Abstract
In most industrial applications, only limited statistical information is available to describe the input uncertainty model due to expensive experimental testing costs. It would be unreliable to use the estimated input uncertainty model obtained from insufficient data for the design optimization. Furthermore, when input variables are correlated, we would obtain non-optimum design if we assume that they are independent. In this paper, two methods for problems with a lack of input statistical information—possibility-based design optimization (PBDO) and reliability-based design optimization (RBDO) with confidence level on the input model—are compared using mathematical examples and an Abrams M1A1 tank roadarm example. The comparison study shows that PBDO could provide an unreliable optimum design when the number of samples is very small. In addition, PBDO provides an optimum design that is too conservative when the number of samples is relatively large. Furthermore, the obtained PBDO designs do not converge to the optimum design obtained using the true input distribution as the number of samples increases. On the other hand, RBDO with confidence level on the input model provides a conservative and reliable optimum design in a stable manner. The obtained RBDO designs converge to the optimum design obtained using the true input distribution as the number of samples increases.
Similar content being viewed by others
References
Ang AH-S, Tang WH (1984) Probability concepts in engineering design, vol I: decision, risk and reliability. Wiley, New York
Annis C (2004) Life prediction isn’t as easy as it looks. J ASTM Int 1(2):3–14
Aughenbaugh JM, Paredis CJJ (2006) The value of using imprecise probabilities in engineering design. J Mech Des 128(4):969–979
Ben-Haim Y, Elishakoff I (1990) Convex models of uncertainty in applied mechanics. Elsevier, Amsterdam
Breitung K (1984) Asymptotic approximations for multinormal integrals. ASCE J Eng Mech 110(3):357–366
Center for Computer-Aided Design, College of Engineering (1999a) DRAW Concept Manual. The University of Iowa, Iowa City, IA
Center for Computer-Aided Design, College of Engineering (1999b) DRAW User Reference. The University of Iowa, Iowa City, IA
Denny M (2001) Introduction to importance sampling in rare-event simulations. Eur J Phys 22:403–411
Du L, Choi KK (2006) Possibility-based design optimization method for design problems with both statistical and fuzzy input data. ASME J 128(4):928–935
Du L, Choi KK (2008) An inverse analysis method for design optimization with both statistical and fuzzy uncertainties. Struct Multidiscip Optim 37(2):107–119
Du X, Sudjianto A, Huang B (2005) Reliability-based design with the mixture of random and interval variables. J Mech Des 127(6):1068–1076
Du L, Choi KK, Youn BD (2006) An inverse possibility analysis method for possibility-based design optimization. AIAA J 44(11):2682–2690
Efron B (1982) The jackknife, the bootstrap, and other resampling plans. SIAM, Philadelphia
Efron B, Tibshirani R (1993) An introduction to the bootstrap. Chapman & Hall/CRC, Boca Raton
Efstratios N, Ghiocel DM, Singhal S (2004) Engineering design reliability handbook. CRC Press, New York
Genest C, Favre A-C (2007) Everything you always wanted to know about copula modeling but were afraid to ask. J Hydrol Eng 12(4):347–368
Haldar A, Mahadevan S (2000) Probability, reliability and statistical methods in engineering design. Wiley, New York
Hasofer AM, Lind NC (1974) An exact and invariant first order reliability format. ASCE J Eng Mech Div 100(1):111–121
Hoel PG (1962) Introduction to mathematical statistics, 3rd edn. Wiley, New York
Hohenbichler M, Rackwitz R (1988) Improvement of second-order reliability estimates by importance sampling. ASCE J Eng Mech 114(12):2195–2199
Huard D, Évin G, Favre AC (2006) Bayesian copula selection. Comput Stat Data Anal 51(2):809–822
