# Structural Damage Identification Using a Modified Directional Bat Algorithm
## 2 Theoretical Background  
In the problem of structural damage identification based on the optimization algorithm, by adjusting the parameters of the finite element model, the response data of the finite element model is close to that of the actual model, so that the structural parameters of the actual model can be obtained. The key is to reduce the error between the measured data and the calculated data as much as possible. In this paper, the modal information data are used to construct the objective function, and the specific process is shown in Figure 1.  
It is well known that the eigenvalue equations of the structural finite element model are as follows:  
\[\Big{(}K-\omega_{i}^{2}M\Big{)}\phi_{i}=0 \tag{1}\]  
Figure 1: The damage identification problem is transformed into the optimization problem. where \(K\) is the global stiffness matrix, \(M\) is the global mass matrix, \(\omega_{i}\) is the \(i\)-th natural frequency and \(\phi_{i}\) is the corresponding mode shape. In this paper, the linear reduction of structural stiffness is used to simulate the structural damage. The damage of each structural element can be expressed by the damage factor of each element; thus, the stiffness matrix of the element is expressed as follows:  
\[K^{d}=\sum_{i=1}^{Nel}(1-\alpha_{i})k_{i}^{e} \tag{2}\]  
where \(k_{i}^{e}\) is the stiffness of the \(i\)-th element in the healthy structure, \(Nel\) is the total number of structural elements and \(K^{d}\) is the overall structural stiffness of the structure to be identified. \(\alpha_{i}\) is the damage degree of \(i\)-th element of the structure to be identified. When \(\alpha_{i}=0\), it means that the structure is healthy; \(\alpha_{i}=1\) means that the structure is completely damaged. In the structural damage identification, the objective function can reflect the difference between real measurement and simulation. Structural modal properties are often used as an index of objective function. The most common modal indicators are the relative error of natural frequency and the modal assurance criterion. Usually, the objective function only uses the first several order modal information of the structure for calculation. In this study, the natural frequency, mode shape and flexibility matrix of the structure are used to form the objective function. The error function of frequency is expressed as follows:  
\[MDLAC_{i}=\frac{\left|\omega_{i}^{m}-\omega_{i}^{e}\right|}{\omega_{i}^{m}} \tag{3}\]  
where \(\omega_{i}^{m}\) and \(\omega_{i}^{e}\) are the \(i\)-th order measured natural frequency and calculated natural frequency, respectively. The modal assurance criterion is used to check the correlation between modes. When the measured mode shape is closer to the calculated mode shape, the modal confidence is closer to 1, otherwise it is 0. The expression of the modal assurance criterion is as follows:  
\[MAC_{i}=\frac{\left|\phi_{i}^{m}\bullet\phi_{i}^{e}\right|^{2}}{\left|\phi_{i }^{m}\bullet\phi_{i}^{m}\right|\left|\phi_{i}^{e}\bullet\phi_{i}^{e}\right|} \tag{4}\]  
where \(\phi_{i}^{m}\) and \(\phi_{i}^{e}\) are the \(i\)-th measured mode shape and calculated mode shape, respectively. It is well known that the stiffness matrix and the flexibility matrix are reciprocal matrices. The decrease of structural stiffness will increase the corresponding flexibility, and the flexibility matrix is more sensitive to damage [40; 41; 42]. The expression of the flexibility assurance criterion is as follows:  
\[MACF_{i}=\frac{\left|F_{i}^{m}\bullet F_{i}^{c}\right|^{2}}{\left|F_{i}^{m} \bullet F_{i}^{c}\right|\left|F_{i}^{c}\bullet F_{i}^{c}\right|} \tag{5}\]  
where \(F_{i}^{m}\) and \(F_{i}^{c}\) are the \(i\)-th order measurement flexibility vector and calculation flexibility vector, respectively.