# A Novel Reliability Analysis Approach under Multiple Failure Modes Using an Adaptive MGRP Model
## 1 Introduction  
Reliability analysis aims at quantifying the reliability state of each performance index with the consideration of various sources of uncertainties, such as sizes, loads, accelerations, and variations in material properties. Generally, it can be divided into \"system reliability analysis\" (SRA) and \"component reliability analysis\" (CRA). The former means considering multiple failure modes, while the latter only considers one single failure mode [1, 2]. For CRA, many methods have been developed to improve its efficiency, accuracy, and engineering applicability. They are divided into three categories according to their models. The first category is the approximate analytical method, which commonly adopts the most probable points (MPPs) to replace the mean point as the Taylor series expansion point of the limit state function [3]. Among them, the first-order reliability method (FORM) and the second-order reliability method (SORM) are the classical analysis methods based on MPP expansion. However, both of them need to normalize the non-normal random variables in the solution process, which increases the nonlinearity of the performance function. Thus, [PERSON] et al. [4] proposed the first-order saddlepoint approximation (FOSPA) method based on the maximum likelihood point model. [PERSON] et al. [5] studied themean value first-order saddlepoint approximation (MVFOSPA) method, which effectively improves the efficiency and robustness of the calculation. Different from the above methods, [PERSON] [6] investigated the two-point estimation method. [PERSON] [7] performed a three-point estimation method considering the first four moments of the input variables. [PERSON] et al. [8], [PERSON] et al. [9], [PERSON] et al. [10], and [PERSON] et al. [11] presented some moment estimation methods, which enrich and improve the development of approximate analytical methods. The second category is the surrogate model-based method, which is widely investigated and used due to its strong fitting ability and small number of calculations. At present, the polynomial response surface method (RSM) [12; 13], artificial neural networks (ANN) [14], radial basis function (RBF) [15], support vector machines (SVM) [16], polynomial chaos expansion (PCE) [17], and single response Gaussian process (SRGP) model (i.e., Kriging model) [18] are the classical surrogate models. Among these methods, the Kriging model has attracted the attention of many scholars because of its strong nonlinear fitting ability and generalization ability. It can be divided into the static surrogate model and the active learning (adaptive) surrogate model. The difference is that the former requires more initial sample points to ensure the approximation accuracy of the surrogate model, and the latter can continuously optimize the sample quality by adding new samples to update the model. Given the high fitting accuracy and efficiency of the adaptive Kriging surrogate model, its initial design point selection, update strategy, point addition scheme, probability evaluation, termination criterion, etc., have gained the most attention. [PERSON] et al. [19] improved the traditional initial design point selection method by rectangular grid sampling. [PERSON] et al. [20] determined the design point selection of the Bucher experimental and Sobol sequence through the comparison of various experimental designs. Furthermore, [PERSON] et al. [21] proposed a single-point adding strategy based on the EI criterion. [PERSON] et al. [22] presented a multi-point update strategy combining the MP criterion and the EI criterion. Moreover, expected feasibility function (EFF) [23], U-function [24], H-function [25], least improvement function (LIF) [26], etc., update strategies have been deeply investigated and applied. The third category is the digital simulation method, which is based on the Monte Carlo (MC) simulation.