Integrated Bayesian Reliability Analysis under Input Variable and Metamodel Uncertainties

Reliability analysis is recently gaining more importance in the structural design process, which is to evaluate reliability due to the associated uncertainties. There are three types of uncertainties: first is the aleatory uncertainty which is irreducible and related with inherent physical randomness that is completely described by a suitable probability model. The second is the statistical uncertainty or the epistemic uncertainty, which results from the lack of knowledge due to the insufficient data, and can be reduced by collecting more information. These two uncertainties are encountered in the input variables such as dimensional tolerances, material properties and loading conditions. The third is the metamodel uncertainty which arises from the approximation of the response function, which is often required in the case of costly computation such as finite element model. In this study, an integrated method for the reliability analysis is proposed that can address all these uncertainties in a single Bayesian framework. Markov Chain Monte Carlo (MCMC) method is employed to facilitate the simulation of the posterior distribution, which is a modern computational method to draw random sequence of parameters that samples the given distribution. Mathematical and engineering examples are used to demonstrate the proposed method.

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

[2]  Sankaran Mahadevan,et al.  Integration of computation and testing for reliability estimation , 2001, Reliab. Eng. Syst. Saf..

[3]  Jun Zhou,et al.  Reliability Estimation and Design with Insufficient Data Based on Possibility Theory , 2004 .

[4]  T. J. Mitchell,et al.  Bayesian Prediction of Deterministic Functions, with Applications to the Design and Analysis of Computer Experiments , 1991 .

[5]  Jay D. Johnson,et al.  A sampling-based computational strategy for the representation of epistemic uncertainty in model predictions with evidence theory , 2007 .

[6]  Z. Qiu,et al.  Comparison between non‐probabilistic interval analysis method and probabilistic approach in static response problem of structures with uncertain‐but‐bounded parameters , 2004 .

[7]  Eduard Hofer,et al.  An approximate epistemic uncertainty analysis approach in the presence of epistemic and aleatory uncertainties , 2002, Reliab. Eng. Syst. Saf..

[8]  A. OHagan,et al.  Bayesian analysis of computer code outputs: A tutorial , 2006, Reliab. Eng. Syst. Saf..

[9]  M. Stein,et al.  A Bayesian analysis of kriging , 1993 .

[10]  Lfp Pascal Etman,et al.  Design and analysis of computer experiments , 1998 .

[11]  A. O'Hagan,et al.  Bayesian calibration of computer models , 2001 .

[12]  K Ahmed Noor.  Perspectives on Nondeterministic Approaches , 2004 .

[13]  Jian Cheng,et al.  Latin Hypercube Sampling in Bayesian Networks , 2000, FLAIRS.

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

[15]  B. Youn,et al.  Inverse Possibility Analysis Method for Possibility-Based Design Optimization , 2006 .

[16]  W. H. Greenwood,et al.  Root Sum Squares Tolerance Analysis with Nonlinear Problems , 1990 .

[17]  K. Choi,et al.  Inverse analysis method using MPP-based dimension reduction for reliability-based design optimization of nonlinear and multi-dimensional systems , 2008 .

[18]  S. Rahman,et al.  A univariate dimension-reduction method for multi-dimensional integration in stochastic mechanics , 2004 .

[19]  R. Rackwitz.  Reliability analysis—a review and some perspectives , 2001 .

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

[21]  B. Youn,et al.  Bayesian reliability-based design optimization using eigenvector dimension reduction (EDR) method , 2008 .

[22]  Thomas A. Cruse,et al.  Confidence Interval Simulation for Systems of Random Variables , 2007 .

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

[24]  Panos Y. Papalambros,et al.  A Bayesian Approach to Reliability-Based Optimization With Incomplete Information , 2006, DAC 2006.

[25]  M. Sasena,et al.  Exploration of Metamodeling Sampling Criteria for Constrained Global Optimization , 2002 .

[26]  Joo-Ho Choi,et al.  Improved dimension reduction method (DRM) in uncertainty analysis using kriging interpolation , 2009 .

[27]  Sankaran Mahadevan,et al.  Validation of reliability computational models using Bayes networks , 2005, Reliab. Eng. Syst. Saf..

[28]  Nando de Freitas,et al.  An Introduction to MCMC for Machine Learning , 2004, Machine Learning.