Application of dimensional analysis to ECT in the era of NDE 4.0

Abstract

This paper introduces Buckingham’s 𝜋 theorem in the context of Non-Destructive Testing & Evaluation (NDT&E). Its application leads to easier problems to handle by reducing the number of variables involved. In this sense, dimensional analysis can provide the foundation for in-line, real-time and low-cost inspection methods that are fully compatible with the requirements of the Industry 4.0 and NDE 4.0 paradigms. In order to show the impact of the Buckingham’s 𝜋 theorem in NDT&E, we consider a practical case of interest, i.e. the simultaneous estimation of thickness and electrical conductivity of metallic plates via Eddy Current Testing. An initial numerical analysis is carried out with the aim to show the metrological performance of the method. The results obtained show that the method combines good accuracy with low computational costs.

Keywords

Dimensional analysis Buckingham’s 𝜋 theorem non-destructive testing & evaluation eddy current testing multi-parameter simultaneous estimation thickness estimation electrical conductivity estimation NDE 4.0

1. Introduction

In the context of Industry 4.0, advanced technologies are revolutionizing the way we evaluate and ensure the safety and reliability of industrial structures and components. In this context, the concept of Non-Destructive Evaluation (NDE) 4.0 is emerging as a relevant methodology in the field of inspection and quality control [1]. The main objective of NDE 4.0 is to identify defects and anomalies without damaging or compromising the integrity of the material or product being tested. This technology is based on the use of advanced tools, intelligent sensors and advanced data analysis to carry out increasingly precise, timely and efficient inspections. In this context, in order to develop more universal and efficient inspection protocols, simplifying the analysis methodologies and guaranteeing the possibility of carrying out in-line and real-time inspections during production, it becomes essential to be able to establish the fundamental relationships between the variables involved in a given problem. A possible methodology capable of optimizing quality control, improving product reliability and increasing operational efficiency in an advanced and connected production context could be the one based on dimensional analysis.

The dimensional analysis is a powerful tool to reduce the complexity of physical problems, by diminishing the number of variables involved. Hence, applying the results from dimensional analysis, complex problems such as the ones in NDT&E can be handled more easily, developing new methods which ensure simple and robust implementation and low-cost operations, according to Industry 4.0 paradigm [1,2]. The original contribution of this work is represented by the application of the Buckingham’s 𝜋 theorem in the context of NDT&E. Basically, Buckingham’s 𝜋 theorem states that a certain physical relationship of n variables can be rewritten in terms of p dimensionless parameters, termed 𝜋 groups, where p = n − k, with k number of physical dimension involved. In this sense, it seems to be the ideal candidate for our purposes.

For the sake of clarity, we focus on a specific case which is the simultaneous estimation of thickness and electrical conductivity of conductive plates by means of Eddy Current Testing (ECT). The specific application is motivated by the increasing demand of methods able to retrieve these two parameters. Indeed, thickness and electrical conductivity have a direct correlation with the important desired feature of final products [3].

In this paper, we derive the relationship in terms of 𝜋 groups between the measured quantity and the physical parameters influencing it. Starting from this relationship, we present a new method for the simultaneous estimation of thickness and electrical conductivity. Moreover, throughout numerical simulation, we analyze the behaviour of the method with respect to measurement noise, to which it appears quite robust. In future works, we will evaluate the performance of the method on experimental data.

The paper is organized as follows. In Section 2 we apply Buckingham’s 𝜋 theorem to the problem of interest and we derive the 𝜋 groups. In Section 3, starting from the results of dimensional analysis, we present the new inversion method. In Section 4 we give some numerical examples to show the effectiveness of the method, while in Section 5 follows conclusions.

2. NdT&E and Buckingham’s 𝜋 theorem

In NdT&E the aim is to retrieve some physical properties of a sample under test in a non-destructive manner. Between the great variety of techniques in NdT&E, the Eddy Current Testing seems to be a very promising technique since it conjugates low-cost set-up and contact-less measurements. The operations are carried out by an Eddy Current Probe (ECP) that, in our case, consists of a single coil producing a time-varying magnetic flux, able to induce eddy currents into the conductive plate and, contemporary, measuring the reaction magnetic flux density of the plate [4,5,6].

