Probabilistic finite element analysis using ANSYS

ProbabilisticfiniteelementanalysisusingANSYSStefanReha,*,Jean-DanielBeleyb,SiddharthaMukherjeeb,EngHuiKhorbaUniversityofAppliedSciences,BerlinerTor21,D-20099Hamburg,GermanybANSYSInc.,275TechnologyDrive,Canonsburg,PA15317,USAAvailableonline24June2005AbstractDrivenbystiffcompetition,industrialmanufacturersfindthemselvesundergrowingpressuretooptimizeapparentlyconflictingtechnicalandfinancialgoalsinanenvironmentofeverincreasingproductcomplex-ity.Inaddition,thischallengeistobemetundertheexistenceofrandomnessanduncertainty,whichtheproductsaresubjectedto.Consequently,findingtherightbalancebetweenconflictinggoalsundertheexis-tenceofuncertaintiesrequirestheuseofprobabilistictools.Toachievethis,ANSYSInc.hasreleasedtwotools,namelytheANSYSProbabilisticDesignSystemandtheANSYSDesignXplorer.Thispaperdescribestheproblemsthatcanbeaddressed,theunderlyingalgorithmsimplementedandmethodologiesofthesemethodsinbothtools.AspecialtopicofthepaperisthediscussionandexplanationoftheVar-iationalTechnology,whichisofferedinbothtools.VariationalTechnologyisahighlyefficientmethodtoprovideaccurate,high-orderresponsesurfacesbasedonasinglefiniteelementanalysis.Thecapabilities,strengthsandweaknessesofthesemethodsarediscussed.Thepossibilitytoreducetheexecutiontimeusingparallelcomputingisdiscussed.Differentmeasurestoassesstheaccuracyandvalidityoftheresultsobtainedwiththedifferentprobabilisticmethodsaregivenspecialattention.Variouscapabilitiestopost-processtheprobabilisticresultsarementioned.Themethodsandthecapabilitiestooptimizemultipleandpossiblyconflictinggoalsarehighlighted.Finally,theapplicationofthesoftwareisillustratedusingvariousindustrialexampleproblems.
2005ElsevierLtd.Allrightsreserved.Keywords:Uncertainty;Reliability;DesignforSixSigma;Reliability-basedoptimization;Robustdesign0167-4730/$-seefrontmatter
2005ElsevierLtd.Allrightsreserved.doi:10.1016/j.strusafe.2005.03.010*Correspondingauthor.Tel.:[phone];fax:[phone].E-mailaddresses:[e-mail](S.Reh),[e-mail](J.-D.Beley),[e-mail](S.Mukherjee),[e-mail](E.H.Khor).(2006)17–43STRUCTURALSAFETY1.IntroductionSincequiteanumberofyears,methodsandtoolstoquantifythereliabilityandqualityofmechanicalproductshavereceivedanever-growinginterestfromindustryaswellasacademia.Dri-venbytheneedtosimultaneouslyreducecosts(manufacturingcosts,warrantycosts,etc.),reducetime-to-market,improveproductqualityandproductreliability,industrialmanufacturersfindthemselveschallengedtooptimizeapparentlyconflictingtechnicalandfinancialgoalsinanenvi-ronmentofeverincreasingproductcomplexity.Inaddition,thischallengeistobemetundertheexistenceofrandomnessanduncertainty,whichtheproductsaresubjectedto,sincetheyareman-ufacturedandoperatedunderreal-lifeconditions.Naturally,optimizationisonlypossibleiftheoptimizationgoalsaswellaspossibleconstraintscanbequantified.Consequently,findingtherightbalancebetweenconflictinggoalsundertheexistenceofuncertaintiesrequirestheuseofprobabi-listictools.Inthiscontext,thefollowinganalysistypesaretypicallyusedtoaddressthisquestion:Deterministicanalysis.Adeterministicanalysisisthetransformationfunctionrepresentingtherelationshipbetweentheinputvariablesinfluencingthebehaviourofaproductandtheresultparameterscharacterizingtheproductbehaviour.Insimplecasestheresultparameterscanbeexpressedasananalyticalfunction,butinrealisticcasestheinput-outputrelationshipisonlygivenalgorithmicallyforexampleusingfiniteelementprogram.Uncertaintyanalysis.Iftheinputvariablesinfluencingthebehaviourofaproductareuncertain,i.e.aresubjectedtoscatter,thentheprimarytaskofanuncertaintyanalysisistoquantifyhowmuchtheresultparameterscharacterizingtheproductbehaviourareaffectedbythoseuncertainties.Reliabilityanalysis.Inordertoquantifythereliabilityofaproductitisusefultocalculatethefailureprobabilityornon-conformanceprobabilitydenotedwithPf.ThereliabilityPsistheprobabilitythattheproductwillsurviveorconformstocertainrequirements,withPs=1
Pf.Reliability-basedoptimization.Asthenameimplies,reliability-basedoptimizationtriestoopti-mizethereliabilityorfailureprobability.Itshouldbenoted,thatimprovingthereliabilityoftenconflictswithothertechnicalandfinancialgoals.Hence,theoptimizationprocessshouldtrytoachieveareasonableandquantifiablebalancebetweenallgoals.Robustdesign.Engineeringproductsarebecomingmoreandmorecomplexandpronetotheeffectsofuncertainty[1].Robustdesigntriestooptimizethedesigntomakeitlesssensitivetounavoidableuncertainties,therebyreducingthevariabilityintheproductbehaviourandmakingitmore
predictable.Achievingthisisanoptimizationproblemusingtheresultsofaprobabilisticanalysisasgoalsandconstraintfunctions.Measurestoquantifyrobustness(orthelackthereof)areforexamplethestandarddeviationorcoefficientofvariation,kurtosis,sig-nal-to-noiseratios[2–4],Shannonsentropy[5]orthefailureprobabilityofparametersdescrib-ingthebehaviourofaproduct.DesignforSixSigma.Theexpression
SixSigmawasfirstdevisedbyMotorola[6]definingthatSixSigmaqualityisgivenifonlyabout3