## 3 Finite Element Simulation
### FEM Results  
#### 3.2.1 Mode Separation  
The interactions between Lamb waves and the damage have been analyzed by many researchers. [PERSON] et al. adopted the wave damage interaction coefficient (WDIC) to investigate the scattering of Lamb waves and demonstrated that the damage in the plate acted as a non-axisymmetric secondary source [33]. In another report, [PERSON] et al. described the 3D interaction (frequency, incident direction, and azimuth direction) of Lamb waves with the damage [34]. [PERSON] et al. adopted a wavenumber filter to filter out the main wavefield generated by the acoustic source. Thus, the wave scattered by the delamination can be clearly observed in the filtered signal [35]. In this paper, Case 1 is taken as an example to describe the interaction between Lamb waves and the damage; moreover, the mode separation is introduced in detail. The snapshots of the Lamb waves are shown in Figure 6. It can be observed that only the A0 mode can be actuated by an ACT with a special incidence angle. However, the S0 mode will be generated when the A0 mode encounters a flat bottom defect. Following this, the incident A0, reflected A0, transmitted A0, and S0 mode will coexist for a long time. It is worth noting that the amplitude of the S0 mode is very low. \begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline \multirow{2}{*}{**Specimen**} & \multirow{2}{*}{**Damage Type**} & \multicolumn{2}{c}{**Excitation**} & \multirow{2}{*}{**Central Position**} & \multirow{2}{*}{**Radius**} & \multirow{2}{*}{**Depth**} & \multirow{2}{*}{**Inclined**} & \multirow{2}{*}{**Width**} & \multirow{2}{*}{**Length**} \\  & & **Source** & & & & & & \\  & & **Position (mm)** & & & & & & \\ \hline Case 1 & One flat bottom hole & (100, 200) & (200, 200) & 5 & 1 & / & / & / \\ Case 2 & One flat bottom hole & (100, 200) & (200, 200) & 5 & 0.5 & / & / & / \\ Case 3 & One flat bottom hole & (100, 200) & (200, 200) & 3.5 & 1 & / & / & / \\ Case 4 & Two flat bottom holes & (100, 200) & No. 1: (200, 215) & 5 & 1 & / & / & / \\  & & No. 2: (200, 185) & 5 & 1 & / & / & / \\ Case 5 & Two flat bottom holes & (100, 200) & No. 1: (200, 210) & 5 & 1 & / & / & / \\  & & No. 2: (200, 190) & 5 & 1 & / & / & / \\ Case 6 & Two flat bottom holes & (200, 100) & No. 1: (200, 215) & 5 & 1 & / & / & / \\ Case 7 & Two flat bottom holes & (200, 100) & No. 1: (200, 210) & 5 & 1 & / & / & / \\ Case 8 & Two flat bottom holes & (200, 100) & No. 1: (200, 210) & 5 & 1 & / & / & / \\ Case 9 & Two flat bottom holes & (200, 100) & No. 1: (200, 210) & 5 & 1 & / & / & / \\ Case 10 & One groove & (100, 200) & (200, 200) & / & 1 & 90 & 4 & 24 \\ Case 11 & One groove & (100, 200) & (200, 200) & / & 1 & 135 & 4 & 24 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Specimen parameters. Figure 5: The excitation signal in the (**a**) time-domain; (**b**) frequency-domain. The Lamb wave propagation was analyzed in detail. The next section will be focused on mode separation. The frequency-thickness product is 0.3 MHz\(\cdot\)mm at the undamaged area. Under this situation, the theoretical wavenumbers of the A0 mode and S0 mode are 0.8102 rad/mm and 0.2312 rad/mm, respectively. The frequency-thickness product is only 0.1 MHz\(\cdot\)mm at the damage. Hence, the theoretical wavenumbers of the A0 mode and S0 mode are 1.3082 rad/mm and 0.2311 rad/mm, respectively. Apparently, the wavenumber of the A0 mode changes more obviously than that of the S0 mode when the thickness changes. Figure 7 shows the mode separation by using 3D FFT and the wavenumber domain filtering technique. The wavenumber spectrum is obtained at a frequency of 200 kHz. The images are given in dB scale so that the wavenumber spectrum can be clearly visualized. The wavenumber spectrum for the pristine plate is shown in Figure 7a. The wavenumber spectrum for the damaged plate is shown in Figure 7b. The high-pass filter in wavenumber domain is shown in Figure 7c. The filtered wavenumber spectrum is shown in Figure 7d.