## 3 Data Completion and Models Correction or Enrichment  
### Model Local Correction and Data Completion  
Efficient diagnoses for structural monitoring are of major relevance in most of engineering applications. Different techniques, based on the analysis of the structure response in the static and dynamic regimes, for identifying the existence of damage, as well as for inferring its location and severity, have been extensively considered in numerous recent works. A short review on these techniques is provided for the sake of completeness in Appendix A. In the present section, we will consider an alternative approach that will identify indirectly the damage from the achieved model local correction. The main advantages of the proposed procedure include the ability to locally repair the model, for instance, in presence of localized damage, and its capacity to complete the collected data. For describing the proposed methodology, we consider a structural model represented by the FEM algebraic system \(\mathbf{K}\mathbf{U}=\mathbf{F}\), where \(\mathbf{K}\) is the stiffness matrix, \(\mathbf{U}\) is the displacement and \(\mathbf{F}\) is the force applied to the structure. We assume that the model's validity was checked in absence of damage. For that purpose, data are collected in a set of points, the sensor's location, corresponding for the sake of simplicity with some nodal positions \(\mathbf{X}_{k}\), with \(k\) defining the sensors set, i.e., \(k\in\mathcal{P}=\{p_{1},\ldots,p_{p}\}\). All measured data are organized in vector \(\mathbf{U}^{m}\). On the other hand, the computed displacements at the sensors locations are grouped in vector \(\tilde{\mathbf{U}}\), a simple extraction from the nodal displacement \(\mathbf{U}\), with \(\tilde{\mathbf{U}}\subset\mathbf{U}\). The model's validation implies that, for a set of applied forces \(\{\mathbf{F}_{1},\ldots,\mathbf{F}_{8}\}\), the corresponding computed nodal displacement \(\{\mathbf{U}_{1},\ldots,\mathbf{U}_{8}\}\), from which the predicted displacements at the sensors locations are extracted leading to \(\{\tilde{\mathbf{U}}_{1},\ldots,\tilde{\mathbf{U}}_{8}\}\), are assumed byy verifying the following:  
\[\texttt{max}_{k=1,\ldots,8}\|\tilde{\mathbf{U}}_{k}-\mathbf{U}_{k}^{m}\|_{2}<e, \tag{11}\]  
with \(e\) being a positive coefficient that is small enough. Thus, we assume that structural model \(\mathbf{K}\) is accurate enough for predicting the nodal displacements related to any loading \(\mathbf{F}\); that is, at the measurement locations, the predicted and the measured displacements remain close enough. Now, we consider that a deviation is observed, \(\|\tilde{\mathbf{U}}-\mathbf{U}^{m}\|_{2}>e\), and is possibly induced by the structure degradation, and we assume in the first approximation that it is localized (that is, affecting a small region \(\omega\) of the entire structure \(\Omega\)), with \(|\omega|\ll|\Omega|\). In that case, it is expected that both the nominal model and the nominal displacement must correct the former (\(\Delta\mathbf{K}\)) and update the last (\(\Delta\mathbf{U}\)); that is:  
\[(\mathbf{K}+\Delta\mathbf{K})(\mathbf{U}+\Delta\mathbf{U})=\mathbf{F}, \tag{12}\]and taking into account that \(\mathbf{K}\mathbf{U}=\mathbf{F}\), the previous system reduces to the following. \[\mathbf{K}\Delta\mathbf{U}+\Delta\mathbf{K}\mathbf{U}+\Delta\mathbf{K}\Delta \mathbf{U}=\mathbf{0}. \tag{13}\]  
The linearized problem reads as follows:  
\[\mathbf{K}\Delta\mathbf{U}+\Delta\mathbf{K}\mathbf{U}\approx\mathbf{0}, \tag{14}\]  
and it requires a parametrization of \(\Delta\mathbf{K}\). A possible approximation of it reads as follows:  
\[\Delta\mathbf{K}=\sum\limits_{e=1}^{\mathtt{E}}a_{e}\mathbf{K}_{e}, \tag{15}\]  
where \(\mathtt{E}\) is the number of elements involved in the mesh that covers domain \(\Omega\), i.e., \(\Omega=\cup_{e=1}^{\mathcal{E}}\Omega_{e}\), where \(\mathbf{K}_{e}\) is the nominal stiffness matrix related to element \(\Omega_{e}\) expressed in the global nodal numbering.