K. K. Choi

Conservative Surrogate Model Using Weighted Kriging Variance for Sampling-Based RBDO

In sampling-based reliability-based design optimization (RBDO) of large-scale engineering applications, the Monte Carlo simulation (MCS) is often used for the probability of failure calculation and probabilistic sensitivity analysis using the prediction from the surrogate model for the performance function evaluations. When the number of samples used to construct the surrogate model is not enough, the prediction from the surrogate model becomes inaccurate and thus the Monte Carlo simulation results as well. Therefore, to count in the prediction error from the surrogate model and assure the obtained optimum design from sampling-based RBDO satisfies the probabilistic constraints, a conservative surrogate model, which is not overly conservative, needs to be developed. In this paper, a conservative surrogate model is constructed using the weighted Kriging variance where the weight is determined by the relative change in the corrected Akaike Information Criterion (AICc) of the dynamic Kriging model. The proposed conservative surrogate model performs better than the traditional Kriging prediction interval approach because it reduces fluctuation in the Kriging prediction bound and it performs better than the constant safety margin approach because it adaptively accounts large uncertainty of the surrogate model in the region where samples are sparse. Numerical examples show that using the proposed conservative surrogate model for sampling-based RBDO is necessary to have confidence that the optimum design satisfies the probabilistic constraints when the number of samples is limited, while it does not lead to overly conservative designs like the constant safety margin approach.

Modified Bayesian Kriging for Noisy Response Problems for Reliability Analysis

Abstract : This paper develops a new modified Bayesian Kriging (MBKG) surrogate modeling method for problems in which simulation analyses are inherently noisy and thus standard Kriging approaches fail to properly represent the responses. The purpose is to develop a method that can be used to carry out reliability analysis to predict probability of failure. The formulation of the MBKG surrogate modeling method is presented, and the full conditional distributions of the unknown MBKG parameters are presented. Using the full conditional distributions with a Gibbs sampling algorithm, Markov chain Monte Carlo is used to fit the MBKG surrogate model. A sequential sampling method that uses the posterior credible sets for inserting new design of experiment (DoE) sample points is proposed. The sequential sampling method is developed in such a way that the newly added DoE sample points will provide the maximum amount of information possible to the MBKG surrogate model, making it an efficient and effective way to reduce the number of DoE sample points needed. Therefore, the proposed method improves the posterior distribution of the probability of failure efficiently. To demonstrate the developed MBKG and sequential sampling methods, a 2-D mathematical example with added random noise is used. It is shown how, with the use of the sequential sample method, the posterior distribution of the probability of failure converges to capture the true probability of failure. A 3-D multibody dynamics (MBD) engineering block-car example illustrates an application of the new method to a simple engineering example for which standard Kriging methods fail.

Design Sensitivity Method for Sampling-Based RBDO With Varying Standard Deviation

Conventional reliability-based design optimization (RBDO) uses the mean of input random variable as its design variable; and the standard deviation (STD) of the random variable is a fixed constant. However, the constant STD may not correctly represent certain RBDO problems well, especially when a specified tolerance of the input random variable is present as a percentage of the mean value. For this kind of design problem, the STD of the input random variable should vary as the corresponding design variable changes. In this paper, a method to calculate the design sensitivity of the probability of failure for RBDO with varying STD is developed. For sampling-based RBDO, which uses Monte Carlo simulation (MCS) for reliability analysis, the design sensitivity of the probability of failure is derived using a first-order score function. The score function contains the effect of the change in the STD in addition to the change in the mean. As copulas are used for the design sensitivity, correlated input random variables also can be used for RBDO with varying STD. Moreover, the design sensitivity can be calculated efficiently during the evaluation of the probability of failure. Using a mathematical example, the accuracy and efficiency of the developed design sensitivity method are verified. The RBDO result for mathematical and physical problems indicates that the developed method provides accurate design sensitivity in the optimization process.

Design Sensitivity Method for Sampling-Based RBDO With Fixed COV

Conventional reliability-based design optimization (RBDO) uses the means of input random variables as its design variables; and the standard deviations (STDEVs) of the random variables are fixed constants. However, the fixed STDEVs may not correctly represent certain RBDO problems well, especially when a specified tolerance of the input random variable is presented as a percentage of the mean value. For this kind of design problem, the coefficients of variations (COVs) of the input random variables should be fixed, which means STDEVs are not fixed. In this paper, a method to calculate the design sensitivity of probability of failure for RBDO with fixed COV is developed. For sampling-based RBDO, which uses Monte Carlo simulation for reliability analysis, the design sensitivity of the probability of failure is derived using a first-order score function. The score function contains the effect of the change in the STDEV in addition to the change in the mean. As copulas are used for the design sensitivity, correlated input random variables also can be used for RBDO with fixed COV. Moreover, the design sensitivity can be calculated efficiently during the evaluation of the probability of failure. Using a mathematical example, the accuracy and efficiency of the developed method are verified. The RBDO result for mathematical and physical problems indicates that the developed method provides accurate design sensitivity in the optimization process.Copyright

Confidence-Based Method for Reliability-Based Design Optimization

An accurate input probabilistic model is necessary to obtain a trustworthy result in the reliability analysis and the reliability-based design optimization (RBDO). However, the accurate input probabilistic model is not always available. Very often only insufficient input data are available in practical engineering problems. When only the limited input data are provided, uncertainty is induced in the input probabilistic model and this uncertainty propagates to the reliability output which is defined as the probability of failure. Then, the confidence level of the reliability output will decrease. To resolve this problem, the reliability output is considered to have a probability distribution in this paper. The probability of the reliability output is obtained as a combination of consecutive conditional probabilities of input distribution type and parameters using Bayesian approach. The conditional probabilities that are obtained under certain assumptions and Monte Carlo simulation (MCS) method is used to calculate the probability of the reliability output. Using the probability of the reliability output as constraint, a confidence-based RBDO (C-RBDO) problem is formulated. In the new probabilistic constraint of the C-RBDO formulation, two threshold values of the target reliability output and the target confidence level are used. For effective C-RBDO process, the design sensitivity of the new probabilistic constraint is derived. The C-RBDO is performed for a mathematical problem with different numbers of input data and the result shows that C-RBDO optimum designs incorporate appropriate conservativeness according to the given input data.Copyright

Development of a Conservative Model Validation Approach for Reliable Analysis

Abstract : Simulation models are approximations of real-world physical systems. Therefore, simulation model validation is necessary for the simulation-based design process to provide reliable products. However, due to the cost of product testing, experimental data in the context of model validation is limited for a given design. When the experimental data is limited, a true output PDF cannot be correctly obtained. Therefore, reliable target output PDF needs to be used to update the simulation model. In this paper, a new model validation approach is proposed to obtain a conservative estimation of the target output PDF for validation of the simulation model in reliability analysis. The proposed method considers the uncertainty induced by insufficient experimental data in estimation of predicted output PDFs by using Bayesian analysis. Then, a target output PDF and a probability of failure are selected from these predicted output PDFs at a user-specified conservativeness level for validation. For validation, the calibration parameter and model bias are optimized to minimize a validation measure of the simulation output PDF and the conservative target output PDF subject to the conservative probability of failure. For the optimization, accurate sensitivity of the validation measure is obtained using the complex variable method (CVM) for sensitivity analysis. As the target output PDF satisfies the user-specified conservativeness level, the validated simulation model provides a conservative representation of the experimental data. A simply supported beam is used to carry out the convergence study and demonstrate that the proposed method establishes a conservatively reliable simulation model.