## 2 Problem Description and General Idea of NPSSO for Sampling  
### Problem Description  
Performance function \(Z\) is the function of state variables \(X=\{x_{1},\,x_{2},\,\ldots,\,x_{n}\}^{\rm T}\) of a system. State variables can be the strength of a component in an engineering structure, the applied load, or any other quantifiable indicators that influence the safety of the system. For instance, performance function of a slope reliability analysis equals \(\rm FS(X)-1\), where \(\rm FS\) is the factor of safety of a slope obtained through slope stability analysis. In this case, state variables could be the strength parameters of each soil layer. As shown in Equation (1), \(Z<0\) denotes the failure of a slope. \(Z>0\) denotes that the slope is stable. \(Z=0\) means the slope is in a limit state. The region where \(Z=0\) was called LSF. LSF divides the variable space into the safe region and the unsafe region (as shown in Figure 1). As described above, finding LSF is the core of MCS-based reliability analysis. \[Z=G(x)=G(x_{1},x_{2},\ldots,x_{n})=\left\{ \begin{array}{l}<0\text{ failure}\\ =0\text{ limit}\\ >0\text{ safety}\end{array} \right. \tag{1}\]  
Figure 1: The reliability problem. For MCS-based reliability analysis, \(P_{f}\) was calculated by Equation (2):  
\[P_{f}=\int I[G(\mathbf{X})<0]f(\mathbf{X})d\mathbf{X}\approx\frac{1}{N_{\text{MCS}}}\sum_{i=1 }^{N_{\text{MCS}}}I[G(\mathbf{X})<0] \tag{2}\]  
where \(P_{f}\) is the probability of failure. \(I[\bullet]\) is an indicator function which equals 1 when \(\bullet>0\), and equals 0 when \(\bullet<0\); \(G(\mathbf{X})\) is the performance function; \(f(\mathbf{X})\) is the joint probability density function of all state variables; \(N_{\text{MCS}}\) is the MCS simulation times. In Equation (2), \(N_{\text{MCS}}\) should be big enough to ensure that the coefficient of variation of \(P_{f}\left(COV(P_{f})\right)\) is smaller than 10% [7]. \[COV(P_{f})=\sqrt{\frac{1-P_{f}}{N_{\text{MCS}}P_{f}}} \tag{3}\]  
In practice, the performance function of a system is mostly implicit. Complex and time-consuming numerical calculations, such as limit equilibrium methods, FEM-based strength reduction methods, or slip surface stress methods, are required. To address this issue, surrogate models were introduced to approximate the actual performance function in Equation (2), as shown in Equation (4). \[P_{f}\approx\frac{1}{N_{\text{MCS}}}\sum_{i=1}^{N_{\text{MCS}}}I[G(\mathbf{X})<0] \approx\frac{1}{N_{\text{MCS}}}\sum_{i=1}^{N_{\text{MCS}}}I[R(\mathbf{X})<0] \tag{4}\]  
Here, \(R(X)\) is the surrogate model of \(G(\mathbf{X})\). As shown in Equation (4), a good approximation quality of \(R(\mathbf{X})\) is the premise of the accuracy of the calculation of \(P_{f}\). The key to the quality of \(R(\mathbf{X})\) lies in DOEs. Further, according to Equation (2) and Figure 1, a good approximation of the LSF is enough for the accuracy of the calculation of \(P_{f}\). In this sense, sampling points near LSF is of the highest priority. SDOEs choose points that have higher uncertainty and near LSF. However, the candidate sampling points are fixed and dependent on the probability model of state variables. Therefore, SDOEs still suffer from the shortcomings as described in the Introduction. How to design a sampling method to make full use of the computational effort is still a problem to be solved. Based on the above analysis, sampling enough points on or near LSF is the key for an accurate calculation of \(P_{f}\). Moreover, distribution uniformity of the sampled points is crucial for calculation efficiency. Therefore, DOEs can be translated to find enough uniformly distributed points on or near LSF. According to Equation (1), LSF is the assemblage of all points that makes \(Z=0\). Therefore, the problem can finally be described as a minimum optimization problem as shown in Figure 2 and Equation (5):  
\[ \begin{array}{rl}\textbf{{Minimize}}&|G(\mathbf{x})|\\ \text{s.t.}&\mathbf{x}\in\Omega\end{array}