Applicability of simulation-assisted probability of detection analysis considering multiple signal features to eddy current testing of weld

Abstract

In this study, a simulation-assisted probability of detection (POD) analysis that considers multiple flaw parameters and signal features is presented, including an evaluation of its applicability to eddy current testing for weld inspection. In the proposed method, both the real and imaginary parts of eddy current signals were considered, unlike conventional methods that only consider signal amplitude. Type 304 stainless steel plates were joined by butt welding, and slits were fabricated on the weld path. The eddy current inspection was conducted using a uniform eddy current probe. Numerical simulations were also performed using the finite element method, with the model imitating the experimental situation. The POD contours were calculated using the proposed method, and they exhibited a reasonable tendency. In addition, the 95% lower confidence intervals of the proposed and conventional PODs were compared and were almost equal with respect to the decision threshold, indicating the applicability of the proposed method.

Keywords

Multi-parameter POD slit butt weld detection capability uniform eddy current stainless steel

1. Introduction

Eddy current testing (ECT) is a widely applied electromagnetic nondestructive testing method for inspecting conductive materials. Although the method has the benefits of non-contact and high speed, it is known that large and ununiform noise will appear when ECT is applied to weld inspection [1]. While many kinds of probes have been developed to reduce noise from welds, it is impossible to eliminate the noise completely [2]. Moreover, because the noise is not constant or uniform, it can result in increased detection uncertainty and improper inspections. Therefore, a probabilistic method is required to quantitatively evaluate the uncertainty of ECT against flaws on welds.

The probability of detection (POD) curve is a typical method for probabilistically evaluating the detection capability of ECT [3,4]. The POD curve enables probabilistic evaluations of the detection capability of NDT by expressing the POD as a function of the flaw size. However, the targets are quite limited because conventional POD can only deal with a single flaw parameter and a single signal feature, requiring many samples. Single and multiple signal features provide important information for distinguishing signals due to flaws from noise. Moreover, it would not usually be reasonable to characterize a flaw using a single flaw parameter. Therefore, a POD considering multiple flaw parameters and signal features is required. A previous study proposed a POD analysis based on the hit/miss approach considering multiple flaw parameters and signal features to evaluate the detection capability of ECT against flaws on the weld [5]. However, the hit/miss approach usually requires a relatively large number of samples, which indicates that the practical applications of the analysis proposed in the study would result in large costs. Therefore, the â-a approach, especially Model-assisted POD [4,6,7], which requires fewer samples than the hit/miss approach, should be promising. Previous studies have proposed â-a analysis considering multiple flaw parameters by expanding the regression analysis to 2-dimensions [8] or by making numerical simulation references of flaw signals [9]. In addition, â-a analysis considering multiple signal features has been proposed [10]. However, no â-a analysis has considered multiple flaw parameters or multiple signal features.

Based on this background analysis, a simulation-assisted POD is proposed that considers multiple flaw parameters and signal features to evaluate the detection capability of ECT against weld flaws. For testing purposes, stainless steel plates joined by butt welding with slits were prepared. Eddy current inspections were then conducted to gather eddy current signals from both flawed and flawless areas. A POD obtained by the proposed method was then compared with that from the conventional method. Finally, the applicability of the proposed method for ECT during weld inspection is discussed.

2. Eddy current testing and its numerical simulation

2.1. Sample preparation

In this study, type 304 stainless steel plates were prepared by joining them with double-v butt welds using manual TIG welding. The width of the weld paths was approximately 10 mm, while the plate length, width, and thickness were approximately 200, 100, and 5 mm, respectively. Then, 20 rectangular slits were introduced parallel to the center of the weld paths by electric discharge machining to imitate weld cracks, as shown in Fig. 1. Each plate had 2 or 4 slits, whose depth and length were measured by a dial depth gauge and calipers. The slit width was 0.5 mm, and the depth and length ranged from 0.04 to 3.41 and from 1 to 20 mm, respectively. The distribution of the flaw depths and lengths is depicted in Fig. 2. The slits were fabricated with no correlation between flaw depth and length.

Fig. 1.

Slits on weld.

Fig. 2.

Distribution of flaw size of slits on weld.

2.2. Eddy current inspection

Eddy current inspections were conducted to gather eddy current signals from 20 slits on welds and 8 flawless areas. A commercial eddy current instrument (Aect-2000 N, Aswan ECT Co., Ltd., Osaka, Japan) was used for the inspections. Uniform eddy current probes (Fig. 3), known to reduce noise from welds [11], was placed to ensure that its exciting eddy current was vertical to the weld path and slits. The probe scanned a 30 mm square area by the XY stage. The scanning pitches were vertical and parallel to the weld path at 0.5 mm and 0.1 mm, respectively. The exciting frequency and lift-off were 200 kHz and 1 mm, respectively, where lift-off was defined as the distance between the bottom of the probe and the highest point of the weld path in the scanning area. The measured signals due to flaws were extracted as signals whose amplitudes were maximum in each measurement. Signals were calibrated as the signal from the slit whose length and depth were 20 and 5 mm is 1 V, 0°.

