## 4 Numerical Results
### Non-Localized Behaviour: Plate  
Now, we consider the same structural model previously addressed (Figure 1). We assume that the structural response of the real plate is described by a linear elastic behavior characterized by a Young's modulus of \(E^{\text{real}}=120\) MPa, whereas the nominal model assumes a higher modulus \(E^{\text{nominal}}=200\) MPa, with the applied traction \(T=10\) KPa. Thirty sensors are placed at the locations shown in Figure 18.  
Figure 16: Top view of \(x\) (**left**), \(y\) (**center**) and \(z\) (**right**) components of the displacement field associated with the nominal solution (**top**); reference solution that takes into account the real damaged region (**middle**) and corrected model solution (**bottom**).  
Figure 17: Top view of the difference in absolute value between the corrected and the reference displacement field, for \(x\) (**left**), \(y\) (**middle**) and \(z\) (**right**) components.  
The model's enrichment described in Section 3.2 is now applied. The solutions (concerning the \(x,y\) and \(z\) components of the displacement field) for the reference, the nominal and the enriched models are compared in Figure 19.  
In Figure 20, the difference in absolute value between the reference displacement field and the one obtained with the enriched model is shown. These results prove the robustness and ability of the proposed procedure for enriching models while completing the collected data.  
Figure 19: Top view of the \(x\) (**left**), \(y\) (**center**) and \(z\) (**right**) components of the displacement field associated to the nominal solution (**top**), the reference solution (**middle**) and the enriched model solution (**bottom**).  
Figure 18: Sensors location expected allowing global model enrichment.  
The errors between the corrected and the reference (the nominal and the reference) displacements field, for \(x\), \(y\) and \(z\) components, are as follows:  
\[\left\|U_{x}^{Em}-U_{x}^{Ref}\right\|_{2}=7.1\times 10^{-4}\;(5.7\times 10^{-3 })\;\;\;,\;\;err_{\%}^{x}=\frac{\left\|U_{x}^{Em}-U_{x}^{Ref}\right\|_{2}}{ \left\|U_{x}^{Ref}\right\|_{2}}\times 100=8.4\%\;(66.7\%);\] \[\left\|U_{y}^{Em}-U_{y}^{Ref}\right\|_{2}=1.5\times 10^{-3}\;(7.2 \times 10^{-2})\;\;\;,\;\;err_{\%}^{y}=\frac{\left\|U_{y}^{Em}-U_{y}^{Ref} \right\|_{2}}{\left\|U_{x}^{Ref}\right\|_{2}}\times 100=1.3\%\;(66.7\%);\] \[\left\|U_{z}^{Em}-U_{z}^{Ref}\right\|_{2}=3.4\times 10^{-4}\;(1.4 \times 10^{-2})\;\;\;,\;\;err_{\%}^{z}=\frac{\left\|U_{x}^{Em}-U_{x}^{Ref} \right\|_{2}}{\left\|U_{x}^{Ref}\right\|_{2}}\times 100=3.4\%\;(66.7\%).\]  
Again in the present case, the quality of the enrichment is strongly dependent of the number and the location of the employed sensors. We performed a study of the relative error between the reference displacement field and the one computed with the enriched model for a different number of sensors placed on the structure, as shown in Figure 21). As expected, the relative error for each displacement component \(\left(\left\|U_{x,y,z}^{Eur}-U_{x,y,z}^{Ref}\right\|_{2}/\left\|U_{x,y,z}^{Ref} \right\|_{2}\right)\) decreases when the number of sensor increases, as Figure 22 proves.  
Figure 21: Top view of sensors location for different configurations.  
Figure 20: Top view of the difference in absolute value between the corrected and the reference displacement field, for \(x\) (**left**), \(y\) (**middle**) and \(z\) (**right**) components.