System Reliability-based Design Optimization Using Copula Estimation Method

Abstract

Stochastic design optimization has been recently developed to predict the reliability of a system and to satisfy a required reliability using stochastic information of design variables. Reliability-based design optimization (RBDO) is one of stochastic design optimization methods, which performs the reliability analysis of a system to solve given optimization problem. Reliability analysis is prerequisite to calculate the reliability at each step in the RBDO process. A number of reliability analysis methods have been established. However, Akaike information criterion has been considered more powerful than other methods and employed in many fields of science and engineering because it enables the reliability analysis to be performed by using data without any assumptions of the distributions. In many engineering problems, even though random variables are correlated, they have often been assumed to be independent because it makes the input uncertainty model much simpler to be handled. In order to perform reliability analysis when correlation exists between input random variables, constructing multivariate distribution for inputs of the system is needed. Another aspect of conventional reliability analysis in previous researches was that the reliability was calculated independently for each response. The target reliability seemed to be satisfied in each reliability constraint in RBDO results while the reliability of system which is the true reliability has lower value than the target that the designer intended to have. Thus, a new concept of reliability analysis should be considered for the whole system and multivariate distribution for outputs of the system should be modeled. In order to cope with correlated input variables and system reliability analysis, statistical model of multivariate distribution for correlated variables is required. Copula is widely used to deal with statistical model under correlated variables in various fields of science and engineering. Copulas are functions that join multivariate distribution using their one-dimensional marginal distributions. The advantage of copula is that more variety of one-dimensional marginal distributions and their combinations can be achieved which multivariate distributions extended from univariate distributions cannot provide. Although many copula estimation methods have been proposed and studied, most of them need some assumptions in the estimation procedure. Thus, a copula estimation method without any necessities of assumptions is required which is expected to construct more accurate statistical models. The first objective of this research is to develop a copula estimation method that overcomes the disadvantages of conventional methods. By copula estimation method, correlation between input variables and output variables can be statistically modelled through copula. In this study, a copula estimation method based on inference function for margins using AIC is proposed. Proposed method consists of two estimation steps. Firstly, it selects the fittest distributions for one-dimensional margins using AIC and then it selects the fittest copula using maximum likelihood estimation. The advantage of this proposed procedure is that no assumptions are required for the statistical models and they can be obtained by only using the given data. Another objective of this research is to suggest a new method of reliability analysis. Conventional reliability analysis considers only the probability of each response independently. However this can lead significant difference between the reliability for a response and the system reliability. Therefore, system reliability analysis (SRA) and system reliability-based design optimization (SRBDO) by using copula estimation are proposed, which can accurately deal with the reliability of system.