Robust design optimization under dependent random variables by a generalized polynomial chaos expansion
New computational methods are proposed for robust design optimization (RDO) of complex engineering systems subject to input random variables with arbitrary, dependent probability distributions. The methods are built on a generalized polynomial chaos expansion (GPCE) for determining the second-moment...
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| Published in | Structural and multidisciplinary optimization Vol. 63; no. 5; pp. 2425 - 2457 |
|---|---|
| Main Authors | , |
| Format | Journal Article |
| Language | English |
| Published |
Berlin/Heidelberg
Springer Berlin Heidelberg
01.05.2021
Springer Nature B.V |
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| Abstract | New computational methods are proposed for robust design optimization (RDO) of complex engineering systems subject to input random variables with arbitrary, dependent probability distributions. The methods are built on a generalized polynomial chaos expansion (GPCE) for determining the second-moment statistics of a general output function of dependent input random variables, an innovative coupling between GPCE and score functions for calculating the second-moment sensitivities with respect to the design variables, and a standard gradient-based optimization algorithm, establishing direct GPCE, single-step GPCE, and multi-point single-step GPCE design processes. New analytical formulae are unveiled for design sensitivity analysis that is synchronously performed with statistical moment analysis. Numerical results confirm that the proposed methods yield not only accurate but also computationally efficient optimal solutions of several mathematical and simple RDO problems. Finally, the success of conducting stochastic shape optimization of a steering knuckle demonstrates the power of the multi-point single-step GPCE method in solving industrial-scale engineering problems. |
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| AbstractList | New computational methods are proposed for robust design optimization (RDO) of complex engineering systems subject to input random variables with arbitrary, dependent probability distributions. The methods are built on a generalized polynomial chaos expansion (GPCE) for determining the second-moment statistics of a general output function of dependent input random variables, an innovative coupling between GPCE and score functions for calculating the second-moment sensitivities with respect to the design variables, and a standard gradient-based optimization algorithm, establishing direct GPCE, single-step GPCE, and multi-point single-step GPCE design processes. New analytical formulae are unveiled for design sensitivity analysis that is synchronously performed with statistical moment analysis. Numerical results confirm that the proposed methods yield not only accurate but also computationally efficient optimal solutions of several mathematical and simple RDO problems. Finally, the success of conducting stochastic shape optimization of a steering knuckle demonstrates the power of the multi-point single-step GPCE method in solving industrial-scale engineering problems. |
| Author | Lee, Dongjin Rahman, Sharif |
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| Keywords | Design sensitivity analysis Second-moment analysis GPCE RDO Stochastic optimization Score functions |
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Wiley-Interscience MATLAB (2019) Version 9.7.0 (R2019b). The MathWorks Inc., Natick, Massachusetts HuangBDuXAnalytical robustness assessment for robust designStruct Multidiscip Optim200734212313710.1007/s00158-006-0068-0 LeeDRahmanSPractical uncertainty quantification analysis involving statistically dependent random variablesAppl Math Model202084324356409121010.1016/j.apm.2020.03.041 ABAQUS (2019) version 2019. Dassault Systèmes Simulia Corp RahmanSA polynomial chaos expansion in dependent random variablesJ Math Anal Appl20184641749775379411410.1016/j.jmaa.2018.04.032 RenXRahmanSRobust design optimization by polynomial dimensional decompositionStruct Multidiscip Optim2013481127148307242410.1007/s00158-013-0883-z BrowderAMathematical analysis: an introduction. 