Ikjin Lee

Metamodeling Method Using Dynamic Kriging for Design Optimization

Metamodeling has been widely used for design optimization by building surrogate models for computationally intensive engineering application problems. Among all the metamodeling methods, the kriging method has gained significant interest for its accuracy.However, in traditional krigingmethods, themean structure is constructed using a fixed set of polynomial basis functions, and the optimization methods used to obtain the optimal correlation parameter may not yield an accurate optimum. In this paper, a new method called the dynamic kriging method is proposed to fit the true model more accurately. In this dynamic kriging method, an optimal mean structure is obtainedusing thebasis functions that are selected bya genetic algorithm from the candidate basis functions based on a new accuracy criterion, and a generalized pattern search algorithm is used to find an accurate optimum for the correlation parameter. The dynamic kriging method generates a more accurate surrogate model than other metamodeling methods. In addition, the dynamic kriging method is applied to the simulation-based design optimization with multiple efficiency strategies. An engineering example shows that the optimal design obtained by using the surrogate models from the dynamic kriging method can achieve the same accuracy as the one obtained by using the sensitivity-based optimization method.

Reduction of Ordering Effect in Reliability-Based Design Optimization Using Dimension Reduction Method

In reliability-based design optimization problems with correlated input variables, a joint cumulative distribution function needs to be used to transform the correlated input variables into independent standard Gaussian variables for the inverse reliability analysis. To obtain a true joint cumulative distribution function, a very large number of data (if not infinite) needs to be used, which is impractical in industry applications. In this paper, a copula is proposed to model the joint cumulative distribution function using marginal cumulative distribution functions and correlation parameters obtained from samples. Using the joint cumulative distribution function modeled by the copula, the transformation and the first-order reliability method can be carried out. However, the first-order reliability method may yield different reliability analysis results for different transformation ordering of input variables. Thus, the most probable-point-based dimension reduction method, which is more accurate than the first-order reliability method and more efficient than the second-order reliability method, is proposed for the inverse reliability analysis to reduce the effect of transformation ordering.

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.

Confidence Level Estimation and Design Sensitivity Analysis for Confidence-Based RBDO

In practical engineering problems, often only limited input data are available to generate the input distribution model. The insufficient input data induces uncertainty on the input distribution model, and this uncertainty will cause us to lose confidence in the optimum design obtained using the reliability-based design optimization (RBDO) method. Uncertainty on the input distribution model requires us to consider the reliability analysis output, which is defined as the probability of failure, to follow a probabilistic distribution. This paper proposes a new formulation for the confidence-based RBDO method and design sensitivity analysis of the confidence level. The probability of the reliability analysis output is obtained with consecutive conditional probabilities of input distribution parameters and input distribution types using a Bayesian approach. The approximate conditional probabilities of input distribution parameters and types are suggested under certain assumptions. The Monte Carlo simulation is applied to practically calculate the output distribution, and the copula is used to describe the correlated input distribution types. A confidence-based RBDO problem is formulated using the derived the distribution of output. In this new formulation, the probabilistic constraint is modified to include both the target reliability and the target confidence level. Finally, the sensitivity of the confidence level, which is a new probabilistic constraint, is derived to support an efficient optimization process. Using accurate surrogate models, the proposed method does not require generation of additional surrogate models during the RBDO iteration; it only requires several evaluations of the same surrogate models. Hence, the efficiency of the method is obtained. For the numerical example, the confidence level is calculated and the accuracy of the derived sensitivity is verified when only limited data are available.Copyright

A Novel Second-Order Reliability Method (SORM) Using Non-Central or Generalized Chi-Squared Distributions

This paper proposes a novel second-order reliability method (SORM) using non-central or general chi-squared distribution to improve the accuracy of reliability analysis in existing SORM. Conventional SORM contains three types of errors: (1) error due to approximating a general nonlinear limit state function by a quadratic function at most probable point (MPP) in the standard normal U-space, (2) error due to approximating the quadratic function in U-space by a hyperbolic surface, and (3) error due to calculation of the probability of failure after making the previous two approximations. The proposed method contains the first type of error only which is essential to SORM and thus cannot be improved. However, the proposed method avoids the other two errors by describing the quadratic failure surface with the linear combination of non-central chi-square variables and using the linear combination for the probability of failure estimation. Two approaches for the proposed SORM are suggested in the paper. The first approach directly calculates the probability of failure using numerical integration of the joint probability density function (PDF) over the linear failure surface and the second approach uses the cumulative distribution function (CDF) of the linear failure surface for the calculation of the probability of failure. The proposed method is compared with first-order reliability method (FORM), conventional SORM, and Monte Carlo simulation (MCS) results in terms of accuracy. Since it contains fewer approximations, the proposed method shows more accurate reliability analysis results than existing SORM without sacrificing efficiency.Copyright

