Efficient variable screening method and confidence-based method for reliability-based design optimization

The objectives of this study are (1) to develop an efficient variable screening method for reliability-based design optimization (RBDO) and (2) to develop a new RBDO method incorporated with the confidence level for limited input data problems. The current research effort involves: (1) development of a partial output variance concept for variable screening; (2) development of an effective variable screening sequence; (3) development of estimation method for a confidence level of a reliability output; and (4) development of a design sensitivity method for the confidence level.

In the RBDO process, surrogate models are frequently used to reduce the number of simulations because analysis of a simulation model takes a great deal of computational time. On the other hand, to obtain accurate surrogate models, we have to limit the dimension of the RBDO problem and thus mitigate the curse of dimensionality. Therefore, it is desirable to develop an efficient and effective variable screening method for reduction of the dimension of the RBDO problem. In this study, it is found that output variance is critical for identifying important variables in the RBDO process. A partial output variance, which is an efficient approximation method based on the univariate dimension reduction method (DRM), is proposed to calculate output variance efficiently. For variable screening, the variables that has larger partial output variances are selected as important variables. To determine important variables, hypothesis testing is used so that possible errors are contained at a user-specified error level. Also, an appropriate number of samples is proposed for calculating the partial output variance. Moreover, a quadratic interpolation method is studied in detail to calculate output variance efficiently. Using numerical examples, performance of the proposed variable screening method is verified. It is shown that the proposed method finds important variables efficiently and effectively.

The reliability analysis and the RBDO require an exact input probabilistic model to obtain accurate reliability output and RBDO optimum design. However, often only limited input data are available to generate the input probabilistic model in practical engineering problems. The insufficient input data induces uncertainty in the input probabilistic model, and this uncertainty forces the RBDO optimum to lose its confidence level. Therefore, it is necessary to consider the reliability output, which is defined as the probability of failure, to follow a probability distribution. The probability of the reliability output is obtained with consecutive conditional probabilities of input distribution type and parameters using the Bayesian approach. The approximate conditional probabilities are obtained under reasonable assumptions, and Monte Carlo simulation is applied to practically calculate the probability of the reliability output. A confidence-based RBDO (C-RBDO) problem is formulated using the derived probability of the reliability output. In the C-RBDO formulation, the probabilistic constraint is modified to include both the target reliability output and the target confidence level. Finally, the design sensitivity of the confidence level, which is the new probabilistic constraint, is derived to support an efficient optimization process. Using numerical examples, the accuracy of the developed design sensitivity is verified and it is confirmed that C-RBDO optimum designs incorporate appropriate conservativeness according to the given input data.