Engineering analyses withintervalsandfuzzysets,applications
Michael Beer 1 / 28 Engineering analyses with intervals and fuzzy sets, applications
EngineeringAnalyseswithIntervals,Fuzzy SetsandImpreciseProbabilitiesEARTHOUAKEANALYSISBridgeovertheStraitofMessinafuzzyinputvariablesZtrumateria s.pacaebaiSall rightsreserveddeadload of theroadconstruction》earthquakeloads(wave propagation)9608103300m7771836273300183+383.00+383.00+118.00+7700元+63.00+54.0052.00sideview[Bontempietal. ][v(t)briadgsirse vibration (middleof span)inutesV(t=ti)=0代t）matidttimetμ=0U2/28MichaelBeer
Michael Beer 2 / 28 EARTHQUAKE ANALYSIS side view [Bontempi et al. ] » material parameters • fuzzy input variables » dead load of the road construction » earthquake loads (wave propagation) v(t) v(t=t1) ~ t1 time t » CPU-time: N × 40 minutes • fuzzy analysis » response surface approximation with neural networks Bridge over the Strait of Messina • fuzzy transverse vibration (middle of span) µ=0 µ=1 µ=0 v v(ti ) μ Engineering Analyses with Intervals, Fuzzy Sets and Imprecise Probabilities
UncertaintyandRobustnessROBUSTNESSMeaning-two perspectives.performance ofa system under exceptional conditions》no substantial lossof serviceability and safetyduetoinappropriate use:abnormalsituations:unforeseen eventsexceptionaloverloadingextremeenvironmentalconditionsinclusion in safety/reliability analysis,reliability-based designperformanceofa systemundernormallyfluctuatingconditions>no noticeable effects onserviceability and safetyduetooccasional orfrequentfluctuations ofenvironmental conditionsandstructuralparametersGENICHI TAGUCHI:"Notjuststrong.Flexible!IdiotProof!Simple!Efficient!Aproduct/processthatproducesconsistent,highlevelperformancedespitebeingsubjectedto a wide range of changing client and manufacturing conditions.separateconsiderationMichaelBeer3/28
Michael Beer 3 / 28 » no noticeable effects on serviceability and safety due to Meaning − two perspectives • performance of a system under exceptional conditions • performance of a system under normally fluctuating conditions » no substantial loss of serviceability and safety due to ▪ inappropriate use ▪ abnormal situations ▪ exceptional overloading ▪ unforeseen events ▪ extreme environmental conditions ▪ occasional or frequent fluctuations of environmental conditions and structural parameters inclusion in safety / reliability analysis, reliability-based design GENICHI TAGUCHI: "Not just strong. Flexible! Idiot Proof! Simple! Efficient! A product/process that produces consistent, high level performance despite being subjected to a wide range of changing client and manufacturing conditions." separate consideration Uncertainty and Robustness ROBUSTNESS
UncertaintyandRobustnessROBUSTNESSSignificance.primary requirementto ensurefaultless operation over a period of time》inclusionof all uncertainty inthenumerical analysis》appropriatemathematical definitionand assessmentof robustness》incorporationinthedesignprocessDefinition and effect on structural designglobalmeasureforthedegreeofindependencebetweenchanges in the whole set of structural and environmental parameters andthe associated range of fluctuations of structural responses or safety measuresU(xx-structural and environmental parametersR(x,zz-structural responses orsafetymeasuresU(ZU(.)-uncertainty,specificmeasure in dependence on uncertainty modelextension:includeanoptionforchangesofthedesignparameterswithin a certainvalue range as uncertainty of x》comfortable decision margins during production/construction work》consideration of coarse specifications in initial design stagesMichaelBeer4/28
