# An Efficient Reliability Method with Multiple Shape Parameters Based on Radial Basis Function
## 4 Case Study
### Dynamic Response of a Nonlinear Oscillator  
Example 2 is the dynamic response of an oscillator (a high-dimensional nonlinear problem), as shown in Figure 7. Its performance function is as follows:  
\[g(m,c_{1},c_{2},r,F_{1},t_{1})=3r-\left|\frac{2F_{1}}{m\omega_{0}^{2}}\sin(\frac {\omega_{0}t_{1}}{2})\right|,\ \omega_{0}=\sqrt{(c_{1}+c_{2})/2} \tag{24}\]  
where \(m,c_{1},c_{2},r,F_{1},t_{1}\) are the six independent random variables obeying the normal distribution. The mean values and standard deviations are shown in Table 3.  
The parameters for MSRBF-MCS are set as \(\varepsilon=10^{-5}\), \(n_{ini}=12,n_{mc}=10^{6}\). Then, example 2 is solved using MCRBF-MCS + G (\(\alpha=1\)), MCRBF-MCS + M (\(\alpha=0.1,\ 0.4,\ 0.7\) and \(1\)),  
\begin{table}
\begin{tabular}{c c c c} \hline \hline
**Model Description** & \(n_{call}\) & \(e_{pr}\)(\%) & \(\hat{P}_{f}\)(\(\times 10^{-3}\)) \\ \hline MCS & \(10^{6}\) & / & 4.416 \\ ARBFM – MCS (\(\alpha=0\),G + M) & 60.1 & 1.13 & 4.466 \\ CVRBF – MCS (\(\alpha=0\)) & 65.4 & 1.44 & 4.480 \\ RBF-GA & 33.3 & 1.10 & 4.367 \\ LIF & 40 & 1.72 & 4.340 \\ Proposed model: & & & \\ MCRBF + G, \(\alpha=1\) & 44.2 & 2.12 & 4.509 \\ MCRBF + M, \(\alpha=1\) & 36.3 & 1.43 & 4.479 \\ MCRBF + IM, \(\alpha=1\) & 36.5 & 19.47 & 3.556 \\ MCRBF + M, \(\alpha=0.1\) & 30.9 & 7.35 & 4.091 \\ MCRBF + M, \(\alpha=0.3\) & 40.8 & 3.23 & 4.273 \\ MCRBF + M, \(\alpha=0.5\) & 35.8 & 4.01 & 4.238 \\ MCRBF + M, \(\alpha=0.7\) & 39.2 & 1.90 & 4.332 \\ MCRBF + M, \(\alpha=0.9\) & 35.3 & 1.57 & 4.347 \\ \hline \hline \end{tabular} Note: The results of the model in this paper are the mean values of 10 simulations.  
\end{table}
Table 2: The results for example 1.  
\begin{table}
\begin{tabular}{c c c} \hline \hline
**Random Variables** & **Mean Value** & **Standard Deviation** \\ \hline \(m\) & 1 & 0.05 \\ \(c_{1}\) & 1 & 0.10 \\ \(c_{2}\) & 0.1 & 0.01 \\ \(r\) & 0.5 & 0.05 \\ \(F_{1}\) & 1 & 0.20 \\ \(t_{1}\) & 1 & 0.20 \\ \hline \hline \end{tabular}
\end{table}
Table 3: The mean values and standard deviations for example 2.  
Figure 7: Dynamic response of a nonlinear oscillator.  
and MCRBF-MCS + IM (\(\alpha=1\)). In addition, the results of example 2 from the proposed model are compared with those from some promising models such as C(\(\mathbf{X}\)) [23], ARBFM-MCS, CVRBF-MCS [25], and RBF-GA [29]. As one can see from Table 4, the model MCRBF-MCS + M (\(\alpha=1\)) provides the best outcomes (\(n_{call}=32.2\), \(e_{pr}=0.07\%\)) of all of the presented models. This result is slightly lower in accuracy than CVRBF-MCS (whose \(e_{pr}=0\)), but it has a smaller \(n_{call}\) than CVRBF-MCS. In terms of convergence curves (as shown in Figure 8), all the presented models converge rapidly with an acceptable prediction accuracy.  
\begin{table}
\begin{tabular}{c c c c} \hline \hline
**Model Description** & \(\mathbf{n_{call}}\) & \(\mathbf{e_{pr}}\)(\%) & \(\hat{\mathbf{P}}_{f}\)(\(\times 10^{-2}\)) \\ \hline MCS & \(10^{6}\) & / & 2.856 \\ ARBFM – MCS (\(\alpha=0\), G + M + IM) & 33.5 & 0.24 & 2.850 \\ CVRBF – MCS (\(\alpha=0\)) & 36.1 & 0 & 2.856 \\ RBF-GA & 20.0 & 1.22 & 2.891 \\ C(X) (\(\alpha=0.3\)) & 55.9 & 0.63 & 2.877 \\ Proposed model: & & & \\ MCRBF + G, \(\alpha=1\) & 40.4 & 1.65 & 2.081 \\ MCRBF + IM, \(\alpha=1\) & 43.5 & 0.18 & 2.851 \\ MCRBF + M, \(\alpha=0.1\) & 27.9 & 1.43 & 2.815 \\ MCRBF + M, \(\alpha=0.4\) & 43.0 & 0.74 & 2.835 \\ MCRBF + M, \(\alpha=0.7\) & 40.2 & 0.07 & 2.854 \\ MCRBF + M, \(\alpha=1\) & 32.2 & 0.07 & 2.854 \\ \hline \hline \end{tabular} Note: The results of the model in this paper are the mean values of 10 simulations.  
\end{table}
Table 4: The results for example 2.  
Figure 8: Convergence curves for different \(\alpha\) in example 2.