# An Improved Monte Carlo Reliability Analysis Method Based on BP Neural Network
## 4 Examples Verification
### Dynamic Response of Nonlinear Oscillator  
Example 3 involves a nonlinear undamped single-degree-of-freedom system, as shown in Figure 7, and its structural performance function is defined as:  
\[G(X)=3R-\left|\frac{2F_{1}}{M\omega_{0}^{2}}\text{sin}\left(\frac{\omega_{0}T_ {1}}{2}\right)\right|,\omega_{0}=\sqrt{\frac{C_{1}+C_{2}}{M}} \tag{15}\]  
where \(X=[R,\ F_{1},\ M,\ C_{1},\ C_{2},\ T_{1}]\) is a random variable with normal distribution, and the distribution parameters are shown in Table 10. Aiming at the reliability problem of six  
\begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Method** & \(\mathbf{N_{\text{FEM}}}\) & \(\mathbf{N_{\text{call}}}\) & \(\mathbf{P_{f}}\) & \(\mathbf{COV_{P_{f}}}\) & \(\mathbf{\varepsilon}\) \\ \hline MCS & \(5\times 10^{7}\) & \(5\times 10^{7}\) & \(2.85\times 10^{-5}\) & 0.0265 & \\ IS & \(19+10^{4}\) & \(5\times 10^{7}\) & \(2.86\times 10^{-5}\) & 0.0239 & 0.35\% \\ AK-IS & \(19+7\) & \(5\times 10^{7}\) & \(2.86\times 10^{-5}\) & 0.0239 & 0.35\% \\ AK-MCS & \(27\) & \(4\times 10^{6}\) & \(2.88\times 10^{-5}\) & 0.0932 & 1.04\% \\ BP-MCS & \(124\) & \(2\times 10^{7}\) & \(3.22\times 10^{-5}\) & 0.0387 & 11.49\% \\ Improved BP-MCS & \(77\) & \(24,155\) & \(2.86\times 10^{-5}\) & 0.0265 & 0.35\% \\ \hline \hline \end{tabular}
\end{table}
Table 8: Reliability calculation results of two-dimensional explicit polynomials.  
\begin{table}
\begin{tabular}{c c c} \hline \hline
**Method** & \(\mathbf{N_{\text{call}}}\) & \(\mathbf{t\ (s)}\) \\ \hline Polynomial & \(5\times 10^{7}\) & 945.7 \\ Direct BP-MCS & \(5\times 10^{7}\) & — \\ Improved BP-MCS & 24,155 & 1335.0 \\ \hline \hline \end{tabular}
\end{table}
Table 9: Example 2 calculation time of failure probability by different methods.  
-dimensional random variables, this paper uses Direct BP-MCS and Improved BP-MCS to calculate the failure probability based on the same BP model and compares the results with those of different methods. The sampling times of MCS are all \(7\times 10^{4}\), and the calculation results are shown in Table 11.  
Table 11 shows that compared with the classical methods such as Kriging, AK-MCS, and AK-MCS-U, the Improved BP-MCS algorithm has greatly reduced the number of sample sets, only 93 groups. The number of calls of the BP model is reduced from \(7\times 10^{4}\) to 37,957, which shows that the improvement effect of the algorithm is reduced when the failure probability is large, and the relative error of the failure probability is 1.55%.  
\begin{table}
\begin{tabular}{c c c c} \hline \hline
**Random Variables** & **Distribution** & **Mean** & **Standard Deviation** \\ \hline \(R\) & Normal & 0.5 & 0.05 \\ \(F_{1}\) & Normal & 1 & 0.2 \\ \(M\) & Normal & 1 & 0.05 \\ \(C_{1}\) & Normal & 1 & 0.1 \\ \(C_{2}\) & Normal & 0.1 & 0.01 \\ \(T_{1}\) & Normal & 1 & 0.2 \\ \hline \hline \end{tabular}
\end{table}
Table 10: Random variable distribution of nonlinear oscillator.  
Figure 6: The Improved BP-MCS sample set from example 2.  
Figure 7: A nonlinear oscillator.  
The calculation time of failure probability of explicit structure performance function, Direct BP-MCS, and Improved BP-MCS are shown in Table 12. When the failure probability is large, the calculation time of the explicit structure performance function is only 3.8 s, while the calculation time of MCS for the implicit structure performance function constructed by BP model is greatly increased, but the Improved BP-MCS still has a large degree of reduction compared with the Direct BP-MCS.