## 4 Numerical Assessment of the Strategy Using 3D and 2D Models
### In the Case of Rectangular Geometry Damage  
In this section, the strategy is carried out in order to investigate the rectangular geometry damage using all **Steps**. Figure 13a,b show a detailed 2D view of rectangular geometry cases. Here, the width (\(W_{d}\)), depth (\(D_{d}\)) and height (\(H_{d}\)) are 2.5 mm, 0.75 mm and 25 mm for rectangular damaged case (a), while 0.9 mm, 1.34 mm and 25 mm dimensions for rectangular damaged case (b), respectively. The rectangular damaged case (a) and (b) are assumed at positions 200 mm and 700 mm on the beam. Numerical cases are meshed and computed in COMSOL Multiphysics in order to obtain the natural frequencies. These frequencies are tabulated in Table 5 for the two rectangular damaged cases. The rectangular geometry damaged case (a) is investigated to find damage properties. Based on the set of frequencies of Table 5, minimizing the FSC yields estimation of position and equivalent severity. Figure 14a shows identified position and severity for rectangular damaged case (a) where the white circle represent 14.37% damage extent at position 0.20 m with a minimum value of 0.011 for the FSC. After obtaining the severity of the damage, it is possible to examine physical damage parameters (width and depth). Figure 14b represents the different possibilities of extracted geometry parameters for 14.37% severity from the damage library (according to Section 4.2). Specifically, the blue  
Figure 12: FSC as a function of tested multiple damage positions and severities (**a**) the white circle indicates that damage positions are 0.45 m and 0.55 m, (**b**) and severities of localized damages are 7% and 2%. Figure 13: 2-D clamped free beam plan with rectangular geometry damaged case (**a**) and case (**b**). \begin{table}
\begin{tabular}{l c c c c c c c} \hline \hline \multirow{2}{*}{**Damaged Cases**} & \multicolumn{8}{c}{**Natural Frequencies (Hz)**} \\ \cline{2-9}  & **1** & **2** & **3** & **4** & **5** & **6** & **7** \\ \hline (a) 3D rectangular (\(D_{d}\) = 0.716 mm, \(W_{d}\) = 2.50 mm) & 4.20 & 26.37 & 73.79 & 144.43 & 238.76 & 356.91 & 498.50 \\ (b) 3D rectangular (\(D_{d}\) = 1.327 mm, \(W_{d}\) = 0.90 mm) & 4.21 & 26.31 & 73.53 & 144.44 & 238.97 & 355.96 & 497.47 \\ \hline \hline \end{tabular}
\end{table} 
Table 5: Numerically measured natural frequencies of rectangular damaged case (a) and (b) with parameters of position and geometry size. dots represent the raw extraction from the library with a 1% tolerance. The red curve is a cubic fit of the raw result. Remember that one value of severity corresponds to two parameters, hence the line. In this section, it is assumed that the width of damage is the actual value, while the depth is a parameter to identify. It is a good way to check the method because if the damage library is reliable, the obtained depth should be the actual depth. In a more general case, it is also possible to assume the _smallest detectable size_ as a width. This makes sense in an experimental context, where a visual inspection would be limited by this smallest detectable size, let us say a hairline thickness. For the rectangular damaged case (a), the assumed width is 2.5 mm. Hence, the result of estimated depth is 0.716 mm, as shown in Figure 14b (black cross). A difference between estimated and actual depth would lie in the tolerance in the library extraction and the curve fitting that is applied. Similarly, the rectangular damaged case (b) is investigated to recover its physical properties. Figure 15a represents the identified position and severity using FSC algorithm. In this case, a 25.35% damage extent is obtained at 0.70 m beam position with 0.012 minimum value. The severity of 25.35% is searched in the damage library, as shown in Figure 15b. Hereafter, the width size is assumed based on the actual value, which is 0.9 mm for the geometry case (b). Based on width, the obtained depth is estimated 1.327 mm from the library. Based on these numerical investigations, the proposed strategy seems reliable to identify the physical geometry of damage.