## 3 Choosing Optimal Parameters for SIPSO Algorithm
### Choosing Optimal Parameter \(b\)  
The ordinary parameters of the SIPSO algorithm are the same as the previous subsection. To choose the optimal value, parameter \(b\) is set as 0.1, 0.3, 0.5, and 0.8, respectively. Each case was calculated 50 times and the average value was taken for comparison, as shown in Table 3.  
Table 3 tabulates the mean optimal function value of the three criterion functions with different values of parameter \(b\). Figure 4 shows the influence of parameter \(b\) on the SIPSO algorithm. As shown in Table 3, with the Rosenbrock criterion function, the parameter \(b\) = 0.5 corresponds to the optimal function value. As shown in Figure 4a, with the Rosenbrock criterion function, the convergence curve of parameter \(b\) = 0.5 is located at the bottom and converges rapidly, which is obviously better than other values. When the Rastrigin criterion function is adopted, parameter \(b\) = 0.5 also corresponds to the optimal function value and the convergence curve of parameter \(b\) = 0.5 is much better than other  
Figure 3: Structural damage detection results under different values of parameter \(a\) of the SIPSO algorithm. (Double damages with 10% damage index under 10% noise level).  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Function** & \(b\) = 0.1 & \(b\) = 0.3 & \(b\) = 0.5 & \(b\) = 0.8 \\ \hline Rosenbrock & 1.0451 & 0.9900 & **0.9995** & 1.0335 \\ Rastrigin & 4.1431 \(\times\) 10\({}^{-8}\) & 1.5700 \(\times\) 10\({}^{-9}\) & **1.0281 \(\times\) 10\({}^{-9}\)** & 2.0728 \(\times\) 10\({}^{-8}\) \\ Griewank & 0.1279 & **0.0644** & 0.0654 & 0.0671 \\ \hline \hline \end{tabular}
\end{table}
Table 3: Mean optimal function value of three criterion functions with different values of parameter \(b\).  
values, as shown in Table 3 and Figure 4b. As shown in Table 3, with the Griewank criterion function, parameter \(b\) = 0.3 corresponds to the optimal function value and parameter \(b\) = 0.5 corresponds to the second-best function value. As shown in Figure 4c, with the Griewank criterion function, when the number of generations is less than 50, the convergence curve of parameter \(b\) = 0.5 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 \(b\) = 0.3 and parameter \(b\) = 0.5 are very close and it is hard to tell which one is better.  
Figure 5 shows the structural damage detection results under different values of parameter \(b\) of the SIPSO algorithm. There are 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 5, the SIPSO algorithm with four values of parameter \(b\) can accurately identify the damage location and degree of damage in element 8 and element 17, but the misjudgment results of each parameter are different. When the values of parameter \(b\) = 0.1,  
Figure 4: Influence of parameter \(b\) on SIPSO algorithm. **(a)** Curve of Rosenbrock function; **(b)** curve of Rastrigin function; **(c)** curve of Griewank function.  
\(b=0.3\), and \(b=0.8\), there are two misjudged-as-damaged sections. When parameter \(b=0.5\), 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 \(b=0.5\) is selected as the optimal parameter \(b\) of the SIPSO algorithm. In this subsection, the optimal parameter \(a\) is adopted by default.