Evidence-theory-based reliability design optimization with parametric correlations

Parametric correlation exists widely in engineering problems. This paper presents an approach of evidence-theory-based design optimization (EBDO) with parametric correlations, which provides an effective computational tool for the structural reliability design involving epistemic uncertainties. According to the existing samples, the most fitting copula function is selected to formulate the joint basic probability assignment (BPA) of the correlated variables. The joint BPA is applied in the constraint reliability analysis, and an approximate technology is given to enhance the efficiency. A decoupling strategy is proposed for transforming the nested optimization of EBDO into a sequential iterative process of deterministic optimization and reliability analysis. The effectiveness of the proposed approach is demonstrated through two numerical examples and an engineering application.

[1]  A. Kiureghian,et al.  Aleatory or epistemic? Does it matter? , 2009 .

[2]  Xiaoqian Chen,et al.  A reliability-based multidisciplinary design optimization procedure based on combined probability and evidence theory , 2013 .

[3]  Ramana V. Grandhi,et al.  Gradient projection for reliability-based design optimization using evidence theory , 2008 .

[4]  Hua Zhang,et al.  A concurrent reliability optimization procedure in the earlier design phases of complex engineering systems under epistemic uncertainties , 2016 .

[5]  B. Y. Ni,et al.  A Vine-Copula-Based Reliability Analysis Method for Structures With Multidimensional Correlation , 2015 .

[6]  Arie Tzvieli.  Possibility theory: An approach to computerized processing of uncertainty , 1990, J. Am. Soc. Inf. Sci..

[7]  Irina Georgescu.  Possibility Theory and the Risk , 2012, Studies in Fuzziness and Soft Computing.

[8]  Xu Han,et al.  Structural reliability analysis using a copula-function-based evidence theory model , 2014 .

[9]  Arthur P. Dempster,et al.  Upper and Lower Probabilities Induced by a Multivalued Mapping , 1967, Classic Works of the Dempster-Shafer Theory of Belief Functions.

[10]  I. Elishakoff,et al.  Combination of probabilistic and convex models of uncertainty when scarce knowledge is present on acoustic excitation parameters , 1993 .

[11]  David E. Stewart,et al.  Rigid-Body Dynamics with Friction and Impact , 2000, SIAM Rev..

[12]  I. Elishakoff,et al.  Probabilistic interval reliability of structural systems , 2008 .

[13]  John E. Mottershead,et al.  Finite Element Model Updating in Structural Dynamics , 1995 .

[14]  Ramana V. Grandhi,et al.  Structural Design Optimization Based on Reliability Analysis Using Evidence Theory , 2003 .

[15]  Fengjiao Guan,et al.  A novel evidence theory model dealing with correlated variables and the corresponding structural reliability analysis method , 2018 .

[16]  R. Nelsen.  An Introduction to Copulas , 1998 .

[17]  Philippe Weber,et al.  Bayesian Networks and Evidence Theory to Model Complex Systems Reliability , 2007, J. Comput..

[18]  R. Fletcher.  Practical Methods of Optimization , 1988 .

[19]  Xiaoping Du,et al.  Sequential Optimization and Reliability Assessment Method for Efficient Probabilistic Design , 2002, DAC 2002.

[20]  John E. Renaud,et al.  Uncertainty quantification using evidence theory in multidisciplinary design optimization , 2004, Reliab. Eng. Syst. Saf..

[21]  W. Dong,et al.  Vertex method for computing functions of fuzzy variables , 1987 .

[22]  Xu Han,et al.  A novel evidence-theory-based reliability analysis method for structures with epistemic uncertainty , 2013 .

[23]  C. Jiang,et al.  A Hybrid Reliability Approach Based on Probability and Interval for Uncertain Structures , 2012 .

[24]  Karl Breitung,et al.  Probability Approximations by Log Likelihood Maximization , 1991 .

