# A Novel Reliability Analysis Approach under Multiple Failure Modes Using an Adaptive MGRP Model
## 4 Case Studies
### A Numerical Series System Analysis  
To verify the applicability and effectiveness of the proposed method, a numerical series system is adopted, and it is compared with a variety of classical combined reliability analysis methods. This numerical series system is composed of four components whose  
Figure 3: Scatter plot of different sampling methods. (**a**) LHS; (**b**) RGS; (**c**) RMQGS. limit state functions are given in Equations (24)-(27) [24]. Among the Equations, \(x_{1}\) and \(x_{2}\) follow the standard normal distribution with their independent and identically distributed. \[G_{1}(x_{1},x_{2})=3+\frac{(x_{1}-x_{2})^{2}}{10}-\frac{(x_{1}+x_{2})}{\sqrt{2}} \tag{24}\]  
\[G_{2}(x_{1},x_{2})=3+\frac{(x_{1}-x_{2})^{2}}{10}+\frac{(x_{1}+x_{2})}{\sqrt{2}} \tag{25}\]  
\[G_{3}(x_{1},x_{2})=(x_{1}-x_{2})+\frac{6}{\sqrt{2}} \tag{26}\]  
\[G_{4}(x_{1},x_{2})=(x_{2}-x_{1})+\frac{6}{\sqrt{2}} \tag{27}\]  
The initial samples are obtained by the RMQGS method according to the range of design variables in Equations (24)-(27), and the true response value of the performance function is solved to build the initial MRGP model, which then calculates the reliability under multiple failure modes and checks its accuracy. Alternate sample points are screened through different active learning functions and convergence criteria, and the best sample points are determined to update the MRGP model, so as to achieve accurate calculation and analysis of reliability under multiple failure modes. In this example, the initial samples of RMQGS are set to 6, and the perfect MRGP model is fitted by the proposed method and its effectiveness is compared by the methods of MRGP + U, MRGP + H, and MRGP + EFF, as shown in Figure 4.  
Figure 4 depicts the fitting accuracy of the MRGP model with different active learning functions. In Figure 4, the black dots are MC simulation sample points; the blue dots are initial samples obtained by LHS or RMQGS; the green lines are the fitting curve of the MRGP model; the red lines are the true limit state surface; the yellow dots are updated samples. It can be seen from the figure that the fitting effects of the four methods are almost the same, which is close to the true limit state surface. However, due to the difference between the active learning function and the termination criterion, the number of true performance function calls and the time-consuming are different. Judging from the number of updated samples, the proposed method only needs 4 updated samples to achieve a convergence state, while the methods of MRGP + U, MRGP + H, and MRGP + EFF need 9, 15, and 12 updated samples, respectively. In terms of sample distribution, the initial samples of the RMQGS method have the advantage of randomness and uniformity. It is the basis for ensuring a high-precision initial MRGP model. Then, although the updated samples of the MRGP + U method and the proposed method are close to the limit state surface, the updated samples obtained by the MRGP + U method are concentrated. The updated samples of the MRGP + H method and MRGP + EFF method are far from the limit state surface, and the sample quality is lower than that of the proposed method. Table 1 shows the reliability comparison results of five different methods under multi-failure models, in which the MC simulation results are used as the accurate solution to verify the reliability accuracy of the other four methods. In Table 1, \(N_{\text{calls}}\) denotes the number of calls to the true performance function; \(P_{f}\) denotes the failure probability, \(\beta\) denotes the reliability index; \(\delta_{\beta}\) denotes the percentage error. From Table 1, it is known that the proposed method has high accuracy and efficiency in dealing with the reliability problem of multiple failure modes. Although the reliability results of the proposed method are similar to those of the MRGP + U method, the number of function calls is reduced by 30%.