Adaptive Neuro-Fuzzy Inference System Model Based on the Width and Depth of the Defect in an Eddy Current Signal

: Non-destructive evaluation (NDE) plays an important role in many industrial ﬁelds, such as detecting cracking in steam generator tubing in nuclear power plants and aircraft. This paper investigates on the effect of the depth of the defect, width of the defect, and the type of the material on the eddy current signal which is modeled by an adaptive neuro-fuzzy inference system (ANFIS). A total of 60 samples of artiﬁcial defects are located 20 mm parallel to the length of the block in each of the three types of material. A weld probe was used to inspect the block. The ANFIS model has three neurons in the input layer and one neuron in the output layer as the eddy current signal. The used design of experiments (DOE) software indicates that the model equations, which contain only linear and two-factor interaction terms, were developed to predict the percentage signal. This signal was validated through the use of the unseen data. The predicted results on the depth and width of defect signiﬁcantly inﬂuenced the percentage of the signal ( p < 0.0001) at the 95% conﬁdence level. The ANFIS model proves that the deviation of the eddy current testing measurement was inﬂuenced by the width and depth of the defect less than the conductivity of the materials.


Introduction
Non-destructive testing and evaluation is the process of assessing the structural integrity of a material or component without causing any physical damage to the test object [1]. Eddy current testing is an effective method to detect fatigue cracks and corrosion in conductive materials because it is cheap and can monitor subsurface defects or defects under insulating coatings without touching the surface of a specimen [2,3].
Many studies investigated the effect of different parameters on eddy current signal. Material permeability and strength of magnetic induction are influenced by the type of material [4]. Thus, more secondary electromagnetic waves are produced in ferrous metals than in non-ferrous metals. Therefore, permeability significantly influences the eddy current defect signal. Crack orientation strongly influences the output of the eddy current probe. Cracks must interrupt the surface eddy current flow to be detected. Defects parallel to the current path will not cause any significant interruption and may not be detected [5][6][7][8]. The response of the pickup coil or receiver coil to an eddy current depends on the conductivity and permeability of the test material and the frequency selected [9].
Artificial intelligence was used in many types of research in eddy current testing. In [10] Postolache proposed a neural network algorithm for fast classification of the aluminum plate defects, such as cracks and holes. Moreover, the discrete wavelet transform allied with an artificial neural network (ANN) was used in [11] to estimate the depth of defect based on eddy current testing ECT data. Morabito and Versaci introduced a fuzzy neural approach to localize holes in conducting plates. However, the limitation is that fuzzy inference systems are effective with a few number of inputs [12]. In 2016 an ANN with the statistical technique of principal component analysis PCA was applied to steam generator data in simultaneous measurement by using PEC. Combining PCA and ANN improves the sensitivity of pulsed eddy current PEC [13].
The eddy current testing was investigated and, statistically, the optimized condition was achieved through response surface methodology (RSM) using central composite design. An adaptive neuro-fuzzy inference system (ANFIS) model was utilized to compare with RSM. The materials of the artificial defect block are mild steel, brass, and copper with dimensions of 420 × 30 × 10 mm. The depth of the defect is between 1 and 2.5 mm from the surface of the artificial defect block, whereas the width of a defect is located between 0.2 and 1 mm. The ANFIS model is used to predict the future behavior of processes, as well as the evaluation of the statistical significance of the effects of the process effects on the desired response.

Materials
Three different materials were used to fabricate calibration blocks (copper, brass, and mild steel). The material conductivities of copper, brass, and mild steel were 99.75%, 23.65%, and 8.48%, respectively, based on International Annealed Copper Standard. The materials were used as calibration blocks with dimensions of 260 mm (length) × 30 mm (width) × 10 mm (height). A total of 20 slots of artificial defects with different depths were produced by performing surface grinding, milling process and electrical discharge machining. AutoCAD (Autodesk, San Rafael, CA, USA) 2004 design software was used to design the artificial defect slots.

