## 2 Cluster-Based Sample Inheritance Statement
### Sample Inheritance  
As aforementioned, quantities of iterations are necessary to promote the closeness of the surrogate model to the real limit state surface when investigating the reliability analysis based on the Kriging model. The mean value of random variables and test points are selected as the checking point and the test point, respectively, when first fitting. The reliability index and checking points can be obtained via model fitting, utilizing test points. Furthermore, the next iteration will be started, selecting the checking point as the new sampling center, and then determining whether the variation coefficient of the reliability index obtained from two iterations can satisfy the exit condition. Apparently, all generated test points have been abandoned, and the existing valid sample information is discarded as well, resulting in low efficiency when calculating the reliability index.  
Taking the above-mentioned gaps into account, we devise a sample inheritance thinking innovatively, implying that part of the test points with high value for fitting will be selected to retain the iteration process following a certain rule, rather than discarding all the generated information. It can promote the model convergence efficiency to a great extent via avoiding the loss of effective information as well as the repeated utilization of redundant information. Significantly, the selecting criterion of test points in the proposed approach is supported by the learning function, which is based on the characteristics of the Kriging model, indicating both the estimation and its error of the predicted response. The calculation of the reliability index is prone to be affected by the model's judgment on the point when the response is close to 0, and the possibility of the response symbol error increases with the increase in error. Therefore, the learning function is the absolute value of the ratio of the response estimate to its error, which can be expressed as follows:  
\[U(S^{*})=\left|\frac{\mu(S^{*})}{\sigma(S^{*})}\right|, \tag{10}\]  
where \(S^{*}\) denotes the test point set. Followed by the proposed approach, the test points oriented to constructing surrogate models become more and more valuable, improving the fitting accuracy and convergence speed noticeably.
the uneven allocation between the sampling center and the edge of the sample space is prone to arise if the sample space is divided by the standard deviation of random variables. Therefore, the clustering method [40] is introduced in this paper. Cluster the newly generated test points and divide them into several clusters. Then, calculate the learning function values based on the model obtained in the last iteration; subsequently, several test points with high learning value are selected into the sample pool. The clustering approach is elaborated as follows, shown in Figure 2.  
**Step 1**: Select \(K\) as the initial centroids;  
**Step 2**: Assign sampling point, \((s_{1}^{*},s_{2}^{*},\ldots,s_{i}^{*})\), to the centroid to form clusters, based on the Euclidean distance between the sampling point and the centroid;  
**Step 3**: Calculate the mean position of the sampling points in each cluster; the calculation converges if the distance between the new centroid and the original centroid, represented by \(Arg\underset{j=1}{\overset{K}{\sum}}\underset{x\in N_{j}}{\sum}\Big{\|} \mathbf{X}-\ u_{j}\Big{\|}^{2}\), is less than the threshold.