MPP-Based Dimension Reduction Method for RBDO Problems with Correlated Input Variables

In reliability-based design optimization (RBDO) problems with correlated input variables, a joint cumulative distribution function (CDF) needs to be obtained to transform, using the Rosenblatt transformation, the correlated input variables into independent standard Gaussian variables for the reliability analysis. However, a true joint CDF requires infinite number of data to be obtained, so in this paper, a copula is used to model the joint CDF using marginal CDFs and correlation parameters obtained from samples, which are available in practical applications. Using the joint CDF modeled by the copula, the transformation can be carried out based on the first order reliability method (FORM), which has been commonly used in reliability analysis. However, the FORM may yield different reliability analysis results with some errors for different transformation ordering of input variables due to the nonlinearities of differently transformed constraint functions. For this, the most probable point (MPP) based dimension reduction method (DRM), which more accurately and efficiently calculates the probability of failure than the FORM and the second order reliability method (SORM), respectively, is proposed to use to reduce the effect of transformation ordering in the inverse reliability analysis, and thus RBDO. To study the effect of transformation ordering on RBDO results, several numerical examples are tested using two different reliability methods, the FORM and DRM.

Sampling-Based RBDO Using Probabilistic Sensitivity Analysis and Virtual Support Vector Machine

In this paper, a sampling-based RBDO method using a classification method is presented. The probabilistic sensitivity analysis is used to compute sensitivities of probabilistic constraints with respect to random variables. Since the probabilistic sensitivity analysis requires only the limit state function, and not the response surface or sensitivity of the response, an efficient classification method can be used for a sampling-based RBDO. The proposed virtual support vector machine (VSVM), which is a classification method, is a support vector machine (SVM) with virtual samples. By introducing virtual samples, VSVM overcomes the deficiency in existing SVM that uses only classification information as their input. In this paper, the universal Kriging method is used to obtain locations of virtual samples to improve the accuracy of the limit state function for highly nonlinear problems. A sequential sampling strategy effectively inserts new samples near the limit state function. In sampling-based RBDO, Monte Carlo simulation (MCS) is used for the reliability analysis and probabilistic sensitivity analysis. Since SVM is an explicit classification method, unlike implicit methods, computational cost for evaluating a large number of MCS samples can be significantly reduced. Several efficiency strategies, such as the hyper-spherical local window for generation of the limit state function and the Transformations/Gibbs sampling method to generate uniform samples in the hyper-sphere, are also applied. Examples show that the proposed sampling-based RBDO using VSVM yields better efficiency in terms of the number of required samples and the computational cost for evaluating MCS samples while maintaining accuracy similar to that of sampling-based RBDO using the implicit dynamic Kriging (D-Kriging) method.© 2012 ASME

System Reliability-Based Design Optimization Using MPP-Based Dimension Reduction Method

The system probability of failure calculation of the series system entails multi-dimensional integration, which is very difficult and numerically expensive. To resolve the computational burden, the narrow bound method, which accounts for the component failures and joint failures between two failure modes, has been widely used. For the analytic calculation of the component probability of failure, this paper proposes to use the most probable point (MPP)-based dimension reduction method (DRM). For the joint probability of failure calculation, three cases are considered based on the convexity or concavity of the performance functions. Design sensitivity analysis for the system reliability-based design optimization (RBDO), which is the major contribution of this paper, is carried out as well. Based on the results of numerical examples, the system probability of failure and its sensitivity calculation show very good agreement with the results obtained by Monte Carlo simulation (MCS) and the finite difference method (FDM).

First-Order Reliability Analysis of Vehicle Safety in Highway Horizontal Curves

This study presents a reliability analysis of vehicle sideslip and rollover in highway horizontal curves, mainly focusing on exit ramps and interchanges. To accurately describe failure modes of a ground vehicle, analytic models for sideslip and rollover are derived considering nonlinear characteristics of vehicle behavior using the commercial software, TruckSim®, with high fidelity. Then, the probability of accident is evaluated using the First-Order Reliability Method (FORM). Furthermore, sensitivity functions of each failure mode are analytically derived to apply FORM.Numerical studies are conducted using a single-unit truck model. The results show that a truck is more likely to rollover than to slip at dry load. To propose practical application of the study, the reliability analysis for the minimum radius recommended by American Association of State Highway and Transportation Officials (AASHTO) at various speeds and bank angles is conducted. The reliability analysis of current design method shows that the method cannot provide the sufficient margin of safety against both of rollover and sideslip when there are deviations from assumed conditions, especially at low speed of vehicles.Copyright