## 4 Simulation Validation
### Localization and Imaging  
With the group velocity and arrival time from signal processing, the breathing crack in the aluminum plate was localized and reconstructed by the ellipse methods. As shown in Figure 12a, the positioning results of the breathing crack based on linear features are not ideal. The elliptical loci from the sensing paths do not fall on the position of the breathing crack, including after removing signals received from the sensing paths AC and BD (Figure 12b). Meanwhile, the positioning results based on nonlinear features are shown in Figure 12c,d. Although many intersections of the elliptical loci create positioning errors (Figure 12c), the breathing crack is accurately identified and localized when the signals received from the sensing paths AC and BD are eliminated (Figure 12d). Furthermore, similar results are obtained when the modified ellipse method is employed, as shown in Figure 12e,f. It can be seen that the center of the breathing crack (highlighted with a red line) is at coordinates (100, 50); there are eight intersections (indicated by blue plus signs) of most of the elliptical loci based on linear features, and all are far away from the coordinates (100, 50). However, there is only one intersection of most of the elliptical loci based on nonlinear features at coordinates (100, 50). In addition, by drawing two vertical lines with \(x=99\) mm and 101 mm and two horizontal lines with \(y=48.5\) mm and 51.5 mm (blue dashed lines in Figure 12f, the intersection of most of the elliptical rings can be localized and highlighted. The damage location defined as the crossover area accurately situates at the actual location of the breathing crack at coordinates (100, 50). Moreover, the distance between the two horizontal lines within the crossover area can accurately estimate the length of the breathing crack (about 3 mm) in the vertical direction. It can be seen that when the length of the breathing crack is smaller than the wavelength of the incident wave at the selected excitation frequency, linear Lamb waves are not sensitive to the breathing crack and have limitations in identifying and locating micro-damages. In contrast, nonlinear Lamb waves can identify the breathing crack. Moreover, the breathing crack can be localized more precisely by the proposed method based on nonlinear S0 modes. This result is in good agreement with the experimental testing, thus verifying the effectiveness of the proposed method. Figure 11: The excitation signal (left ordinate) and (**a**) the Lamb wave signal (right ordinate) and (**b**) the Lamb wave signal with a band pass filters (right ordinate) received from the sensing path AB in the intact and the current state. In the simulation, the ellipse method based on nonlinear features for damage localization is almost as accurate as the modified ellipse method. This is because the group velocity at fundamental frequency is not significantly different from that at double frequency, namely, the group velocity remains essentially unchanged before and after interacting with the breathing crack. Thus, both imaging methods can be applied to identify and localize the breathing crack accurately. In order to further confirm the performance of the modified ellipse method, the breathing crack set at a random position (118.75, 70), not situated in any propagation paths, was identified and localized using the proposed method. With the signals received from all six sensing paths, the identification result of the breathing crack based on nonlinear features is shown in Figure 13a. The intersection of most of the elliptical rings is enclosed by two vertical lines with \(x=117.7\) mm and 119.1 mm and two horizontal lines with \(y=68.3\) mm and 72.3 mm (blue dashed lines in Figure 13a). It is found that the crossover area precisely  
Figure 12: Simulated localization and imaging of the breathing crack ((**a**) red line) with ellipse method using linear Lamb waves (**a,b**) and nonlinear Lamb waves (**c,d**), and with modified ellipse method using nonlinear Lamb waves (**e,f**).