[25]  F. O. Hoffman,et al.  Propagation of uncertainty in risk assessments: the need to distinguish between uncertainty due to lack of knowledge and uncertainty due to variability. , 1994, Risk analysis : an official publication of the Society for Risk Analysis.

[26]  Haiyan Zhao,et al.  Decision-theoretic rough fuzzy set model and application , 2014, Inf. Sci..

[27]  Z. Mourelatos,et al.  A Design Optimization Method Using Evidence Theory , 2006, DAC 2005.

[28]  C. Jiang,et al.  A decoupling approach for evidence-theory-based reliability design optimization , 2017 .

[29]  Anne-Catherine Favre,et al.  Bayesian copula selection , 2006, Comput. Stat. Data Anal..

[30]  H. Akaike.  A new look at the statistical model identification , 1974 .

[31]  Kari Sentz,et al.  Combination of Evidence in Dempster-Shafer Theory , 2002 .

[32]  Katsuichiro Goda,et al.  Statistical modeling of joint probability distribution using copula: Application to peak and permanent displacement seismic demands , 2010 .

[33]  Madan M. Gupta.  Fuzzy set theory and its applications , 1992 .

[34]  Efstratios Nikolaidis,et al.  Engineering Design Reliability Handbook , 2004 .

[35]  Xue Han,et al.  First and second order approximate reliability analysis methods using evidence theory , 2015, Reliab. Eng. Syst. Saf..

[36]  Dian-Qing Li,et al.  Impact of copulas for modeling bivariate distributions on system reliability , 2013 .

[37]  Tao Yourui,et al.  A Reliability-Based Multidisciplinary Design Optimization Method with Evidence Theory and Probability Theory , 2017 .

[38]  K. K. Choi,et al.  Reliability-based design optimization of problems with correlated input variables using a Gaussian Copula , 2009 .

[39]  Glenn Shafer,et al.  A Mathematical Theory of Evidence , 2020, A Mathematical Theory of Evidence.

[40]  C. Jiang,et al.  Correlation analysis of non-probabilistic convex model and corresponding structural reliability technique , 2011 .

[41]  Andy Hart,et al.  Application of Uncertainty Analysis to Ecological Risks of Pesticides , 2010 .

[42]  S. Kodiyalam,et al.  Structural optimization using probabilistic constraints , 1992 .

[43]  Kyung K. Choi,et al.  Identification of marginal and joint CDFs using Bayesian method for RBDO , 2009 .

[44]  Achintya Haldar,et al.  Probability, Reliability and Statistical Methods in Engineering Design (Haldar, Mahadevan) , 1999 .

[45]  Xiaoping Du,et al.  Uncertainty Analysis With Probability and Evidence Theories , 2006, DAC 2006.

[46]  M. Sklar.  Fonctions de repartition a n dimensions et leurs marges , 1959 .

[47]  Robert E. Melchers,et al.  Structural Reliability: Analysis and Prediction , 1987 .

[48]  Z. Kang,et al.  Non-probabilistic reliability-based topology optimization of geometrically nonlinear structures using convex models , 2009 .

[49]  Baoding Liu.  Uncertain Risk Analysis and Uncertain Reliability Analysis , 2010 .

[50]  Eugene Komaroff,et al.  Increase in scrotal temperature in laptop computer users. , 2005, Human reproduction.

[51]  H. Joe.  Multivariate Models and Multivariate Dependence Concepts , 1997 .

[52]  Kalyanmoy Deb,et al.  An evolutionary algorithm based approach to design optimization using evidence theory , 2013 .

[53]  Frank E. Grubbs.  An Introduction to Probability Theory and its Applications , 1967 .

[54]  William F. Caselton,et al.  Decision making with imprecise probabilities: Dempster‐Shafer Theory and application , 1992 .

[55]  Ronald L. Wasserstein,et al.  Monte Carlo: Concepts, Algorithms, and Applications , 1997 .