figure 3).
Figure 2: Schematic representation of the developed iterative algorithm and the respective
graphic representation [6].
Figure 3: Schematic representation of the developed algebraic algorithm [6].
For the second limitation, in [15] was proposed a strategy with a multi-tone and interpolation combination. In particular, a multisine excitation signal approach was applied to collect the
data onto a proper set of frequencies then appropiate techniques were applied in order to interpolate the data at all the frequencies required to locate accurately the minimum of the response
(see Figure 4). The combination of the multisine signal to allocate efficiently the measurement frequencies with the data interpolation results in a reduction of the number of required
measurements and, in the ultimate analysis, of the overall measurement time.
Figure 4: Graphic representation of the developed strategy with a multi-tone and interpolation
combination [15].
A suitable experimental set-up was realized to allow the excitation and acquisition of the
signal of interest and to show experimentally the performance obtained with the proposed methods [6,15]. In Figure 5 is represented the block diagram experimental set-up (more information
are available in [6]). Tests were carried out on samples with different conductivities and thicknesses as shown in Table 1.
Figure 5: Schematic block diagram of the experimental set-up [7].
Table 1: Characteristics of the considered case studies [6].
Name code
Metal alloy
#a
#b
#c
#d
#e
#f
EN AW-1050A
EN AW-1050A
EN AW-1050A
EN AW-1050A
EN AW-1050A
2024T3
Nominal
thickness
[mm]
0.469
1.035
1.969
2.912
3.994
2.003
Electrical
conductivity
[MS/m]
35.4
35.3
35.0
35.3
34.5
18.8
Plate
dimensions
[cm x cm]
13 x 18
25 x 25
25 x 50
25 x 25
25 x 25
20 x 20
Thickness estimation performance has been compared in terms of the Relative Thickness
Error (RTE), defined as:
∆e − ∆a
RT E =
· 100
(2)
∆a
where ∆a is the actual thickness of the sample and ∆e is the estimated thickness. Figure 6 (a)
summarizes the obtained experimental results for the configurations of Table 1 applying the
iterative and the algebraic algorithms.
As expected, the smallest RTEs were observed by adopting the best α value related to the
considered nominal thickness, with RTEs lower than 1% except for the sample #a, for which
an RTE of 1.35% was found. For the original approach of [5], the RTE significantly worsened,
rising to 14%. The smallest error (2.68%) is obtained for a thickness equal to 1 mm. On the
other hand, the new approaches yield an excellent performance. Indeed, the RTEs are always
lower than 2.5% and, in several cases, are comparable with the “ideal” values obtained by using
the a priori knowledge of α(∆), i.e. the thought experiment.
About the experimental performance obtained applying the developed strategy with a multitone and interpolation combination, different kind of interpolating polynomial have been analyzed. In particular were analyzed classic interpolating polynomial and piecewise interpolating
polynomial: Fourth Order Interpolating Polynomial (FOIP), Fourth Order Chebyshev Interpolating Polynomial (FOCIP),Modified Akima Interpolator Polynomial (MAIP) and Piecewise
Cubic Hermite Interpolator Polynomial (PCHIP).
Figure 6: Obtained experimental results applying (a) the developed iterative and the algebraic
algorithms and (b) the developed strategy with a multi-tone and interpolation combination.
From the results shown in Figure 6 (b), it is possible to note how the maximum RTE is about
4% (accuracies in line with the reference methods). In the other hand, the measurement time
for a typical situation was reduced from 13 to 2.66 seconds with a similar level of accuracy to
that represented in [5].
3
Introduction of dimensional analysis in Nondestructive Testing and Evaluation
Problems related to NDT&E typically involve several variables. As a matter of fact, the result
of an NDT&E test depends on (i) the probe’s parameters (geometry, materials, ...), (ii) the geometrical and the physical characteristics of the sample under testing (e.g. thickness, electrical
conductivity, magnetic permeability etc.) and (iii) the parameters describing the geometrical relationship between the probe and the sample under testing (e.g. lift-off, tilt angle etc.). From the
experience gathered in this context [6, 15], the number of variables involved and the correlated
nature of these variables (e.g., excitation frequency or electrical conductivity and thickness of
the sample) make an NDT&E problem difficult to handle.