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Wiley series in probability and mathematical statistics1993New YorkWiley ShinSChoBRDevelopment of a sequential optimization procedure for robust design and tolerance design within a bi-objective paradigmEng Optim20084011989100910.1080/03052150802148910 ToropovVFilatovAPolynkinAMultiparameter structural optimization using FEM and multipoint explicit approximationsStruct Multidiscip Optim19936171410.1007/BF01743169 MourelatosZLiangJA methodology for trading-off performance and robustness under uncertaintyJ Mechan Design2006128485686310.1115/1.2202883 RosenblattMRemarks on a multivariate transformationAnn Math Statist1952234704724952510.1214/aoms/1177729394 ElishakoffIHaftkaRFangJStructural design under bounded uncertainty—optimization with anti-optimizationComput Struct19945361401140510.1016/0045-7949(94)90405-7 MarlerRTAroraJSThe weighted sum method for multi-objective optimization: new insightsStruct Multidiscip Opt2010416853862261084410.1007/s00158-009-0460-7 NhaVTShinSJeongSHLexicographical dynamic goal programming approach to a robust design optimization within the pharmaceutical environmentEur J Oper Res20132292505517305424910.1016/j.ejor.2013.02.017 |
| References_xml | – reference: NhaVTShinSJeongSHLexicographical dynamic goal programming approach to a robust design optimization within the pharmaceutical environmentEur J Oper Res20132292505517305424910.1016/j.ejor.2013.02.017 – reference: MiettinenKNonlinear multiobjective optimization, vol 122012BerlinSpringer Science & Business Media – reference: ParkGLeeTKwonHHwangKRobust design: an overviewAIAA J200644118119110.2514/1.13639 – reference: RahmanSA polynomial chaos expansion in dependent random variablesJ Math Anal Appl20184641749775379411410.1016/j.jmaa.2018.04.032 – reference: BashiriMMoslemiAAkhavan NiakiSTRobust multi-response surface optimization: a posterior preference approachInt Trans Oper Res202027317511770405806610.1111/itor.12450 – reference: ChenWAllenJTsuiKMistreeFProcedure for robust design: minimizing variations caused by noise factors and control factorsJ Mechan Design Trans ASME1996118447848510.1115/1.2826915 – reference: LeeDRahmanSPractical uncertainty quantification analysis involving statistically dependent random variablesAppl Math Model202084324356409121010.1016/j.apm.2020.03.041 – reference: RahmanSStochastic sensitivity analysis by dimensional decomposition and score functionsProbab Eng Mechan200924327828710.1016/j.probengmech.2008.07.004 – reference: LeeSChenWKwakBRobust design with arbitrary distributions using gauss-type quadrature formulaStruct Multidiscip Optim2009393227243252506210.1007/s00158-008-0328-2 – reference: BrowderAMathematical analysis: an introduction. Undergraduate texts in mathematics1996BerlinSpringer10.1007/978-1-4612-0715-3 – reference: OnoSYoshitakeYNakayamaSRobust optimization using multi-objective particle swarm optimizationArtificial Life Robot200914217410.1007/s10015-009-0647-4 – reference: NatafADetermination des distributions de probabilités dont les marges sont donnéeśC R Acad Sci, Paris196225542431391870109.11904 – reference: RosenblattMRemarks on a multivariate transformationAnn Math Statist1952234704724952510.1214/aoms/1177729394 – reference: DuXChenWTowards a better understanding of modeling feasibility robustness in engineering designJ Mechan Design2000122438539410.1115/1.1290247 – reference: MarlerRTAroraJSThe weighted sum method for multi-objective optimization: new insightsStruct Multidiscip Opt2010416853862261084410.1007/s00158-009-0460-7 – reference: ShinSChoBRDevelopment of a sequential optimization procedure for robust design and tolerance design within a bi-objective paradigmEng Optim20084011989100910.1080/03052150802148910 – reference: RamakrishnanBRaoSA general loss function based optimization procedure for robust designEng Optim199625425527610.1080/03052159608941266 – reference: SundaresanSIshiiKHouserDRA robust optimization procedure with variations on design variables and constraintsEng Opt+ A35199524210111710.1080/03052159508941185 – reference: HuangBDuXAnalytical robustness assessment for robust designStruct Multidiscip Optim200734212313710.1007/s00158-006-0068-0 – reference: Taguchi G (1993) Taguchi on robust technology development: bringing quality engineering upstream. ASME Press series on international advances in design productivity, ASME Press – reference: RenXRahmanSRobust design optimization by polynomial dimensional decompositionStruct Multidiscip Optim2013481127148307242410.1007/s00158-013-0883-z – reference: Stephens R, Fatemi A, Stephens RR, Fuchs H (2000) Metal fatigue in engineering. Wiley-Interscience – reference: YaoWChenXLuoWVan ToorenMGuoJReview of uncertainty-based multidisciplinary design optimization methods for aerospace vehiclesProg Aerosp Sci201147645047910.1016/j.paerosci.2011.05.001 – reference: JinRDuXChenWThe use of metamodeling techniques for optimization under uncertaintyStruct Multidiscip Optim20032529911610.1007/s00158-002-0277-0 – reference: MourelatosZLiangJA methodology for trading-off performance and robustness under uncertaintyJ Mechan Design2006128485686310.1115/1.2202883 – reference: XiuDKarniadakisGEThe Wiener-Askey polynomial chaos for stochastic differential equationsSIAM J Sci Comput200224619644195105810.1137/S1064827501387826 – reference: RubinsteinRShapiroADiscrete event systems: sensitivity analysis and stochastic optimization by the score function method. Wiley series in probability and mathematical statistics1993New YorkWiley – reference: Lee I, Choi KK, Noh Y, Zhao L, Gorsich D (2011) Sampling-based stochastic sensitivity analysis using score functions for rbdo problems with correlated random variables. J Mech Des 133(2) – reference: WienerNThe homogeneous chaosAm J Math1938604897936150735610.2307/2371268 – reference: MATLAB (2019) Version 9.7.0 (R2019b). The MathWorks Inc., Natick, Massachusetts – reference: BhushanMNarasimhanSRengaswamyRRobust sensor network design for fault diagnosisComput Chem Eng2008324-51067108410.1016/j.compchemeng.2007.06.020 – reference: ABAQUS (2019) version 2019. Dassault Systèmes Simulia Corp – reference: ChatterjeeTChakrabortySChowdhuryRA critical review of surrogate assisted robust design optimizationArchiv Comput Methods Eng2019261245274389517510.1007/s11831-017-9240-5 – reference: BusbridgeISome integrals involving hermite polynomialsJ Lond Math Soc1948231351412738010.1112/jlms/s1-23.2.135 – reference: CramerAMSudhoffSDZiviELEvolutionary algorithms for minimax problems in robust designIEEE Trans Evol Comput200813244445310.1109/TEVC.2008.2004422 – reference: ChenWWiecekMMZhangJQuality utility—a compromise programming approach to robust designJ Mech Des1999121217918710.1115/1.2829440 – reference: RahmanSRenXNovel computational methods for high-dimensional stochastic sensitivity analysisInt J Numer Methods Eng20149812881916321433810.1002/nme.4659 – reference: ToropovVFilatovAPolynkinAMultiparameter structural optimization using FEM and multipoint explicit approximationsStruct Multidiscip Optim19936171410.1007/BF01743169 – reference: RahmanSExtended polynomial dimensional decomposition for arbitrary probability distributionsJ Eng Mechan-ASCE2009135121439145110.1061/(ASCE)EM.1943-7889.0000047 – reference: NohYChoiKDuLReliability-based design optimization of problems with correlated input variables using a gaussian copulaStruct Multidiscip Opt200938111610.1007/s00158-008-0277-9 – reference: CREO (2016) version 4.0. PTC – reference: LiGRabitzHD-MORPH regression: application to modeling with unknown parameters more than observation dataJ Math Chem20104810101035272633910.1007/s10910-010-9722-2 – reference: ShenDEBraatzRDPolynomial chaos-based robust design of systems with probabilistic uncertaintiesAIChE J20166293310331810.1002/aic.15373 – reference: ZamanKMcDonaldMMahadevanSGreenLRobustness-based design optimization under data uncertaintyStruct Multidiscip Optim201144218319710.1007/s00158-011-0622-2 – reference: ElishakoffIHaftkaRFangJStructural design under bounded uncertainty—optimization with anti-optimizationComput Struct19945361401140510.1016/0045-7949(94)90405-7 |
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