Engineering analyseswithimpreciseprobabilities,applications
Michael Beer 1 / 23 Engineering analyses with imprecise probabilities, applications
ImpreciseProbabilitiesFUZZYPROBABILITIESINRELIABILITYANALYSISFuzzy parametersFailure probability.structural parametersprobabilisticmodelparametersacceptablePfμ(x)μ(P.)acceptable11PparameterintervalXSensitivityof Pw.r.t.x(probabilisticmodelchoice)mappinga;aαiX→PX00coarse specificationsofdesignparameters & probabilisticmodelsattentionto/excludemodeloptionsleadingtolargeimprecisionofPacceptableimprecisionof parameters & probabilisticmodelsindicationstocollectadditionalinformationdefinitionofqualityrequirementsμ)notimportant,butrobustdesignanalysis with variousintensities of imprecision2/23MichaelBeer
Michael Beer 2 / 23 FUZZY PROBABILITIES IN RELIABILITY ANALYSIS • structural parameters Fuzzy parameters Failure probability X ~ acceptable Pf Pf 0 αi µ(x) 1 x Pf ~ • probabilistic model parameters acceptable parameter interval Sensitivity of Pf w.r.t. x (probabilistic model choice) 0 αi µ(Pf) 1 mapping X ~ Pf → ~ coarse specifications of design parameters & probabilistic models attention to / exclude model options leading to large imprecision of Pf acceptable imprecision of parameters & probabilistic models indications to collect additional information definition of quality requirements robust design µ(.) not important, but analysis with various intensities of imprecision Imprecise Probabilities
ImpreciseProbabilitiesIMPLEMENTINGEPISTEMICUINCERTAINTYSubjectiveprobabilitiesBayesian updateprior distribution[f(x[0)] 9。 (0)[f (x, 0) . g (0) deposterior distribution》approachresultfrom"insidetheepistemicuncertaintyImpreciseprobabilities个F(x).set-theoreticalmodelsforindeterminacy/imprecision(imprecise data,vague conditions orcopulas etc.)F(x)e.g.,set-valuedparametersF(×) = ([F(x),F(×)) Vx)in probabilistic modelsX》approachresultfrom"outsidetheepistemicuncertaintychoicedepending onavailableinformationand purpose of the analysisapproaches not competing but complementaryandcanbecombined(e.g.setof priors,updatewithimprecisedata)Michael Beer3/23
Michael Beer 3 / 23 IMPLEMENTING EPISTEMIC UNCERTAINTY ( ) ( ) ( ) ( ) ( ) Θ = Θ Θ = ∏ θ⋅ θ θ = ∫ ∏ θ⋅ θθ 1 n n i i 1 X , ,X 1 n n i i 1 fx g f x , ,x fx g d posterior distribution prior distribution » approach result from “inside the epistemic uncertainty” Subjective probabilities • Bayesian update Imprecise probabilities • set-theoretical models for indeterminacy / imprecision choice depending on available information and purpose of the analysis » approach result from “outside the epistemic uncertainty” F x F x ,F x x ( ) = { li ui i ( ) ( ) ∀ } e.g., set-valued parameters in probabilistic models F(x) F(x) ~ x (imprecise data, vague conditions or copulas etc.) approaches not competing but complementary and can be combined (e.g. set of priors, update with imprecise data) Imprecise Probabilities
ConceptualcomparisonRELIABILITYANALYSIS-EXAMPLE1Fixedjacketplatformimprecisioninthemodelsfor》wave,dragandiceloadswindload》corrosion》jointsoftubularmembers》foundation》possibledamage4/23MichaelBeer