Inspection the Calibration Blocks Using a Weld Probe
A weld (differential) probe was used to inspect the materials. The locator menu was first adjusted to the appropriate settings, as shown in Table 1. The positive and negative index points were indicated on the probe by maximizing the 1 mm notch in the D50 reference block, which is used to calibrate the weld probe, as shown in Figure 1.
Appl. Sci. 2017, 7, 668 2 of 12 current depends on the conductivity and permeability of the test material and the frequency selected [9]. Artificial intelligence was used in many types of research in eddy current testing. In [10] Postolache proposed a neural network algorithm for fast classification of the aluminum plate defects, such as cracks and holes. Moreover, the discrete wavelet transform allied with an artificial neural network (ANN) was used in [11] to estimate the depth of defect based on eddy current testing ECT data. Morabito and Versaci introduced a fuzzy neural approach to localize holes in conducting plates. However, the limitation is that fuzzy inference systems are effective with a few number of inputs [12]. In 2016 an ANN with the statistical technique of principal component analysis PCA was applied to steam generator data in simultaneous measurement by using PEC. Combining PCA and ANN improves the sensitivity of pulsed eddy current PEC [13].
The eddy current testing was investigated and, statistically, the optimized condition was achieved through response surface methodology (RSM) using central composite design. An adaptive neuro-fuzzy inference system (ANFIS) model was utilized to compare with RSM. The materials of the artificial defect block are mild steel, brass, and copper with dimensions of 420 × 30 × 10 mm. The depth of the defect is between 1 and 2.5 mm from the surface of the artificial defect block, whereas the width of a defect is located between 0.2 and 1 mm. The ANFIS model is used to predict the future behavior of processes, as well as the evaluation of the statistical significance of the effects of the process effects on the desired response.

Materials
Three different materials were used to fabricate calibration blocks (copper, brass, and mild steel). The material conductivities of copper, brass, and mild steel were 99.75%, 23.65%, and 8.48%, respectively, based on International Annealed Copper Standard. The materials were used as calibration blocks with dimensions of 260 mm (length) × 30 mm (width) × 10 mm (height). A total of 20 slots of artificial defects with different depths were produced by performing surface grinding, milling process and electrical discharge machining. AutoCAD (Autodesk, San Rafael, CA, USA) 2004 design software was used to design the artificial defect slots.

Inspection the Calibration Blocks Using a Weld Probe
A weld (differential) probe was used to inspect the materials. The locator menu was first adjusted to the appropriate settings, as shown in Table 1. The positive and negative index points were indicated on the probe by maximizing the 1 mm notch in the D50 reference block, which is used to calibrate the weld probe, as shown in Figure 1.    Figure 2 shows the images of a commercial ECT system and test specimens. The accurate gain, frequency, and velocity were considered to inspect all materials. All results were used to compare the defect signal of the width on different block materials and to measure the variations of the eddy current between the brass block, copper block, and mild steel block.   Figure 2 shows the images of a commercial ECT system and test specimens. The accurate gain, frequency, and velocity were considered to inspect all materials. All results were used to compare the defect signal of the width on different block materials and to measure the variations of the eddy current between the brass block, copper block, and mild steel block.

Proposed ANFIS Model
The MATLAB Neural Network Toolbox R2013a (MathWorks, Natick, MA, USA) was used to train and optimize ANNs. ANFIS is an adaptive network which allows the implementation of a neural network topology with fuzzy logic [14,15]. An ANFIS study compiles these two methods and utilizes the characteristics of both methods. In addition, ANFIS gathers the neural network and fuzzy logic and can address nonlinear and complex problems [16]. ANFIS is a class of adaptive multilayer feed-forward networks, which are functionally equivalent to a fuzzy inference system. The Takagi-Sugeno fuzzy inference system, which contains a five-membership and simple schematic of the proposed ANFIS model, is shown in Figure 3. The proposed model and all tests were implemented using MATLAB. The error function used is a function of the logistic sigmoid and standard total of squared error function.
The model inputs include the type of material, the width of the crack, and the depth of the crack. The model outputs include the percentage of the signal. The training data were collected experimentally and normalized to obtain the values. The formula used is as follows: Low is the minimum normalized data value, which is equal to 0.1, and minimum is the minimum data value. High is the maximum normalized data value, which is equal to 0.9, and the maximum is the maximum data value [17].