In this context, dimensional analysis is a mathematical tool for analyzing problems involving physical quantities [16]. Dimensional analysis can be used to simplify complex equations
by highlighting the fundamental quantities that describe a problem. Among the various tools
available, the Buckingham’s π theorem represents a widely used tool in the literature [17, 18].
It states that a certain physical relation of n variables can be rewritten in terms of p dimensionless parameters, defined as π groups, where p = n − k, with k number of physical dimensions
involved [19].
This research activity brought the application of the Buckingham π theorem to the NDT&E
community for the first time [7, 20].
In particular, application of the ECT for the estimation of the thickness and the electrical
conductivity has been analyzed. This specific application was chosen because the thickness estimation methods usually presented in the literature require a priori knowledge of the electrical
conductivity of the metallic sample [5], which is not always known. In addition, the electrical
conductivity of metallic material is a good feature to estimate the mechanical characteristics.
In particular, the research activity has given to the NDT&E community these major contributions: (i) a proper relationship in terms of dimensionless groups among the measured quantity
Figure 7: Eddy Current Probe placed on a conductive plate with its geometrical characteristics
[7].
and the physical variables affecting it, in a frequency domain ECT experiment. It was assumed
that the measured quantity is the self or mutual impedance of the probe. This assumption is
not restrictive of the generality of the method; (ii) a method together with its algorithm counterpart for the simultaneous estimation of the thickness and electrical conductivity by using
dimensionless groups.
3.1
Buckingham’s π theorem in the Eddy Current framework
In literature, it is possible to find various kinds of Eddy Current Probe (ECP) which differ
in shape and materials as well as different kind of operations, i.e. single frequency or multimodal measurements. In order to show the effectiveness of the proposed method we focus on
a specific case, but all the reasoning below can be applied with few irrelevant changes to other
cases. Specifically, for a double-coil ECP with a non-magnetic core, the measured quantity is
∆Ż = Żm,plate − Żm,air , i.e. the difference between the mutual-impedance of the coil when it
is placed on the plate and when it is in air. The measured quantity, for a prescribed angular
frequency ω, depends on several physical parameters
∆Ż
= f (ω, σ , ν0 , ∆h, D, t, L0 , θ ) ,
N1 N2
(3)
where σ is the electrical conductivity of the plate, ν0 the magnetic reluctance of the vacuum,
∆h the thickness of the plate, D a reference length for the probe, l0 the lift-off,θ the tilting
of the probe w.r.t to the normal to the plate and N1 -N2 the number of turns of the excitation
and receiver coil respectively. Furthermore, in the dimensionless vector t = (r1 /D, h/D) all the
normalized geometrical parameters of the ECP are grouped, see Figure 7. The normalization
length D is chosen equal to the external radius r2 of the probe.
The Buckingham’s π theorem allows us to reduce the number of variables required to characterize the measured quantity. Applying this theorem, can be stated that there exist p = n − k
dimensionless groups π1 , π2 , . . . , π p such that the original scalar equation can be cast in the
form
π1 = F (π2 , . . . , π p ) .
(4)
For the case of interest, all the variables can be expressed in terms of k = 3 fundamental
dimensions that are: length, time and resistance. For this reason, Equation (3) can be recast in
terms of p = 6 π groups, that are listed in Table 2.
Since the aim is to retrieve the electrical conductivity and the thickness of the sample under
test, given the probe, i.e. given θ , l0 and t, π4 , π5 and π6 are assumed to be known and we focus
on π1 , π2 and π3 . Specifically, the relationship between dimensionless groups is
r
∆Żν0
ωσ ∆h
=F D
,
.
