# An Efficient Reliability Method with Multiple Shape Parameters Based on Radial Basis Function
## 1 Introduction  
The reliability analysis of structures is of great importance for modern equipment and systems [1; 2; 3; 4; 5]. Generally, the reliability state of a structure is described by its performance function, i.e., the structure is safe when its performance function is greater than 0, fails when its performance function is less than 0, and is in its limited state when its performance function is equal to 0 [6; 7; 8; 9]. In practical engineering, the structural performance function is frequently very complex [10; 11]. To obtain a simpler alternative for the limit state function (LSF), the traditional reliability analysis methods carries out a first-order or second-order Taylor expansion at the most probable point (MPP), such as the first-order reliability mothed (FORM) [6; 12] and second-order reliability method (SORM) [13; 14]. Nevertheless, large prediction error occurs when the performance function is of great nonlinearity, as the failure probabilities obtained by FORM or SORM are subjected to the nonlinearity of the structural performance function. The Monte Carlo simulation (MCS) [15; 16] is frequently used in structural reliability analysis, because of its conceptual simplicity and irrelevance to the abovementioned nonlinearity. However, MCS is limited by the simulation (finite element analysis (FEA) in general) time, because a sufficiently large sample size is essential for the accuracy of MCS [16]. The surrogate model is a simpler approximation of the structure's response. Therefore, MCS with surrogate models can avoid the massive computational cost caused by multiple FEsimulations [17]. To further reduce the computational cost and ensure prediction accuracy, MCS is often combined with adaptive models. Different from the general metamodel, the adaptive model finds its training samples iteration by iteration. Generally, the framework of reliability analysis based on MCS with an adaptive model consists of four steps:  
1. Initialize the design of experiment (DoE);
2. Generate the metamodel according to DoE;
3. Perform MCS based on the metamodel to estimate the failure probability;
4. Check the stopping criterion: if satisfied, end the iteration, and output the failure probability; otherwise, add new samples into DoE through the learning function and go back to (2). Currently, there are mainly four available surrogate techniques, namely response surface methodology (RSM) [18], artificial neural networks (ANN) [19], support vector machines (SVM) [20], and the Kriging model [21]. Among them, the Kriging model is frequently employed in the above framework. This is because the prediction variance provided by the Kriging model can represent the local uncertainty of prediction to some degree. Many Kriging-based adaptive models have been proposed to address the reliability analysis problem. Specifically, [PERSON] et al. proposed AK - MCS + EFF and AK - MCS + U, in which the learning functions \(U(\mathbf{X})\) and \(EFF(\mathbf{X})\) aim to find sample points that are simultaneously close to the LSF and have a high local uncertainty [21]. Furthermore, by searching the line between the best sample point found by \(U(\mathbf{X})\) and its nearest sample point with different signs in the MC population, [PERSON] et al. [22] obtained a better new sample than that using \(U(\mathbf{X})\). Besides the prediction's value and variance, the location of the new sample is also an important factor to consider. To make the sample points prone to being located in the sensitive region, the learning function \(C(\mathbf{X})\) proposed by [PERSON] et al. takes the distance from the candidate sample point to the mean point into account, i.e., sample points with a smaller above-mentioned distance are easier to be selected for sequential sampling [23]. Unlike the Kriging model, the structural reliability research based on the RBF model is relatively less. This is because RBF model does not provide prediction variance.