# An Adaptive Kriging-Based Fourth-Moment Reliability Analysis Method for Engineering Structures
## 3 Results
### Reliability Analysis with Explicit State Function  
In order to verify the effectiveness of the method proposed in this paper, four types of explicit nonlinear state functions are set up by considering two-parameter input variables \((x,y)\) as follows:  
1. Exponential: \(g_{a}(x,y)=e^{x}-y\);
2. Power law: \(g_{b}(x,y)=x^{2}-y\);
3. Trigonometric: \(g_{c}(x,y)=cosx-siny\);
4. Segmented functional: \(g_{d}(x,y)= \begin{array}{ll}x-siny&x\geq 0\\ -x-siny&x<0\end{array} \)  
Based on the above four nonlinear state functions, using the method proposed in this paper, 1000 groups of \(x,y\sim N(0,1)\) are selected to construct the input sample set, and the initial training set is selected to train the adaptive Kriging agent model by selecting the first 25 groups; the corresponding Kriging-predicted response surfaces and training points of the four types of state functions are shown in Figure 2, and it can be seen that the response surfaces of the various functions are well fitted. The response surfaces corresponding to the exponential, power, and trigonometric state functions are depicted in (a), (b), and (c), respectively. It is evident that these response surfaces exhibit a relatively smooth nature. Additionally, the response surface of the segmented state function, illustrated in (d), aligns well with the sample points despite the significant local variability of the surface. The reliability results calculated using the AK-FM method proposed in this paper are compared with those calculated using the MCS method with a total number of samples of \(10^{6}\), as shown in Table 1. Since the MCS method is calculated directly by corresponding to the explicit state function, it can be recognized as an exact solution. Most of the explicit state functions in the reliability analysis of mechanical products can be represented by the above four nonlinear functions, so the above results are sufficient to show the accuracy and validity of the proposed method applied to the explicit state function reliability analysis problem. As can be seen from Table 1, on the one hand, the two methods do not have much error in the calculation results for four different types of state functions, but the proposed method requires a significantly smaller sample size than the MCS method. On the other hand, for the continuous state functions of exponential, power law, and trigonometric type, the number of updates of the Kriging model is smaller, and the model's training speed is very fast; for the segmented state functions, the Kriging model needs more updates, which indicates that more samples are needed to achieve a certain prediction accuracy for the complex state functions, even though the number of calls of the state functions in this paper is still much less than that of the MCS method. An intriguing question is whether the AK-FM proposed in this paper can obtain more accurate reliability results as the U function threshold increases. Therefore, taking the reliability calculation with the trigonometric state function as an example, different U constraint values are selected as the convergence judgment conditions and calculated using the method proposed in this paper; in order to ignore the inherent error between the MCS and the fourth-moment method, the calculated results of the fourth-moment method with a sample size of \(10^{6}\) are used as the theoretical true value for this comparison, and the comparison results are shown in Figure 3.  
A known conclusion is that the larger the U function, the better the Kriging model is fitted around the state function [34]. However, as can be seen in Figure 3, the error between the reliability results and the theoretical truth is not monotonically decreasing because an inaccurate Kriging model can also incorrectly produce reliability results when the value of the U function is less than a certain level (e.g., less than 2).