N1 N2 ωD
2ν0 D
(5)
For given θ , l0 and t, Equation (3) becomes
∆Ż
= f˜ (ω, σ , ν0 , ∆h, D) ,
N1 N2
(6)
where f˜ is equal to f but with prescribed θ , l0 and t. Comparing Equation (5) and (6) it is
possible to appreciate the impact of Buckingham’s π theorem. In fact, starting from a complex
function of five real-valued parameters, we obtain a relationship involving only two real-valued
parameters. In other words, function F can be easily characterized numerically or experimentally while the same task requires much more effort with respect to the function f . Furthermore,
the function F can be represented in the plane (π2 , π3 ) and the inverse problem of interest is
traced back to an analysis in this plane. This is the fundamental point leading to the inversion
method presented in the next section.
Assuming the functional form in (5), retrieving the electrical conductivity and the thickness
of the plate means to solve the following equation
F(π2 , π3 ) = π̄1 ,
(7)
where π̄1 is the measured quantity at a prescribed angular frequency ω. Since the measured
quantity (∆Ż) and the unknowns (σ , ∆h) are not mixing in the π groups, this task can be easily
done by means of level curves of π̄1 . Specifically, it is possible to represent any real-valued
function of π̄1 in the plane (π2 , π3 ). For example, we choose as features the real part Re{π̄1 },
the imaginary Im{π̄1 } , the modulus |π̄1 | and the phase π̄1 of π̄1 (see Figure 8).
The solution of (7) is evaluated by putting the level curves of at least two features on the
same (π2 , π3 ) plane. All the curves intersect in one point (π2∗ , π3∗ ). Hence, it is trivial to compute
the unknowns as
2ν0 π2∗ 2
σ=
, ∆h = Dπ3∗ .
(8)
ω
D
The method has been presented from the underlying concept to a successful experimental
validation carried out on metallic plates with different thicknesses (from 1 to 4 mm) and electrical conductivities (from 17 to 58 MS/m) shown in Table 3 adopting the experimental set-up
represented in Figure 5.
The obtained experimental set-up are shown in Figure 9. From the results, it can be note that
the maximum relative error in thickness estimantion is about 4% while in electrical conductivity
estimation is about 2.5% The method can be applied to either single or multi-frequency data and
its negligible computational cost makes it suitable for industrial in-line inspections.
Table 2: Dimensionless groups, arising from Buckingham’s π theorem, for the case of interest.
0
π̄1 = N1∆NŻν
2 ωD
π groups
q
ωσ
π2 = D 2ν0 π3 = ∆h
D
π4 = lD0
π5 = t
π6 = θ
Figure 8: Level curves of Re{π̄1 }, Im{π̄1 }, |π̄1 | and π̄1 [20].
Table 3: Analyzed sample.
Name code
#a
#b
#c
#d
#e
#f
4
Metal alloy
Aluminium (2024-T3)
Copper
Aluminium (6061-T6)
Aluminium (AW-1050A)
Aluminium (AW-1050A)
Aluminium (AW-1050A)
Electrical conductivity [MS/m]
17.66
58.50
28.23
35.27
35.44
35.91
Thickness [mm]
2.03
0.98
1.97
1.03
2.93
3.98
Conclusions
The proposed research activities in Non-Destructive Testing, particularly in NDE 4.0, have resulted in solutions for thickness measurements, electrical and magnetic parameters of metallic
materials, and corrosion detection using Eddy Current methods. Two solutions for estimating
metallic material thickness were proposed. The first optimized existing methods from scientific
literature to meet industrial requirements, resulting in three combined methods achieving the
needed accuracy (4%) over a broader thickness range (0.5 - 4 mm) and in less than 3 seconds.
The second solution introduced dimensional analysis to NDT&E, reducing variables and ensuring excellent measurement performance for thickness and electrical conductivity estimation.
Future developments include extending Buckingham’s π theorem to multi-layer structures,
unknown liftoff problems, and magnetic materials. Application on anisotropic materials and
data fusion strategies for corrosion studies are also considered, along with exploring electrical
signatures in artificial intelligence algorithms.
5
Statement
The presented work is based on the PhD thesis of Alessandro Sardellitti entitled “Design and development of sensors, measurement systems, and measurement methods in the NDE 4.0 framework. ”, which was submitted at University of Cassino and Southern Lazio in December 2023.
Figure 9: Obtained experimental results for (a) thickness and (b) electrical conductivity estimation.
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