Proposed ANFIS Model
The MATLAB Neural Network Toolbox R2013a (MathWorks, Natick, MA, USA) was used to train and optimize ANNs. ANFIS is an adaptive network which allows the implementation of a neural network topology with fuzzy logic [14,15]. An ANFIS study compiles these two methods and utilizes the characteristics of both methods. In addition, ANFIS gathers the neural network and fuzzy logic and can address nonlinear and complex problems [16]. ANFIS is a class of adaptive multilayer feed-forward networks, which are functionally equivalent to a fuzzy inference system. The Takagi-Sugeno fuzzy inference system, which contains a five-membership and simple schematic of the proposed ANFIS model, is shown in Figure 3. The proposed model and all tests were implemented using MATLAB. The error function used is a function of the logistic sigmoid and standard total of squared error function.
The model inputs include the type of material, the width of the crack, and the depth of the crack. The model outputs include the percentage of the signal. The training data were collected experimentally and normalized to obtain the values. The formula used is as follows: Low is the minimum normalized data value, which is equal to 0.1, and minimum is the minimum data value. High is the maximum normalized data value, which is equal to 0.9, and the maximum is the maximum data value [17].

Experimental Designs
The central composite design (CCD) was employed in the optimization study of the eddy current testing. The depth of defect, width of defect, and type of material were the independent variables (process factors) selected to optimize the percentage of the signal (response). Furthermore, RSM is used to generate the mathematical equation for the actual functional relationship between the dependent parameter (y) and the independent parameters. The first-order model is considered for linear function models of the independent parameters: If the system has curvature, then the second-order model should be considered: where y is the response (dependent variable, i.e., the percentage of the signal), b0 is the constant regression coefficient, bi and bii are the linear and quadratic regression coefficients, respectively, and bij is the regression coefficients of the two-factor interactions (i, j, = 1,2,3). xi, xj are the process factors (independent variables, i.e., type of material, depth of defect and width of defect). The fit of the model equation was evaluated by ANOVA, revealing the statistically-significant process factors with the confidence level of 95% (p-value < 0.05). The developed second-order polynomial equation was then modified by eliminating the insignificant terms. The CCD consisted of 60 experiments including five centre points, to identify the error. The design was executed with Design Expert software.

Inspection Results for Mild Steel Blocks
The effect of the depth of the defect and the width of the defect with mild steel on the eddy current testing signal could be detected by the weld probe. The inspection was conducted by using a frequency of 100 kHz. The amplitude was set to 100% FSH, and the gain was set to 50 dB (approximate). Table 2 shows the signal of the eddy current testing measurements for mild steel with different depths and widths.

Experimental Designs
The central composite design (CCD) was employed in the optimization study of the eddy current testing. The depth of defect, width of defect, and type of material were the independent variables (process factors) selected to optimize the percentage of the signal (response). Furthermore, RSM is used to generate the mathematical equation for the actual functional relationship between the dependent parameter (y) and the independent parameters. The first-order model is considered for linear function models of the independent parameters: If the system has curvature, then the second-order model should be considered: where y is the response (dependent variable, i.e., the percentage of the signal), b 0 is the constant regression coefficient, b i and b ii are the linear and quadratic regression coefficients, respectively, and b ij is the regression coefficients of the two-factor interactions (i, j, = 1,2,3). x i , x j are the process factors (independent variables, i.e., type of material, depth of defect and width of defect). The fit of the model equation was evaluated by ANOVA, revealing the statistically-significant process factors with the confidence level of 95% (p-value < 0.05). The developed second-order polynomial equation was then modified by eliminating the insignificant terms. The CCD consisted of 60 experiments including five centre points, to identify the error. The design was executed with Design Expert software.

Inspection Results for Mild Steel Blocks
The effect of the depth of the defect and the width of the defect with mild steel on the eddy current testing signal could be detected by the weld probe. The inspection was conducted by using a frequency of 100 kHz. The amplitude was set to 100% FSH, and the gain was set to 50 dB (approximate). Table 2 shows the signal of the eddy current testing measurements for mild steel with different depths and widths.

