The Experts below are selected from a list of 204 Experts worldwide ranked by ideXlab platform
Kyung K. Choi - One of the best experts on this subject based on the ideXlab platform.
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reliability based design optimization of problems with correlated input variables using a gaussian copula
Structural and Multidisciplinary Optimization, 2009Co-Authors: Kyung K. Choi, Liu DuAbstract:The reliability-based design optimization (RBDO) using performance measure approach for problems with correlated input variables requires a Transformation from the correlated input random variables into independent standard normal variables. For the Transformation with correlated input variables, the two most representative Transformations, the Rosenblatt and Nataf Transformations, are investigated. The Rosenblatt Transformation requires a joint cumulative distribution function (CDF). Thus, the Rosenblatt Transformation can be used only if the joint CDF is given or input variables are independent. In the Nataf Transformation, the joint CDF is approximated using the Gaussian copula, marginal CDFs, and covariance of the input correlated variables. Using the generated CDF, the correlated input variables are transformed into correlated normal variables and then the correlated normal variables are transformed into independent standard normal variables through a linear Transformation. Thus, the Nataf Transformation can accurately estimates joint normal and some lognormal CDFs of the input variable that cover broad engineering applications. This paper develops a PMA-based RBDO method for problems with correlated random input variables using the Gaussian copula. Several numerical examples show that the correlated random input variables significantly affect RBDO results.
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The Use of Copulas and MPP-Based Dimension Reduction Method (DRM) to Assess and Mitigate Engineering Risk in the Army Ground Vehicle Fleet
2008Co-Authors: David Lamb, David Gorsich, Kyung K. ChoiAbstract:Abstract : 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 inverse reliability analysis. However, a true joint CDF requires infinite number of test data to be obtained, so in this paper, a copula is used, which models a joint CDF only using marginal CDFs and limited data. Then, the inverse reliability analysis can be carried out using the joint CDF modeled by the copula and the first order reliability method (FORM), which has been commonly used in the inverse reliability analysis. However, because of the nonlinear Rosenblatt Transformation, the FORM may yield inaccurate reliability analysis results. To resolve the problem, this paper proposes to use the most probable point (MPP)-based dimension reduction method (DRM) for more accurate inverse reliability analysis and RBDO. As an example of the proposed method, an RBDO study of an M1A1 Abrams tank roadarm is carried out.
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MPP-Based Dimension Reduction Method for RBDO Problems with Correlated Input Variables
12th AIAA ISSMO Multidisciplinary Analysis and Optimization Conference, 2008Co-Authors: Kyung K. ChoiAbstract: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.
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Selection of Copula to Generate Input Joint CDF for RBDO
Volume 1: 34th Design Automation Conference Parts A and B, 2008Co-Authors: Kyung K. Choi, Liu DuAbstract:For RBDO problems with correlated input variables, it is necessary to obtain the input joint distribution (CDF, cumulative distribution function). Then Rosenblatt Transformation is used to transform the correlated input variables into the independent standard normal variables for the purpose of inverse reliability analysis. However, in practical industry RBDO problems, often only the marginal CDFs and paired samples are available from limited experimental data. In this paper, a copula, which is a link between a joint CDF and marginal CDFs, is proposed to generate an input joint CDF from these marginal CDFs and paired samples. To identify the right copula from limited data, Bayesian method is proposed to use in this paper. Using Bayesian method, the number of samples required to properly identify the right copula is investigated for different types of copulas and for different correlation coefficients. A real industry problem is used to show how a copula can be identified from the limited experimental data.Copyright © 2008 by ASME
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Reliability Based Design Optimization with Correlated Input Variables
SAE Technical Paper Series, 2007Co-Authors: Kyung K. Choi, Liu DuAbstract:Reliability-based design optimization (RBDO), which includes design optimization in design space and inverse reliability analysis in standard normal space, has been recently developed under the assumption that all input variables are independent because it is difficult to construct a joint probability distribution function (PDF) of input variables with limited data such as the marginal PDF and covariance matrix. However, since in real applications, it is common that some of the input variables are correlated, the RBDO results might contain a significant error if the correlation between input variables for RBDO is not considered. In this paper, Rosenblatt and Nataf Transformations, which are the most representative Transformation methods and have been widely used in the reliability analysis, have been studied and compared in terms of applicability to RBDO with correlated input variables. It is identified that Nataf Transformation is one of copulas and more applicable than Rosenblatt Transformation. Using numerical examples, it is also shown that the correlation of input variables significantly affects the RBDO results.
