# An Efficient Reliability Method with Multiple Shape Parameters Based on Radial Basis Function
## 3 The Proposed Approach: MSRBF-MCS
### Reliability Analysis Procedure Based on MSRBF-MCS  
In this section, based on the learning function in Equation (17), MSRBF-MCS is proposed to carry out the probabilistic reliability analysis. In order to present the framework of the method clearly, a simple system with sufficient nonlinearity is taken as a demonstration of the main steps of MSRBF-MCS. Assuming that the system's performance function and distribution of input variables are as follows:  
\[g(x_{1},x_{2})=\cos({x_{1}}^{2})-2x_{2}+5,\;x_{1}\;\text{and}\;x_{2}\;\sim N(0,1) \tag{21}\]  
Then, the steps of MSRBF-MCS approach and the solving process of this demo is as follows:  
1. Generate the initial DoE with Latin hypercube sampling (LHS). In the demo, \(n_{ini}=8\). 2. Initialize the number of iterations \(k=0\), the kernel function type, shape parameters, adjustment coefficient \(\alpha\), and stopping criterion constant \(\varepsilon\). For this demo, \(\mathbf{c}=[0.2,0.4,0.6,0.8]\), \(\alpha=0.5,\varepsilon=1\times 10^{-5}\), and the kernel function type is \"Multiquadric\". 3. Build RBF models with different shape parameters for multiple predictions, calculate the weight coefficient \(q_{i}\) by cross-validation, and obtain the metamodel's prediction (i.e., \(mean^{\text{\tiny{tr}}}\)) and its variance (i.e., \(sigma\)), and then formulate the learning function \(L(\mathbf{X})\). 4. Set \(k=k+1\) and perform MCS to obtain the predicted failure probability \(\hat{p}_{f}^{k}\) of the \(k\)-th iteration. For MCS in the demo, the initial population size is set as \(n_{mc}=1\times 10^{5}\). 5. Check if \(k\geq 2\) and the stopping criterion (Equation (18)) are satisfied at the same time. If satisfied, proceed to the next step; otherwise, pick the sample point from the MC population with a minimum \(L(\mathbf{X})\) and add it into DoE. Then, go back to step 3 and proceed. 6. Check the \(COV\). If \(COV<5\%\) proceed to next step; otherwise, enlarge the size of the MC population by \(n_{mc}=n_{mc}\times 10\) and go back to the step 6. 7. End the iteration and output \(\hat{p}_{f}^{k}\) as the predicted failure probability. All steps of MSRBF-MCS are also shown in Figure 2. In the demonstration, step 1 to step 3 of the solving process is shown in Figure 3. In Figure 3a, the hollow dots are generated by LHS as the initial DoE; the curves \(c=0.2\), \(c=0.4\), \(c=0.6\), and \(c=0.8\) are the LSFs of corresponding RBF models; and the curve \(mean^{w}\) is the LSF of the weighted combination of RBF models corresponding to \(c=0.2\), \(c=0.4\), \(c=0.6\), and \(c=0.8\). In order to formulate the \(mean^{w}\), cross-validation of the RBF models is performed, and the corresponding \(\textbf{PRESS}=[2.01,1.82,1.64,1.47]\), \(\textbf{q}=[0.21,\ 0.24,\ 0.26,\ 0.29]\). Figure 3b shows the contour map of the weighted prediction variance \(sigma^{w}\). As the predictions of the RBF model are unbiased at the known sample points, \(sigma^{w}\) increases significantly when the distance from DoE becomes bigger. This fact indicates that, if only \(sigma^{w}\) is examined, there will be a tendency for the learning function to select sample points far away from DoE, which may result in newly added sample points not being located in a sensitive area. Figure 2: The flowchart of MSRBF-MCS. The sequential sampling process (corresponding to step 6-9) of MSRBF-MCS is shown in Figures 4 and 5. In Figure 4, the triangular dots represent newly added sample points. Denote \(n_{add}\) as the number of newly added sample points, then Figure 4a-c shows the LSFs of the corresponding surrogate models, of which \(n_{add}=4,\ 8\) and \(12\), respectively. Figure 4d shows the convergence curve of the sequential sampling. The iteration ends when \(n_{add}=12\), then the times of calling performance function \(n_{call}=20\) (\(n_{call}=n_{add}+n_{ini}\)). From this we can see the fast convergence of the proposed approach. The \(\hat{p}_{f}\) by MSRBF-MCS and \(\hat{p}_{f\_mc}\) by direct MCS are \(0.2340\) and \(0.2323\), respectively.