In literature, it is possible to find various kinds of ECP which differ in shape and materials as well as different kind of operations, i.e. single frequency or multi-modal 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 single-coil ECP with a non-magnetic core, the measured quantity is ${\rm\Delta}{\dot{Z}}={\dot{Z}}_{\mathit{plate}}-{\dot{Z}}_{\mathit{air}}$ , i.e. the difference between the self-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 $\begin{eqnarray}\displaystyle \frac{{\rm\Delta}{\dot{Z}}}{N^{2}}=f({\omega},{\sigma},{\nu}_{0},{\rm\Delta}h,D,\mathbf{t},L_{0},{\theta}), & & \displaystyle\end{eqnarray}$ (1) where 𝜎 is the electrical conductivity of the plate, 𝜈₀ the magnetic reluctance of the vacuum, Δh 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 N the number of turns of the coil. Furthermore, in the dimensionless vector t = (R_int∕D, H∕D) all the normalized geometrical parameters of the ECP are grouped, see Fig. 1. The normalization length D is chosen equal to the external radius R_ext of the probe.

Fig. 1.

Eddy Current Probe placed on a conductive plate with its geometrical characteristics.

Equation (1) shows as ${\rm\Delta}{\dot{Z}}$ depends on 9 variables: 5 scalar and real-valued parameters, a vector t of four scalar and real-valued variables and a complex quantity ${\rm\Delta}{\dot{Z}}/N^{2}$ . Hence, the evaluation of f by experimental data or numerical simulation is a very demanding task, given the high number of variables involved.

The Buckingham’s 𝜋 theorem allows us to reduce the number of variables required to characterize the measured quantity.

TheoremA . (Buckingham’s 𝜋 Theorem [7]).

Let a physical problem involving n dimensional scalar variables be modeled by a scalar equation of the type: q₁ = f (q₂, …, q_n) and let the physical dimensions of all variables be expressed in term of a set of k fundamental dimensions D₁, …, D_k: $\dim q_{i}=D_{1}^{a_{i1}}\times \cdots \times D_{k}^{a_{ik}},i=1,\ldots ,n$ .

Then there exist p = n − k dimensionless groups 𝜋₁, 𝜋₂, …, 𝜋_p such that the original scalar equation can be cast in the form $\begin{eqnarray}\displaystyle {\pi}_{1}=F({\pi}_{2},\ldots ,{\pi}_{p}). & & \displaystyle\end{eqnarray}$ (2)

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, Eq. (1) can be recast in terms of p = 6𝜋 groups, that are listed in Table 1.

Table 1

Dimensionless groups, arising from Buckingham’s 𝜋 theorem, for the case of interest

𝜋 groups
$\bar{{\pi}}_{1}=\displaystyle \frac{{\rm\Delta}{\dot{Z}}{\nu}_{0}}{N^{2}{\omega}D}$	${\pi}_{2}=D\sqrt{\displaystyle \frac{{\omega}{\sigma}}{2{\nu}_{0}}}$	${\pi}_{3}=\displaystyle \frac{{\rm\Delta}h}{D}$	${\pi}_{4}=\displaystyle \frac{L_{0}}{D}$	𝜋₅= t	𝜋₆= 𝜃

Since the aim is to retrieve the electrical conductivity and the thickness of the plate, 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 $\begin{eqnarray}\displaystyle \frac{{\rm\Delta}{\dot{Z}}{\nu}_{0}}{N^{2}{\omega}D}=F\left(D\sqrt{\frac{{\omega}{\sigma}}{2{\nu}_{0}}},\frac{{\rm\Delta}h}{D}\right). & & \displaystyle\end{eqnarray}$ (3) For given 𝜃, L₀ and t, Eq. (1) becomes $\begin{eqnarray}\displaystyle \frac{{\rm\Delta}{\dot{Z}}}{N^{2}}=\tilde{f}({\omega},{\sigma},{\nu}_{0},{\rm\Delta}h,D), & & \displaystyle\end{eqnarray}$ (4) where $\tilde{f}$ is equal to f but with prescribed 𝜃, L₀ and t. Comparing Eqs (3) and (4) 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.

