## 4 Numerical Results  
### Localized Behaviour: Plate  
In this section, we consider the structural model illustrated in Figure 1, which consists of a plate of dimensions \(L_{x}=1\) m, \(L_{y}=2\) m and \(L_{z}=0.1\) m that is fully clamped on its left boundary, free on the other boundaries and subjected to a uniform force distribution on its right side \(\mathbf{T}=(0.4\times 10^{5},0)\) Pa. The plate's behaviour is assumed to be linearly elastic, with the Young modulus is \(E=2\) GPa and the Poison coefficient is \(\
u=0.3\). 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 2, which serves as support for the approximation of the different mechanical fields: displacement, strains and stresses. The elements located in the damaged region are highlighted in red in Figure 2; here, [PERSON]'s modulus is reduced to \(E^{\prime}=0.1\cdot E\). The sensors location is assumed, for the sake of simplicity, to coincide with some nodes of the mesh, particularly the blue ones in Figure 2.  
After solving the minimization problem (16), the elements in which \(|a_{e}|>\epsilon=0.7\texttt{min}\{a_{e}\}\) are identified and highlighted in red in Figure 3. 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 4 and 5. The proposed procedure exhibits an excellent accuracy for correcting the model whi  
Figure 1: Structural model considered for illustrating the local model correction methodology. Figure 4: 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 3: 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 2: Damaged region (red) and location of the 30 sensors (nodes highlighted in blue). The errors between the corrected and the reference (the nominal and the reference) displacements field, for \(x\), \(y\) and \(z\), components are as follows. Of course the quality of this correction technique is highly related to the number and the location of the sensors. We performed a study on the relative errors between the reference displacement solution and the one computed from the corrected model for a different number of sensors placed on the structure, as shown in Figure 6. As expected, the relative error for the different components of the displacement field \(\left(\left\|U_{x,y,z}^{Corr}-U_{x,y,z}^{Ref}\right\|_{2}/\left\|U_{x,y,z}^{Ref} \right\|_{2}\right)\) decreases when the number of sensors increases, as Figure 7 proves. Figure 5: 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. Figure 6: Top view of sensors location for different configurations. The main drawback related to this methodology is the fact that, at the first iteration of the procedure described in Section 3.1, the corrected matrix parametrization considers all elements of the mesh. Thus, when the problem is large (the mesh is too refined), the minimization problem to solve (16) becomes computationally very expensive. To avoid such problems, one could consider a coarser parametrization of the stiffness matrix correction involving larger subdomains where the correction is eventually performed, and then it is refined by zooming in the identified region and so on until reaching an accurate enough localization.