## 6 Numerical Experiments
### Results of Crack Identification  
After trial and errors, PSO was adopted with the swarm size of 50, the maximum swarm of 100, and acceleration constants of 1.5, and criterions of \(C_{1}\geq 0.99\) and \(C_{2}\geq 0.95\) were used to determine if the updating process should be stopped. Figure 9 and Table 3 are the update process and identification results of the under-test samples in Figure 7. Especially, in Figure 9b, sample 6 was selected as a representative case to show the \(\mathbf{Y_{*}(x^{*})}\) evaluated at the optimal solution \(x^{*}\) in current iteration compared with \(\mathbf{Y_{Target}}\) of actual parameters. As the samples in the initial sampling plan and the samples added to update the surrogate model have the resolution of 2 elements in crack location and \(0.05D\) in crack depth, when the crack location and depth identification errors fall in [\(-\)2,2] and [\(-\)0.05,0.05] respectively, the identification results are considered to be accurate. From Figure 9, one can see all the samples were identified quite effectively. The samples located in the inside area were identified through updating for no more than 3 times, while for samples at the boundary, numbered 2, 3, and 6, needed updating for 4 to 5 times. As for the identification accuracy, from Table 3, one can see that all the crack parameters of the under-test samples were accurately identified, except the crack location of sample 1, which has been identified as the 32 th element. The error is thought to be acceptable, considering the resolution of crack locations. In addition, only after one updating step, the update process was stopped. The identification accuracy can be improved if more severe criterions are set, but it will lead to more updating steps. So, there is a \begin{table}
\begin{tabular}{c} **Crack** \\ **Times** \\ **(C1,C2)** \\ \end{tabular} \\ \hline
1 & 34 & 0.1 & 32 & 0.1 & 1 & (1, 0.96) \\
2 & 46 & 0.05 & 46 & 0.05 & 4 & (1, 1) \\
3 & 50 & 0.35 & 50 & 0.35 & 5 & (1, 1) \\
4 & 28 & 0.25 & 28 & 0.25 & 2 & (1, 1) \\
5 & 20 & 0.4 & 20 & 0.4 & 2 & (0.99, 0.99) \\
6 & 12 & 0.3 & 12 & 0.3 & 5 & (1, 1) \\
7 & 42 & 0.2 & 42 & 0.2 & 3 & (1, 1) \\
8 & 38 & 0.45 & 38 & 0.45 & 2 & (1, 1) \\
9 & 16 & 0.15 & 16 & 0.15 & 3 & (1, 1) \\
10 & 24 & 0 & 28 & 0 & 1 & (1, 1) \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Crack identification results of the under-test samples. Figure 9: Update process of the under-test samples. (**a**) Identification process for all samples; (**b**) identification process and corresponding value of identified feature for sample 6.  
trade-off between identification accuracy and efficiency. In view of all the identification results and the error factors, it can be concluded that the proposed crack identification method based on Kriging surrogate model performs quite well for the above unpolluted samples. To study the robustness of the identification method, each sample was now added with noise. The noise-polluted response \(\mathbf{y}_{\text{N}}\) can be expressed as [61]:  
\[\mathbf{y}_{\text{N}}=\mathbf{y}+L_{\text{N}}\sigma_{\mathbf{y}}\mathbf{s} \tag{20}\]  
where \(\text{L}_{\text{N}}\) is a number within \((0,1)\) that represents the noise level, which was selected as 5% in this work; \(\sigma_{\mathbf{y}}\) is the standard deviation of original response \(\mathbf{y}\) after noise is added; and \(\mathbf{s}\) is an N-length vector of normally distributed random numbers with zero mean and unit variance. Figure 10 and Table 4 show the update process and identification results of the under-test samples with 5% noise. Compared with the update process in Figure 9, from Figure 10 one can see that much more updating times are required for the samples at the boundary numbered 2, 3 and 6 with 5% noise, while the updating times for the non-boundary area are more or less the same. On the identification accuracy of the results in Table 4, there were some errors in samples 2 and 8.