# A Novel Reliability Analysis Approach under Multiple Failure Modes Using an Adaptive MGRP Model
## 4 Case Studies
### A Numerical Parallel System Analysis  
A parallel system with multiple failure regions is considered to identify the efficiency and accuracy of the proposed method. The reliability problem is defined by Equations (28)-(31) [36]. \[P_{f}=\Pr[G_{1}(\boldsymbol{x})\leq 0\cap G_{2}(\boldsymbol{x})\leq 0\cap G_{3}( \boldsymbol{x})\leq 0] \tag{28}\]  
\[G_{1}(x_{1},x_{2})=8{x_{2}}^{2}-8{x_{1}}^{2}+\left({x_{1}}^{2}+{x_{2}}^{2} \right)^{2} \tag{29}\]  
\[G_{2}(x_{1},x_{2})=2{x_{1}}^{2}-2{x_{2}}^{2}+\left({x_{1}}^{2}+{x_{2}}^{2} \right)^{2} \tag{30}\]  
\[G_{3}(x_{1},x_{2})=8{x_{2}}^{2}-8{x_{1}}^{2}-\left({x_{1}}^{2}+{x_{2}}^{2} \right)^{2} \tag{31}\]  
\(x_{1}\) and \(x_{2}\) follow the normal distributions with zero mean and unit standard deviation, and the RMQGS method is employed to produce initial samples to build the initial MRGP model. The adaptive MRGP model is then built by the proposed update strategy and termination criterion and compared with the methods of MRGP + U, MRGP + H, and \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Method** & \(N_{\text{calls}}\) & \(P\)/\(10^{-3}\) & \(\beta\) & \(\delta_{\beta}\)/\% \\ \hline MC simulation & \(10^{6}\) & 4.52 & 2.6105 & – \\ Proposed method & 10 & 4.50 & 2.6121 & 0.0613 \\ MRGP + U & 15 & 4.45 & 2.6159 & 0.2069 \\ MRGP + H & 21 & 4.40 & 2.6197 & 0.3524 \\ MRGP + EFF & 18 & 4.35 & 2.6236 & 0.5018 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Results of example 4.2 with different methods. Figure 4: MRGP models and its samples distribution for the series system. (**a**) MRGP + U; (**b**) MRGP + H; (**c**) MRGP + EFF; (**d**) Proposed method. MRGP + EFF, as shown in Figure 5. Finally, the accuracy and efficiency of different methods for the results of reliability analysis are compared in Table 2.  
Figure 5 illustrates the active learning process upon which the MRGP models are built with four different methods. In Figure 5, the initial samples (blue dots) generated by the RMQGS method can cover most of the design space to be efficient, while the samples generated by the LHS method are more concentrated. In general, the MRGP + U method and the proposed method can achieve the sample points added in the vicinity of the limit state, while the updated samples of the MRGP + H method and MRGP + EFF method have high uncertainty. In terms of overall fitting accuracy, the proposed method is lower, but it has high fitting accuracy for the failure domain and requires the least number of sample points, which is beneficial to the reliability analysis. The results in Table 2 illustrate the efficiency and accuracy of each method, for which \(10^{6}\) samples of an MC simulation population are classified. An MC simulation is performed on the true limit state surface with a population of \(10^{6}\) samples and taken as the reference for the probability of multiple failure models \(P_{f}\). From Table 2, it is known that the proposed method requires 27 calls to the composite performance function and leads to a relatively good estimation probability of failure. However, the MRGP + U method involves an excessive number of calls (123) to the composite performance function and leads to a \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Method** & \(N_{\text{calls}}\) & \(P_{f}10^{-3}\) & \(\beta\) & \(\delta_{\beta}\)/\% \\ \hline MC simulation & \(10^{6}\) & 0.19242 & 0.8690 & – \\ Proposed & 27 & 0.19260 & 0.8684 & 0.0690 \\ MRGP + U & 123 & 0.19340 & 0.8654 & 0.4143 \\ MRGP + H & 51 & 0.19230 & 0.8695 & 0.0575 \\ MRGP + EFF & 52 & 0.19180 & 0.8713 & 0.2647 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Results of example 4.3 with different methods. Figure 5: MRGP models and its samples distribution for the parallel system. (**a**) MRGP + U; (**b**) MRGP + H; (**c**) MRGP + EFF; (**d**) Proposed method. relatively poor estimation result. The cause of this imprecision may be that the reliability method cannot effectively be sampled in the failure domain or the quality of sample points is low.