## 6 Numerical Application
### Computational Efficiency in Reliability Analysis of Continuum Structure
It is important to highlight that the failure probability and computational cost obtained through AK-MCS reflect the results achieved using the Kriging model with 100 DoE points obtained during the enrichment process, reaching the prescribed maximum added points for all cases in this study. In contrast, the proposed method identified the Kriging model with approximately 38 DoE points. Consequently, the increase in computational cost is not significantly influenced by the mesh sizes in the proposed framework. It is worth noting that, for relatively small mesh sizes, SORM and AK-MCS exhibit faster computational times, given that the active learning process in the proposed method, involving the initial DoE, is relatively expansive. However, as the mesh size increases, resulting in a substantial rise in the computational cost for finite element analysis, there is an overall increase in computation time. Nevertheless, in the proposed method, once the Kriging model is established and trained through the hybrid active learning process, the reliability evaluation process becomes computationally effective and efficient. Figure 13: (**a**) A discretized continuum beam structure, depicting its geometry, loading, and boundary conditions, and (**b**) illustration of the distribution of random variables, specifically the three groups \((n_{E}=3;E_{1},E_{2},E_{3})\) of modulus of elasticity, within the beam. To examine the influence of material uncertainty on the failure probability, various material groups (\(E_{i},i=1,...,5\)) are introduced. Specifically, five beam structures, discretized with the same mesh using 1000 polygonal elements, are considered, each assigned a  
Figure 14: (**a**) Illustration of the assignment of random variables (\(E_{1},E_{2},E_{3}\)) to polygonal elements. (**a**) \(N_{elem}=500\), (**b**) \(N_{elem}=2000\), (**c**) \(N_{elem}=5000\), and (**d**) \(N_{elem}=20,000\). Figure 15: (**a**) Comparison of failure probabilities, \(P_{f}\), obtained by the proposed method (HAK), FORM, SORM, and AK-MCS for varying degrees of freedom (DoF). (**b**) Normalized computational time, with normalization performed with respect to the computation time for SORM with 5000 elements. The horizontal axis (DoF) is plotted using a base-10 logarithmic scale, corresponding to 500, 1000, 2000, 3000, 5000, 10,000, and 20,000 polygonal elements, respectively. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Random Variables** & **Distribution** & **Mean** & **Moments** \\ \hline \(E_{i},\ i=1,...,n_{E}\) & Lognormal & 29,000 ksi & \((0.05\cdot i)\) - 29,000 ksi \\ \(F_{1}\) & Normal & 3 kips & 0.45 kips \\ \(F_{2}\) & Lognormal & 4 kips & 0.6 kips \\ \(F_{3}\) & Gamma & 5 kips & 0.75 kips \\ \hline \hline \end{tabular}
\end{table} 
Table 8: Distribution type and parameters of the random variables considered for the reliability analysis of the continuum structure. different set of \(E_{i}\) values, as depicted in Figure 16a. The study utilizes the same distribution types for force and modulus of elasticity random variables in Table 8. Figure 16b illustrates the changes in failure probability corresponding to varying numbers of modulus of elasticity groups assigned to polygonal elements. As anticipated, the failure probability increases with the number of material property random variables. This escalation is primarily attributed to the higher standard deviations of modulus of elasticity for a greater number of elasticity groups. Furthermore, due to the nonlinearity of the limit-state function, FORM exhibits a less accurate estimate of the failure probability.