Input model uncertainty and reliability-based design optimization with associated confidence level

The objective of this study is to develop new methods for modeling of input uncertainty of correlated variables and to carry out reliability-based design optimization (RBDO) using the identified input uncertainty model with associated confidence level. The proposed research involves: (1) use of copulas to model joint CDFs of input variables; (2) use of the Bayesian method to identify marginal cumulative distribution functions (CDF) and joint CDF of input variables using limited experimental data; (3) reduction of the transformation ordering effect on the inverse reliability analysis using the most probable point (MPP)-based dimension reduction method (DRM); and (4) assessment of the confidence level of the input model uncertainty, and implementation of the confidence level in RBDO to offset the inexact quantification of the input uncertainties due to limited data.

It has been well documented that many random variables such as material properties are correlated, but the correlation has not been considered in RBDO because modeling the joint CDF of correlated variables is known to be difficult. In this study, a copula is introduced to model a joint CDF of input variables. The copula requires marginal CDFs and correlation parameters, which can be obtained in real applications, so it is possible to model a joint CDF. Once the joint input CDF is modeled using a copula, the input variables can be transformed to independent Gaussian variables using Rosenblatt transformation for the inverse reliability analysis.

This study proposes a method to identify correct marginal and joint CDFs (copulas) of input variables. In practical applications, since only limited experimental data are available, it is challenging task to correctly identify the marginal and joint CDF of input variables using the limited data. In this study, a Bayesian method is proposed to identify the marginal and joint CDFs of input variables that best describe given data among candidates. The performance of the proposed method is investigated and compared with an existing method, the goodness-of-fit (GOF) test.

Using the identified input model, the transformation from original input variables into independent Gaussian variables is carried out, and then the first-order reliability method (FORM), which has been commonly used in reliability analysis, is carried out. However, when the input variables are correlated with non-elliptical copulas, the FORM may yield different reliability analysis results with some errors for different transformation orderings of input variables due to the nonlinearities of the transformed constraint functions. For this, the MPP-based DRM, which more accurately and efficiently calculates the probability of failure than the FORM and the second-order reliability method (SORM), respectively, is used to reduce the effect of transformation ordering in the inverse reliability analysis and, thus, RBDO.

However, when the number of experimental data is limited, the estimated input joint CDF will be inaccurate, which will lead to inaccurate RBDO result. Thus, a method to assess the confidence level of the input model uncertainty in RBDO is developed, and the input model with confidence level is implemented for RBDO.