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Alternative Model Order Reduction in Elastic Multibody Systems
Alternative Model Order Reduction in Elastic Multibody Systems
Alternative Model Order Reduction in Elastic Multibody Systems
Alternative Model Order Reduction in Elastic Multibody Systems
Alternative Model Order Reduction in Elastic Multibody Systems
Alternative Model Order Reduction in Elastic Multibody Systems
Alternative Model Order Reduction in Elastic Multibody Systems
Alternative Model Order Reduction in Elastic Multibody Systems
Alternative Model Order Reduction in Elastic Multibody Systems
Alternative Model Order Reduction in Elastic Multibody Systems
Alternative Model Order Reduction in Elastic Multibody Systems
Alternative Model Order Reduction in Elastic Multibody Systems
Alternative Model Order Reduction in Elastic Multibody Systems
Alternative Model Order Reduction in Elastic Multibody Systems
Alternative Model Order Reduction in Elastic Multibody Systems
Alternative Model Order Reduction in Elastic Multibody Systems
Alternative Model Order Reduction in Elastic Multibody Systems
Alternative Model Order Reduction in Elastic Multibody Systems
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Alternative Model Order Reduction in Elastic Multibody Systems

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  • 1. Institute of Engineeringand Computational MechanicsUniversity of Stuttgart, GermanyProf. Dr.-Ing. Prof. E.h. Peter Eberhard6th European ATCTurin, April 22-24, 2013Alternative Model Order Reduction inElastic Multibody SystemsPhilip Holzwarth, Peter Eberhard
  • 2. Institute of Engineeringand Computational MechanicsUniversity of Stuttgart, GermanyProf. Dr.-Ing. Prof. E.h. Peter EberhardExample: FE StructureFEM-model of a structure11873 nodes 4986 elementsabout 35000 elastic degrees of freedomgoal is controllower plate assumed to be rigidmodelled as point massconnected with CERIG command to rodshole in upper plate is interface to remaining partof the structuremodelled with spider web of beamsdiameter 1 mmYoungs modulus 1018 N/m2density 100 kg/m3
  • 3. Institute of Engineeringand Computational MechanicsUniversity of Stuttgart, GermanyProf. Dr.-Ing. Prof. E.h. Peter EberhardComparison with ModalReductionFFEFF)()()()()()(HHHHH21212121 Fd)(d)(Q22E  HHQKrylov (173) 2.06 10-7Krylov+gram(35/173)5.16 10-6POD (36/24) 2.64 10-5modal (40) 7.27 10-3alternative reduction methodsshow better results
  • 4. Institute of Engineeringand Computational MechanicsUniversity of Stuttgart, GermanyProf. Dr.-Ing. Prof. E.h. Peter EberhardOutline motivation model order reduction in elastic MBS – Why is this animportant step to obtain a good model? different methods to obtain reduced flexible bodies examples large systems (industrial application) software package Morembs summary
  • 5. Institute of Engineeringand Computational MechanicsUniversity of Stuttgart, GermanyProf. Dr.-Ing. Prof. E.h. Peter EberhardBasis of ElasticMultibody Systemsmultibody systemelastic bodydiscretizationfinite element,finite difference,...continuumelastic multibody systemrigid bodybearings andcoupling elements p bodiesf degrees of freedomq reaction forceCreduction of theelastic degreesof freedommodels are getting largerand more detailedmany degrees of freedomFE-models have to be reducedwith the floating frame of referenceformulation linear model orderreduction is possible
