## 3 Illustrative Examples
### Structural/Mechanical Examples
\begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Algorithms** & **HL-RF** & **STM** & **HLRF-BFGS** & **RHL-RF** & **FSL** & **AFSL** \\ \hline \(A_{1}\)(m\({}^{2}\)) & \(8.30\times 10^{-3}\) & \(8.30\times 10^{-3}\) & \(8.30\times 10^{-3}\) & \(8.30\times 10^{-3}\) & \(8.30\times 10^{-3}\) & \(8.30\times 10^{-3}\) \\ \(A_{2}\)(m\({}^{2}\)) & \(1.29\times 10^{-3}\) & \(1.29\times 10^{-3}\) & \(1.29\times 10^{-3}\) & \(1.29\times 10^{-3}\) & \(1.29\times 10^{-3}\) & \(1.29\times 10^{-3}\) \\ \(A_{3}\)(m\({}^{2}\)) & \(5.75\times 10^{-3}\) & \(5.75\times 10^{-3}\) & \(5.75\times 10^{-3}\) & \(5.75\times 10^{-3}\) & \(5.75\times 10^{-3}\) & \(5.75\times 10^{-3}\) \\ \(P\)(N) & \(4.85\times 10^{5}\) & \(4.85\times 10^{5}\) & \(4.85\times 10^{5}\) & \(4.85\times 10^{5}\) & \(4.85\times 10^{5}\) & \(4.85\times 10^{5}\) \\ \(L\)(m) & \(9.33\) & \(9.33\) & \(9.33\) & \(9.33\) & \(9.33\) & \(9.33\) \\ \(E\)(MPa) & \(6.75\times 10^{4}\) & \(6.75\times 10^{4}\) & \(6.75\times 10^{4}\) & \(6.75\times 10^{4}\) & \(6.75\times 10^{4}\) & \(6.75\times 10^{4}\) \\ \(\beta\) & \(1.1464\) & \(1.1464\) & \(1.1464\) & \(1.1464\) & \(1.1464\) & \(1.1464\) \\ \(P_{f}\) & \(0.1258\) & \(0.1258\) & \(0.1258\) & \(0.1258\) & \(0.1258\) & \(0.1258\) \\ Iterations & 7 & \(115\) & \(127\) & \(20\) & \(9\) & \(7\) \\ \hline \hline \end{tabular}
\end{table} 
Table 5: Convergent results of different algorithms for Example 4. **Example 5**. : _A two-degree of freedom dynamic system is shown in Figure 9. According to the force capacity of the secondary spring, the LSF is expressed as [34]_  
\[G_{5}=F_{s}-K_{s}\times P\Big{(}E\Big{[}x_{s}^{2}\Big{]}\Big{)}^{1/2} \tag{29}\]  
_where \(F_{s}\) is the force, \(P=3\) is the peak factor and mean-square relative displacement of the secondary spring \(E\Big{[}x_{s}^{2}\Big{]}\) is computed by_  
\[E\Big{[}x_{s}^{2}\Big{]}=\frac{\pi S_{0}}{4\xi_{s}\omega_{s}}\Bigg{[}\frac{\xi _{a}\xi_{s}}{\xi_{p}\xi_{s}\Big{(}4\xi_{a}^{2}+\theta^{2}\Big{)}+\gamma\xi_{a }^{2}}\times\frac{\Big{(}\xi_{p}\omega_{p}^{3}+\xi_{s}\omega_{s}^{3}\Big{)} \omega_{p}}{4\xi_{a}\omega_{a}^{4}}\Bigg{]} \tag{30}\]  
_where \(S_{0}\) is intensity of the white noise, \(\gamma=M_{s}/M_{p}\), \(\omega_{p}=\sqrt{K_{p}/M_{p}}\omega_{s}=\sqrt{K_{s}/M_{s}}\), \(\omega_{a}=\Big{(}\omega_{p}+\omega_{s}\Big{)}/2\), \(\xi_{a}=\Big{(}\xi_{p}+\xi_{s}\Big{)}/2\) and \(\theta=\Big{(}\omega_{p}-\omega_{s}\Big{)}/\omega_{a}\). The information of eight random variables with lognormal distribution is given in Table 6._  
The number of iterations, \(G\)-evaluations, \(\
abla G\)-evaluations, and reliability index for all algorithms are listed in Table 7. The iterative processes of reliability index are shown in Figure 10. Using the MCS with \(10^{6}\) samples, the failure probability is obtained as \(5.04\times 10^{-3}\) (\(\beta=2.5733\)). The HL-RF algorithm has failed to converge. It yields the periodic solutions of {3.2644, 3.6503}. The other algorithms converge to the stable results as \(\beta=2.1002\), \(P_{f}=1.79\times 10^{-2}\) and \(X^{*}=\)(1.02, 9.98 \(\times\) 10\({}^{-3}\), 1.06, 1.04 \(\times\) 10\({}^{-2}\), 2.67 \(\times\) 10\({}^{-2}\), 1.16 \(\times\) 10\({}^{-2}\), 13.61, 104.19). However, the proposed AFSL algorithm is more efficient than the other algorithms. \begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline
**Random Variable** & \(M_{p}\) & \(M_{s}\) & \(K_{p}\) & \(K_{s}\) & \(\xi_{p}\) & \(\xi_{s}\) & \(F_{s}\) & \(S_{0}\) \\ \hline
**Mean** & 1 & 0.01 & 1 & 0.01 & 0.05 & 0.02 & 15 & 100 \\
**Standard** & 0.1 & 0.001 & 0.2 & 0.001 & 0.02 & 0.01 & 1.5 & 10 \\
**Deviation** & & & & & & & & \\ \hline \hline \end{tabular}
\end{table} 
Table 6: Statistics of random variables in Example 5.