## 6 Numerical Application
### Computational Efficiency in Reliability Analysis of Continuum Structure  
A continuum structure is considered as a continuous medium for analytical purposes, especially in the field of structural engineering. This concept is prominently applied when modeling and analyzing deformable bodies like beams, plates, and solids, treating them as if they were infinitely divisible. Typically, the continuum structure is discretized using finite elements for numerical analysis, with the accuracy of the finite element analysis being influenced by the mesh size. To evaluate the computational efficiency of the proposed framework in the reliability analysis of a continuum structure with varying mesh sizes, the MBB beam discretized with polygonal elements is investigated. Unstructured polygonal meshes are constructed using Voronoi tessellations, and polygonal finite element analysis is conducted for computing structural response and reliability analysis. Figure 13 illustrates the discretization of the beam with 500 polygonal elements (\(N_{elem}=500\)), including its loading and boundary conditions. Consider that the point loads \(F_{1},F_{2}\), and, \(F_{3}\) are random variables following different types of marginal distributions, as summarized in Table 8. Furthermore, three random variables (\(E_{1},E_{2},E_{3}\)), each following the lognormal distribution with the different second moment, are randomly assigned to polygonal elements shown in Figure 13b. Consequently, each finite element is associated with one modulus of elasticity random variable. During the assembly of the stiffness matrix in the finite element setting, each element's stiffness matrix is influenced by the assigned modulus of the elasticity random variable. The three groups of polygonal elements, categorized based on the assigned random variable of modulus of elasticity, have a similar number of elements. For  
Figure 12: Probability of failure for the truss structure problem under varying levels of imposed noise variances. (**a**) AK-MCS, (**b**) proposed HAK. a comprehensive computational study, the continuum domain is discretized with different numbers of polygonal elements (500, 1000, 2000, 3000, 5000, 10,000, 20,000 finite elements). The three moduli of elasticity are randomly but uniformly assigned over the entire domain, ensuring a similar distribution of elements for each modulus. Figure 14 illustrates the distribution of the modulus of elasticity. The limit-state function is expressed in terms of compliance, which is a scalar value obtained from the product of the displacement vector and the applied force vector:  
\[g(\mathbf{x})=\Big{\{}C_{max}-\mathbf{u}(\mathbf{x})^{T}\mathbf{f}(\mathbf{x}) \leq 0\Big{\}};\ \ \mathbf{K}(\mathbf{x})\mathbf{u}(\mathbf{x})=\mathbf{f}(\mathbf{x}) \tag{43}\]  
where \(\mathbf{K},\mathbf{f}\) and \(\mathbf{u}\) denote the stiffness matrix, force vector, and displacement vector, respectively. The threshold value \(C_{\max}\) is set to 1.0 in this problem. Figure 15 presents a comparison of failure probabilities obtained by the proposed method (HAK), FORM, SORM, and AK-MCS. Three independent HAKs and AK-MCSs were performed, and the failure probabilities obtained were averaged. An increase in the failure probability is observed across all approaches. Notably, as the number of finite elements increases in the linearized elasticity problem, more accurate analysis results are achieved. Specifically, the compliance increases and converges to a more accurate value with finer mesh discretization. Consequently, an increase in the failure probability is anticipated. However, it is essential to acknowledge that the distribution of random variables of modulus of elasticity for each different number of finite elements (as shown in Figure 14) may also influence the failure probability, potentially impacting the results illustrated in Figure 15. Figure 15 illustrates the normalized computational time. The normalization is performed with time obtained using the SORM for \(N_{Elem}=5000\) (the degrees of freedom (DoF) is 19,320).