3. Inversion method for simultaneous estimation of electrical conductivity and thickness

Assuming the functional form in (3), retrieving the electrical conductivity and the thickness of the plate means to solve the following equation $\begin{eqnarray}\displaystyle F({\pi}_{2},{\pi}_{3})=\bar{{\pi}}_{1}, & & \displaystyle\end{eqnarray}$ (5) where $\bar{{\pi}}_{1}$ is the measured quantity at a prescribed angular frequency 𝜔. Since the measured quantity ( ${\rm\Delta}{\dot{Z}}$ ) and the unknowns (𝜎, Δh) are not mixing in the 𝜋 groups, this task can be easily done by means of level curves of $\bar{{\pi}}_{1}$ . Specifically, it is possible to represent any real-valued function of $\bar{{\pi}}_{1}$ in the plane (𝜋₂, 𝜋₃). For example, we choose as features the real part $\text{Re}\{\bar{{\pi}}_{1}\}$ , the imaginary $\text{Im}\{\bar{{\pi}}_{1}\}$ , the modulus $|\bar{{\pi}}_{1}|$ and the phase $\angle \bar{{\pi}}_{1}$ of $\bar{{\pi}}_{1}$ (see Fig. 2).

Fig. 2.

Level curves of $\text{Re}\{\bar{{\pi}}_{1}\}$ , $\text{Im}\{\bar{{\pi}}_{1}\}$ , $|\bar{{\pi}}_{1}|$ and $\angle \bar{{\pi}}_{1}$ .

The solution of (5) 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 $({\pi}_{2}^{\ast },{\pi}_{3}^{\ast })$ . Hence, it is trivial to compute the unknowns as $\begin{eqnarray}\displaystyle {\sigma}=\frac{2{\nu}_{0}}{{\omega}}\left(\frac{{\pi}_{2}^{\ast }}{D}\right)^{2},\qquad {\rm\Delta}h=D{\pi}_{3}^{\ast }. & & \displaystyle \nonumber\end{eqnarray}$

4. Numerical examples

In this Section, a preliminary application of the method in a numerical environment is carried out. We consider three different samples, whose thickness and electrical conductivity are reported in Table 2, that are: common copper, aluminium for automotive application [8] and a different aluminium, aerospace applications [9] and common brass.

In order to observe the self-impedance trends, the analysis is carried out numerically by means of the Dodd and Deeds semi-analytical model [10]. The results of the model are corrupted by synthetic noise. Specifically, let ${\rm\Delta}{\dot{Z}}_{n}$ the noisy version of the simulated self-impedance, we consider ${\rm\Delta}{\dot{Z}}_{n}={\rm\Delta}{\dot{Z}}+N$ , where N is the Gaussian Noise characterized by mean equal to 0 and standard deviation equal to 𝜎_x. As the work shows the application of the method on the various features of 𝜋₁ (related to self-impedance measured by the eddy current probe), the standard deviation 𝜎_x is chosen starting from the typical accuracy values of LCR meters [11,12].

Figure 3 shows the intersection points obtained through the features $\text{Re}\{\bar{{\pi}}_{1}\}$ , $\text{Im}\{\bar{{\pi}}_{1}\}$ , $|\bar{{\pi}}_{1}|$ and $\angle \bar{{\pi}}_{1}$ considering different noise levels equal to 0. 1%,0.25% and 1% (e.g. the best performances achievable by [11]) for fifty numerical tests. As it can be seen, as the noise level value increases, the intersection points tend to have greater variability. The effect of the noise levels considered is also shown in Fig. 4. It can be seen that the intersection distribution in the scatter plot on normalized (Δh, 𝜎) plane tends to cover a greater area as the measurement performance worsens.

Table 2
Performance achieved with the considered noise levels

Cases Noise levels [%] 𝜖_{Δh, mean} [%] std_{𝜖_{Δ h, mean}}[%] 𝜖_{𝜎, mean}[%] std_{𝜖_{𝜎, mean}}[%]

