## 2 Theory of the Inverse Analysis of Structural Damage
### Formulation of the Objective Function  
The aforementioned deterministic mathematical approach that studies the sensitivity of modal features is vulnerable to incomplete data and noisy measurements, where any simple variation of modal parameters might lead to wrong identification. Therefore, a stochastic optimization method should be employed. Moreover, it is well-known that the structural damage identification problem is an ill-posed problem with a damage parameters region that may have multiple local optima. In this case, the traditional optimizers could be stuck in the local minima, which may deliver wrong damage identification results. Hence, it is essential to develop robust and powerful objective functions and algorithms, which could tackle the complex damage identification problem. Therefore, the use of more than one modal property contributes to improve the overall damage identification, where each property helps to discover more underlying damage information. Thus, the MSEn subobjective can be first developed based on Equations (23), (25), and (27) as:  
\[F_{1}=min\ \left(\sum\limits_{i=1}^{N}\frac{MSEn_{i}^{D}-MSEn_{i}}{MSEn_{i}} \right). \tag{36}\]  
Secondly, the global \(MKEn\) of the intact structure and the \(MKEn^{D}\) of the damaged structure can be derived using the modal frequencies, mode shapes, and global mass matrix of the structure, and, based on Equations (4)-(6), the \(MKEn\) sub-objective function can be written as:  
\[F_{2}=\min\left(\sum\limits_{i=1}^{N}\frac{MKEn_{i}^{D}-MKEn_{i}}{MKEn_{i}} \right). \tag{37}\]  
Using the abovementioned damage indicator, the effect of large values of modal frequencies related to higher modes of vibration can be reduced, which makes the damage identification process faster. In order to further improve damage deductibility when dealing with incomplete data and noisy conditions, the effect of mode shapes can be separated from the global mass and stiffness matrices. Therefore, another sub-objective, which takes advantage of the modal assurance criterion (\(MAC\)) of the mode shapes, can be written as:  
\[F_{3}=\min(1-mean(diag(MAC))), \tag{38}\]  
where, \(diag\) represents the diagonal elements of \(MAC\) matrix that measures the correlation between the analytical and experimental mode shapes; the \(MAC\) matrix can be written as:  
\[MAC_{i}=\frac{\left|\phi_{i}^{DT}\phi_{i}\right|^{2}}{\left|\phi_{i}^{T}\phi_ {i}\right|\left|\phi_{i}^{DT}\phi_{i}^{D}\right|}. \tag{39}\]  
The MAC matrix [5] is composed of diagonal and off-diagonal elements. When measuring the correlation between two mode shapes, two correlated mode shapes should result in a MAC matrix having diagonal elements of 1, and off-diagonal elements of less than 0.1 (typically zeros). This feature is used in this sub-objective to update the FE model based on minimizing the difference between the value of one and the mean diagonal elements of the MAC matrix. When the difference is zero, there is a complete correlation between the two mode shapes; i.e., the FE model matches the current condition of the structure. Based on Equations (36)-(38), the overall objective function of the considered optimization problem can be written as:  
\[ \begin{split}\min F=\min&\left(\sum\limits_{i=1}^{N }\left(\frac{MSEn^{D}-MSEn_{i}}{MSEn_{i}}\right)+\sum\limits_{i=1}^{N}\left( \frac{MKEn_{i}^{D}-MKEn_{i}}{MKEn_{i}}\right)\right.\\ &\left.+(1-mean(diag(MAC)))\right).\end{split}  \tag{40}\]As observed in the literature [6; 7; 8; 9; 10; 11; 12; 13; 14; 15; 16; 17; 18; 19; 20; 21; 22; 23; 24; 25; 26; 27; 28; 29; 30; 31; 32], most simulated damage scenarios are modeled as a reduction of stiffness. The reduction of stiffness can explain most commonly occurring structural damage types, such as corrosion, cracks, defects, erosion, structural element links, bolt loosening, etc. Usually, mass-related structural damage is not studied as it is mostly associated with loss of material or structural elements, which can be easily observed without a damage identification algorithm.