Michael Beer 4 / 23 RELIABILITY ANALYSIS − EXAMPLE 1 Fixed jacket platform imprecision in the models for » wave, drag and ice loads » wind load » corrosion » joints of tubular members » foundation » possible damage • Conceptual comparison
Conceptual comparisonRELIABILITYANALYSIS-EXAMPLE1Probabilisticmodel(AfterR.E.Melchers)c(t,E) =b(t,E)·f(t,E)+(t,E)(time-dependentcorrosiondepth,uniform)》c(t,E)-corrosiondepth》f(t,E)-meanvaluefunction》b(t,E)-biasfunction》e(t,E)-uncertainty function(zero mean Gaussian white noise)>E-collection of environmental and material parameterscorrosionaModeling ?rstransition2Lvg1.0b(t,E)f(t,E)180APCsC0.ta2dimensionless exposure period t/t,timephase23MichaelBeer5/23
Michael Beer 5 / 23 Probabilistic model (After R.E. Melchers) » c(t,E) − corrosion depth » f(t,E) − mean value function » b(t,E) − bias function » ε(t,E) − uncertainty function (zero mean Gaussian white noise) » E − collection of environmental and material parameters • ctE btE f tE tE ( , , , ) = ⋅ +ε ( ) ( ) ( ) ca cs ro ta ra rs 1 2 3 4 AP corrosion phase time transition (t,E) f(t,E) b(t,E) Modeling ? (time-dependent corrosion depth, uniform) RELIABILITY ANALYSIS − EXAMPLE 1 Conceptual comparison
Conceptual comparisonRELIABILITYANALYSIS-EXAMPLE1Modelsforthebiasfactormodel yariants2Boundedrandomvariable-betadistribution-q==1asupieio..·-21.5-9=r=3CaseII(x- a)9-1 (b -x)r-11CaseIa<x<bB(q,r)0.5(b-a)withXrepresentingarandombiasb(t,E)0xReliabilityanalysisintervalandfuzzysetselectonevalueb(t,E)=x个μ(x)calculateP,viaMonteCarlo simulationxpdfforP》betadistribution("overall"sensitivityof P)a》intervalintervalforP,(bounds)》fuzzy set fuzzysetforP0(set of intervals with various intensities ofX"incremental"sensitivitiesofP.)imprecision6/23MichaelBeer
Michael Beer 6 / 23 Models for the bias factor • ( ) ( ) ( ) ( ) ( ) − − + − − ⋅− = ⋅ − q 1 r 1 X qr1 1 xa bx f x Bqr, b a with X representing a random bias b(t,E) 0 0.5 1 1.5 2 0 1 probability density X q=r=1 q=r=2 q=r=3 model variants , axb ≤ ≤ x X ~ αi 1 0 µ(x) • interval and fuzzy set Reliability analysis • select one value b(t,E) = x • calculate Pf via Monte Carlo simulation » beta distribution pdf for Pf » interval interval for Pf (bounds) » fuzzy set fuzzy set for Pf (set of intervals with various intensities of imprecision "incremental" sensitivities of Pf) ("overall" sensitivity of Pf) Case I Case II Bounded random variable − beta distribution RELIABILITY ANALYSIS − EXAMPLE 1 Conceptual comparison
Conceptual comparisonRELIABILITYANALYSIS-EXAMPLE1Fixedjacketplatformdimensions.loads,environment》T=15°C,t=5a》height:142m》top:27X54m》random:waveheight,current,yield stress,》bottom:56X.70mandcorrosiondepthc(t,E)》pdf/intervalforimpreciseparametersReliabilityanalysisMonteCarlosimulationwithimportancesamplingandresponsesurfaceapproximationIntervalprobabilityNpdf=2000Nopti=114-pdf 19.60pdf29.739.49intervalb(.) = 1.0Npf=50007.08.09.0P, [10-7]MichaelBeer7/23
Michael Beer 7 / 23 dimensions » height: 142 m » top: 27 X 54 m » bottom: 56 X 70 m • Monte Carlo simulation with importance sampling and response surface approximation • loads, environment » T = 15°C, t = 5 a » random: wave height, current, yield stress, and corrosion depth c(t,E) » pdf / interval for imprecise parameters • Reliability analysis Interval probability 7.0 8.0 9.0 10.0 Beta (q=r=1) Beta (q=r=2) Interval Pf (×10-7) 9.49 9.60 9.73 7.0 8.0 9.0 Pf [10−7] 9.73 9.49 9.60 pdf 1 pdf 2 interval b(.) = 1.0 Fixed jacket platform Npdf = 2000 Nopti = 114 NPf = 5000 RELIABILITY ANALYSIS − EXAMPLE 1 Conceptual comparison
