# An Efficient Reliability Method with Multiple Shape Parameters Based on Radial Basis Function
## 5 Conclusions  
In this paper, an adaptive reliability analysis method called MSRBF-MCS is proposed. MSRBF-MCS utilizes multiple shape parameters to generate different RBF models for multiple predictions. Then, the weighted mean and standard deviation of the multiple predictions is used as the estimation of the surrogate model and the measurement of local uncertainty, respectively. Additionally, a distance control function \(dc(X)\) that is likely to avoid the clustering problem and the \"not being located in a sensitive region\" problem of sample points is presented. The performance of MSRBF-MCS is further investigated with three benchmark problems. The results indicate the following:  
1. Compared with MSRBF-MCS + G and MSRBF-MCS + IM, the performance of MSRBF-MCS + M is often more stable and accurate;
2. The proposed method has good stability, i.e., for different adjustment coefficients, kernel functions, or systems with different failure probabilities, the proposed method tends to provide acceptable results;
3. Compared with models in the literature, the model proposed in this paper has the characteristics of a high prediction accuracy and efficiency.  
Funding acquisition, Y.G.; Methodology, P.Y.; Writing--original draft, W.D.; Writing--review & editing, J.M. All authors have read and agreed to the published version of the manuscript.  
This research was funded by Natural Science Foundation of China (Grant No. 11572233), and Pre-Research Foundation (Grant No. 61400020106) as well as the Fundamental Research Funds for the Central Universities.  
\begin{table}
\begin{tabular}{c c c c} \hline \hline  & **Model Description** & \(n_{call}\) & \(e_{pr}(\%)\) & \(\hat{P}_{f}(\times 10^{-2})\) \\ \hline \(q\)’s coefficient of & MCS & \(1.5\times 10^{4}\) & / & 3.128 \\ variation equals to 0.05 & RBF-LHS & 500 & / & 0 \\  & MCRBF + M, \(\alpha=0.5\) & 89 & 0.12 & 3.124 \\ \(q\)’s coefficient of & MCS & \(1.5\times 10^{4}\) & / & 17.589 \\ variation equals to 0.10 & RBF-LHS & 500 & 35.5 & 27.292 \\  & MCRBF + M, \(\alpha=0.5\) & 78 & 0.39 & 27.401 \\ \(q\)’s coefficient of & MCS & \(1.5\times 10^{4}\) & / & 26.737 \\ variation equals to 0.15 & RBF-LHS & 500 & 50.3 & 40.113 \\  & MCRBF + M, \(\alpha=0.5\) & 73 & 0.5 & 26.872 \\ \hline \hline \end{tabular} Note: The results of the model in this paper are the mean values of 10 simulations.  
\end{table}
Table 6: The results for example 3.  
\begin{table}
\begin{tabular}{c c c} \hline \hline
**Random Variables** & **Mean Value** & **Coefficient of Variation** \\ \hline Uniform load \(q\) (MPa) & 0.49 & 0.05, 0.1, 0.15 \\ Elastic modulus of carbon fiber \(E_{c}\) (MPa) & \(2.1\times 10^{5}\) & 0.05 \\ Elastic modulus of epoxy resin \(E_{c}\) (MPa) & \(7.3\times 10^{4}\) & 0.05 \\ Elastic modulus of glass fiber \(E_{g}\) (MPa) & \(8.7\times 10^{4}\) & 0.05 \\ Poisson’s ratio of carbon fiber \(\mu_{c}\) & 0.307 & 0.05 \\ Poisson’s ratio of epoxy resin \(\mu_{c}\) & 0.22 & 0.05 \\ Poisson’s ratio of glass fiber \(\mu_{g}\) & 0.22 & 0.05 \\ \hline \hline \end{tabular}
\end{table}
Table 5: The mean values and standard deviations for example 3.