Michael Beer 4 / 28 Significance • primary requirement to ensure faultless operation over a period of time » inclusion of all uncertainty in the numerical analysis » appropriate mathematical definition and assessment of robustness » incorporation in the design process » comfortable decision margins during production / construction work » consideration of coarse specifications in initial design stages Definition and effect on structural design global measure for the degree of independence between changes in the whole set of structural and environmental parameters and the associated range of fluctuations of structural responses or safety measures • ( ) ( ) ( ) = U R , U x x z z x − structural and environmental parameters z − structural responses or safety measures U(.) − uncertainty, specific measure in dependence on uncertainty model extension: include an option for changes of the design parameters within a certain value range as uncertainty of x • Uncertainty and Robustness ROBUSTNESS
UncertaintyandRobustnessROBUSTNESSInclusion in the design processthree-criteria optimizationproblem》CriterionI-optimum structural design"in themean"traditional objectivesuchasminimummassorcostC, = T(x,z) = Maxformulated asa maximizationproblem》CriterionII-maximumrobustnessC = R(x,z) = Maxminimumuncertainty ofthecomputational resultswith respectto the uncertaintyof the input quantities》 Criterion III-maximumrange of possible values forthe input quantitiesCm = V(x,z) → Maxmaximumdecisionmarginsforthespecificationofthefinaldesignandsubsequentrevisionsandmaximumcapabilitiesforcompensatingchangesof input guantitiesduring production oroperation》overall optimizationproblem-optimumrobustandflexibledesignVrnVm-weightingfactorsVi·C,+W·Cn+Wm·C=MaxMichael Beer5/28
Michael Beer 5 / 28 Inclusion in the design process • three-criteria optimization problem » Criterion I − optimum structural design "in the mean" C T , Max I = ⇒ (x z) traditional objective such as minimum mass or cost, formulated as a maximization problem » Criterion II − maximum robustness C R , Max II = ⇒ (x z) minimum uncertainty of the computational results with respect to the uncertainty of the input quantities » Criterion III − maximum range of possible values for the input quantities C V , Max III = ⇒ (x z) maximum decision margins for the specification of the final design and subsequent revisions and maximum capabilities for compensating changes of input quantities during production or operation » overall optimization problem − optimum robust and flexible design ψ ⋅ +ψ ⋅ +ψ ⋅ ⇒ I I II II III III C C C Max ψψ ψ − weighting factors I II III , , Uncertainty and Robustness ROBUSTNESS
RobustnessAssessmentandRobustDesign via ClusterAnalysisCRASHWORTHINESSANALYSISVancomponent,stochasticanalysis,failureprobabilityFiniteElementmodelstochasticparameters>20.000shellelements》four sheet thicknesses,strainrateabsorbing boxtime historyMonteCarlosimulationstonewall force frontbumpertranslationVelocitvin x-directi maxF.0 m/smaxFcourtesyofDaimler AGandDYNAmoreGmbHt6/28MichaelBeer
Michael Beer 6 / 28 CRASHWORTHINESS ANALYSIS Van component, stochastic analysis, failure probability stochastic parameters » four sheet thicknesses, strain rate Finite Element model • » 20,000 shell elements • front bumper absorbing box translation velocity in x-direction v = 10 m/s . • Monte Carlo simulation • time history − stonewall force max F max F max F F t courtesy of Daimler AG and DYNAmore GmbH Robustness Assessment and Robust Design via Cluster Analysis
Robustness Assessment and RobustDesign via ClusterAnalysisCRASHWORTHINESSANALYSISStochastic structural response-maximum stonewallforceresponsesurfaceapproximationwithneuralnetworks》 182deterministiccomputationsa1hour histogram》trainingwith 1oofunctionalvalues(committeemachine)》verificationwith82functionalvaluespdf f(max F)》MonteCarlosimulationwithresponsesurfacehn(max F) 4definedlimit (maxF)imitf(maxF)probabilityofexceedanceP [(max F) > (max F)imitl < Plimitmax FMichaelBeer7/28
