## 4 Numerical Assessment of the Strategy Using 3D and 2D Models
### Damage Library Using 3D and 2D Model Correlation  
In this section, 2D and 3D FE models are correlated using FSC minimization in order to obtain a damage library. The damage library is built for rectangular geometry damage. First, the library is built in midpoint (0.5 m) of a cantilever beam. The process detailed in **Step 1** (see Figure 4) is implemented. In order to build a damage library, a beam embedding a rectangular damage is simulated using 3D FE, while the correlation between the two FE models permits us to obtain the corresponding 2D bending stiffness reduction. The physical parameters of the rectangular geometry \begin{table}
\begin{tabular}{l l} \hline \hline
**Beam Properties** & **Value** \\ \hline Length (\(L\)) & 1000 mm \\ Width (\(W\)) & 25 mm \\ Thickness (\(T\)) & 5.3 mm \\ Young’s modulus (\(E\)) & 210 GPa \\ Mass density (\(\rho\)) & 7850 kg/m\({}^{3}\) \\ Poisson’s ratio (\(v\)) & 0.33 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Beam dimensions and steel properties. \begin{table}
\begin{tabular}{l r r r r} \hline \hline \multirow{3}{*}{**Mode**} & \multirow{3}{*}{\(f_{experimental}\)**(Hz)**} & \multicolumn{3}{c}{**3D FE updated**} & \multicolumn{2}{c}{**2D FE Updated**} \\  & & \multicolumn{1}{c}{\(E\) = 188.90 GPa**} & \multicolumn{2}{c}{\(E\) = 189.26 GPa**} \\ \cline{3-5}  & & \(f_{3D}\) **(Hz)** & **Error (\%)** & \(f_{2D}\) **(Hz)** & **Error (\%)** \\ \hline
1 & 4.23 & 4.21 & 0.48 & 4.20 & 0.71 \\
2 & 26.33 & 26.36 & 0.11 & 26.35 & 0.08 \\
3 & 73.26 & 73.81 & 0.75 & 73.77 & 0.69 \\
4 & 142.34 & 144.63 & 1.58 & 144.56 & 1.54 \\
5 & 240.12 & 239.04 & 0.46 & 238.96 & 0.49 \\
6 & 355.67 & 357.03 & 0.38 & 356.97 & 0.36 \\
7 & 499.66 & 498.56 & 0.22 & 498.57 & 0.22 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Eigenfrequencies of healthy cantilever beam. are then varied and the process is repeated. In the library, stiffness reduction is performed in a single element while the width and depth of the rectangular damage are varied from 0.01 mm to 10 mm and 0.01 mm to 4.0 mm, respectively. Each item from the damage library is a triplet: width and depth of the rectangular damage in the 3D model and bending stiffness reduction in the 2D model. Figure 5 shows the surface view of the damage library built at midposition on the beam. The colorbar represents severity associated to the size of damage. The result of the simulation is, as expected, a smooth surface. Here, the library also illustrates a monotonic response to the size of damages. Interestingly, the width of damage contributes to the decrement of severity to a larger extent compared to its depth. Notice that the damage severity here ranges from 1 to 99%. It is not in the scope of this paper to assess whether or not it is relevant to model a deep damage with linear assumptions and an important stiffness reduction. In the rest of the study, the damage dimensions will be kept small and severity will be kept less than 50%. The effect of beam position for building a library is investigated. A library is built by choosing a different beam position, with a damage positioned at 0.3 m. Again, a range of sizes for the rectangular damage is computed and the stiffness is reduced in a selected element during 2D and 3D correlation. Figure 6a,b shows contour view of two different damage libraries which are built at 0.30 m and 0.50 m beam positions, respectively. Note that Figure 6b is simply another view of the results seen in Figure 5. The colorbar represents the percentage of severity corresponding to the depth and width of the damage. The evolution of severity with the geometry of damage is similar for the two positions. In fact, the maximum difference of severity between the two position is 1.8% on the entire domain. Moreover, Figure 6c shows the severity equivalent to a given damage for a range of positions along the beam. Two geometries are shown here: the blue curve is associated to a 1 mm \(\times\) 1.5 mm damage, and the red curve to a 1 mm \(\times\) 2.5 mm damage.