# An Improved Monte Carlo Reliability Analysis Method Based on BP Neural Network
## 2 Calculation Theory of the BP-MCS Method
### BP-MCS Calculation Method  
Monte Carlo Simulation (MCS) mainly generates a large number of sample points and determines whether sample points fail by calculating the structural performance function values of each sample point. Finally, the frequency of failure points in all sample points is taken as the failure probability of the structure, namely:  
\[\lim_{N_{\text{MC}}\rightarrow+\infty}\left|\frac{N_{f}}{N_{\text{MC}}}-P_{f} \right|<\varepsilon \tag{1}\]  
According to [PERSON]'s law of large numbers, when the total number of samples \(N_{\text{MC}}\) is large enough, the frequency of failure points converges to the failure probability \(P_{f}\), and the difference between the two is less than any small positive number \(\varepsilon\). According to the coefficient of variation (\(COV\)) test of failure probability, \(N_{\text{MC}}\) must be large enough to calculate a convincing and reasonable failure probability when MCS is used. The influence of \(N_{\text{MC}}\) on calculation accuracy is mainly controlled by the coefficient of variation [15]:  
\[COV_{P_{f}}=\sqrt{\frac{1-P_{f}}{P_{f}N_{\text{MC}}}} \tag{2}\]  
\(COV\) reflects the deviation of the calculated results from the actual values. The larger the \(N_{\text{MC}}\), the smaller the deviation and the more accurate the result. After the calculation of failure probability is completed, the coefficient of variation test should be carried out to ensure that there are enough sample points, and the general coefficient of variation can be taken as \(COV=0.05\)[15]. For actual engineering structures, the failure probability is generally very small, usually on the order of \(10^{-4}\) or even smaller [27]. According to Equation (2), at least \(4\times 10^{6}\) finite element analyses are needed to obtain a more accurate failure probability, and the calculation cost is very high.  
In the traditional BP-MCS method, when judging whether the sample point is failed, it is put into the BP model instead of the finite element model so as to shorten the calculation time of each sample point and improve the calculation efficiency. However, when using MCS to calculate the failure probability, it is still necessary to call the BP model to calculate at least \(4\times 10^{6}\) times. In general, the traditional polynomial response surface has an explicit structural performance function, so its failure probability calculation time is shorter. However, for the implicit structural performance function constructed by intelligent algorithms such as the BP neural network, the calculation time of failure probability is relatively long. With the increase of the MCS sample number, the calculation time also doubles, especially when the failure probability is small. Therefore, how to reduce the number of BP model  
Figure 1: BP neural network model.  
calls is the key to reducing the calculation time of BP-MCS failure probability, and to build a BP model near the failure surface with high prediction accuracy is the premise to improve the calculation accuracy of failure probability.