###### Abstract  
In active learning reliability methods, an approximation of limit state function (LSF) with high precision is the key to accurately calculating the failure probability (\(P_{j}\)). However, existing sampling methods cannot guarantee that candidate samples can approach the LSF actively, which lowers the accuracy and stability of the results and causes excess computational effort. In this paper, a novel candidate samples-generating algorithm was proposed, by which a group of evenly distributed candidate points on the predicted LSF of performance function (either the real one or the surrogate model) could be obtained. In the proposed method, determination of LSF is considered as an optimization problem in which the absolute value of performance function was considered as objective function. After this, a normal search particle swarm optimization (NPSPO) was designed to deal with such problems, which consists of a normal search pattern and a multi-strategy framework that ensures the uniform distribution and diversity of the solution that intends to cover the optimal region. Four explicit performance functions and two engineering cases were employed to verify the effectiveness and accuracy of NPSPO sampling method. Four state-of-the-art multi-modal optimization algorithms were used as competitive methods. Analysis results show that the proposed method outperformed all competitive methods and can provide candidate samples that evenly distributed on the LSF. system reliability analysis; limit state function; regional-modal optimization problems; particle swarm optimization algorithm; normal search +
Footnote †: journal: Applied Sciences  
unacceptable level. Therefore, it is necessary to achieve an accurate reliability analysis with a much lower calculation cost [8]. Accordingly, methods incorporating a surrogate model with MCS were proposed [9]. The core of these methods is the design of experiments (DOEs) that help to establish the surrogate model which characterize the performance function. The MCS was then performed on the surrogate model, instead of the actual performance function, to calculate the probability of failure. In these methods, computational cost was reduced to the number of training samples obtained from DOEs. The key to guaranteeing the accuracy of the surrogate MCS method lies in the goodness-of-fit of the surrogate model, which fundamentally depends on the selection of the training samples. In the early stage of DOEs research, a sampling method called one-shot design (OSD) was often used. In OSD, to improve the convenience of sample acquisition, a large number of training samples were obtained at once in the design space according to a certain sampling method (such as Latin hypercube sampling (LHS)) [10; 11; 12]. New random samples may be added depending on the MCS calculation results. This kind of sampling method is simple and straightforward but aimless. Since P\({}_{f}\) was calculated as \(N_{x\in 0}/N_{\text{mcs}}\), i.e., the ratio of MCS realization times the value of surrogate model is smaller than 0 compared to the total realization times, it is not necessary for surrogate model to be accurate everywhere. An accurately predicted limit state function (LSF) is enough. Therefore, the ultimate objective of the established surrogate model is to locate the LSF that divide the variable space into safe and unsafe subareas [7]. In this case, different positions in variable space should be valued differently. Sampled points that have closer distances to the LSF indicates higher potential contributions. OSD method is therefore inefficient in that the sample process is totally random. Sequential design of experiments (SDOEs) was then invented to improve the calculation efficiency. In SDOEs, training samples are selected based on its potential value. A large sampling population is firstly initialized. A small number of samples (such as ten samples) are selected to establish the initial surrogate model.