Liu Du - One of the best experts on this subject based on the ideXlab platform.
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reliability based design optimization of problems with correlated input variables using a gaussian copula
Structural and Multidisciplinary Optimization, 2009Co-Authors: Kyung K. Choi, Liu DuAbstract:The reliability-based design optimization (RBDO) using performance measure approach for problems with correlated input variables requires a Transformation from the correlated input random variables into independent standard normal variables. For the Transformation with correlated input variables, the two most representative Transformations, the Rosenblatt and Nataf Transformations, are investigated. The Rosenblatt Transformation requires a joint cumulative distribution function (CDF). Thus, the Rosenblatt Transformation can be used only if the joint CDF is given or input variables are independent. In the Nataf Transformation, the joint CDF is approximated using the Gaussian copula, marginal CDFs, and covariance of the input correlated variables. Using the generated CDF, the correlated input variables are transformed into correlated normal variables and then the correlated normal variables are transformed into independent standard normal variables through a linear Transformation. Thus, the Nataf Transformation can accurately estimates joint normal and some lognormal CDFs of the input variable that cover broad engineering applications. This paper develops a PMA-based RBDO method for problems with correlated random input variables using the Gaussian copula. Several numerical examples show that the correlated random input variables significantly affect RBDO results.
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Selection of Copula to Generate Input Joint CDF for RBDO
Volume 1: 34th Design Automation Conference Parts A and B, 2008Co-Authors: Kyung K. Choi, Liu DuAbstract:For RBDO problems with correlated input variables, it is necessary to obtain the input joint distribution (CDF, cumulative distribution function). Then Rosenblatt Transformation is used to transform the correlated input variables into the independent standard normal variables for the purpose of inverse reliability analysis. However, in practical industry RBDO problems, often only the marginal CDFs and paired samples are available from limited experimental data. In this paper, a copula, which is a link between a joint CDF and marginal CDFs, is proposed to generate an input joint CDF from these marginal CDFs and paired samples. To identify the right copula from limited data, Bayesian method is proposed to use in this paper. Using Bayesian method, the number of samples required to properly identify the right copula is investigated for different types of copulas and for different correlation coefficients. A real industry problem is used to show how a copula can be identified from the limited experimental data.Copyright © 2008 by ASME
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Reliability Based Design Optimization with Correlated Input Variables
SAE Technical Paper Series, 2007Co-Authors: Kyung K. Choi, Liu DuAbstract:Reliability-based design optimization (RBDO), which includes design optimization in design space and inverse reliability analysis in standard normal space, has been recently developed under the assumption that all input variables are independent because it is difficult to construct a joint probability distribution function (PDF) of input variables with limited data such as the marginal PDF and covariance matrix. However, since in real applications, it is common that some of the input variables are correlated, the RBDO results might contain a significant error if the correlation between input variables for RBDO is not considered. In this paper, Rosenblatt and Nataf Transformations, which are the most representative Transformation methods and have been widely used in the reliability analysis, have been studied and compared in terms of applicability to RBDO with correlated input variables. It is identified that Nataf Transformation is one of copulas and more applicable than Rosenblatt Transformation. Using numerical examples, it is also shown that the correlation of input variables significantly affects the RBDO results.
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Reliability Based Design Optimization with Correlated Input Variables Using Copulas
Volume 6: 33rd Design Automation Conference Parts A and B, 2007Co-Authors: Kyung K. Choi, Liu DuAbstract:For the performance measure approach (PMA) of RBDO, a Transformation between the input random variables and the standard normal random variables is necessary to carry out the inverse reliability analysis. For reliability analysis, Rosenblatt and Nataf Transformations are commonly used. In many industrial RBDO problems, the input random variables are correlated. However, often only limited information such as the marginal distribution and covariance could be practically obtained, and the input joint probability distribution function (PDF) is very difficult to obtain. Thus, in literature, most RBDO methods assume all input random variables are independent. However, in this paper, it is found that the RBDO results can be significantly different when the input variables are correlated. Thus, various Transformation methods are investigated for development of a RBDO method for problems with correlated input variables. It is found that Rosenblatt Transformation is impractical for problems with correlated input variables due to difficulty of constructing a joint PDF from the marginal distributions and covariance. On the other hand, Nataf Transformation can construct the joint CDF using the marginal distributions and covariance, and thus applicable to problems with correlated random input variables. The joint CDF is Nataf model, which is called a Gaussian copula in the copula family. Since the Gaussian copula can describe a wide range of the correlation coefficient, Nataf Transformation can be widely used for various types of correlated input variables. In this paper, Nataf Transformation is used to develop a RBDO method for design problems with correlated random input variables. Numerical examples are used to demonstrate the proposed method. Also, it is shown that the correlated random input variables significantly affect the RBDO results.
