## 4 Structural Damage Identification Based on SIPSO Algorithm
### Single Damage Identification Case  
In this condition, there is one damaged section located at element 17. The damage index is set as 10% and 40% and the noise level is set as 0, 5%, and 10%, respectively.  
As shown in Figure 7a, when the damage index is set as 10% and the noise level is set as 0, the PSO algorithm can identify the damage at element 17 except for two obvious misjudged-as-damaged sections at element 2 and element 5. On the contrary, the identification results of the SNPSO, the SAPSO, and the SIPSO algorithm are very accurate. Namely, the identification damage index of the three improved PSO algorithms is 10%, which is equal to the actual damage index. In addition, none of these three algorithms has a misjudged-as-damaged section.  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Function** & **PSO** & **SNPSO** & **SAPSO** & **SIPSO** \\ \hline Rosenbrock & 1.1758 & 1.0075 & 1.0016 & **0.9995** \\ Rastrigin & 5.9698 \(\times\) 10\({}^{-2}\) & 1.0281 \(\times\) 10\({}^{-9}\) & 2.1207 \(\times\) 10\({}^{-2}\) & **7.8939 \(\times\) 10\({}^{-10}\)** \\ Griewank & 0.3926 & 0.0654 & 0.0688 & **0.0651** \\ \hline \hline \end{tabular}
\end{table}
Table 4: Mean optimal function value of three criterion functions with different PSO-based algorithms.  
Figure 6: Comparison of convergence characteristics of four PSO-based algorithms. (**a**) Curve of Rosenbrock function; (**b**) curve of [PERSON] function; (**c**) curve of Griewank function.  
Figure 7: _Cont._As shown in Figure 7, when the damage index is set as 10% and the noise level is set as 5%, the PSO algorithm can identify the damage at element 17 except for three misjudged-as-damaged sections at element 2, element 8, and element 9. On the contrary, the identification results of the SNPSO, the SAPSO, and the SIPSO algorithm are much better with only one misjudged-as-damaged section at element 2.  
As shown in Figure 7, when the damage index is set as 10% and the noise level is set as 10%, the PSO algorithm and the SAPSO algorithm can identify the damage at element 2.  
Figure 7: Single damage detection results of four PSO-based algorithms: (**a**) 10% damage index without noise; (**b**) 10% damage index with 5% noise level; (**c**) 10% damage index with 10% noise level; (**d**) 40% damage index with 5% noise level; (**e**) 40% damage index with 10% noise level.  
17 except for four misjudged-as-damaged sections at element 2, element 3, element 11, and element 12. On the contrary, the identification results of the SNPSO and the SIPSO algorithm are much better with only one misjudged-as-damaged section at element 2.  
As shown in Figure 7d, when the damage index is set as 40% and the noise level is set as 5%, all four PSO-based algorithms have one misjudged-as-damaged section at element 2. As shown in Figure 7e and Table 5, when the damage index is set as 40% and the noise level is set as 10%, the identification results of the SNPSO and the SIPSO algorithm are the best with only one misjudged-as-damaged section at element 2. On the contrary, the other two PSO-based algorithms have two misjudged-as-damaged sections.  
Considering the numerical simulation results of the five conditions with a single damage section, the identification results of the PSO algorithm are the worst while the three improved PSO algorithms are all effective compared with the PSO algorithm. In addition, as shown in Figure 7 and Table 5, the SNPSO algorithm and the SIPSO algorithm are the best to be used for single damage section identification, with only one misjudgent element in four conditions containing noise interference.