## 3 Choosing Optimal Parameters for SIPSO Algorithm
### Choosing Optimal Parameter a  
Ordinary parameters of the SIPSO algorithm are set as follows: the number of particle swarms is set as 30 and the maximum number of generations is set as 200. In Equation (5), \(\omega_{max}\)= 0.9 and \(\omega_{min}\)= 0.4. To choose the optimal value, parameter \(a\) is set as 1, 1.5, 2, and 2.5, respectively. Each case is calculated 50 times and the average value is taken for comparison as shown in Table 2.  
Table 2 tabulates the mean optimal function value of the three criterion functions with different values of parameter \(a\). Figure 2 shows the influence of parameter \(a\) on the SIPSO algorithm. As shown in Table 2, with the Rosenbrock criterion function, the parameter \(a\) \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Function** & \(a=1\) & \(a=1.5\) & \(a=2\) & \(a=2.5\) \\ \hline Rosenbrock & 1.0061 & 1.0048 & **0.9995** & 0.9982 \\ Rastrigin & 2.1169 \(\times\) 10\({}^{-2}\) & \(-\)3.7135 \(\times\) 10\({}^{-4}\) & **1.0281 \(\times\) 10\({}^{-9}\)** & 1.2809 \(\times\) 10\({}^{-7}\) \\ Griewank & 0.0673 & 0.0656 & 0.0654 & **0.0638** \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Mean optimal function value of three criterion functions with different values of parameter \(a\). corresponds to the optimal function value. As shown in Figure 2a, with the Rosenbrock criterion function, the convergence curve of parameter \(a=2\) is located at the bottom and converges rapidly, which is obviously better than other values. When the Rastrigin criterion function is adopted, the parameter \(a=2\) also corresponds to the optimal function value, and the convergence curve of parameter \(a=2\) is much better than other values, as shown in Table 2 and Figure 2b. As shown in Table 2, with the Griewank criterion function, the parameter \(a=2.5\) corresponds to the optimal function value and the parameter \(a=2\) corresponds to the second-best function value. As shown in Figure 2c, with the Griewank criterion function, when the number of generations is less than 50, the convergence curve of parameter \(a=2\) is located at the bottom and converges rapidly, which is better than other values. In addition, when the number of generations is more than 50, the convergence curves of parameter \(a=2\) and parameter \(a=2.5\) are very close and it is hard to tell which one is better. Figure 2: Influence of parameter \(a\) on SIPSO algorithm. (**a**) Curve of Rosenbrock function; (**b**) curve of Rastrigin function; (**c**) curve of Griewank function. The most direct and effective method to select the optimal parameter value \(a\) of the SIPSO algorithm is to apply it to structural damage identification. In this study, each damage identification case was calculated five times and the average value was taken as the final identification result. The calculation principle is consistent with this in subsequent studies. Figure 3 shows the structural damage detection results under different values of parameter \(a\) of the SIPSO algorithm. There were two damaged sections, namely element 8 and element 17. The damage index is set as 10% and the noise level is also set as 10%. As shown in Figure 3, the SIPSO algorithm with four values of parameter \(a\) can accurately identify the damage location and degree of damage in element 8 and element 17, but the misjudgment results of each parameter are significantly different. When parameter \(a\) = 1, there are four misjudged-as-damaged sections and the identification result is the worst among the four parameters. When parameter \(a\) = 1.5 and \(a\) = 2.5, there are two misjudged-as-damaged sections. When parameter \(a\) = 2, there is only one misjudged-as-damaged section and the identification result is the best among the four parameters. After comprehensively considering the results of the three criterion functions and structural damage identification, parameter \(a\) = 2 is selected as the optimal parameter \(a\) of the SIPSO algorithm.