## 5 Numerical Example
### Series System with Two Failure Regions  
The efficacy of the proposed method is evaluated in the case involving multiple failure regions. To illustrate this, a series system with three design points is considered, and the corresponding limit-state function is defined as per the formulation presented in [38,46]. \[g(x_{1},x_{2})=\min\biggl{\{} \begin{matrix}(c-1)-x_{2}+\exp(-x_{1}^{2}/10)+(x_{ 1}/5)^{4}\\ c^{2}/2-x_{1}x_{2}\end{matrix} \biggr{\}} \tag{40}\]  
where \(x_{1}\) and \(x_{2}\) denote uncorrelated standard normal random variables and \(c\) is an integer parameter. This limit-state function is known to have three design points: \(\mathrm{x}_{1}^{*}=(0,c)^{T}\), \(\mathrm{x}_{2}^{*}=(c/\sqrt{2},c/\sqrt{2})^{T}\), and \(\mathrm{x}_{3}^{*}=(-c/\sqrt{2},-c/\sqrt{2})^{T}\). In this study, various traditional learning functions, including the \(U_{LF}\) function in Equation (31) [35], expected feasibility function (EFF) [47], and constrained min-max (CMM) function [40], are employed to construct a Kriging model and estimate the failure probability of the series system. Their performances are compared with the results obtained from the proposed HAK approach. As part of the assessment, EFF and \(U_{LF}\) are employed both with and without \(K\)-means clustering algorithms, serving as active learning functions for performance evaluation. Figure 7 illustrates the final experiment designs and Kriging predictions using several learning algorithms, each initiated with two randomly generated initial DoE points (Case A and Case B). In Case A, the EFF and CMM approaches encounter challenges in accurately establishing the Kriging model, leading to imprecise failure probability estimates, possibly attributed to the presence of multiple failure points. In contrast, the proposed HAK approach adeptly constructs the Kriging model, demonstrating improved accuracy. \(U_{LF}\) exhibits relatively better performance when compared to EFF and CMM in this scenario. In Case B, both HAK and the Kriging model with the \(U_{LF}\) have estimate failure probabilities close to the reference values from MCS. The Kriging model with EFF and CMM identifies both failure regions but exhibits a less precise estimate of failure probability. Moreover, as shown in Figure 7c,e, Kriging models based on \(U_{LF}\) and EFF without the \(K\)-means clustering algorithm demonstrate less accurate predictions of the limit-state function compared to those with the \(K\)-means clustering scheme. Table 2 presents the estimated failure probabilities by the proposed HAK method, MCS, and metamodel-based importance sampling (Meta-IS) [38]. \begin{table}
\begin{tabular}{c c c c c} \hline
**Case** & **Method** & \(\mathbf{P_{f}}\) & \(\Delta_{\mathbf{P_{f}}}\) & \(\mathbf{\beta}\) & \(\mathbf{N_{eall}}\) \\ \hline \multirow{4}{*}{\(c=4\)} & FORM & \(1.31\times 10^{-4}\) & - & 3.651 & 192 \\  & SORM & \(7.89\times 10^{-5}\) & - & 3.778 & 204 \\  & MCS & \(1.51\times 10^{-4}\) & 0.037 & 3.615 & 5 \(\times 10^{6}\) \\  & AK-MCS & \(1.37\times 10^{-4}\) & 0.049 & 3.6393 & 42 \((10^{6})\) \\  & HAK & \(1.57\times 10^{-4}\) & 0.023 & 3.604 & 25 \\ \hline \multirow{4}{*}{\(c=5\)} & FORM & \(2.07\times 10^{-6}\) & - & 4.604 & 349 \\  & SORM & \(1.05\times 10^{-6}\) & - & 4.743 & 361 \\ \cline{1-1}  & MCS & \(2.08\times 10^{-6}\) & 0.102 & 4.618 & 5 \(\times 10^{7}\) \\ \cline{1-1}  & AK-MCS & - & - & - & \\ \cline{1-1}  & HAK & \(2.25\times 10^{-6}\) & 0.030 & 4.587 & 25 \\ \hline \end{tabular}
\end{table} 
Table 1: _Cont._ \begin{table}
\begin{tabular}{c c c c} \hline
**Method** & \(\mathbf{P_{f}}\) & \(\Delta_{\mathbf{P_{f}}}\) & \(\mathbf{N_{eall}}\) \\ \hline HAK & \(3.51\times 10^{-3}\) & 0.018 & 50 \\ MCS & \(3.47\times 10^{-3}\) & 0.017 & \(10^{6}\) \\ Meta-IS [38] & \(3.54\times 10^{-3}\) & \(<\)0.05 & 644 \\ \hline \end{tabular}
\end{table} 
Table 2: Comparative analysis of results obtained using HAK and various other reliability analysis methods for the series system example.