## 3 Illustrative Examples
### Structural/Mechanical Examples  
**Example 3**. : _A vehicle side-impact with the performance function is expressed as [26]_  
\[G_{3}=0.489X_{3}X_{7}+0.843X_{5}X_{6}-0.0432X_{9}X_{10}+0.0556X_{9}X_{11}+0.000786 X_{11}^{2}-0.75 \tag{26}\]  
_where \(X_{i}(i=1,\cdots,11)\) are normal variables, for \(X_{1}\sim X_{7}\), \(\mu=1\), and \(\sigma=0.05\), for \(X_{8}\sim X_{9}\), \(\mu=0,3\) and \(\sigma=0.006\), and for \(X_{10}\sim X_{11}\), \(\mu=0\), and \(\sigma=10\)._  
Results of the HL-RF, STM, HLRF-BFGS, RHL-RF, FSL, and the proposed AFSL algorithms are listed in Table 3. The iterative processes of reliability index are shown in Figure 7. The failure probability based on the MCS with \(10^{7}\) samples is \(1.24\times 10^{-4}\) (\(\beta=3.6643\)). According to Table 3 and Figure 7, the reliability indexes of all algorithms converge to 3.4975, except the HL-RF and HLRF-BFGS algorithms. The HL-RF algorithm produces the periodic solutions of {1.2957, 3.1757, 2.0351, 1.9180}. The number of iterations, \(G\)-evaluations and \(\
abla G\)-evaluations prove that the proposed AFSL algorithm is more efficient than other algorithms. **Example 4**. : _A ten-bar truss structure is shown in Figure 8. Based on the tip displacement at Node 3, its LSF is given by [27]_  
\[G_{4}=d_{0}-D \tag{27}\] \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Algorithms** & **Iterations** & \(G\)**-Evaluations** & \(\
abla G\)**-Evaluations** & \(\beta\) & \(P_{f}\) \\ \hline HL-RF & – & – & – & – & – \\ STM & 114 & 114 & 114 & 3.4975 & \(2.35\times 10^{-4}\) \\ HLRF-BFGS & – & – & – & – & – \\ RHL-RF & 120 & 239 & 120 & 3.4975 & \(2.35\times 10^{-4}\) \\ FSL & 64 & 64 & 64 & 3.4975 & \(2.35\times 10^{-4}\) \\ Proposed AFSL & 36 & 39 & 36 & 3.4975 & \(2.35\times 10^{-4}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Results of different algorithms for Example 3. Figure 7: Iterative processes of reliability index for Example 3. _where \(d_{0}=0.11\)(m) is the allowable value, and the maximum displacement \(D\) is computed by_  
\[D=\frac{PL}{A_{1}A_{2}E}\left\{ \begin{array}{l}4\sqrt{2}A_{1}^{3}\big{(}24A_{1 }^{2}+A_{3}^{2}\big{)}+A_{3}^{3}\big{(}7A_{1}^{2}+26A_{2}^{2}\big{)}\\ \hline 4A_{2}^{2}\big{(}8A_{1}^{2}+A_{3}^{2}\big{)}+4\sqrt{2}A_{1}A_{2}A_{3}(3A_{1 }+4A_{2})+A_{1}A_{3}^{2}(A_{1}+6A_{2})\\ \hline 4A_{1}A_{2}A_{3}\big{(}20A_{1}^{2}+76A_{1}A_{2}+10A_{3}^{2}+25\sqrt{2}A_{1}A _{3}+29\sqrt{2}A_{2}A_{3}\big{)}\\ \hline 4A_{2}^{2}\big{(}8A_{1}^{2}+A_{3}^{2}\big{)}+4\sqrt{2}A_{1}A_{2}A_{3}(3A_{1 }+4A_{2})+A_{1}A_{3}^{2}(A_{1}+6A_{2})\end{array} \right\} \tag{28}\]  
The relevant parameters and their distribution are described in Table 4.  
The results of the HL-RF, STM, HLRF-BFGS, RHL-RF, FSL, and the proposed AFSL algorithms are given in Table 5. It can be seen that all algorithms converge to the point \(X^{*}=(8.30\times 10^{-3},\,1.29\times 10^{-3},\,5.75\times 10^{-3},\,4.85 \times 10^{5},\,9.33,\,6.75\times 10^{4})\) with \(P_{f}=0.1258\) (\(\beta=1.1464\)). The result is close to \(P_{f}=0.1345\) (\(\beta=1.1055\)), which is computed by MCS with \(10^{5}\) samples. The number of iterations of the HL-RF and proposed AFSL algorithms is 7, which is more efficient than the other algorithms in this example. \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Variable** & **Distribution** & **Mean** & **Coefficient of Variation** & **Dimension** \\ \hline \(A_{1}\) & Normal & \(8.39\times 10^{-3}\) & 0.05 & m\({}^{2}\) \\ \(A_{2}\) & Normal & \(1.29\times 10^{-3}\) & 0.05 & m\({}^{2}\) \\ \(A_{3}\) & Normal & \(5.81\times 10^{-3}\) & 0.05 & m\({}^{2}\) \\ \(P\) & Gumbel & \(4.45\times 10^{5}\) & 0.1 & N \\ \(L\) & Normal & 9.14 & 0.05 & m \\ \(E\) & Log-normal & \(6.89\times 10^{4}\) & 0.05 & MPa \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Statistics of random variables in Example 4. Figure 8: Ten-bar truss structure.