## 4 Numerical Assessment of the Strategy Using 3D and 2D Models
### Localization and Quantification of Single Damage  
In this section, the proposed methodology (see Figure 4) is tested by considering numerical damaged cases (see Figure 7), i.e., a 2D damaged FE model is used to generate a set of frequencies, and **Step 2** is used to find position and severity based on this set. In given cases, the damage is introduced at different beam positions: (a) 0.15 m near the clamped end (b) and 0.80 m close to the free end of the beam with 10% and 5% stiffness reduction, respectively. The first seven frequencies for each case were used in the FSC computation. They are plotted in Table 3. Note that differences between the two cases are very small. The minimization of the FSC (see Equation (8)) is carried out to localize and estimate damage. The two parameters considered in the procedure are hence the position and severity of the stiffness reduction.  
In Figure 8a, FSC is shown in color levels as a function of position and severity of the damage for case (a) schematized in Figure 7a. Note that this color map is only an illustration of the variation of FSC on the whole range of parameters, the definitive result being provided by the minimization algorithm. We notice several dark areas (low value of FSC) that correspond to frequency shifts between healthy and updated damaged model that, overall, are close to the actual frequency shifts. The dark area for positions close to the free end of the beam is especially interesting: it means that from the FSC point view, any stiffness reduction close to the free end is quite similar to the investigated case. The white circle indicates the set of parameters that minimized the FSC: location and severity values of damage are obtained at \(x\) abscissa 0.15 m with 10%, respectively. The zero value of FSC for this set of parameters indicates that the tried damaged case and the unknown case are similar from FSC point of view. The minimum value is likely to indicate the parameters of damaged case, especially if it is zero or close to zero. Figure 8b shows a slice view of the FSC as a function of severity only in the range [1, 90]% at absissa 0.15 m. On this view, it is more clear that the FSC reaches zero for a severity of 10%. Figure 9 also indicates results for another single damaged case (b) (see Figure 7b) where the minimum value of FSC is obtained with 4% estimated damage at \(x\) abscissa 0.80 m, respectively. Again, a slice view at absissa 0.8 m is shown in Figure 9b, which represents estimated severity for single damaged case (b). Thus, minimization of FSC provides accurate values for location and estimated severity of the damage. In terms of CPU times, the algorithm takes 5-10 s on a Intel Core i7 (2.6 GHz) using MATLAB's _particleswarm_ for a given case.  
\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{2-7}  & **1** & **2** & **3** & **4** & **5** & **6** & **7** \\ \hline (a) 10\% at position 0.15 m & 4.20 & 26.34 & 73.77 & 144.50 & 238.76 & 356.62 & 498.23 \\ (b) 4\% at position 0.80 m & 4.20 & 26.34 & 73.74 & 144.48 & 238.88 & 356.94 & 498.56 \\ \hline \hline \end{tabular}
\end{table}
Table 3: Numerically measured natural frequencies of single damaged cases (a) and (b) with parameters of position and severity.  
Figure 7: Single-damaged cases set in 2D beam, where positions and stiffness reduction are: (**a**) 0.15 m with 10% ; (**b**) and at 0.8 m with 4%.