# Edge Effect Analysis and Edge Defect Detection of Titanium Alloy Based on Eddy Current Testing
## 3 The Defect Detection Capability Evaluation
### Simulation Scheme
#### Optimization of Coil Parameters  
During the analysis above, three distance responses were defined: \(D_{half},D_{30\%}\) and \(DIS\). \(D_{half}\) and \(D_{30\%}\) represent the scope of influence of edge effect, while DIS represents the distance between the edge and dip caused by the defect. The smaller \(D_{half}\) and \(D_{30\%}\) and the larger \(DIS\), the lower the possibility of the dip caused by defect mixing with edge effect, so a new response was defined with a combination of the three responses above:  
\[SPA=DIS-\frac{\left(D_{half}+D_{30\%}\right)}{2} \tag{14}\]  
For defect detection, especially edge defect detection, it would be desirable if _SPA_ and _RATIO_ became larger simultaneously so that the dip caused by edge defect could be easily identified.  
In order to find a set of coil parameters that optimizes the capability of edge defect detecting, an approach making use of desirability functions to optimize multiple responses was applied [26]. The approach converted responses _SPA_ and _RATIO_ into their individual desirability functions \(d_{1}\) and \(d_{2}\) that vary from 0 to 1:  
\[d_{i}\;(i=1,\;2)=\begin{cases}0&y{<}L_{i}\\ \frac{y_{i}-L_{i}}{T_{i}-L_{i}}&L_{i}\leq y\leq T_{i}\\ 1&y>T_{i}\end{cases} \tag{15}\]  
where \(y_{i}\) is the value of the response. \(L_{i}\) and \(T_{i}\) are the lower and upper limits of the response, respectively, and were set to the maximum and minimum values of the response in previous simulation,  
\begin{table}
\begin{tabular}{c c c c} \hline \hline
**Group** & **Mean** & **Standard Deviation** & **Mean Standard Error** \\ \hline Control & 3.2474 & 0.1073 & 0.0246 \\ Experimental 1 & 5.2368 & 0.1116 & 0.0256 \\ \hline \hline \end{tabular}
\end{table}
Table 11: Comparation results of _DIS_.  
\begin{table}
\begin{tabular}{c c c c} \hline \hline
**Group** & **Mean** & **Standard Deviation** & **Mean Standard Error** \\ \hline Control & 0.2525 & 0.0866 & 0.0199 \\ Experimental 1 & 0.2544 & 0.0793 & 0.0182 \\ Experimental 2 & 0.3099 & 0.1174 & 0.0391 \\ \hline \hline \end{tabular}
\end{table}
Table 12: Comparation results of _RATIO_.  
shown in Table 5. If the value of response comes closer to its target \(T_{i}\), then \(d_{i}\) comes closer to 1. The coil parameters' upper and lower limits during optimization are also shown in Table 13.  
The coil parameters were then chosen to maximize the overall desirability  
\[D=\left(d_{1}*d_{2}\right)^{1/2} \tag{16}\]  
Table 14 shows the top 5 solutions with the highest overall desirability. It can be seen that under the constraints of each parameter, an EM coil with a height of 1 mm, an outer diameter of 1 mm, and excitation frequency of 100 kHz lifted 0.1 mm off the test piece was the best one in detecting edge defect.  
It should be noted that in Section 3 only one specific kind of defect was studied, so theoretically all the regression models constructed in Section 3 are only available for the specific defect. However, it is reasonable that the best solution can be applied to all rectangular defects of similar size.