The DOE Analysis
Design Expert 7 software was used to investigate the impact of the simulated runs on the responses. Numerical data revealed that the percentage of the signal is varied between 0.38 and 1. Table 3 shows that the value of total determination R 2 is 0.8808% (close to 1), which indicates that the quadratic model reasonably fits the numerical data and can represent signal percentage in terms of the independent parameters [18]. The significance of each parameter was determined (p < 0.05) using the p-value. Table 3 shows that the analysis of variance (ANOVA) of the independent parameters, depth of the defect, and width of the defect were significant (p < 0.05). Additionally, the type of material, the interaction impact of depth of defect with the type of material, width of the defect with the type of material, depth of the defect with width of the defect, quadratic terms of the depth of the defect, and quadratic terms of the width of the defect were not significant, given that the p-values are 0.0004, 0.6257, 0.0254, 0.0020, 0.0002, and 0.3040, respectively. Thus, non-significant terms were eliminated, and the optimization process was repeated until all terms become significant.

The DOE Analysis
Design Expert 7 software was used to investigate the impact of the simulated runs on the responses. Numerical data revealed that the percentage of the signal is varied between 0.38 and 1. Table 3 shows that the value of total determination R 2 is 0.8808% (close to 1), which indicates that the quadratic model reasonably fits the numerical data and can represent signal percentage in terms of the independent parameters [18]. The significance of each parameter was determined (p < 0.05) using the p-value. Table 3 shows that the analysis of variance (ANOVA) of the independent parameters, depth of the defect, and width of the defect were significant (p < 0.05). Additionally, the type of material, the interaction impact of depth of defect with the type of material, width of the defect with the type of material, depth of the defect with width of the defect, quadratic terms of the depth of the defect, and quadratic terms of the width of the defect were not significant, given that the p-values are 0.0004, 0.6257, 0.0254, 0.0020, 0.0002, and 0.3040, respectively. Thus, non-significant terms were eliminated, and the optimization process was repeated until all terms become significant.

The DOE Analysis
Design Expert 7 software was used to investigate the impact of the simulated runs on the responses. Numerical data revealed that the percentage of the signal is varied between 0.38 and 1. Table 3 shows that the value of total determination R 2 is 0.8808% (close to 1), which indicates that the quadratic model reasonably fits the numerical data and can represent signal percentage in terms of the independent parameters [18]. The significance of each parameter was determined (p < 0.05) using the p-value. Table 3 shows that the analysis of variance (ANOVA) of the independent parameters, depth of the defect, and width of the defect were significant (p < 0.05). Additionally, the type of material, the interaction impact of depth of defect with the type of material, width of the defect with the type of material, depth of the defect with width of the defect, quadratic terms of the depth of the defect, and quadratic terms of the width of the defect were not significant, given that the p-values are 0.0004, 0.6257, 0.0254, 0.0020, 0.0002, and 0.3040, respectively. Thus, non-significant terms were eliminated, and the optimization process was repeated until all terms become significant.

The DOE Analysis
Design Expert 7 software was used to investigate the impact of the simulated runs on the responses. Numerical data revealed that the percentage of the signal is varied between 0.38 and 1. Table 3 shows that the value of total determination R 2 is 0.8808% (close to 1), which indicates that the quadratic model reasonably fits the numerical data and can represent signal percentage in terms of the independent parameters [18]. The significance of each parameter was determined (p < 0.05) using the p-value. Table 3 shows that the analysis of variance (ANOVA) of the independent parameters, depth of the defect, and width of the defect were significant (p < 0.05). Additionally, the type of material, the interaction impact of depth of defect with the type of material, width of the defect with the type of material, depth of the defect with width of the defect, quadratic terms of the depth of the defect, and quadratic terms of the width of the defect were not significant, given that the p-values are 0.0004, 0.6257, 0.0254, 0.0020, 0.0002, and 0.3040, respectively. Thus, non-significant terms were eliminated, and the optimization process was repeated until all terms become significant.