  • 6. Institute of Engineeringand Computational MechanicsUniversity of Stuttgart, GermanyProf. Dr.-Ing. Prof. E.h. Peter Eberhardindustrial practicefine discretizationmany dof (e.g.10 million)Modeling Elasticity withthe FEMcontinuum formulationPDEspatial discretizationODEfinite element methodlinear model order reductionreduced FE equation of motionwithwith𝐌e ∙ 𝐪e + 𝐃e ∙ 𝐪e + 𝐊e ∙ 𝐪e = 𝐡e𝐌e = 𝐕T ∙ 𝐌e ∙ 𝐕, …𝐡e = 𝐕T∙ 𝐁e ∙ 𝐮eprojection matrixdim 𝐪e ≪ dim 𝐪e , 𝐪e ≈ 𝐕 ∙ 𝐪e𝐕 ∈ ℝn×mfinite element methodFE equation of motioninput/output aspectdefine input or control matrixdefine output/observation matrix𝐌e ∙ 𝐪e + 𝐃e ∙ 𝐪e + 𝐊e ∙ 𝐪e = 𝐡e𝐌e ∙ 𝐪e + 𝐃e ∙ 𝐪e + 𝐊e ∙ 𝐪e = 𝐁e ∙ 𝐮e𝐲 = 𝐂e ∙ 𝐪e𝐁e𝐂e
  • 7. Institute of Engineeringand Computational MechanicsUniversity of Stuttgart, GermanyProf. Dr.-Ing. Prof. E.h. Peter EberhardModel ReductionTechniquesmodel reduction techniquesused for elastic bodiescondensation static condensation(Guyan, constraint modes) dynamic condensationmodal truncation free-free modes fixed-interface modescomponent mode synthesis Hurty/Craig-Bampton method Craig-Chang method ...Component Mode Synthesisstatic moment-matching viaPadé-type approximationmoment-matching withKrylov subspaces Arnoldi, Lanczos iterative methods (IRKA,MIRIAM), adaptive methods (SOAGA) ...tangential interpolationinterpolation methodsbalanced truncation second order balancing Lyapunov balancing stochastic balancing bounded real balancing …frequency weighted balancedtruncationProper Orthogonal Decomposition(POD)Gramian-based methodsBlock-Krylov vectors as componentmodes(Extended) Singular ValueDecomposition Model OrderReduction (E)SVDMORLaguerre-based model reduction RK-ICOP…hybrid methods
  • 8. Institute of Engineeringand Computational MechanicsUniversity of Stuttgart, GermanyProf. Dr.-Ing. Prof. E.h. Peter Eberhardapproachapproximation of the nodal displace-ment vector by a linear combination ofthe dominant eigenvectors (normal modes)quadratic eigenvalue problemprojection matricesproblem: how to select important normal modes?standard: sorting by eigenfrequencyand experience of the userModal Approximation/Truncation3rd6th30th, ..., ...𝐪 𝐞 ≈ 𝛗𝑖 ∙𝐫𝐢=𝟏𝐪e,𝑖= 𝚽 ∙ 𝐪e 𝚽T=−λ𝑖2𝐌e + 𝐊e ∙ 𝛗𝑖 = 0
  • 9. Institute of Engineeringand Computational MechanicsUniversity of Stuttgart, GermanyProf. Dr.-Ing. Prof. E.h. Peter Eberhardapproachdefinition of boundary nodes, wherefurther components are connectedconstraint modes for all boundary coordinates (as in Guyan condensation)unit displacement of one boundary while others held fixedadditional fixed-interface normal modesfor the inner part of the systemprojection matrices𝐪e,i𝐪e,b= 𝚽k 𝚿c ∙𝐪e,i𝐪e,bCMS/ Craig-Bamptonbig improvements to modal truncationinter-component compatibilityexact static responsemovement of boundary dofs isexplicitly availableerror𝜀[−] frequency f [Hz]normal modes for internaldynamics are selectedby their frequency
  • 10. Institute of Engineeringand Computational MechanicsUniversity of Stuttgart, GermanyProf. Dr.-Ing. Prof. E.h. Peter Eberhardapproachpartitioning into boundary and internal dofsecond order LTI system: definition of input and output, transfer matrixtransfer matrix at s = 0, using Schur complementstatic matching of the transfer function for reduced and original system ifGuyan-Condensation𝐌ii 𝐌ib𝐌bi 𝐌bb⋅𝐪i𝐪b+𝐊ii 𝐊ib𝐊bi 𝐊bb⋅𝐪i𝐪b=𝐡i𝐡b𝐡i𝐡b=𝐁i𝐁b⋅ 𝐮, 𝐲 = 𝐂i 𝐂b ⋅𝐪i𝐪b𝐇 s = 𝐂e ∙ s2𝐌e + 𝐊e−1∙ 𝐁e𝐇 0 = 𝐂i ⋅ 𝐊ii−1⋅ 𝐁i + 𝐂b ⋅ 𝐊bb−1⋅ 𝐁b with 𝐊bb = 𝐊bb − 𝐊bi ⋅ 𝐊ii−1⋅ 𝐊ib𝐂b = 𝐂b − 𝐂i ⋅ 𝐊ii−1⋅ 𝐊ib𝐁b = 𝐁b − 𝐊bi ⋅ 𝐊ii−1⋅ 𝐁i= 𝟎𝐕 =𝐈ii−𝐊ii−1⋅ 𝐊ibKrylov subspace methods:numerically robust extension of this concept to• arbitrary combinations of matching frequencies• derivatives of 𝐇(s)
  • 11. Institute of Engineeringand Computational MechanicsUniversity of Stuttgart, GermanyProf. Dr.-Ing. Prof. E.h. Peter Eberhardtransfer matrixseries expansion with expansion pointmoment matching of original andreduced system is achieved implicitlywhen using the Krylov subspacewithMoment Matching viaKrylov-Subspaces𝐇 s = 𝐂e ∙ s2 𝐌e + s𝐃e + 𝐊e−1 ∙ 𝐁e𝐇 s = 𝐓0σ+ 𝐓1σs − σ + ⋯ + 𝐓∞σs − σ ∞𝐓jσ: moments of the transfer functionσ𝒦r 𝐌, 𝐑 =span 𝐑, 𝐌 ∙ 𝐑, ⋯ , 𝐌r−1 ∙ 𝐑 ⊆ colsp{𝐕}𝐌 = 𝐊e−1∙ 𝐌e𝐑 = 𝐊e−1∙ 𝐁eσ‖𝐇f‖Ffrequency f [Hz]
