## 4 Numerical Results
### Localized Behaviour: Plate with Hole  
In this section, we consider the structural model illustrated in Figure 8 consisting of a plate of dimensions \(L_{x}=1.5\) m, \(L_{y}=3\) m and \(L_{z}=0.15\) m, with a hole in the middle of radius \(R=0.5\) m fully clamped on its left boundary, free on the other boundaries and subjected to a uniform force distribution on its right side \(\mathbf{T}=(2\times 10^{3},0,0)\) Pa. The plate's behaviour is assumed to be linearly elastic, with a Young's modulus of \(E=2\) MPa and a Poison coefficient of \(v=0.3\).  
This configuration corresponds to the nominal structural system, whereas the real system is assumed to contain a damaged area. The plate is equipped with a finite element mesh, depicted in Figure 9, which serves as a support for the approximation of different mechanical fields: displacement, strains and stresses. The elements located in the damaged region are highlighted in red in Figure 9; here, the Young's modulus is reduced to \(E^{\prime}=0.1\cdot E\). The sensor's location is assumed, for the sake of simplicity, to coincide with some nodes of the mesh, particularly the blue ones in Figure 9.  
Figure 8: Schematic representation of the problem.  
Figure 7: Representation of the relative error \(\left(\left\|U^{Corr}_{x,y,z}-U^{Ref}_{x,y,z}\right\|_{2}/\left\|U^{Ref}_{x,y,z }\right\|_{2}\right)\) in percentage, with respect to the number of sensors for the three direction (blue for x-direction, red for y-direction and yellow for the z-direction). In particular, the result for the 6 different configurations (Figure 6) are specified with some marker and connected with straight lines.  
After solving the minimization problem (16), the elements in which \(|a_{e}|>\epsilon=0.25\ \texttt{min}\{a_{e}\}\) are identified and highlighted in red in Figure 10. It can be observed that the region in which the damage is located in is identified; however, the exact damage distribution remains poorly represented.  
With both the identified damaged elements and the identified reduced mechanical properties (reduced element stiffness), the solution of the mechanical problem was solved and compared with the reference solution, as depicted in Figures 11 and 12. The proposed procedure exhibits an excellent accuracy for correcting the model while completing the data.  
The errors between the corrected and the reference (the nominal and the reference) displacements field, for \(x,y\) and \(z\) components, are as follows:  
* \(\left\|U_{x}^{Corr}-U_{x}^{Ref}\right\|_{2}=4.3\times 10^{-3}\ (2.6\times 10^{-2})\), \(err_{\%}^{x}=\frac{\left\|U_{x}^{Corr}-U_{x}^{Ref}\right\|_{2}}{\left\|U_{x}^ {Ref}\right\|_{2}}\times 100=3.1\%\ (19.4\%)\);
* \(\left\|U_{y}^{Corr}-U_{y}^{Ref}\right\|_{2}=2.5\times 10^{-3}\ (2.1\times 10^{-2})\), \(err_{\%}^{y}=\frac{\left\|U_{x}^{Corr}-U_{x}^{Ref}\right\|_{2}}{\left\|U_{x}^ {Ref}\right\|_{2}}\times 100=9.6\%\ (85.1\%)\);
* \(\left\|U_{z}^{Corr}-U_{z}^{Ref}\right\|_{2}=3.7\times 10^{-3}\ (5.6\times 10^{-4})\), \(err_{\%}^{z}=\frac{\left\|U_{x}^{Corr}-U_{x}^{Ref}\right\|_{2}}{\left\|U_{x}^ {Ref}\right\|_{2}}\times 100=49.7\%\ (7.5\%)\).  
Figure 10: Identified elements affected by the damage (in red) at the first iteration (\(q=1\)) of the algorithm proposed in Section 3.1. (**a**) Top view; (**b**) bottom view.  
Figure 9: Damaged region (red) and location of the 18 sensors (nodes highlighted in blue). (**a**) Top view; (**b**) bottom view.