The DOE Analysis
Design Expert 7 software was used to investigate the impact of the simulated runs on the responses. Numerical data revealed that the percentage of the signal is varied between 0.38 and 1. Table 3 shows that the value of total determination R 2 is 0.8808% (close to 1), which indicates that the quadratic model reasonably fits the numerical data and can represent signal percentage in terms of the independent parameters [18]. The significance of each parameter was determined (p < 0.05) using the p-value. Table 3 shows that the analysis of variance (ANOVA) of the independent parameters, depth of the defect, and width of the defect were significant (p < 0.05). Additionally, the type of material, the interaction impact of depth of defect with the type of material, width of the defect with the type of material, depth of the defect with width of the defect, quadratic terms of the depth of the defect, and quadratic terms of the width of the defect were not significant, given that the p-values are 0.0004, 0.6257, 0.0254, 0.0020, 0.0002, and 0.3040, respectively. Thus, non-significant terms were eliminated, and the optimization process was repeated until all terms become significant.

The DOE Analysis
Design Expert 7 software was used to investigate the impact of the simulated runs on the responses. Numerical data revealed that the percentage of the signal is varied between 0.38 and 1. Table 3 shows that the value of total determination R 2 is 0.8808% (close to 1), which indicates that the quadratic model reasonably fits the numerical data and can represent signal percentage in terms of the independent parameters [18]. The significance of each parameter was determined (p < 0.05) using the p-value.  Table 3 shows that the analysis of variance (ANOVA) of the independent parameters, depth of the defect, and width of the defect were significant (p < 0.05). Additionally, the type of material, the interaction impact of depth of defect with the type of material, width of the defect with the type of material, depth of the defect with width of the defect, quadratic terms of the depth of the defect, and quadratic terms of the width of the defect were not significant, given that the p-values are 0.0004, Appl. Sci. 2017, 7, 668 6 of 12 0.6257, 0.0254, 0.0020, 0.0002, and 0.3040, respectively. Thus, non-significant terms were eliminated, and the optimization process was repeated until all terms become significant.
The implementation of RSM provides the following regression equation, which can be considered an empirical relationship between the percentage of the signal and the independent parameters.
where y is the percentage of the signal, X 1 is the type of material, X 2 is the depth of the defect, and X 3 is the width of the defect. Figure 4 shows the response surface 3D plots with contour plots on their bases for the percentage of the signal as a function of the depth of the defect and width of the defect. Figure 4a shows the results for brass, and Figure 4b shows the results for mild steel. Figure 4c shows the results for copper. In general, these plots are useful to visualize the effects of the process factors and their interactions on the percentage of the signal and the optimal process conditions. The designs provided the results, where the percentage of the signal increased with the increase in depth and width of the defect and conductivity of the material. Figure 4a shows that, for brass, the influence of the depth of the defect on the percentage of the signal was insignificant when the width of the defect was narrow. The influence of the depth of the defect on the percentage of the signal was significant when the width was increased. where y is the percentage of the signal, X1 is the type of material, X2 is the depth of the defect, and X3 is the width of the defect. Figure 4 shows the response surface 3D plots with contour plots on their bases for the percentage of the signal as a function of the depth of the defect and width of the defect. Figure 4a shows the results for brass, and Figure 4b shows the results for mild steel. Figure 4c shows the results for copper. In general, these plots are useful to visualize the effects of the process factors and their interactions on the percentage of the signal and the optimal process conditions. The designs provided the results, where the percentage of the signal increased with the increase in depth and width of the defect and conductivity of the material. Figure 4a shows that, for brass, the influence of the depth of the defect on the percentage of the signal was insignificant when the width of the defect was narrow. The influence of the depth of the defect on the percentage of the signal was significant when the width was increased.