Y Zhao - One of the best experts on this subject based on the ideXlab platform.
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third moment standardization for structural reliability analysis
Journal of Structural Engineering-asce, 2000Co-Authors: Y ZhaoAbstract:First- and second-order reliability methods are generally considered to be among the most useful for computing structural reliability. In these methods, the uncertainties included in resistances and loads are generally expressed as continuous random variables that have a known cumulative distribution function. The Rosenblatt Transformation is a fundamental requirement for structural reliability analysis. However, in practical applications, the cumulative distribution functions of some random variables are unknown, and the probabilistic characteristics of these variables may be expressed using only statistical moments. In the present study, a structural reliability analysis method with inclusion of random variables with unknown cumulative distribution functions is suggested. Normal Transformation methods that make use of high-order moments are investigated, and an accurate third-moment standardization function is proposed. Using the proposed method, the normal Transformation for random variables with unknown cumulative distribution functions can be realized without using the Rosenblatt Transformation. Through the numerical examples presented, the proposed method is found to be sufficiently accurate to include the random variables with unknown cumulative distribution functions in the first- and second-order reliability analyses with little extra computational effort.
Can Xu - One of the best experts on this subject based on the ideXlab platform.
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Mapping-Based Hierarchical Sensitivity Analysis for Multilevel Systems With Multidimensional Correlations
Journal of Mechanical Design, 2020Co-Authors: Can XuAbstract:Abstract Hierarchical sensitivity analysis (HSA) of multilevel systems is to assess the effect of system’s input uncertainties on the variations of system’s performance through integrating the sensitivity indices of subsystems. However, it is difficult to deal with the engineering systems with complicated correlations among various variables across levels by using the existing hierarchical sensitivity analysis method based on variance decomposition. To overcome this limitation, a mapping-based hierarchical sensitivity analysis method is proposed to obtain sensitivity indices of multilevel systems with multidimensional correlations. For subsystems with dependent variables, a mapping-based sensitivity analysis, consisting of vine copula theory, Rosenblatt Transformation, and polynomial chaos expansion (PCE) technique, is provided for obtaining the marginal sensitivity indices. The marginal sensitivity indices can allow us to distinguish between the mutual depend contribution and the independent contribution of an input to the response variance. Then, extended aggregation formulations for local variables and shared variables are developed to integrate the sensitivity indices of subsystems at each level so as to estimate the global effect of inputs on the response. Finally, this paper presents a computational framework that combines related techniques step by step. The effectiveness of the proposed mapping-based hierarchical sensitivity analysis (MHSA) method is verified by a mathematical example and a multiscale composite material.
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A Vine Copula-Based Hierarchical Framework for Multiscale Uncertainty Analysis
Journal of Mechanical Design, 2019Co-Authors: Can XuAbstract:Abstract Uncertainty analysis is an effective methodology to acquire the variability of composite material properties. However, it is hard to apply hierarchical multiscale uncertainty analysis to the complex composite materials due to both quantification and propagation difficulties. In this paper, a novel hierarchical framework combined R-vine copula with sparse polynomial chaos expansions is proposed to handle multiscale uncertainty analysis problems. According to the strength of correlations, two different strategies are proposed to complete the uncertainty quantification and propagation. If the variables are weakly correlated or mutually independent, Rosenblatt Transformation is used directly to transform non-normal distributions into the standard normal distributions. If the variables are strongly correlated, the multidimensional joint distribution is obtained by constructing R-vine copula, and Rosenblatt Transformation is adopted to generalize independent standard variables. Then, the sparse polynomial chaos expansion is used to acquire the uncertainties of outputs with relatively few samples. The statistical moments of those variables that act as the inputs of next upscaling model can be gained analytically and easily by the polynomials. The analysis process of the proposed hierarchical framework is verified by the application of a 3D woven composite material system. Results show that the multidimensional correlations are modeled accurately by the R-vine copula functions, and thus uncertainty propagations with the transformed variables can be done to obtain the computational results with consideration of uncertainties accurately and efficiently.
