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This paper proposes a novel probabilistic approach for multidisciplinary design optimization (MDO) under uncertainty, especially for systems with feedback coupled analyses with multiple coupling variables. The proposed approach consists of four components: multidisciplinary analysis, Bayesian network, copula-based sampling, and design optimization. The Bayesian network represents the joint distribution of multiple variables through marginal distributions and conditional probabilities, and updates the distributions based on new data. In this methodology, the Bayesian network is pursued in two directions: (1) probabilistic surrogate modeling to estimate the output uncertainty given values of the design variables, and (2) probabilistic multidisciplinary analysis (MDA) to infer the distributions of the coupling and output variables that satisfy interdisciplinary compatibility conditions.

This paper discusses the treatment of uncertainties corresponding to relatively few samples of random-variable quantities. The importance of this topic extends beyond experimental data uncertainty to situations involving uncertainty in model calibration, validation, and prediction. With very sparse samples it is not practical to have a goal of accurately estimating the underlying variability distribution (probability density function, PDF). Rather, a pragmatic goal is that the uncertainty representation should be conservative so as to bound a desired percentage of the actual PDF, say 95% included probability, with reasonable reliability. A second, opposing objective is that the representation not be overly conservative; that it minimally over-estimate the random-variable range corresponding to the desired percentage of the actual PDF. The presence of the two opposing objectives makes the sparse-data uncertainty representation problem an interesting and difficult one.

An accurate and efficient Monte Carlo simulation (MCS) method is developed in this paper for limit state-based reliability analysis, especially at system levels, by using a response surface approximation of the failure indicator function. The Moving Least Squares (MLS) method is used to construct the response surface of the indicator function, along with an Optimum Symmetric Latin Hypercube (OSLH) as the sampling technique. Similar to MCS, the proposed method can easily handle implicit, highly nonlinear limit-state functions, with variables of any statistical distributions and correlations. However, the efficiency of MCS can be greatly improved. The method appears to be particularly efficient for multiple limit state and multiple design point problem. A mathematical example and a practical example are used to highlight the superior accuracy and efficiency of the proposed method over traditional reliability methods.

An efficient simulation-based method is proposed for the reliability analysis of a vehicle body-door subsystem with respect to an important quality issue -- wind noise. A nonlinear seal model is constructed for the automotive wind noise problem and the limit state function is evaluated using finite element analysis. Existing analytical as well as simulation-based methods are used to solve this problem. A multi-modal adaptive importance sampling method is then developed for reliability analysis at system level. It is demonstrated through this industrial application problem that the multi-modal adaptive importance sampling method is superior to existing methods in terms of efficiency and accuracy. The method can easily handle implicit limit-state functions, with variables of any statistical distributions.

This paper proposes new methods to assess the validity of reliability prediction models through a Bayesian approach. The concept of Bayesian hypothesis testing is extended to system-level problems where full-scale testing is impossible. Component-level validation results are used to derive a system-level validation measure. This derivation depends on the knowledge of interrelationships between component modules. Bayes networks are used for the propagation of validation information from the component -level to system-level. Validation of reliability prediction model for a single degree of freedom oscillator under high-cycle fatigue and fatigue life prediction of a helicopter rotor hub is illustrated for this purpose.

In this paper, an efficient method for the reliability analysis of systems with nonlinear limit states is described. It combines optimization-based and simulation-based approaches and is particularly applicable for problems with highly nonlinear and implicit limit state functions, which are difficult to solve by conventional reliability methods. The proposed method consists of two major parts. In the first part, an optimization-based method is used to search for the most probable point (MPP) on the limit state. This is achieved by using adaptive response surface approximations. In the second part, a multi-modal adaptive importance sampling method is proposed using the MPP information from the first part as the starting point. The proposed method is applied to the reliability estimation of a vehicle body-door subsystem with respect to one of the important quality issues -- the door closing effort.

This paper develops a multiobjective optimization methodology for automotive door quality design under uncertainty, in which the tradeoffs between two competing objectives need to be considered. Two important quality issues, door closing effort and wind noise, are of concern and the corresponding probabilities of unsatisfactory performance are considered in the optimization. Model-based reliability analysis methods are used to compute these probabilities. Both component and system-level reliability analyses need to be performed in RBDO. While a first order reliability method (FORM) is found adequate for the reliability estimation with respect to door closing effort, an adaptive Monte Carlo simulation method is found suitable for reliability analysis of the wind noise problem with multiple limit states. An efficient decoupled RBDO approach is used to solve the multiobjective optimization and the Pareto frontier is generated for decision-making.

A general methodology for fatigue reliability analysis and inspection of railroad wheels is proposed in this paper. Both fatigue crack initiation and crack propagation life are included in the proposed methodology using previously developed multiaxial fatigue models by the authors. The life prediction is validated with field data. A methodology for calculating the optimized inspection schedule of railroad wheels is then developed using the reliability methodology. The optimized inspection scheduling methodology combines clustering analysis, to identify critical samples, reliability analysis, to calculate the expected life of the critical samples, and reliability-based optimization into an overall methodology which optimizes the inspection schedule. The proposed methodology minimizes the number of wheels inspected while at the same time maintaining or exceeding the desired reliability of the population.