Fig. 3.

Uniform eddy current probe.

2.3. Numerical simulation

Numerical simulations were conducted using the commercial finite element method software COMSOL Multiphysics v5.20 with the AC/DC module. Because welding shape has uncertainties in their width, thickness and electromagnetic properties, the specimen was modeled as flat plates made of stainless steel, whose relative permeability and electrical conductivity were 1 and 1.35 M S/m, respectively. The slits were modeled as rectangular air regions, with a width of 0.5 mm, lengths of 1, 3, 5, 10, 15, and 20 mm, and depths of 1, 2, 3, 5, and 10 mm. Simulated signals due to flaw were extracted as signals whose amplitudes were maximum in each calculation. Signals were calibrated as the signal from the slit whose length and depth were 20 and 5 mm is 1 V, 0°.

3. POD analysis

3.1. The main flow of POD analysis considering multiple signal features

In this study, POD analysis was employed that considered multiple signal features to expand the concept of the reference [9] to 2 signal features. There are two candidates of them: real and imaginary parts, and amplitude and phase of the signals. This study selected real and imaginary parts as signal features for POD analysis because the estimation of the following Eq. (1) is not suitable for phase. Further, the signal distribution was estimated using signals obtained from the numerical simulations: $\begin{eqnarray}\displaystyle \text{} & \text{} & \displaystyle \boldsymbol{V }=V^{\text{sim}}\boldsymbol{\boldsymbol{\it\varepsilon}}_{\mathbf{1}}+\boldsymbol{\boldsymbol{\it\varepsilon}}_{\mathbf{2}}\end{eqnarray}$ (1) $\begin{eqnarray}\displaystyle \text{} & \text{} & \displaystyle \boldsymbol{V }=\left[\begin{array}{@{}c@{}}V_{\text{real}}\\ V_{\text{imag}}\end{array}\right],\quad V^{\text{sim}}=\left[\begin{array}{@{}cc@{}}V_{\text{real}}^{\text{sim}} & 0\\ 0 & V_{\text{imag}}^{\text{sim}}\end{array}\right].\end{eqnarray}$ (2) Here, V , V_real, V_imag are signals due to flaws and the real and imaginary parts of signals due to flaws, while V^sim, V_real^sim, V_imag^sim refer to simulated signals due to flaws and the real and imaginary parts of simulated signals due to flaws, respectively. Terms 𝜺₁ and 𝜺₂ are error terms which follow binormal distributions whose means are 𝝁₁ and 𝝁₂ and covariance matrixes are 𝛴₁ and 𝛴₂, respectively as $\begin{eqnarray}\displaystyle \text{} & \text{} & \displaystyle \boldsymbol{\boldsymbol{\it\varepsilon}}_{\mathbf{1}}\sim N_{2}(\boldsymbol{\boldsymbol{\it\mu}}_{\mathbf{1}},{\Sigma}_{1}),\quad \boldsymbol{\boldsymbol{\it\varepsilon}}_{2}\sim N_{2}(\boldsymbol{\boldsymbol{\it\mu}}_{\mathbf{2}},{\Sigma}_{2})\end{eqnarray}$ (3) $\begin{eqnarray}\displaystyle \text{} & \text{} & \displaystyle \boldsymbol{\boldsymbol{\it\mu}}_{\mathbf{1}}=\left[\begin{array}{@{}c@{}}{\mu}_{1,r}\\ {\mu}_{1,i}\end{array}\right],\quad \boldsymbol{\boldsymbol{\it\mu}}_{\mathbf{2}}=\left[\begin{array}{@{}c@{}}{\mu}_{2,r}\\ {\mu}_{2,i}\end{array}\right],\quad {\Sigma}_{1}=\left[\begin{array}{@{}cc@{}}{\sigma}_{1,r}^{2} & {\rho}_{1}{\sigma}_{1,r}{\sigma}_{1,i}\\ {\rho}_{1}{\sigma}_{1,r}{\sigma}_{1,i} & {\sigma}_{1,i}^{2}\end{array}\right],\quad {\Sigma}_{2}=\left[\begin{array}{@{}cc@{}}{\sigma}_{2,r}^{2} & {\rho}_{2}{\sigma}_{2,r}{\sigma}_{2,i}\\ {\rho}_{2}{\sigma}_{2,r}{\sigma}_{2,i} & {\sigma}_{2,i}^{2}\end{array}\right],\end{eqnarray}$ (4) where μ_{t, r}, μ_{t, i}, ${\sigma}_{t,rr}^{2}$ , ${\sigma}_{t,ii}^{2}$ and 𝜌_t (t = 1,2) are means and variances of real and imaginary parts of error terms and correlation coefficients of real and imaginary parts of error terms. Therefore, V also follows a binormal distribution with mean 𝝁 = V^sim𝝁₁ + 𝝁₂ and covariance matrix 𝛴 = V^sim𝛴₁(V^sim)^T +𝛴 ₂. Then, the signal due to the kth flaw follows a binormal distribution whose mean of 𝝁(d_k, l_k) = V^sim(d_k, l_k)𝝁₁ + 𝝁₂ and the covariance matrix of 𝛴(d_k, l_k) = V^sim(d_k, l_k)𝛴₁(V^sim(d_k, l_k))^T +𝛴 ₂ as its depth and length are d_k and l_k, respectively. Then, the log-likelihood was calculated as $\begin{eqnarray}\displaystyle \log L\text{} & =\text{} & \displaystyle \sum _{k=1}^{M_{l}}\log \left(\iint _{S_{l}}{\phi}(\boldsymbol{V }|\boldsymbol{\boldsymbol{\it\mu}}(d_{k},l_{k}),{\Sigma}(d_{l},l_{k}))dv_{r}dv_{i}\right)\nonumber\\ \displaystyle \text{} & \text{} & \displaystyle \quad +\sum _{k=M_{l}+1}^{M}\log ({\phi}(\boldsymbol{V }_{k}|\boldsymbol{\boldsymbol{\it\mu}}(d_{k},l_{k}),{\Sigma}(d_{l},l_{k}))),\end{eqnarray}$ (5) where V _k, S_l, V_r, M_l, M, v_r, v_i are the kth measured signal, left censor area defined as an ellipse on the impedance plane, number of measured flaws under the left censor area, number of all measured flaws (M = 20 in this study), and real and imaginary parts of the eddy current signals. Term 𝜙, which is the probability density function of signals due to flaws, is expressed as follows: $\begin{eqnarray}\displaystyle {\phi}(\boldsymbol{X}|\boldsymbol{\boldsymbol{\it\mu}},{\Sigma})=\frac{1}{2{\pi}(\det {\Sigma})^{\frac{1}{2}}}\exp \left(-\frac{1}{2}(\boldsymbol{X}-\boldsymbol{\boldsymbol{\it\mu}})^{\text{T}}{\Sigma}^{-1}(\boldsymbol{X}-\boldsymbol{\boldsymbol{\it\mu}})\right). & & \displaystyle\end{eqnarray}$ (6) Terms 𝝁₁, 𝝁₂, 𝛴₁, 𝛴₂ were estimated by the maximum likelihood estimation method, using measured and simulated signals. In this study, the log-likelihood was maximized using the particle swarm method. The POD was calculated as the probability that the signal was out of the decision threshold region, which is defined as the ellipse area on the impedance plane where signals inside the decision threshold region were undetected. $\begin{eqnarray}\text{POD}(d,l)=1-\iint _{S_{\text{dec}}}{\phi}(\boldsymbol{V }|\boldsymbol{\boldsymbol{\it\mu}}(d,l),{\Sigma}(d,l))dS_{r}dS_{i},\end{eqnarray}$ (7) where l, d and S_dec are the flaw length, depth, and decision threshold regions, respectively.

