## 4 Numerical Assessment of the Strategy Using 3D and 2D Models
### Localization and Quantification of Double Damage  
For highlighting the potential of the method, it is applied to localize and estimate double damage based on the given strategy (**Step 2**) from Figure 4. Numerical cases with two damaged areas are proposed in Figure 10. Two damaged cases are investigated: (a) at 0.3 m and 0.7 m beam positions with 40% and 30% stiffness reduction (see Figure 10a), (b) and near the clamped area of the beam at 0.10 m and 0.20 m with 7% and 2% stiffness reduction (see Figure 10b), respectively.  
Figure 8: Frequency Shift Coefficient (FSC) as a function of tested position and severity, where (**a**) the white circle indicates 10% severity and 0.15 m position identified by the algorithm (**b**) slice view of 10% estimated damage.  
Figure 10: Double damage cases set in a 2D beam (**a**) at 0.3 m and 0.7 m with 40% and 30% stiffness reduction, (**b**) at 0.10 m and 0.20 m with 7% and 2% stiffness reduction, respectively.  
Figure 9: FSC as a function of tested position and severity, where (**a**) the white circle indicates 4% severity and 0.80 m position identified by the algorithm (**b**) slice view of 4% estimated damage.  
The test was initiated by obtaining a set of frequencies from each numerical case. The natural frequencies are given in Table 4 for damaged cases. Damage identification is done by updating a model embedding two damages, thanks to the FSC. As with single damage, the first seven natural frequencies of the damaged structure were used as an input for the FSC. In double damage identification, there are four parameters (position 1, position 2, severity 1 and severity 2) to identify. Evolution of FSC with parameters for the two studied cases is plotted on Figures 11 and 12. Because it is difficult to represent a variable of four parameters, the plot is split in two in these examples: First, in Figures 11a and 12a, for the actual severities for the two damages (40 and 30% in Figure 11a, 7 and 2% in Figure 12a), the FSC is represented as a function of positions only. Second, in Figures 11b and 12b, for the actual positions of the damages (0.30 and 0.70 m in Figure 11b, 0.45 and 0.55% in Figure 12b), the FSC is represented as a function of severities only. The minimization of FSC gives the following results (indicated as white circles in Figures 11 and 12): for case (a), the severities are 30% and 40% and positions are 0.30 m and 0.70 m. For case (b), the severities are 7% and 2% for two damage localized at 0.10 m and 0.20 m). In the two cases, the estimation correspond to the values given in Figure 10. In other words, the algorithm is employed satisfactorily to identify both locations and severities of double damage.  
Note that the same number of frequencies has been used in the single and the double damage cases. The numerical study that is described here demonstrates that seven frequencies are sufficient. Further work needs to be done to study the effect of adding more natural frequencies.  
\begin{table}
\begin{tabular}{l c c c c c c c} \hline \hline \multirow{2}{*}{**Damaged Cases**} & \multicolumn{6}{c}{**Natural Frequencies (Hz)**} \\ \cline{2-9}  & **1** & **2** & **3** & **4** & **5** & **6** & **7** \\ \hline (a) 40\% at 0.50 m and 30\% at 0.70 m & 4.18 & 26.25 & 73.00 & 144.01 & 238.72 & 353.61 & 495.02 \\ (b) 7\% at 0.10 m and 2\% at 0.20 m & 4.20 & 26.33 & 73.76 & 144.53 & 238.90 & 356.84 & 498.32 \\ \hline \hline \end{tabular}
\end{table}
Table 4: Numerically computed natural frequencies of double damage cases (a) and (b) with parameters of position and severity.  
Figure 11: FSC as a function of tested multiple damage positions and severities (**a**) the white circle indicates that damage positions are 0.30 m and 0.70 m, (**b**) and severities of localized damage are 40% and 30%.