## 1 Introduction  
The reliability of mechanical structures is of great significance in enhancing the stability of products. Based on traditional time-invariant reliability models, the degradation phenomena of internal materials are usually not considered, and the calculated structural reliability is a constant value. The complexity and intelligence of the mechanical system give the system performance obvious time-variant dynamic characteristics. Complex engineering requirements cannot be met through traditional static reliability analysis methods. However, temporal variables, spatial variables, and the correlation of structural performance between time points are considered in time-variant reliability (TVR) analysis methods. Trends in the reliability of the system over the life cycle can be reflected to some extent [1, 2]. The main approaches to solving TVR structural problems are the approximate analytical method [3, 4], the numerical simulation method [5, 6], and the surrogate model method [7, 8]. Jiang et al. [9, 10] discretized stochastic processes, and the TVR problem was transformed into a traditional time-invariant reliability problem. However, the accuracy and efficiency are seriously affected by the size of discrete time intervals. To address this limitation, Zhang et al. [11] applied the Kriging surrogate model to approximate the trajectory of the maximum possible point (MPP), referred to as AMPPT. Zhang et al. [12] improved training efficiency by applying the active Kriging surrogate model to the traditional stochastic process discretization method. Meng et al. [4] proposed a semi-analytic extreme value method to approximate the limit state function (LSF) at the MPP. However, searching for the MPP requires calling a large number of time-variant LSF, which is less efficient. Meanwhile, the method cannot address multiple MPPs or highly nonlinear and complex LSFs. Wu et al. [13] proposed a system reliability method by combining the envelope method and the second-order reliability method (SORM). Gong and Frangopol [14] proposed a new method for analyzing TVR based on the first-order reliability method (FORM). In addition, Wang and Chen [15] proposed a stochastic process transformation method considering both random variables and stochastic process parameters for solving whole life cycle reliability variation. The main numerical simulation methods are Monte Carlo Simulation (MCS) [16], Subset Simulation (SS) [17], and Importance Sampling (IS) [6], etc. The advantages of the numerical simulation method are high accuracy and simple operation. Moreover, the dimensionality problem and complexity of LSF need not be taken into account. However, the guarantee of computational accuracy requires a large number of samples, which leads to inefficient computation [18]. The surrogate model method approximates the actual model by constructing a function and applies the constructed function to solve TVR, which greatly improves computational efficiency. Hu et al. [19] proposed an effective TVR analysis method based on a fast integration approach and surrogate model method. Classical approaches include the double-loop approach [20; 21] and the single-loop approach. Wang et al. [22] proposed a coupled extreme value response surface method based on double-loop substitution. Hu and Du [21] improved the coupled extreme value response surface method to construct a more efficient adaptive Kriging-MCS hybrid model. The method based on a double-loop surrogate model needs to find extreme value in the inner loop and use the result to perform reliability analyses in the outer loop, which is inefficient. In order to improve efficiency, the single-loop method is proposed to transform the TVR problem into a classification problem through the adaptive surrogate model, e.g., single-loop Kriging (SILK) [23; 24]. With the surrogate model method, high nonlinearities, high dimensionality, and implicit performance functions can be handled well. However, only reliability over a particular time interval can be obtained.