## 4 Building an Active-Learning-Based Reliability Analysis Platform
### Active-Learning Platform Construction and Reliability Analysis  
According to the previous subsection, active learning is an iterative process, where each iteration necessitates invoking the PCK metamodel and the learning function, and then solving the FEM. Therefore, the PCK metamodel function and the learning function are established in MATLAB, which are convenient to be called at any time. Iterations are facilitated by the \"while\" function. The active learning process with multi-software interaction is shown in Algorithm 1. The sample size in step 2 can be set to \(10^{6}\); steps 6-8 are automated batch processing of multi-software, where step 6 is pre-processing using ANSA to modify the model parameters to \(X_{\text{opt}}\), step 7 is solving using OptiStruct and step 8 is post-processing using META to export the response file. In step 13, the reliability analysis is performed based on the PCK metamodel. If the LF-based convergence criterion is used, the failure probability can be calculated after the metamodel update is complete. Algorithm 1 can be used as an active-learning reliability analysis platform, which is used to analyze the reliability of the bending and torsional stiffness of the BIW. The iterative process of the LF-based (U-learning function) convergence condition is illustrated in Figure 9. The initial DOE sample size is 20. Initially, the metamodels for bending and torsional stiffness are established. The error in the torsional stiffness metamodel is statistically found to be larger than in the bending stiffness metamodel, so the active-learning reliability analysis is firstly carried out for torsional stiffness, and the convergence condition is reached after iterating nine samples. Subsequently, 29 samples are used as initial samples to analyze the bending stiffness. After iterating through 2 additional samples, convergence is achieved, totaling 11 new samples used in the process. Figure 9: Iterative process of LF-based convergence criterion: (**a**) Active-learning iterative process for torsional stiffness; (**b**) Active-learning iterative process for bending stiffness. The iterative process based on bounds on \(P_{\text{f}}\) is shown in Figure 10. The initial sample size is still 20; after adding 47 new samples, the metamodel of torsional stiffness reaches convergence, and, using these 67 samples as the initial samples to update the metamodel of bending stiffness, as can be seen from Figure 10b, the accurate metamodel is established with only 1 new sample. The U-learning function is used as the convergence criterion and is not constrained by variance-based criteria (bounds on \(P_{\text{f}}\)). As a result, although the error interval of the failure probability does not converge effectively, it remains within approximately 10%, meeting the general accuracy requirement and reducing the number of iterations. Without the application of active learning, HyperStudy suggests conducting a DOE with 132 samples. Following the DOE, response surfaces are constructed. The \(R\)-square (coefficient of determination) for bending and torsional stiffnesses are 0.9998 and 0.9997, respectively, indicating high accuracy. Reliability analysis is performed using the MC method, and the average of the 20 simulations is used as the benchmark value for the failure probability of each response. The results are shown in Table 3. The failure probability of the bending stiffness \(P_{\text{b}}\) is \(1.53\times 10^{-4}\), and the failure probability of the torsional stiffness \(P_{\text{t}}\) is \(3.33\times 10^{-4}\). The results of the PCK metamodel are close to these, which verifies the accuracy  
Figure 10: Iterative process based on the bounds on \(P_{\text{f}}\) convergence criterion: (**a**) Active-learning iterative process for torsional stiffness; (**b**) Active-learning iterative process for bending stiffness. and efficiency of the scheme in this paper.