Δh = 1 [mm],𝜎 = 58 [MS/m] 0.10 0.49 0.10 0.46 0.11

0.25 0.60 0.34 0.57 0.36

1 0.91 0.57 1.38 0.50

Δh = 3 [mm],𝜎 = 20 [MS/m] 0.10 0.17 0.05 0.12 0.04

0.25 0.29 0.18 0.27 0.21

1 2.64 3.81 2.21 3.23

Δh = 5 [mm],𝜎 = 17 [MS/m] 0.10 0.13 0.31 0.06 0.14

0.25 0.74 0.98 0.41 0.39

1 2.69 2.27 1.39 1.19

Δh = 2 [mm],𝜎 = 35 [MS/m] 0.10 0.16 0.11 0.20 0.14

0.25 0.40 0.07 0.41 0.07

1 2.90 4.86 3.30 4.85

Δh = 4 [mm],𝜎 = 16 [MS/m] 0.10 0.07 0.01 0.28 0.33

0.25 0.31 0.27 0.72 0.71

1 1.20 0.75 2.52 1.77

Cases	Noise levels [%]	𝜖_{Δh, mean} [%]	std_{𝜖_{Δ h, mean}}[%]	𝜖_{𝜎, mean}[%]	std_{𝜖_{𝜎, mean}}[%]
Δh = 1 [mm],𝜎 = 58 [MS/m]	0.10	0.49	0.10	0.46	0.11
	0.25	0.60	0.34	0.57	0.36
	1	0.91	0.57	1.38	0.50
Δh = 3 [mm],𝜎 = 20 [MS/m]	0.10	0.17	0.05	0.12	0.04
	0.25	0.29	0.18	0.27	0.21
	1	2.64	3.81	2.21	3.23
Δh = 5 [mm],𝜎 = 17 [MS/m]	0.10	0.13	0.31	0.06	0.14
	0.25	0.74	0.98	0.41	0.39
	1	2.69	2.27	1.39	1.19
Δh = 2 [mm],𝜎 = 35 [MS/m]	0.10	0.16	0.11	0.20	0.14
	0.25	0.40	0.07	0.41	0.07
	1	2.90	4.86	3.30	4.85
Δh = 4 [mm],𝜎 = 16 [MS/m]	0.10	0.07	0.01	0.28	0.33
	0.25	0.31	0.27	0.72	0.71
	1	1.20	0.75	2.52	1.77

Fig. 3.

Intersection points obtained on 𝜋₂, 𝜋₃ plane considering a set of fifty numerical tests at 500 (red), 1000 (blue), 1500 (green), 2000 (magenta), 2500 (cyan) Hz and different noise levels.

Fig. 4.

Intersection points obtained on 𝜎, Δh plane considering a set of fifty numerical tests at 500 (red), 1000 (blue), 1500 (green), 2000 (magenta), 2500 (cyan) Hz and different noise levels.

In order to realize a quantitative analysis, in Table 2 we report the results of the application of the presented method in the different cases. Specifically, we report the mean absolute relative error of the estimated quantity (𝜀_mean) and the estimated relative standard deviations on overall absolute relative errors (std_{𝜀_mean}) on a set of fifty numerical tests at 500, 1000, 1500, 2000, 2500 Hz.

As it can be seen, the method appears to be quite robust with respect to noise. Indeed, the values of 𝜀_mean and std_{𝜀_mean} are reasonably low even for increasing 𝜎_x and compatible with the performance in terms of measurement time, in-line inspections related to the paradigm of the Industry 4.0.

5. Conclusion

In this paper, we introduce the Buckingham’s 𝜋 theorem for the first time in NDT&E through a concrete case of study related to Eddy Current Testing for the simultaneous estimation of thickness and electrical conductivity of metallic plates. Due to dimensional analysis, we are able to reduce the complexity of the problem, reducing the number of involved variables. In this specific case, Buckingham’s 𝜋 theorem allows us to bring back the original problem to a two-dimensional problem. Consequently, we develop a new inversion method, based on the results of dimensional analysis, that has very low computational cost and that appears quite robust with respect to noise. The method has been presented from the concept to a successful numerical validation carried out on different thicknesses (from 1 to 5 mm) and electrical conductivities (from 17 to 58 MS/m) scenarios. In addition, different noise levels were implemented in order to observe the performance of the method. In the worst considered case, the relative estimation errors were shown to be no greater than 2.7% for thickness estimation and 2.2% for electrical conductivity estimation. The evident advantages of the method related to the low computational cost (linked to the reduction of the study variables involved), good accuracy and low measurement times, make it to be perfectly integrated into Industry 4.0 and the NDE 4.0 concept.

Future activities will be related to the complete experimental measurement campaign in order to validate and characterize the metrological performance in terms of accuracy shown in this work.

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