NumericallyefficientsolutionRELIABILITYANALYSIS-EXAMPLE2Multi-storey buildingmodel》8,200finiteelements,66,300dof》244randomvariablesand5intervals(parametersofsomerv)reliabilityanalysis》component failure》line sampling》intervalanalysiswithqlobaloptimization》distributed computingAnalysistypeSequentialParallel~13>6daysDirectMonteCarlo Simulationhours10000samplesMethodSpeed-up1h32min7.8minAdvancedMCS-LineSampling100samples11.7Distributed computing~100AdvancedSimulationAdvancedSimulation+>1000Distributedcomputing8/23MichaelBeer
Michael Beer 8 / 23 Multi-storey building • model » component failure • reliability analysis » line sampling » interval analysis with global optimization » distributed computing » 8,200 finite elements, 66,300 dof » 244 random variables and 5 intervals (parameters of some rv) Numerically efficient solution RELIABILITY ANALYSIS − EXAMPLE 2
NumericallyefficientsolutionRELIABILITYANALYSIS-EXAMPLE2High dimensional problems,line samplingglobaloptimizationproblemP,= inf h. (5,p)dop-distributionparametersx,p (x)-randomvariablesx -intervalsp,=supJh.(5,p)dx,p2r(x)2dependson intervalsxmapintervals xto augmentedprobabilityspaceQ××→0: ×→ne C=(h,(n)=)exploittopological properties of for line samplingsamplingdirection-Vgoptimal points(p",x")=y"(-Vg),(p',x)=y'(-Vg)P,=J h (5,p)d2p,=了ha(E,p)d2,2r(x)2 (x)Michael Beer9/23
Michael Beer 9 / 23 High dimensional problems, line sampling global optimization problem exploit topological properties of Θ for line sampling map intervals x to augmented probability space • p – distribution parameters ξ – random variables x – intervals ( ) ( ) f f d x p x p h pd Ω = ξΩ ∫ , inf , ( ) ( ) f f d x p x p h pd Ω = ξΩ ∫ , sup , Ωf depends on intervals x ! Ω× → Θ X : • • sampling direction −∇g ( ) ( ) uu u optimal points px g , , = ψ −∇ ( ) ( ) u f u f d x p h pd Ω = ξΩ ∫ , , ( ) ( ) ll l px g , = ψ −∇ ( ) ( ) l f l f d x p h pd Ω = ξΩ ∫ , xh x →η∈ = η µ σ µ = ℂx n xx x { ( ; , ) } RELIABILITY ANALYSIS − EXAMPLE 2 Numerically efficient solution
NumericallyefficientsolutionEXAMPLE:ROBUSTRELIABILITYASSESSMENTMulti-storeybuilding-componentfailureanalysis·structureeimpreciseprobabilisticinput》8,200finite elements,》488fuzzyparametersfor66,300 dof244fuzzyrandomvariables0.90.80.7f000.3± 7.5 %025tolerance rangem0.075P,(1-e)PeP,(1+e)SV#UnitsProb.dist.Description卫=pe[1-e,l+e]1Columns'strengthN(μ, )GPaμc = 0.1c=0.01Unif(a, D)2-193Sections'sizeac= 0.36bc=0.44mYoung's modulusGPa194-212LN(m，)Ue=12.25mc=35Vc=6.2510-2kg/dm3Material's density213-231LN(,)me=2.5Ve=6.2510-4LN(m,)Poisson's ratio232-244mc=0.25Michael Beer10/23
Michael Beer 10 / 23 EXAMPLE: ROBUST RELIABILITY ASSESSMENT Multi-storey building – component failure analysis • structure » 8,200 finite elements, 66,300 dof » 488 fuzzy parameters for 244 fuzzy random variables • imprecise probabilistic input ± 7.5 % tolerance range Numerically efficient solution