Lee I, Choi KK, Du L, Gorsich D (2008) Inverse analysis method using MPP-based dimension reduction for reliability-based design optimization of nonlinear and multi-dimensional systems. Comput Methods Appl Mech Eng 198(1):14–27
Meggiolaro MA, Castro JTP (2004) Statistical evaluation of strain-life fatigue crack initiation predictions. Int J Fatigue 26(5):463–476
Mourelatos ZP, Zhou J (2005) Reliability estimation and design with insufficient data based on possibility theory. AIAA J 43(8):1696–1705
Nelsen RB (1999) An introduction to copulas. Springer, New York
Nikolaidis E, Cudney HH, Chen S, Haftka RT, Rosca R (2004) Comparison of probability and possibility for design against catastrophic failure under uncertainty. J Mech Des 126:386–394
Noh Y, Choi KK, Du L (2008) Reliability based design optimization of problems with correlated input variables using copulas. Struct Multidiscip Optim. doi:10.1007/s00158-008-0277-9
Noh Y, Choi KK, Lee I (2009) Reduction of transformation ordering effect in RBDO using MPP-based dimension reduction method. AIAA J 47(4):994–1004
Noh Y, Choi KK, Lee I (2010) Identification of marginal and joint CDFs using the Bayesian method for RBDO. Struct Multidiscip Optim 40(1):35–51
Noh Y, Choi KK, Lee I (2011a) Reliability-based design optimization with confidence level under input model uncertainty due to limited test data. J Struct Multidiscip Optim 43(4):443–458
Noh Y, Choi KK, Lee I (2011b) Reliability-based design optimization with confidence level for non-Gaussian distributions using bootstrap method. J Mech Des 133(9):91001
Picheny V, Kim NH, Haftka RT (2010) Application of bootstrap method in conservative estimation of reliability with limited samples. Struct Multidiscip Optim 41(2):205–217
Rahman S, Wei D (2006) A univariate approximation at most probable point for Higer-order reliability analysis. Int J Solids Struct 43:2820–2839
Rosenblatt M (1952) Remarks on a multivariate transformation. Ann Math Stat 23:470–472
Ross TJ (2010) Fuzzy logic with engineering application, 3rd edn. Wiley, Chichester
Socie DF (2003) Seminar notes: “Aspects of Fatigue”. URL:https://www.efatigue.com. Accessed 1 Jan 2011
Swanson Analysis System Inc. (1989) ANSYS engineering analysis system user’s manual, vol I, II. Houston, PA
Tu J, Choi KK (1999) A new study on reliability-based design optimization. J Mech Des 121(4):557–564
Tu J, Choi KK, Park YH (2001) Design potential method for reliability-based system parameter design using adaptive constraint evaluation. AIAA J 39(4):667–677
Wu W, Rao SS (2004) Interval approach for the modeling of tolerances and clearances in mechanism analysis. J Mech Des 126(4):581–592
Youn BD, Choi KK, Du L (2005) Enriched Performance Measure Approach (PMA+) for reliability-based design optimization. AIAA J 43(4):874–884
Zadeh LA (1965) Fuzzy sets. Inform Control 8(12):338–353
Zhou J, Mourelatos ZP (2008) Design under uncertainty using a combination of evidence theory and a bayesian approach. In: Proceedings of the third international workshop on reliable engineering computing (REC). NSF Workshop on Imprecise Probability in Engineering Analysis & Design, Savannah, Georgia
Acknowledgments
Research is supported by the Automotive Research Center, which is sponsored by the U.S. Army Tank Automotive Research, Development and Engineering Center (TARDEC). This research was also partially supported by the World Class University Program through the National Research Foundation of Korea (NRF) grant funded by the Ministry of Education, Science and Technology (Grant Number R32-2008-000-10161-0 in 2009).
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Lee, I., Choi, K.K., Noh, Y. et al. Comparison study between probabilistic and possibilistic methods for problems under a lack of correlated input statistical information. Struct Multidisc Optim 47, 175–189 (2013). https://doi.org/10.1007/s00158-012-0833-1
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00158-012-0833-1