## 4 Structural Damage Identification Based on SIPSO Algorithm
### Double Damages Identification Case  
In this condition, there are two damaged sections located at element 8 and element 17. The damage index is set as 10% and 40% and the noise level is set as 5% and 10%, respectively.  
As shown in Figure 8a and Table 6, when the damage index is 10% and the noise level is set as 5%, the PSO algorithm can identify the damages at element 8 and element 17 except for seven obvious misjudged-as-damaged sections. In this condition, the SNPSO algorithm has two misjudged-as-damaged sections and the SAPSO algorithm has three misjudged-as-damaged sections. The identification result of the SIPSO algorithm is the best, with only one misjudged-as-damaged section at element 2.  
As shown in Figure 8b, when the damage index is set as 10% and the noise level is set as 10%, the PSO algorithm can identify the damages at element 8 and element 17 except for eight misjudged-as-damaged sections. In this condition, the SNPSO algorithm has two misjudged-as-damaged sections and the SAPSO algorithm has five misjudged-as-damaged sections. The SIPSO algorithm is the best with only one misjudged-as-damaged section at element 2.  
\begin{table}
\begin{tabular}{c c c c c c c} \hline \hline
**Condition** & **Damage Index** & **Noise Level** & **PSO** & **SNPSO** & **SAPSO** & **SIPSO** \\ \hline
6 & 10\% & 5\% & 7 & 2 & 3 & **1** \\
7 & 10\% & 10\% & 8 & 2 & 5 & **1** \\
8 & 40\% & 5\% & 3 & 1 & 3 & **1** \\
9 & 40\% & 10\% & 3 & 1 & 3 & **1** \\ \hline \hline \end{tabular}
\end{table}
Table 6: Numbers of misjudgments in damage detection of four PSO-based algorithms with double damages.  
\begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Condition** & **Damage Index** & **Noise Level** & **PSO** & **SNPSO** & **SAPSO** & **SIPSO** \\ \hline
1 & 10\% & 0 & 2 & **0** & 0 & **0** \\
2 & 10\% & 5\% & 3 & **1** & 1 & **1** \\
3 & 10\% & 10\% & 4 & **1** & 4 & **1** \\
4 & 40\% & 5\% & 1 & **1** & 1 & **1** \\
5 & 40\% & 10\% & 2 & **1** & 2 & **1** \\ \hline \hline \end{tabular}
\end{table}
Table 5: Numbers of misjudgments in damage detection of four PSO-based algorithms with single damage.  
Figure 8: Double damages detection results of four PSO-based algorithms: (**a**) 10% damage index with 5% noise level; (**b**) 10% damage index with 10% noise level; (**c**) 40% damage index with 5% noise level; (**d**) 40% damage index with 10% noise level.  
As shown in Figure 8c, when the damage index is set as 40% and the noise level is set as 5%, the PSO algorithm and the SAPSO algorithm can identify the damages at element 8 and element 17 except for three misjudged-as-damaged sections. The SNPSO algorithm and the SIPSO algorithm are the best with only one misjudgment element. As shown in Figure 8d, when the damage index is set as 40% and the noise level is set as 10%, the PSO algorithm and the SAPSO algorithm have three misjudged-as-damaged sections, and the SNPSO algorithm and the SIPSO algorithm have one misjudged-as-damaged section.  
Considering the numerical simulation results of the four conditions with double damage sections, the identification results of the PSO algorithm are the worst while those of the SIPSO algorithm are the best among the four PSO-based algorithms. When the SIPSO algorithm is used for double damage sections identification, the number of misjudgment elements is only one in all conditions. Moreover, the SNPSO algorithm and the SAPSO algorithm have some improvements compared with the PSO algorithm.