제한된 상관 데이터와 멀티모달 데이터를 고려한 신뢰성 기반 최적설계

Abstract

Numerous methods of reliability analysis such as simulation-based method, most probable point-based method, and moment-based method have been developed to estimate reliability of a system under uncertainty of variables. These methods have often assumed the uncertainty of variables to be univariate and unimodal continuous distribution such as normal distribution. Therefore, when the uncertainty of variables is observed as multivariate and multimodal distribution, accuracy of estimated reliability by using the methods could not be guaranteed. In practice, the uncertainty of variables is often given as correlated and multimodal limited data. Therefore a new method of reliability analysis which can estimate the reliability under correlated and multimodal limited data is required. To consider correlation, multimodality and limited data for reliability analysis, conventional methods for reliability analysis such as sampling-based method, most probable point-based method, and moment-based method have been improved. Since sampling-based method such as Monte Carlo simulation (MCS) estimates the reliability by counting the number of samples which satisfy target performance, correlated and multimodal data can be directly considered by adopting multivariate and multimodal distributions. However, MCS requires large number of samples for accurate reliability. Therefore MCS is not frequently used as a method for reliability analysis in engineering fields. Most probable point-based method has been improved to consider the uncertainty of correlated variables. Correlated data is modeled by copula which connects marginal distributions, then reliability analysis is carried out by using most probable point-based method such as first order reliability method. However, since limit state function has to be approximated by Taylor’s expansion into linear function and the distribution of response has to be assumed as normal distribution, multimodal data cannot be considered in the method. Moment-based method considers correlation, multimodality and limited data, individually. Copula and maximum entropy principle have been adopted for modelling of correlated data and multimodal data, respectively. Confidence levels for parameters of distribution and Bayesian approach have been adopted for limited data. However, an integrated method to consider correlated data, multimodal data and limited data simultaneously has not been suggested. In this dissertation, to consider correlated and multimodal limited data simultaneously, a new method of reliability analysis is proposed. The proposed method contains three methods of reliability analysis for correlated data, multimodal data and limited data, respectively. Copula is adopted to estimate a model for correlated data. Maximum likelihood estimation of candidate copulas is used for selecting the fittest copula under correlated data. Based on the fittest copula, the reliability is estimated by using Akaike information criterion (AIC). Finite mixture model (FMM) is adopted to estimate a multimodal distribution for multimodal data. By defining candidate FMMs and selecting the fittest FMM by AIC, the reliability then can be estimated by the fittest FMM. Copula and FMM are estimated by using correlated and multimodal data, respectively. Therefore, accuracy of the estimated reliability cannot be guaranteed definitely when small number of data is used, i.e. limited data are given. To assess the uncertainty of estimated reliability, binomial test of nonparametric hypothesis test is adopted. Based on the binomial test, “reliability-confidence” is defined as probability that the estimated reliability is larger than the specified target reliability. By integration of the three methods, the proposed method of reliability analysis under correlated and multimodal limited data is suggested. On the other hand, in reliability-based design optimization (RBDO), reliability constraint is formulated by the reliability to consider the uncertainty of variables. Therefore, RBDO usually provides more conservative optimum design than deterministic design optimization (DDO). Likewise, reliability-confidence based design optimization (RCBDO) using the proposed method provides more conservative optimum design than RBDO but considers the reliability and the uncertainty of the estimated reliability due to limited data. Finally, RCBDO considers both the reliability and the reliability-confidence when correlated and multimodal limited data are given. Examples of reliability analysis and RCBDO are carried out by using the proposed method. To validate the proposed method, mathematical examples of reliability analysis are performed. The three methods are compared with MCSs for correlated data, multimodal data and limited data, respectively. Then, the proposed method for reliability-confidence is examined with respect to the number of data. Mathematical example of tubular column design is carried out. Results of the proposed method are discussed with respect to the target reliability-confidence and the number of data. Comparisons of the proposed method with previous methods are given to validate the proposed method for correlated data, multimodal data and limited data, respectively. RCBDOs for engineering applications, combat vehicle design, motor controller, manganese nodule collector and lower control arm design are performed. The results show that the proposed method considers the reliability under correlated and multimodal data as well as the reliability-confidence under limited data.