  • 12. Institute of Engineeringand Computational MechanicsUniversity of Stuttgart, GermanyProf. Dr.-Ing. Prof. E.h. Peter Eberhardadvantagesa priori errorbound existsweighting of a certainfrequency rangedisadvantageonly efficiently possiblefor small modelsMOR with FrequencyWeighted Gramian MatricesapproachGramian matrices provide an energyinterpretation of the system’s statescontrollability Gramian matrixobservability Gramian matrixbalanced representationHankel singular valuestruncation of states represented bysmall singular valuesBalanced Truncationfrequency weighted reductionusage of frequency weighted Gramianmatricesvery good approximation of a specificfrequency range2-step approachPOD𝐏𝐐σi = λi(𝐏 ∙ 𝐐)35302013error𝜀[−]frequency f [Hz]
  • 13. Institute of Engineeringand Computational MechanicsUniversity of Stuttgart, GermanyProf. Dr.-Ing. Prof. E.h. Peter EberhardNumerical Examples –Frequency Domaincrankshaftexhausterror𝜀[−]frequency f [Hz]error𝜀[−]frequency f [Hz]
  • 14. Institute of Engineeringand Computational MechanicsUniversity of Stuttgart, GermanyProf. Dr.-Ing. Prof. E.h. Peter EberhardNumerical Examples –Time Domaintime t [s]accelerationa[mm/s2]meanerrorcalculationtime[s]errortime t [s]large improvement on mean errorwith shorter calculation timeKrylov (50)POD (50) Gram (50)CMS (110)
  • 15. Institute of Engineeringand Computational MechanicsUniversity of Stuttgart, GermanyProf. Dr.-Ing. Prof. E.h. Peter Eberhard- Preprocessorto reduce elastic bodiesMOREMBS contains implementationsof converters for a wide variety of FE-programs and numerous reductionmethods. The software is available asa Matlab-based version (MatMorembs)as well as one in C++ (Morembs++).MOREMBS(Model Order Reduction ofElastic Multibody Systems)- users inresearch & industryDepartementWerktuigkundeLUT Metal Technology,Faculty of TechnologyVDLAB (VehicleDynamicsLaboratory)UniversitätKasselInstitut fürMechanikcooperationand projectpartners
  • 16. Institute of Engineeringand Computational MechanicsUniversity of Stuttgart, GermanyProf. Dr.-Ing. Prof. E.h. Peter EberhardFE-Softwareworkflow inimport from Ansys, Abaqus,Permas, …reduction with traditional andmodern methodsexport to Neweul-M2, RecurDyn,Simpack, VL.Motion, …advantagesusage ofstandard FEM programsstandard MKS programsmodern reduction methodsinstead of only modal methodspreserving the familiar processchainMKS-Software
  • 17. Institute of Engineeringand Computational MechanicsUniversity of Stuttgart, GermanyProf. Dr.-Ing. Prof. E.h. Peter EberhardMorembs in theHyperWorks ProcessChainprocesscontrolfull bodyreducedbodyFE SolverRADIOSSABAQUSLS-DYNANASTRANANSYS…MBSMotion SolveAdamsSIMPACK…making use of HyperWorks’various interfaces(sketch, current project with Altair)
  • 18. Institute of Engineeringand Computational MechanicsUniversity of Stuttgart, GermanyProf. Dr.-Ing. Prof. E.h. Peter EberhardSummaryadvantages of alternative MORbetter, more reliable results(guaranteed error bounds)no mode selection by handnecessaryshorter computation timesautomated algorithms availablemany examples with industrialrelevancechallenges and current topicsmany inputscoupled bodiesmoving loadsbreak squealuncertaintiesindustrial applicability,interfaces…advanced MOR techniques• improve results if computationaleffort is the same• speed up calculations for thesame quality of resultsThank you for your attention!

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