An Adaptive Two-Stage Framework for AK-MCS in Reliability Analysis

A Kriging model enhanced integral method for reliability analysis

Time-Variant Reliability Analysis via Copula Failure Correlation and Advanced Kriging

In practical engineering, mechanical structures are affected by time-dependent uncertainties and by correlated failure modes. This study proposes an enhanced time-variant reliability analysis framework by coupling a mixed Archimedean Copula model with an adaptive Kriging model driven by the maximum expected prediction error (MEPE) learning function. The mixed Copula combines Gumbel, Clayton and Frank components, so that upper-tail, lower-tail and nearly symmetric dependence can be represented in one model. The MEPE function combines Kriging prediction variance and leave-one-out cross-validation error through a dynamic balance factor, which guides early global exploration and later refinement near the most probable point trajectory. For the planar three-bar structure, the mixed-Copula failure probability at t = 7 is 0.4078, close to the direct-MCS benchmark 0.4059, with a relative error of 0.47%; the number of original limit-state evaluations decreases from 1.00 × 106 to 326, corresponding to a 99.97% reduction. For the two-rod parallel structure, the mixed-Copula failure probability at T = 2 is 0.0987, close to the direct-MCS benchmark 0.0992, with a relative error of 0.50%; the number of original limit-state evaluations decreases to 284. The calculated upper- and lower-tail dependence coefficients of the mixed Copula are 0.1345 and 0.2985 for the planar three-bar structure and 0.0003 and 0.3709 for the two-rod parallel structure. These quantitative results show that the proposed framework can describe nonlinear failure dependence more flexibly than a single Copula while retaining high computational efficiency.

Cross-term extreme learning machine for accurate reliability analysis

To address the challenge of balancing accuracy and efficiency in existing neural network-assisted structural reliability analysis methods for low-sample-size and high-dimensional problems, this study proposes a simple and highly efficient deep network model that significantly improves prediction accuracy while retaining the rapid training advantage of extreme learning machine (ELM). Furthermore, the optimal configuration of the model's key parameters is determined, providing a more efficient and accurate technical tool for quantifying uncertainties in engineering structures. This study proposes a structural reliability method based on the cross-term extreme learning machine (CTELM) model. This model adds a cross-term layer to the ELM, calculating cross-terms by pairwise combinations of hidden layer nodes. Least squares estimation (LSE) is used for rapid training, and a considering cross node (CCN) is introduced to balance accuracy and computational efficiency. Furthermore, comparative experiments are conducted using three high-dimensional engineering cases, focusing on the impact of different training sample sizes, the number of hidden nodes and the CCN value on model performance analysis. The results show that CTELM achieves dual optimization in accuracy and efficiency: in scenarios with low training sample size (N < 100), its reliability index error is consistently lower than other models, with an error of only 0.4% when N = 100 in the composite wood-aluminum beam case; its training efficiency is comparable to ELM and superior to artificial neural networks (ANN) and deep neural network, which rely on backpropagation (BP) training, with training time in the cantilever beam case being less than 1/3 of that of ANN. Therefore, the proposed CTELM can achieve high-precision prediction without a large number of training samples, making up for the performance deficiencies of traditional models in low-sample, high-dimensional scenarios. The core innovation of this study lies in proposing an improved ELM framework based on a cross-term layer. By adding a cross-term layer between the hidden and output layers, the model's expressive power is expanded through the nonlinear combination of hidden nodes, overcoming the limitation of traditional ELM relying on linear mapping of a single hidden layer. A cross-node selection mechanism (CCN) is designed to avoid the surge in computational cost caused by fully connected cross-terms, balancing model complexity and practicality.

Parameter Estimation of Shifted Mixture Models for Software Reliability Assessment

This paper proposes a novel approach for predicting software reliability using shifted mixtures of Weibull, lognormal and Gompertz distributions. By integrating these three distributions with shift parameters, we aim to model the complex, multi-phase failure processes observed in software systems more accurately than traditional models. One way to get accurate solutions in software reliability assessments is to adopt the challenges of parameter estimation of proposed shifted models that arise due to the numerous parameters involved in the models. We employ expectation–maximization (EM), particle swarm optimization (PSO) and differential evolution (DE) algorithms, which provide not only a granular view of failure mechanisms but also guide effective maintenance and reliability enhancement strategies. A comparative analysis of proposed models using an actual failure data set is presented. Numerical experiments illustrate the performance of proposed approach and use some goodness of fit tests to determine the best-fitting shifted mixture model. We show that the method can automatically select the better mixture model for the data set, and validate the performance indices for the software reliability prediction.

A Bayesian Design-Use-Stage Data Fusion Framework for Reliability Assessment Under Uncertainty: A Case Study From Machine Tool Rotary Table Manufacturing Enterprise

Reliability is a fundamental performance metric for products. However, for capital-intensive products, accurate reliability assessment remains challenging because of scarce, heterogeneous, and uncertain data. To address this issue, this article proposes a Bayesian design-use-stage data fusion framework for product reliability assessment. The framework integrates machining error data and lifetime data of similar products from the design stage, as well as precision testing data and field lifetime data from the use stage. In the design stage, an initial precision reliability model is first constructed based on component machining error data and the error accumulation mechanism. Then, the Weibull distribution is used to describe the uncertainty in the lifetime data of similar products. Finally, the lifetime data of similar products are fused through Bayesian updating to revise the initial precision reliability model. In the use stage, the reliability model obtained from the design stage is used as prior information. The Wiener process is used to describe the uncertainty and time evolution characteristics of precision testing data. Based on this process, the precision testing data are converted into approximate lifetime data. The approximate lifetime data are then used to update the prior model. Finally, field lifetime data are further fused to revise the updated model. A case study on a CNC rotary table is used to verify the effectiveness of the proposed framework. By integrating heterogeneous models constructed from multisource data into a unified Bayesian structure, the proposed framework enables continuous and stage based reliability assessment under uncertainty.

MODELING AND RELIABILITY ANALYSIS OF MULTI-STRESS COUPLED ACCELERATED LIFE MODEL BASED ON FUZZY INFORMATION FUSION

To address the challenges in identifying and accurately characterizing coupling effects in existing multi-stress acceleration models, this paper adopts fuzzy mathematics to characterize uncertainties in both empirical knowledge and data, and proposes a multi-stress accelerated life evaluation method that explicitly accounts for coupling effects. The method first adopts the Distance Correlation Coefficient (DCOR) technique to establish a preliminary screening framework for coupling effects. The objective statistical analysis results are then integrated with subjective empirical judgments via the Fuzzy Number–Entropy Weight–TOPSIS fusion method, yielding an importance ranking of the coupling effects. The selected and optimized acceleration model is subsequently embedded into the log-likelihood function of the Weibull distribution, and the Particle Swarm Optimization (PSO) algorithm is utilized to estimate model parameters and predict service life. Case study results demonstrate that, compared with the traditional Pearson correlation coefficient method, the proposed method exhibits a broader applicability under nonlinear and non-monotonic conditions and significantly improves the prediction accuracy of product service life under constant stress conditions, thereby validating the effectiveness of the model.