## 3 Numerical Validation
### Numerical Results  
Table 3 lists the first three boundary modes of the nine cases of Model A. Significant changes in boundary modes are observed with damage in the structural sealant. For case 2-5, the first boundary mode of edge-b changes significantly as damage severity increases, while the boundary modes of undamaged edges-a, c, and d remain mostly unchanged. In cases 6-9, damage on edge-d causes notable displacement increases. For example, in cases 6 and 9, the displacement of edge-d increases by 7.6% and 50%, respectively, compared to the undamaged case. However, the displacement changes on edges-a and c are minimal.  
Interestingly, the second and third boundary modes of undamaged edges-a and c change significantly with increasing damage severity on edges-b and d. For the damaged edge b and d, the higher boundary modes are not always changed with the increase in the  
\begin{table}
\begin{tabular}{c c c c c c c c c c c c} \hline \hline \multirow{2}{*}{**Model**} & \multirow{2}{*}{**Edge**} & \multirow{2}{*}{**Damage**} & \multicolumn{8}{c}{**Case**} \\ \cline{3-14}  & & & **1** & **2** & **3** & **4** & **5** & **6** & **7** & **8** & **9** \\ \hline \multirow{4}{*}{A} & \multirow{2}{*}{E\({}_{\text{b}}\)} & Location & - & 30–32 & 30–34 & 30–36 & 30–38 & 30–38 & 30–38 & 30–38 & 30–38 \\  & & Severity & 0 & 5\% & 8.3\% & 11.6\% & 15\% & 15\% & 15\% & 15\% & 15\% \\ \cline{1-1}  & \multirow{2}{*}{E\({}_{\text{d}}\)} & Location & - & - & - & - & - & 10–12 & 10–14 & 10–16 & 10–18 \\ \cline{1-1}  & & Severity & 0 & 0 & 0 & 0 & 0 & 5\% & 8.3\% & 11.6\% & 15\% \\ \hline \multirow{4}{*}{B} & \multirow{2}{*}{E\({}_{\text{b}}\)} & Location & - & 28–30 & 28–32 & 28–34 & 28–36 & 28–36 & 28–36 & 28–36 & 28–36 \\  & & Severity & 0 & 5\% & 8.3\% & 11.6\% & 15\% & 15\% & 15\% & 15\% & 15\% \\ \cline{1-1}  & \multirow{2}{*}{E\({}_{\text{c}}\)} & Location & - & - & - & - & 38–40 & 38–42 & 38–44 & 38–46 \\ \cline{1-1}  & & Severity & 0 & 0 & 0 & 0 & 0 & 3.8\% & 6.3\% & 8.8\% & 11.3\% \\ \hline \hline \end{tabular}
\end{table}
Table 2: Location and length of damage points.  
\begin{table}
\begin{tabular}{c c c c c c c c c c} \hline \hline
**Mode** & **Edge a** & **Edge b** & **Edge c** & **Edge d** \\ \hline \hline
1 st & 0.2 & 0.2 & 0.2 & 0.2 & 0.2 & 0.2 & 0.2 \\
1 st & 0.2 & 0.2 & 0.  
damage severity, for instance, the second boundary mode of edge-b and the third boundary mode of edge-d. This can be attributed to two reasons: (1) higher boundary modes are more sensitive to local damage and when adjacent edges are damaged, the boundary modes of undamaged edges may also change. (2) When the damage is located at the modal node position, the boundary modes will not change regardless of the extent of the damage.  
Table 4 lists the first three BMCs for Model A. Due to the influence of noise, the BMC shows some discreteness, and the same situation will occur in the actual measurement. In this case, the influence of noise can be reduced by averaging multiple measurements. Compared with boundary modes, the BMC is more sensitive to the damage. In the first-order boundary mode, the BMC of edge-b at the damage location is more than 10 times larger than that at non-damage. As the damage severity increases, the BMC at the damage location increases significantly, while the BMC at the undamaged location shows minimal change. Therefore, damage can be better identified by the changes in BMC. The results for Model B are similar and will not be repeated here.