Analyzing generalized failure probability function considering probabilistic inputs and fuzzy state is crucial for capturing how variations in distribution parameters within their design regions affect structural safety levels. The existing single-loop importance sampling method can avoid the highly time-consuming problem resulting from redundant reliability analysis and the large sample pool in high reliability requirements for direct double-loop Monte Carlo method. However, this method still cannot handle the time-consuming issues of constructing unified importance sampling density and low sampling efficiency. To address this, this paper constructs an explicit and easy to sample expression of the unified importance sampling density and proposes its progressive stratification strategy, which render the importance samples of constructing unified importance sampling density cover the regions with higher contribution to generalized failure probability, and enhance the computational efficiency by reducing the estimate variance. Compared to the existing single-loop importance sampling density method, the main innovation of the proposed method is the pro-gressive stratification strategy adopted when constructing unified importance sampling density, which alleviates the optimization requirement with high additional computational cost, thus enhancing the efficiency of exploring target fuzzy failure domain to construct unified importance sampling density and estimate the generalized failure probability function in high-dimensional and high-reliability scenarios. This superiority is fully validated by the examples presented in this paper.

Turbine disk is one of the main components of aero-engine. Once a critical failure occurs, it will lead to serious consequences. On the basis of fully considering the uncertain factors affecting high-low cycle complex fatigue life of turbine disks, co-simulation platform of reliability optimization with high-low cycle compound fatigue life for turbine disk was designed by MATLAB. The uncertain factors affecting high-low cycle complex fatigue life of turbine disks are full considered in the platform. Based on the common requirements of lifetime function and lifetime reliability analysis limit state function, a strategy of sharing training sample points in adaptive construction of lifetime function Kriging model and lifetime reliability limit state surface Kriging model in the process of optimization iteration was proposed. Meanwhile, a learning function for constructing Kriging model of lifetime function was proposed. The high-low cycle complex fatigue life reliability optimization of turbine disk center and mortise were completed using the co-simulation platform. The results show that the local maximum stress of the optimal result is significantly reduced, the average life-cycle is increased, and the reliability constraints are satisfied.

In order to ensure the safe service of aero structure, it is of great significance to evaluate the reliability lifetime under the constraint of extremely small target failure probability. However, the computational efficiency of the existing reliability lifetime analysis methods is difficult to meet the requirements of reliability lifetime analysis under high reliability requirements in engineering effectively. For this issue, a sequential stratified importance sampling method based on the first failure instant is proposed to solve reliability lifetime. Firstly, a sequential stratified exploration strategy for the rare failure domain with the extremely small target failure probability is established, which transforms the exploration problem of the rare failure domain into a gradual exploration problem of a series of failure domains with large probabilities, and it can effectively reduce the difficulty of obtaining the rare failure domain information. Secondly, the method for hierarchically constructing the explicit rule importance sampling density function is proposed to reduce the difficulty and computational complexity of obtaining the importance sample in the rare failure domain, which improves the computational efficiency for solving the reliability lifetime. Finally, in order to reduce the number of model evaluations, the Kriging surrogate model is embedded into the proposed sequential stratified importance sampling method, and an adaptive update strategy guided by misjudgment of the first failure instant is designed, which improve the efficiency of the sequential stratified importance sampling method to solve the reliability lifetime under the constraint of the extremely small target failure probability. The results show that, for the test function, the proposed method reduces the number of model evaluations and computational time by up to 45.4% and 99.6%, respectively, compared with the state-of-the-art methods. For a certain type of aero-engine turbine shaft structure, the proposed method reduces the number of model evaluations and computational time by up to 40.2% and 90.7%, respectively, compared with the state-of-the-art methods.

Based on a surrogate model of the performance function with an adaptive learning strategy, the metamodel-based importance sampling (Meta-IS) method can approximate the optimal importance sampling probability density function (IS-PDF) for estimating failure probabilities, making it an efficient approach for reliability analysis. However, when dealing with extremely small failure probabilities, estimating the normalization factor in the IS-PDF becomes computationally expensive for Meta-IS. To mitigate the computational burden, a meta-model-based double importance sampling (Meta-IS₂) method for estimating extremely small failure probabilities is proposed. The hierarchical weighted clustering strategy is designed to construct an IS-PDF for estimating the normalization factor. The feasibility of the proposed method is verified with instants. The results show that, under equivalent accuracy, the computational efficiency of the proposed method is no less than that of the existing Meta-IS method. Furthermore, for the cases with extremely small failure probabilities, the proposed method significantly outperforms that of the existing Meta-IS method.

It is important to determine the safety lifetime of Multi-mode Time-Dependent Structural System (MTDSS). However, there is still a lack of corresponding analysis methods. Therefore, this paper establishes MTDSS safety lifetime model firstly, and then proposes a Kriging surrogate model based method to estimate safety lifetime. The first step of proposed method is to construct the Kriging model of MTDSS performance function by using extremum learning function. By identifying possible extremum mode of MTDSS, the performance function of MTDSS can be equivalently transformed into the one of Single-mode Time-Dependent Structure (STDS). The second step is to use the Advanced First Failure Instant Learning Function (AFFILF) to train the Kriging model constructed in the first step, so that the convergent Kriging model can identify the possible First Failure Instant (FFI) of STDS. Then safety lifetime can be searched quickly by dichotomy search. By using AFFILF, the minimum instant that the state is not accurately identified by the current Kriging model is selected as the training point, which avoids the unnecessary calculation which may be introduced into the existing First Failure Instant Learning Function (FFILF). In addition, the Candidate Sample Pool (CSP) reduction strategy is also adopted. By adaptively deleting the random candidate sample points whose FFI have been accurately identified by the current Kriging model, the training efficiency is further improved. Three cases show that the proposed method is accurate and efficient.