An advanced mixed-degree cubature formula for reliability analysis

Introduction to Bayesian Estimation and Copula Models of Dependence / Arkady Shemyakin, Alexander Kniazev

In reliability-based robust design optimization (RBRDO) formulation, the product quality loss function is minimized subject to probabilistic constraints. Since the quality loss function is expressed in terms of the first two statistical moments, mean and variance, three methods have been recently proposed to accurately and efficiently estimate the moments: the univariate dimension reduction method (DRM), performance moment integration (PMI) method, and percentile difference method (PDM). In this paper, a reliability-based robust design optimization method is developed using DRM and compared to PMI and PDM for accuracy and efficiency. The numerical results show that DRM is effective when the number of random variables is small, whereas PMI is more effective when the number of random variables is relatively large.

Dimension Reduction Method for Reliability-Based Robust Design Optimization

The failure probability function (FPF), aiming to decouple the nested-loop reliability-based design optimization solution into a single-loop optimization problem, has attracted a great deal of interest from designers and researchers. It is defined as a function of failure probability with respect to the design parameter. Among the estimation methods for the FPF, the Bayes theorem based on probability distribution function methods is competitive. It transforms the FPF as the estimation of the conditional joint probability density function (PDF) of design parameters on the failure event and the augmented failure probability. The augmented failure probability can be estimated by Monte Carlo simulation, while for the joint multi-dimensional PDF, the existing estimation methods are unavailable. To alleviate this issue, a novel FPF estimation method is proposed by Bayes theorem and copula. In the proposed method, the FPF is derived as a product of the conditional copula density and the augmented failure probability, in which the vine copula is employed to disassemble the multi-dimensional conditional copula density into several bivariate copula density functions, and they can be completed by the existing PDF estimation methods. In contrast to the existing Bayes theorem–based estimation methods for the FPF, the proposed method interprets the FPF as the dependence function between design parameters and the augmented failure probability in terms of copula, which involuntarily breaks the limitation of the multi-dimensional design parameters. In addition, the adaptive Kriging surrogate model is embedded in the proposed method to improve the efficiency of the proposed method. The presented examples demonstrate the efficiency and accuracy of the proposed method.

Bayes theorem–based and copula-based estimation for failure probability function

A novel single-loop meta-model importance sampling with adaptive Kriging for time-dependent failure probability function

Conditional copula which measures the conditional dependence among variables, possesses a special position in copula field. In this article, based on Bayes theorem, we derive three kinds of conditional copula functions as the product of the corresponding conditional copula density functions and the corresponding unconditional copula functions (or the cumulative distribution functions). Then, a novel nonparametric method for estimating these conditional copula functions is proposed by the classification of the Monte Carlo Simulation (MCS) samples and by the kernel density estimation. In contrast to other estimation methods for conditional copula functions, the proposed method needs only a set of samples without any parameter or distribution assumption, or other complicated operators (such as estimation of the weights, integral operator, etc.). Therefore, the proposed nonparametric method reduces the computational complexity and possesses more universality for estimating the conditional copula functions. A 2-dimensional normal copula function, a numerical example, a structural system reliability analysis considering the common cause failure and an astrophysics model based on real data are employed to validate the effectiveness of the proposed method. Results show that the proposed nonparametric method is accurate and practical well.

/pdf/a-novel-nonparametric-estimation-for-conditional-copula-1ijfogblcl.pdf

A Novel Nonparametric Estimation for Conditional Copula Functions Based on Bayes Theorem

https://iopscience.iop.org/article/10.1088/1757-899X/751/1/012058/pdf

Copula-based Estimation Method of the Expected Value Function for Robust Design Optimization

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Bayes theorem–based and copula-based estimation for failure probability function

A Novel Nonparametric Estimation for Conditional Copula Functions Based on Bayes Theorem

Copula-based Estimation Method of the Expected Value Function for Robust Design Optimization