# An Adaptive Kriging-Based Fourth-Moment Reliability Analysis Method for Engineering Structures
## 4 Discussion  
In this paper, we propose an adaptive Kriging-based fourth-moment method (AK-FM) for an efficient reliability analysis under limited sample conditions. When the distribution of the random variable is difficult to determine due to limited available samples, we adopt the fourth-moment method to construct a fourth-order moment reliability index. To establish a surrogate model between the moments of the random variables and the moments of the  
\begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Time Point** & **Method** & **Initial Sample Size** & **Increased Sample Size** & **U Function Value** & **Number of State Function Predictions** & **Reliability** \\ \hline \multirow{3}{*}{t = 1040 h} & AK-FM & 10 & 4 & 3.2139 & 1000 & 0.7862 \\  & AK-FM & 10 & 4 & 3.2139 & 10,000 & 0.7907 \\  & AK-MCS & 10 & 4 & 3.2139 & 10\({}^{6}\) & 0.7955 \\ \hline \multirow{3}{*}{t = 1050 h} & AK-FM & 10 & 4 & 4.3016 & 1000 & 0.3143 \\  & AK-FM & 10 & 4 & 4.3016 & 10,000 & 0.3269 \\  & AK-MCS & 10 & 4 & 4.3016 & 10\({}^{6}\) & 0.3288 \\ \hline \multirow{3}{*}{t = 1060 h} & AK-FM & 10 & 4 & 3.2908 & 1000 & 0.0416 \\  & AK-FM & 10 & 4 & 3.2908 & 10,000 & 0.0442 \\ \cline{1-1}  & AK-MCS & 10 & 4 & 3.2908 & 10\({}^{6}\) & 0.0448 \\ \hline \hline \end{tabular}
\end{table}
Table 4: Reliability comparison at different time points.  
Figure 7: Comparison of smooth time-varying fatigue reliability curves of 6016.  
complex state function, we employ the Kriging method and update the Kriging model based on the U function. We use the adaptive Kriging model to generate state function prediction samples, approximating the first four moments of the state function, and utilize the fourth-moment method for the reliability analysis. The main contribution of our method is that it overcomes the challenge of computing or incurring high costs for the moments of complex state functions, which is a limitation of existing fourth-moment methods. Additionally, our method inherits the efficiency advantage of the moment method over the simulation method compared to the existing AK-MCS method, and it can be applied in time-varying reliability analysis. Several numerical examples demonstrate that our proposed method achieves a comparable computational accuracy to the MCS method or the AK-MCS method, even with an inadequate sample size. Furthermore, it significantly improves the computational efficiency for a reliability analysis of engineering structures with complex state functions.  
Methodology, S.E.; software (MATLAB R2017a and Abaqus/CAE 2020), S.E.; validation, S.E.; formal analysis, S.E.; investigation, S.E.; resources, S.E.; data curation, B.X.; writing--original draft preparation, S.E.; writing--review and editing, S.E.; visualization, S.E.; supervision, S.E.; project administration, Y.W.; funding acquisition, F.L. All authors have read and agreed to the published version of the manuscript.  
This research was supported by the National Key Research and Development Program of China (Grant No. 2019 YFB2004400) and the National Key Laboratory of Science and Technology on Helicopter Transmission (Grant No. HTL-O-21K02).  
Not applicable.  
The original contributions presented in the study are included in the article, further inquiries can be directed to the corresponding author.  
The authors declare no conflicts of interest.