# Structural Reliability Analysis Using Stochastic Finite Element Method Based on Krylov Subspace
## 4 Conclusions  
An SFE-KS method has been proposed in this work specifically for the static reliability analysis of structures with the aim of enhancing the calculation efficiency. The initial step involves preprocessing the static response equation, taking into account randomness, to diminish the condition number of the coefficient matrix. Following this, a Krylov subspace is constructed utilizing the preprocessed static response equation. Subsequently, the static displacement resulting from random sampling is formulated as a linear combination of subspace basis vectors, facilitating swift solutions. Ultimately, statistical metrics and failure probabilities are determined based on the static responses garnered from numerous random samples. Three numerical examples are presented to evaluate the performance of the proposed SFE-KS method. The classical finite element method, which is referred to as complete analysis, and the existing NSFE method are used as references to validate the computational accuracy and efficiency of the proposed method. The main conclusions drawn from the comparative study are as follows: (1) In terms of the computational efficiency, both the number of flops and the computation time demonstrate that the proposed method achieves the highest computational efficiency. For example 1, the calculation time of the SFE-KS method is about 17% of that of the complete analysis and 87% of that of the NSFE algorithm. For example 2, the calculation time of the SFE-KS approach is about 8% of that of the complete analysis and 68% of that of the NSFE algorithm. For example 3, the calculation time of the SFE-KS approach is about 25% of that of the complete analysis and \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Method** & **Complete Analysis** & **NSFE (with \(m\) = 4)** & **SFE-KS (with \(m\) = 2)** \\ \hline Time, s & 5.7450 & 2.8350 & 1.4110 \\ \hline \hline \end{tabular}
\end{table} 
Table 13: Comparison of calculation time for example 3 when the standard deviation is 0.2 times the mean. Figure 11: Vertical displacement of 3000 samples for example 3 when the standard deviation is 0.2 times the mean. 50% of that of the NSFE algorithm. (2) In terms of the calculation accuracy, the calculation results of the SFE-KS method are closer to the exact solution obtained using the complete analysis than those of the NSFE method. Especially for cases where the standard deviation of the random variable is relatively large, the proposed method exhibits significantly better computational accuracy than the NSFE method. For instance, when the standard deviation of the elastic modulus is 0.2 times the mean value, the computational errors of the failure probability using the NSFE and SFE-KS methods in example 1 are approximately 13.8% and 9.0%, while in example 3, the computational errors of the NSFE and SFE-KS methods are approximately 8.8% and 4.4%, respectively. For example 2, when the standard deviation of the cross-sectional area is 0.1 times the mean value, the computational errors of the failure probability using the NSFE and SFE-KS methods are approximately 5.8% and 1.9%, respectively. (3) Considering both the calculation time and accuracy, this new stochastic finite element method is more suitable for the reliability analysis of large-scale engineering structures, particularly in situations where discreteness is prominent. Conceptualization, J.H. and Q.Y. ; methodology, Q.Y. ; software, H.C.; validation, J.H., Q.Y. and H.C.; formal analysis, H.C.; investigation, J.M. ; resources, H.C.; data curation, J.M. ; writing--original draft preparation, Q.Y. ; writing--review and editing, J.H. ; visualization, Q.Y. ; supervision, project administration and funding acquisition, J.H. All authors have read and agreed to the published version of the manuscript. This research was supported by the Zhejiang Provincial Natural Science Foundation of China (No. LGF22E080021; No. LY24E080010).