## 3 An Improved Objective Function Based on SVDM
### Substructural Virtual Distortion Method (SVDM) in Frequency Domain  
The SVDM [39] can be used to quickly compute the frequency response of the FE model of the damaged structure, without the numerically costly process of the re-analysis of the entire structure. During the optimization, it is applied to calculate the frequency response of the damaged structure (as defined by the modified damage factors) with a high computational efficiency.  
[PERSON] and [PERSON] [39] studied the SVDM in detail and deduced the respective formulas. The FRF of the FE model of the damaged structure is equivalently modeled as a superposition of the FRF of the undamaged structure and the frequency responses to certain virtual distortions that model the substructural damages. The frequency response \(h_{aq}(\mathbf{\mu},\overline{\omega}_{i})\) can be computed as shown in Equations (6) and (7):  
\[h_{aq}(\mathbf{\mu},\overline{\omega}_{i})=h_{aq}^{L}(\overline{\omega}_{i})+\sum _{i,j}D_{aij}(\overline{\omega}_{i})\gamma_{ij}^{0}(\overline{\omega}_{i}) \tag{6}\]  
\[(1-\mu_{i})\gamma_{ij}^{L}(\omega)=\gamma_{ij}^{0}(\omega)-(1-\mu_{i})\times \sum_{k,l}D_{ijkl}^{\gamma\gamma}(\omega)\gamma_{kl}^{0}(\omega) \tag{7}\]  
where \(D_{aij}(\omega)\) is the frequency response of the undamaged structure at the \(\alpha th\) sensor to the unit impulse distortion \(\mathbf{\psi}_{ij}\delta(t)\); \(D_{ijkl}^{\gamma\gamma}(\omega)\) is the \(j\)-th actual virtual distortion coefficient of the \(i\)-th substructure of the undamaged structure to the unit impulse distortion \(\mathbf{\psi}_{kl}\delta(t)\); \(\gamma_{ij}^{L}(\omega)\) and \(\gamma_{ij}^{0}(\omega)\) are the actual distortion coefficient and virtual distortion coefficient of the undamaged structure, respectively [39].  
Figure 2: The frequency response of the damaged structure.  
In Equations (6) and (7), \(h_{aq}^{L}(\omega)\)\(D_{aij}(\omega)\) and \(D_{ijkl}^{\gamma\gamma}(\omega)\) represent the characteristics of the original undamaged structure, and they can be computed in advance under the unit impulse excitation using the theoretical FE model. Therefore, the frequency response \(h_{aq}(\omega)\) of the damaged structure can be computed quickly by first solving Equation (7) for \(\gamma_{ij}^{0}(\omega)\) and then substituting the result into Equation (6). Note that the size of Equation (7) is related to the total number of the damage factors (substructures) instead of the total number of all structural DOFs. In this way, the time-consuming structural re-analysis in the traditional approach is avoided.  
During the optimization procedure for the purpose of damage identification, the main numerical effort is to calculate \(h_{aq}(\mathbf{\mu},\overline{\omega_{i}})\) for given damage factors, i.e., to solve Equation (7). Therefore, the number of damage factors to be identified and the number of the considered virtual distortions have a large influence on computational efficiency. In order to reduce the number of virtual distortions, only the crucial virtual distortions are considered. They are selected by analyzing the correlation between the actual distortions and the structural frequency response [39].