###### Abstract
Then, active learning function (ALF) is used to add new sampled points from the large sample population iteratively. During iteration process, P\({}_{f}\) was calculated based on the continuously updated surrogate model through MCS until the iterative termination condition is met. Usually, the purpose of ALF is to select the points those are close to LSF or with greater uncertainty. This is the key in guaranteeing the accuracy and efficiency of the SDOEs method. The Kriging model is able to obtain both the predicted value and prediction variance. Therefore, the SDOEs-based Kriging-MCS method is often used in reliability analysis [1; 8; 13]. Using the knife variance method, [PERSON] et al. [14] successfully improve the application range of SDOEs method by adding the ability of solving the variance of non-sample points to surrogate model other than Kriging model. Furthermore, application of different models such as radial based neural network [15], Support vector regression model [16; 17; 18], and Bayesian regression network [19] were also investigated. Among them, the SDOEs method, that uses support vector machine (SVM) as a surrogate model, has attracted much attention. This is mainly because the calculation of P\({}_{f}\) can be regarded as a binary classification problem in which LSF serve as a classification boundary, while SVM is good at classification of sample data. Accordingly, in their study, [PERSON] et al. [20; 21] introduce the pool-based ALF and build the surrogate model based on the adaptive SVM model. Despite these substantial investigations, SDOEs still suffer from a shortcoming, which is that the candidate samples are randomly generated in each iteration. This makes the results rely heavily on the initial sample population [14]. Different initial population may lead to different spent time of sampling process and different results, which restricted the accuracy and stability of the method. To overcome the above problems, a novel optimization-based method was proposed in this study to produce candidate samples with high potential value during the iteration of SDOEs. Since LSF is a sub-area in variable space where value of the performance function equal zero, by taking the absolute value of the performance function and using it as the objective function, the process of finding LSF can be treated as a minimization problem with a continuous region as solution. Usually, based on the number of solutions, optimization problems can be categorized as unimodal optimization problems (UMOPs) and multimodal optimization problems (MMOPs). Optimization problems that have a continuous region as the optimal solution have barely been investigated. In our study, we named this kind of optimization problem as Regional-modal optimization problems (RMOPs) [22]. Furthermore, an innovative particle swarm optimization using a normal search pattern called normal search particle swarm optimization (NPSSO) was introduced for the sampling process to reach an accurate and comprehensive approximation of LSF. The proposed algorithm replaces the normally used point search pattern in optimization algorithms with a normal search pattern to consider the continuity of the solution. Further, a multi-strategy framework with three components was introduced to improve the performance of NPSSO, so as to increase the precision and integrity of the found LSF. The finding of LSF is regarded as an optimization process, therefore the sampling process is purposive, so the sampled points can approach LSF with higher accuracy. The remainder of the paper is organized as follows. In Section 2, problem definition and the related theories are stated. In Section 3, NPSSO-based sampling method for approximating the LSF is described. Experiments and results analysis are demonstrated in Sections 4 and 5. In Section 6, our conclusions are described.