The Effect of the Depth of the Defect and Width of the Defect on the Response
With mild steel, the increase in the percentage of the signal was evident, as shown in Figure 4b. The interactive effect of the size of the depth and size of the width on the percentage of signal was similar to that of brass, except that the obtained percentage of a signal was lower with brass. This result was explained by the negative effect of the type of material on the percentage of the signal formation. With the copper material, the percentage of signal continued to increase, as shown in Figure 4c. The effect of the depth of defect on the percentage of the signal was more significant at wide than at narrow defect widths.   With mild steel, the increase in the percentage of the signal was evident, as shown in Figure 4b. The interactive effect of the size of the depth and size of the width on the percentage of signal was similar to that of brass, except that the obtained percentage of a signal was lower with brass. This result was explained by the negative effect of the type of material on the percentage of the signal formation. With the copper material, the percentage of signal continued to increase, as shown in Figure 4c. The effect of the depth of defect on the percentage of the signal was more significant at wide than at narrow defect widths. Figure 5 shows the effect of the defect width and type of material on the signal output at a defect depth of 2.5 mm. The material type and defect width show significant effect on output signal, and the interaction of both shows an insignificant effect.  Figure 5 shows the effect of the defect width and type of material on the signal output at a defect depth of 2.5 mm. The material type and defect width show significant effect on output signal, and the interaction of both shows an insignificant effect.  To benchmark the accuracy of the present RSM models, Figure 7 compared the predicted values of the percentage of the signal, which were obtained using the regression models with the   To benchmark the accuracy of the present RSM models, Figure 7 compared the predicted values of the percentage of the signal, which were obtained using the regression models with the

ANFIS Simulation Results
Sixty experimental data records were used to verify the performances of the ANFIS models. To improve the ANFIS model, approximately 75% of the data were used for training and the remainder for testing the performance. In Table 4, the training and testing experimental data are specified at the different run numbers. Figure 8 shows that the training optimized structure was selected as ANFIS, based on the percentage error minimum at 25 epochs.

ANFIS Simulation Results
Sixty experimental data records were used to verify the performances of the ANFIS models. To improve the ANFIS model, approximately 75% of the data were used for training and the remainder for testing the performance. In Table 4, the training and testing experimental data are specified at the different run numbers. Figure 8 shows that the training optimized structure was selected as ANFIS, based on the percentage error minimum at 25 epochs.   The relative error results of ANFIS model are shown in Figure 9 for training data, where the percentage error (ER) for the input test variable and the average error (AER) are estimated as follows: The relative error results of ANFIS model are shown in Figure 9 for training data, where the percentage error (ER) for the input test variable and the average error (AER) are estimated as follows:  The ANFIS models for training and testing data are shown in Figure 10. The numerical and predicted values of the average percentage of the signal using the ANFIS model are in good agreement with the R-squared value of 0.9983. Table 5 shows that the maximum percentage error and average percentage error were approximately 0.000998 and 0.000493, respectively.  The ANFIS models for training and testing data are shown in Figure 10. The numerical and predicted values of the average percentage of the signal using the ANFIS model are in good agreement with the R-squared value of 0.9983. Table 5 shows that the maximum percentage error and average percentage error were approximately 0.000998 and 0.000493, respectively.

Conclusions
In short, the processing variables of eddy current testing have been experimentally studied in detail. Based on the RSM model, the most effective factor on the percentage of the signal is the depth of the defect in the present study. The ANFIS model is developed to predicate the percentage signal of the eddy current testing measurements. The ANFIS models can be used effectively to estimate the characteristics of defects. The study results can be summarized as follows: • For the percentage signal, the maximum error for trained and tested values is 0.023%, and the average error is 0.000493 in the ANFIS model. • The prediction of the averaged percentage signal with the ANFIS models is in good agreement with the experimental result; it also has a smaller error.
The results prove that material conductivity and the size of cracks directly affect the eddy current signal.

Conclusions
In short, the processing variables of eddy current testing have been experimentally studied in detail. Based on the RSM model, the most effective factor on the percentage of the signal is the depth of the defect in the present study. The ANFIS model is developed to predicate the percentage signal of the eddy current testing measurements. The ANFIS models can be used effectively to estimate the characteristics of defects. The study results can be summarized as follows:

•
For the percentage signal, the maximum error for trained and tested values is 0.023%, and the average error is 0.000493 in the ANFIS model.

•
The prediction of the averaged percentage signal with the ANFIS models is in good agreement with the experimental result; it also has a smaller error.
The results prove that material conductivity and the size of cracks directly affect the eddy current signal.