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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 proposes a methodology to estimate errors in computational models and to include them in reliability-based design optimization (RBDO). Various sources of uncertainties, errors and approximations in model form selection and numerical solution are considered. The solution approximation error is quantified based on the model itself while the model form error is quantified based on the comparison of model prediction with physical observations using an interpolated resampling approach. The error in reliability analysis is also quantified and included in the RBDO formulation. The proposed methods are illustrated through numerical examples.

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 develops a reliability-based methodology for the evaluation of stiffness degradation of spot-welded joints under high mileage. A global-local finite element analysis is used, with the loads on the detailed three-dimensional joint model coming from finite element analysis of the entire car model with proving ground loads. Probabilistic fatigue crack propagation analysis is developed for multi-axial variable amplitude loading history on the joint. Multiple spot welds contribute to the stiffness of the joint. Hence the problem is addressed through system reliability techniques. The effect of spot-weld separation on joint stiffness, and on global vehicle stiffness, is incorporated. This results in the computation of the statistics of vehicle stiffness degradation with mileage.

This paper investigates a wide range of statistical methods for application in model validation under uncertainty. Hypothesis testing methods are explored first and an interval-based testing is found to be more practically useful for model validation than the commonly used point null hypothesis testing. Also, a more direct approach is proposed by formulating model validation as a reliability estimation problem. The proposed methods are illustrated and compared using numerical examples.

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.

An essential step in predicting the output of a complicated system is the calibration of model parameters, but this step cannot be conducted directly if the data at the system-level are not available. In such a situation, a reasonable route is to quantify the system model parameters using tests at lower levels of complexity which share the same model parameters with the system, and propagate the results through the computational model at the system level. For such a multi-level problem, this paper proposes a methodology to quantify the uncertainty in the system-level model parameters by integrating model calibration, model validation and sensitivity analysis at different levels. The proposed approach considers the validity of the models used for parameter estimation at lower levels, as well as the relevance at the lower level to the prediction at the system level.