## 3 Results and Discussion
### Analysis of the Results Using an Elliptical Based Signal Processing Method  
This section focuses on the application of the elliptical-based signal processing method to localizing the damage [21]. The underlying concept is the intersection of the ellipses that correspond to wave reflections. The region where a large number of ellipses intersects identifies the damage location. Optimum sensor positioning for damage localization is considered. Six cases with different sensor positioning are shown in Table 7. The sensor placements for cases 1 and 5 were taken from research on the optimal sensor placement assisted with a genetic algorithm reported in [22]. In all of the cases, the scanning points were placed at a distance of 10 cm.  
For the elliptical-based method, a velocity profile was determined because the velocity can be angle-dependent. The steps performed to obtain the velocity profile are summarized with the equation:  
\[\text{Velocity}\ [\text{m}/\text{s}]\ =\ \frac{\text{delx}}{\text{delt}} \tag{2}\]  
where delx is the distance between the scanning points (grid points in the case of SLDV and nodal points in the case of SEM) and delt = TOA2 \(-\) TOA1, where TOA represents the wave time of arrival at a given scanning point. The calculation of delt between two points located at 0 degrees of the signals of cases 4 and 6 are shown in Figure 4. Figure 4 shows experimental signals and Figure 4 shows the numerical SEM signals. Table 8 shows the velocity values obtained for cases 4 and 6. The velocity profile was calculated by taking these velocity values and fitting them to an elliptical function. This gives the entire 360\({}^{\circ}\) plot (Figure 5). Because the 0 and 90\({}^{\circ}\) values are similar, the fit results in an almost perfect  
\begin{table}
\begin{tabular}{c c c c} \hline \hline
**Case** & **Test** & **Sensor Placement** & **No. of Sensing Points** \\ \hline
1 & Experimental & Quadrant I, II, III, IV & 8 \\
2 & Experimental & Quadrant I, II & 4 \\
3 & Experimental & Quadrant III, IV & 4 \\
4 & Experimental & Quadrant II & 6 \\
5 & Numerical & Quadrant I, II, III, IV & 8 \\
6 & Numerical & Quadrant II & 6 \\ \hline \hline \end{tabular}
\end{table}
Table 7: Case models for elliptical approach with experimental and numerical data.  
Figure 3: (**a**) Mesh arrangement and PZT position in SEM; (**b**) hole of \(\mathcal{O}\) 0.5 cm detected at 29 μs for 150 kHz excitation.  
circular profile. These velocity profiles were used for the elliptical-based method.  
Initially, the elliptical method was applied to the complete experimental case 1 with 2 BVIDs and a hole as shown in Figure 6a. The intersection of white dotted lines in all of the figures with the results indicates the region where the damage map maximum is identified. The result for case 1 shows that only the hole location was indicated, as seen in Figure 6b. Scanning points in quadrants I and II (case 2) identified the energy convergence zone near to BVID 1, as shown in Figure 6c. Scanning points in quadrants III and IV (case 3) identified the energy convergence zone near to BVID 2, as shown in Figure 6d. Different sensor  
Figure 4: delt calculation between two points (**a**) SLDV measurement; (**b**) numerical SEM.  
Figure 5: Velocity profile obtained.  
\begin{table}
\begin{tabular}{c c c} \hline \hline
**Cases Studied** & **Group Velocity [m/s] at 0\({}^{\circ}\)** & **Group Velocity [m/s] at 90\({}^{\circ}\)** \\ \hline Case 4 & 2048 & 2108 \\ Case 6 & 2410 & 2365 \\ \hline \hline \end{tabular}
\end{table}
Table 8: Group velocities obtained from SLDV, SEM measurement.  
arrangements in case 4 (quadrant II-Figure 6e) indicated the location of the hole (Figure 6f). Table 9 provides the error estimation of cases 1-4 with the experimental data.  
Figure 6: Experimental case: (**a**) case 1 quadrant I–IV arrangement of sensors; (**b**) elliptical method on case 1; (**c**) elliptical method on case 2; (**d**) elliptical method on case 3; (**e**) case 4 quadrant II arrangement of sensors; (**f**) elliptical method on case 4.