## 2 Particle Swarm Optimization with Sigmoid Increasing Inertia Weight Algorithm
### Damage Simulation of Rigid Frame Structure  
The structure could be expressed as a dynamic system composed of stiffness matrix, mass matrix, and damping matrix. The equation of motion of the structure at unit \(i\) can be expressed as,  
\[(\mathbf{k}-\omega_{i}^{2}\mathbf{m})\Phi_{i}=[0] \tag{8}\]  
where \(\mathbf{k}\) and \(\mathbf{m}\) are the global stiffness matrix and the mass matrix of the structure, \(\Phi_{i}\) and \(\omega_{i}\) are the \(i\)-th mode shape and natural frequency of the structure. In general, for structural damage identification, the stiffness reduction in the structure is considered while the structural mass change is ignored, then,  
\[\mathbf{k}_{d}=\sum_{i}^{n}(1-x_{i})k_{ih} \tag{9}\]  
where \(k_{d}\) is the structural stiffness of the damaged system, \(k_{ih}\) is the \(i\)-th stiffness of the intact system, \(n\) presents the number of total finite elements, \(x\) is the \(i\)-th stiffness damage coefficient, the expression of damage element stiffness can be expressed as,  
\[x=1-\frac{k_{id}}{k_{ih}} \tag{10}\]  
where \(k_{ih}\) and \(k_{id}\) are the stiffness of the \(i\)-th element with intact and damaged conditions, respectively. Structural damage detection is performed to determine the location and degree of damage. In this paper, structural damage detection is transformed into a parameter constrained optimization problem. The damaged conditions are set and the dynamic parameters corresponding to the damaged condition are established. Then, the measured dynamic parameters of the structure are compared with the set damaged condition. If the structure is damaged, the stiffness of the structure will be decreased. In this study, the stiffness change of the structure is taken as the parameter representing structural damage identification. The finite element model of the damaged structure is established by the relationship ratio of the frequency and mode shape before and after the structure is damaged. The objective function is as follows,  
\[f(x)=\sum_{i=1}^{s}\Bigl{(}\Bigl{(}1-MAC\Bigl{(}\varnothing_{i}^{h},\varnothing _{i}^{d}\Bigr{)}\Bigr{)}+ER\Bigl{(}\omega_{i}^{h},\omega_{i}^{d}\Bigr{)}\Bigr{)} \tag{11}\]  
\[MAC\Bigl{(}\varnothing_{i}^{h},\varnothing_{i}^{d}\Bigr{)}=\frac{\left| \varnothing_{i}^{HT},\varnothing_{i}^{d}\right|^{2}}{\left|\varnothing_{i}^{ HT},\varnothing_{i}^{h}\right|\left|\left|\varnothing_{i}^{dT},\varnothing_{i}^{d} \right|} \tag{12}\]  
\[ER\Bigl{(}\omega_{i}^{h},\omega_{i}^{d}\Bigr{)}=\left|\frac{\omega_{i}^{h}- \omega_{i}^{d}}{\omega_{i}^{h}}\right|\times 100\% \tag{13}\]  
_MAC_--modal assurance criterion;\(ER\)--frequency change rate;  
\(\left(\mathscr{D}_{i}^{h},\varnothing_{i}^{d}\right)\)--vibration modes of the structure under intact and damaged conditions;  
\(\left(\omega_{i}^{h},\omega_{i}^{d}\right)\)--frequencies of the structure under intact and damaged conditions;  
\(s\)--modal number of structure;  
\(x\)--damage value of element stiffness. where the value of \(s\) is five, and the variation range of \(x\) is 0 to 0.9 in this study. Then, the structural damage detection can be described as a typical constrained optimization problem by obtaining the minimum value of Equation (11). The objective function can be solved iteratively by the proposed algorithm. For the PSO-based algorithm, the fitness value is to obtain the minimum value of Equation (11). When the objective function reaches the minimum value, the corresponding \(x\) can be obtained and defined as a damage index. As shown in Figure 1, a finite element model of a two-story rigid frame structure is adopted in this study. The height and width of each layer of the rigid frame is 1.41 m. The total element number of the structure is 18 and the degrees of freedom number (DOF) is 16. The structural physical parameters of the beam and the column are listed in Table 1.  
We assume that the nodes are rigid connections and that there is no need to consider the damage of the nodes.