Michael Beer 7 / 28 Stochastic structural response − maximum stonewall force • response surface approximation with neural networks max F defined limit (max F)limit hn(max F) f(max F) » 182 deterministic computations á 1 hour histogram » training with 100 functional values (committee machine) » verification with 82 functional values » Monte Carlo simulation with response surface pdf f(max F) probability of exceedance P [(max F) > (max F)limit] < Plimit CRASHWORTHINESS ANALYSIS Robustness Assessment and Robust Design via Cluster Analysis
RobustnessAssessmentandRobustDesignviaClusterAnalysisCONCEPTOEROBUSTSTRUCTURALDESIGNEvaluation of simulation results (arbitrary computational model)resultsdesignparameters》e.g.systemresponsee.g.structuralparameters》e.g.safety level》e.g.distributionparameters￥2uZo analyzed pointsfuzzypermissible pointsoanalysisnon-permissiblepoints00with00aαXmCα-level00COoptimizationo0DO01OESperm_z.design constraintinverse problem:structural designclusteridentificationalternative,imprecisedesignvariantsMichaelBeer8/28
Michael Beer 8 / 28 CONCEPT OF ROBUST STRUCTURAL DESIGN Robustness Assessment and Robust Design via Cluster Analysis α 0 1 perm_z z Evaluation of simulation results (arbitrary computational model) fuzzy analysis x1 x2 design parameters » e.g. structural parameters » e.g. distribution parameters • results » e.g. system response » e.g. safety level • analyzed points permissible points non-permissible points inverse problem: structural design design constraint cluster identification alternative, imprecise design variants with α-level optimization µ(z)
RobustnessAssessment and RobustDesign via ClusterAnalysisDETERMINATIONOEDESIGNVARIANTSClusteranalysis-groupingof similarobjectsanalysisofstructures/patternsindatasets/pointsetsdetermination of "favorable"value ranges of the input quantitiesdeterministicclusteranalysisfuzzy cluster analysise.g.k-medoid methode.g.fuzzy-c-meansmethodμc>0.0μc ≥ 0.25X2similaritydissimilarityrepresentativeobjectsLkFCXX1J=(μk)d()→MINJ=(k)MINVx = 2(μk.) ×) (2(μk.)FMichaelBeer9/28
Michael Beer 9 / 28 Cluster analysis − grouping of similar objects • analysis of structures/patterns in data sets / point sets determination of ”favorable” value ranges of the input quantities deterministic cluster analysis e.g. k-medoid method • fuzzy cluster analysis e.g. fuzzy-c-means method • C3 C2 C1 dissimilarity similarity x1 x2 ( ) k k i kiC J d r x MIN ∈ = ∑ ∑ , → representative objects rk FC3 FC2 FC1 µc > 0.0 C1 C2 C3 µc ≥ 0.25 x1 x2 ( ) ( ) q 2 k i k i k i J =µ → ∑ ∑ , d v x MIN , (( ) ) ( ( ) ) 1 q q k i k i k i i i v x − = µ ⋅⋅ µ ∑ ∑ , , Robustness Assessment and Robust Design via Cluster Analysis DETERMINATION OF DESIGN VARIANTS
RobustnessAssessmentandRobustDesignviaClusterAnalysisDETERMINATIONOFDESIGNVARIANTSCluster analysis-quality measures for assessing the partitioningoptimumnumbernofclustersmulti-criteriaoptimizationcompromisesolution.deterministicclusteranalysis.fuzzyclusteranalysisSC^SDsilhouetteASPPCNcoefficient SCseparationnormalizeddegree SDpartitionaveragecoefficientPCNseparationASP61262248161020ncncaik -bi.k14SC&max[a,k,bik]nnPCN = 11n,d(,x),bik=min[ak]axCMichaelBeer10/28
Michael Beer 10 / 28 • optimum number nC of clusters multi-criteria optimization compromise solution • deterministic cluster analysis 2 6 12 16 20 silhouette coefficient SC 12 average separation ASP SC ASP nC = ∈ − = ∑ ∑ C k n ik ik k1 iC C k ik ik 1 1 a b SC n C a b , , max , , , ( ) ∈ ∈ ≠ = ∑ = k k i k i j i k j k ijC ji j C k 1 a dx x b a C , , , , , , , min 2 4 66 8 10 • fuzzy cluster analysis separation degree SD SD PCN normalized partition coefficient PCN nC = =− − µ ∑ ∑ − nC C 2 k i k1 i C i n 1 PCN 1 1 n1 n , Cluster analysis − quality measures for assessing the partitioning Robustness Assessment and Robust Design via Cluster Analysis DETERMINATION OF DESIGN VARIANTS