# A Novel Reliability Analysis Approach under Multiple Failure Modes Using an Adaptive MGRP Model
## 1 Introduction
MC simulation is usually used as a standard to verify the effectiveness of other methods because of its simple calculation process and no restrictions on the form of functional functions, dimensions of basic variables, and distribution types, but has low computational efficiency. For this purpose, some sampling methods have been developed, such as important sampling (IS) [27], line sampling (LS) [28], directional sampling (DS) [29], and subset simulation (SS) [30], etc. Some scholars often combine digital simulation methods with surrogate models to facilitate engineering applications and reduce time spent. [PERSON] et al. [24] reported an AK-MCS method, which combines the SRGP model and the MC simulation procedure. [PERSON] et al. [31] combined the SRGP model and the SS procedure to solve the problem of small failure probability reliability. In addition, the methods of AK-IS, Meta-IS-AK, etc., provide a new idea for dealing with CRA. For SRA, it requires calling the limit state functions of multiple failure modes repeatedly, which seriously affects the computational efficiency. Especially in practical engineering problems, each limit state function has to be calculated with a time-consuming computer-aided program, such as dynamics simulations, finite element analysis, etc. Therefore, many efforts have been devoted to minimizing the number of the limit state function calls to achieve a realistic and acceptable state. Generally, methods proposed for CRA can be applied to SRA, and they can still be divided into three categories according to the characteristics of SRA. The first category is called the boundary method, which aims to determine the narrow bound of system failure probability through low-order joint probability. The second-order narrow boundary method was first proposed by [PERSON] [32]. [PERSON] et al. [33] employed it to conduct a multiple load case reliability analysis. Then, [PERSON] [34] derived the third narrow boundary method. [PERSON] [35] developed linear programming based on bounds methods. These methods quantify the correlation between failure modes through low-order joint probability, which can be used to deal with multiple failure mode problemsin series and parallel, but the computational efficiency is low. The second category is called the digital simulation method, which is the same as CRA. As an efficient solution method, the IS, LS, DS, and SS methods developed for a single failure model can all be extended to solve the reliability analysis of multiple failure model problems. The third category is called the surrogate model method, in which the construction process is different from the CRA. Although the single failure mode surrogate model can be used for multiple failure modes, it fails to consider the correlation of each failure mode. Thus, the surrogate model for multiple failure modes must be a composite model that considers the correlation of each failure mode, but this usually increases the number of function calls and the computational cost. To overcome this defect, [PERSON] et al. [36] proposed an AK-SYS method based on the adaptive Kriging surrogate model. [PERSON] et al. [37] investigated the improvement of the SRA method, which combines the active learning Kriging (ALK) model with MC simulation. [PERSON] et al. [38] studied the AK-SYSi method by using a refined U learning function. [PERSON] et al. [39] proposed the MRGP method to directly build a surrogate model with multiple output variables. [PERSON] et al. [40] introduced the MRGP model into the reliability and importance analysis and combined the strategy of the AK-MCS method to analyze the system reliability. Although many achievements in the field of SRA have been made, a vital issue about how to determine the better initial design samples or optimal size of candidate samples or improve termination criterion needs to be addressed. It not only determines the fitting accuracy of the performance function of each failure mode but also helps to improve the efficiency of multiple failure mode reliability analysis.