The 95% confidence intervals were calculated using the bootstrap method [12] and the number of bootstrap samples was 1000.

3.2. The setting of decision threshold and censors

The decision threshold regions were selected based on noise distribution, which is the probability distribution of the noise defined as signals due to the flawless area. The noise was fitted to a binormal distribution by maximum likelihood estimation using the Nelder–Mead method. Then, an ellipse containing a certain probability of noise could be calculated using the following equation: $\begin{eqnarray}(\boldsymbol{X}-\boldsymbol{\boldsymbol{\it\mu}})^{T}{\Sigma}^{-1}(\boldsymbol{X}-\boldsymbol{\boldsymbol{\it\mu}})=D^{2},\end{eqnarray}$ (8) where X , 𝝁, 𝛴, D are variables of the probability distribution, average, covariance matrix, and Mahalanobis’ generalized distance, respectively. The ellipse contains a 99% probability of noise distribution when D² is approximately 9. This means it is possible to control the size of the decision threshold regions to change D². In addition, the decision threshold regions are large enough to decrease the probability of failure to <1% [4] when D² ≥ 9. Therefore, the decision threshold regions were set as ellipses with D² = 9, 40, and 60. The decision thresholds with a single signal feature were set as the maximum amplitude of each ellipse. The decision thresholds and decision threshold regions are displayed with the probability density function of the noise distribution in Fig. 4.

