# An Efficient Reliability Method with Multiple Shape Parameters Based on Radial Basis Function
## 1 Introduction
In fact, the RBF model has many merits such as applicability to high dimensional problems, exponential convergence speed, and a high prediction accuracy [24]. Hence, in order to make up for the above-mentioned inability of RBF model, multiple predictions by RBF are performed to obtain prediction variance. Some of the similar studies are as follows: [PERSON] et al. [25] presented two adaptive models named CVRBF-MCS and ARBFM-MCS for the structural reliability analysis, in which different RBF models are launched for multiple predictions by applying different subsets of DoE or different kernel functions. Furthermore, by conducting multiple predictions, [PERSON] et al. used the efficient global optimization (EGO) as the learning function for sequential sampling based on the RBF model and achieved satisfactory results for some benchmark problems [26]. In addition, for sequential sampling based on the RBF model, using the MPP of the current surrogate model as the newly added sample is also an effective approach. [PERSON] et al. [27] conducted a performance measure approach (PMA) to find the MPP of the RBF model as the new sample point of the sequential sampling. To improve the efficiency, the adaptive RBF model presented by [PERSON] can find \(2m\) (\(m\) is the dimension of random input) additional newly added sample points in each iteration by using points near the MPP [28]. Differing from [PERSON]'s and [PERSON]'s methods, the RBF-GA proposed by [PERSON] et al. used MCS rather than the reliability index to calculate the structure failure probability, and the potential MPP was found using the genetic algorithm (GA) [29]. However, because of the \"most probable\" characteristic of the MPP, sequential sampling methods must utilize specialized distance control strategies to avoid the clustering problem, when taking MPP as the new sample for sequential sampling. Although the above-mentioned RBF-based adaptive models have proven to be promising for structural reliability analysis, they do not consider the effect of shape parameters on the search for new samples for sequential sampling. In fact, predictions using the RBF model with different shape parameters can vary widely [24]. To address this problem, an adaptive model, which based on a combination of the RBF model and MCS, is proposed. For the proposed model (MSRBF-MCS), different shape parameters are used to launch multiple predictions. The effect of the shape parameters on sequential sampling is considered by calculating the weighted variance of the multiple predictions. Furthermore, the 2-norm of the MC population in the standard normal space is investigated using the proposed learning function to ensure new samples are in the sensitive region (see the sensitive region in Figure 1). Moreover, to make the clustering problem of new samples difficult, a distance control strategy that encourages a larger minimum inter-sample distance is adopted. Finally, the proposed method is verified with three benchmark problems and a particle engineering problem with good results. The remainder of this paper is organized as follows. Section 2 shows the relevant basic theories of the RBF model. The proposed adapted metamodel is proposed in Section 3. In Section 4, the proposed model is validated by three academic examples and one practical example. Finally, the concluding remarks are given in Section 5.