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A Novel Hierarchical Framework for Uncertainty Analysis of Multiscale Systems Combined Vine Copula With Sparse PCE
Volume 2B: 45th Design Automation Conference, 2019Co-Authors: Can XuAbstract:Abstract Uncertainty analysis is an effective methodology to acquire the variability of composite material properties. However, it is hard to apply hierarchical multiscale uncertainty analysis to the complex composite materials due to both quantification and propagation difficulties. In this paper, a novel hierarchical framework combined R-vine copula with sparse polynomial chaos expansions is proposed to handle multiscale uncertainty analysis problems. According to the strength of correlations, two different strategies are proposed to complete the uncertainty quantification and propagation. If the variables are weakly correlated or mutually independent, Rosenblatt Transformation is used directly to transform non-normal distributions into the standard normal distributions. If the variables are strongly correlated, multidimensional joint distribution is obtained by constructing R-vine copula, and Rosenblatt Transformation is adopted to generalize independent standard variables. Then the sparse polynomial chaos expansion is used to acquire the uncertainties of outputs with relatively few samples. The statistical moments of those variables that act as the inputs of next upscaling model, can be gained analytically and easily by the polynomials. The analysis process of the proposed hierarchical framework is verified by the application of a 3D woven composite material system. Results show that the multidimensional correlations are modelled accurately by the R-vine copula functions, and thus uncertainty propagations with the transformed variables can be done to obtain the computational results with consideration of uncertainties accurately and efficiently.
Tamas Turanyi - One of the best experts on this subject based on the ideXlab platform.
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Investigation of the effect of correlated uncertain rate parameters via the calculation of global and local sensitivity indices
Journal of Mathematical Chemistry, 2018Co-Authors: É. Valkó, Á. Busai, Alison S Tomlin, T. Varga, Tamas TuranyiAbstract:Applications of global uncertainty methods for models with correlated parameters are essential to investigate chemical kinetics models. A global sensitivity analysis method is presented that is able to handle correlated parameter sets. It is based on the coupling of the Rosenblatt Transformation with an optimized Random Sampling High Dimensional Model Representation method. The accuracy of the computational method was tested on a series of examples where the analytical solution was available. The capabilities of the method were also investigated by exploring the effect of the uncertainty of rate parameters of a syngas–air combustion mechanism on the calculated ignition delay times. Most of the parameters have large correlated sensitivity indices and the correlation between the parameters has a high influence on the results. It was demonstrated that the values of the calculated total correlated and final marginal sensitivity indices are independent of the order of the decorrelation steps. The final marginal sensitivity indices are meaningful for the investigation of the chemical significance of the reaction steps. The parameters belonging to five elementary reactions only, have significant final marginal sensitivity indices. Local sensitivity indices for correlated parameters were defined which are the linear equivalents of the global ones. The results of the global sensitivity analysis were compared with the corresponding results of local sensitivity analysis for the case of the syngas–air combustion system. The same set of reactions was indicated to be important by both approaches.
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Investigation of the effect of correlated uncertain rate parameters on a model of hydrogen combustion using a generalized HDMR method
Proceedings of the Combustion Institute, 2016Co-Authors: É. Valkó, Alison S Tomlin, T. Varga, Tamas TuranyiAbstract:Abstract The High Dimensional Model Representation (HDMR) method has been applied in several previous studies to obtain global sensitivity indices of uncorrelated model parameters in combustion systems. However, the rate parameters of combustion models are intrinsically correlated and therefore uncertainty analysis methods are needed that can handle such parameters. A generalized HDMR method is presented here, which uses the Rosenblatt Transformation on a correlated model parameter sample to obtain a sample of independent parameters. The method provides a full set of both correlated and marginal sensitivity indices. Ignition delay times predicted by an optimized hydrogen–air combustion model in stoichiometric mixtures near the three explosion limits are investigated with this new global sensitivity analysis tool. The sensitivity indices which account for all the correlated effects of the rate parameters are shown to dominate uncertainties in the model output. However, these correlated indices mask the individual influence of parameters. The final marginal uncorrelated sensitivity indices for individual parameters better indicate the change of importance of homogeneous gas phase and species wall-loss reactions as the pressure is increased from above the first explosion limit to above the third limit. However, these uncorrelated indices are small and whilst they provide insights into the dominant chemical and physical processes of the model over the range of conditions studied, the correlations between parameters have a very significant effect. The implications of this result on model tuning will be discussed.
