# A Novel Reliability Analysis Approach under Multiple Failure Modes Using an Adaptive MGRP Model
## 2 Random Moving Quadilateral Grid Sampling Method  
Stratified sampling aims to determine the random value of each variable according to the given variable interval and the number of levels of each variable. Assume that \(\Omega\) is a design space with the random variables \(\mathbf{x}\) and the number of samples is \(N\). The \(\Omega\) can be divided into \(i\) stratified regions by stratified sampling method, and the space \(p_{i}\) of each stratified region can be described by its probability.  
\[p_{i}=P\big{(}x_{ij}\in\Omega_{i}\big{)}\ \ j=1,2,\cdots,n_{i} \tag{1}\]  
where \(i\) is the number of stratified regions, \(j\) is the number of sample points, and \(x_{ij}\) is the random sample in the \(i\)th stratified region, \(j=1,2,\cdots,n_{i}\). If \(i=1\), the random sampling space will degenerate to the total space.  
A vital issue about randomness and uniformity needs to be addressed in the field of initial sample point quality based on the stratified sampling method. Although Latin hypercube sampling (LHS) [41; 42] and rectangular grid sampling (RGS) [19] can accurately select the initial sample points in the design space, it is prone to the situation that the sample points are too close or distributed on the boundary, resulting in the reduction of sampling efficiency or the loss of information. Thus, an RMQGS method is proposed based on RGS.  
If the dimension of the random variables of the system is \(m\), its value range can be defined as \(l_{j^{\prime}}\leq x_{j^{\prime}}\leq u_{j^{\prime}}\), \(j^{\prime}=1,2,\cdots,m\), and \(q_{j^{\prime}}\) represents the level of the \(j^{\prime}\)-dimensional variables. Then, the RMQGS method can be described as follows:  
Step 1: The original design space is randomly reduced according to the dimension and level of variables, which is provided as follows:  
\[\hat{u}_{j^{\prime}}=u_{j^{\prime}}-\alpha\frac{u_{j^{\prime}}-l_{j^{\prime}}} {q_{j^{\prime}}-1},j^{\prime}=1,2,\cdots,m \tag{2}\]  
where \(u_{j^{\prime}}\) and \(l_{j^{\prime}}\) are the upper and lower limits of the \(j^{\prime}\)-dimensional variables, and \(\alpha\) is random reduction coefficient related to the mean value of design variables, \(\alpha=1/(\Gamma+rand)\), \(rand\in[0,1]\), \(\Gamma\in[1,+\infty)\) is an integer.  
Step 2: Grid sampling in a randomly reduced design space, which is provided as follows:  
\[x_{j^{\prime}}^{(i)}=l_{j^{\prime}}+k_{j^{\prime}}^{(i)}\frac{\hat{u}_{j^{ \prime}}-l_{j^{\prime}}}{q_{j^{\prime}}-1} \tag{3}\]  
where \(k_{j^{\prime}}^{(i)}\) is the coefficient related to the levels of the design variable, \(k_{j^{\prime}}^{(i)}=1,2,\cdots,q_{j^{\prime}}-1\), and \(i\) is the number of sample points.  
Step 3: A random move is added to the column element \(x_{i}\) and row element \(x_{j}\) of the sample point \(x_{ij}\), which is provided as follows:  
\[d_{j^{\prime}}=\frac{\gamma_{ij^{\prime}}}{\lambda}\frac{u_{j^{\prime}}-l_{j^ {\prime}}}{q_{j^{\prime}}-1} \tag{4}\]  
where \(\gamma_{ij^{\prime}}\) is the coefficient of motion, which follows a uniform distribution, and \(\lambda\) is the coefficient of random moving, \(\lambda=\Gamma+rand\).  
Step 4: Any two adjacent random moving sample points should satisfy the minimum distance, which is provided as follows:  
\[d=\underset{1\leq j^{\prime}\leq m}{min}\left[\frac{u_{j^{\prime}}-l_{j^{ \prime}}}{2\left(q_{j^{\prime}}-1\right)}\left(1-\frac{1}{q_{j^{\prime}}-1} \right)\right] \tag{5}\]  
Step 5: The sample points with random moving in the randomly reduced design space are obtained, which is provided as follows:  
\[x_{j^{\prime}}^{(i)}=l_{j^{\prime}}+k_{j^{\prime}}^{(i)}\frac{\hat{u}_{j^{ \prime}}-l_{j^{\prime}}}{q_{j^{\prime}}-1}+\frac{\gamma_{ij^{\prime}}}{\lambda }\frac{u_{j^{\prime}}-l_{j^{\prime}}}{q_{j^{\prime}}-1} \tag{6}\]