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