Fig. 4.

Probability density function of noise and decision thresholds.

The left censors were set considering single and multiple signal features to eliminate the effect of measured signals masked by the noise. All the measured signals, simulated signals, and noise are depicted in Fig. 5. This revealed that some of the measured signals (expressed as red crosses) were masked by the noise. Therefore, the left censor was set considering multiple signal features as an ellipse with D² = 40, and one considering a single signal feature was set with the signal amplitude equal to 0.2 V. Then, the measured signals under the left censors were shown as the red crosses in Fig. 5. It was confirmed that these left censors succeeded in eliminating the signals masked by the noise. The right censor was not set because the limitation of the ECT instrument was large enough to collect all measured signals properly. Figure 5 also reveals that the phases of the signals due to flaws were different from those of the noise. Therefore, the necessity to consider multiple signal features to evaluate detection capability correctly in this dataset was reconfirmed.

Fig. 5.

Noise, measured and simulated flaw signals on the complex plane.

3.3. Comparison among POD contours

Figure 6 presents the decision thresholds, decision threshold regions, and the probability density function of a signal due to a flaw whose length and depth was 2 mm, which was calculated following the method shown in Section 3.1. Figure 6(b) indicates that the proposed method was able to deal with more complex decision threshold regions (shown as the ellipse on the impedance plane), while Fig. 6(a) demonstrates that this could not be achieved using the conventional method. It was also expected that the proposed POD would be higher than the conventional one because half of the signal was out of the decision threshold region in Fig. 6(b), while the signal was completely inside the decision threshold in Fig. 6(a).

Fig. 6.

PDF of signal due to a flaw (Length 2 mm, Depth 2 mm): (a) single signal feature, (b) multiple signal features.

Figure 7 depicts the POD contours when POD = 0.1, 0.5 and 0.9 with three types of D²: 9, 40, and 60. The black and red lines were the conventional and proposed POD contours, respectively. All the POD contours gradually decreased to 0 when either flaw parameter approached 0. Therefore, the tendency of all POD contours was reasonable from the viewpoint of eddy current inspection. The proposed and conventional POD contours were compared, and the proposed PODs were >0.9, whereas the conventional POD contours were 0.1. This clear difference between the proposed and conventional PODs was confirmed, as the proposed PODs were higher than the conventional ones in all types of D², as shown in Fig. 6.

Fig. 7.

POD contours calculated by POD analyses using different decision thresholds, black line: POD considering single signal feature, red line: POD considering multiple signal features.

Figure 8(a)–(c) displays the POD contours with 95% lower confidence intervals where D² = 9. The 95% lower confidence intervals when POD = 0.1 and 0.9 were a little a bit wider than the conventional ones, although they were narrower in POD = 0.5. Small differences were confirmed between the proposed and conventional 95% lower confidence intervals, which meant that these PODs had almost equal validities. In contrast, Fig. 8(d)–(f) reveals that the proposed 95% lower confidence intervals were wider than the conventional ones when D² = 60, which meant that the proposed PODs had larger uncertainty than the conventional ones. This decrease in validities could have been caused by the increment in the number of signal features in the proposed method.

Fig. 8.

POD contours with 95% lower confidence intervals.

4. Conclusion

In this study, a simulation-assisted POD analysis was proposed that considers multiple flaw parameters and signal features. Further, the applicability of the proposed method for ECT against flaws on welds was evaluated. The ECT was conducted for the inspection of slits on the welded joints of stainless steel plates, and a numerical simulation was performed to simulate the eddy current signals due to these slits on a stainless steel plate. The simulation-assisted POD analysis, considering single and multiple signal features, was carried out with three combinations of decision thresholds and decision threshold regions. A comparison between the probability density function of signals due to a flaw (calculated using the proposed and conventional methods) proved that the proposed method was able to deal with more complex decision threshold regions, which the conventional method was unable to achieve. All the proposed PODs revealed a reasonable tendency, which proofed the applicability of the proposed method to the ECT against slits on the weld. A clear difference between proposed and conventional PODs was also found, as the proposed PODs were higher than the conventional ones in all combinations of decision thresholds and decision threshold regions. This difference revealed the effect of the number of signal features on the POD analysis. The 95% lower confidence intervals derived from the proposed and conventional methods were also compared. The proposed confidence intervals were equal to the conventional one when D² = 9 and wider than the conventional one when D² = 60. Therefore, this study revealed the applicability of the proposed method to ECT for weld flaws and the advantage of the proposed method. However, further investigation is necessary to reveal the relationship between samples, decision thresholds, and confidence intervals.

Footnotes

Acknowledgements

This work was supported in part by a Grant-in-Aid for JSPS Fellows (Grant Number: 22J10460).

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