Structure-preserving Integrators in Nonlinear Structural Dynamics and Flexible Multibody Dynamics PDF
Preview Structure-preserving Integrators in Nonlinear Structural Dynamics and Flexible Multibody Dynamics
CISM International Centre for Mechanical Sciences 565 Courses and Lectures Peter Betsch Editor Structure-preserving Integrators in Nonlinear Structural Dynamics and Flexible Multibody Dynamics International Centre for Mechanical Sciences CISM International Centre for Mechanical Sciences Courses and Lectures Volume 565 Series editors The Rectors Friedrich Pfeiffer, Munich, Germany Franz G. Rammerstorfer, Vienna, Austria Elisabeth Guazzelli, Marseille, France The Secretary General Bernhard Schrefler, Padua, Italy Executive Editor Paolo Serafini, Udine, Italy Theseriespresentslecturenotes,monographs,editedworksandproceedingsinthe field of Mechanics, Engineering, Computer Science and Applied Mathematics. Purpose of the series is to make known in the international scientific and technical community results obtained in some of the activities organized by CISM, the International Centre for Mechanical Sciences. More information about this series at http://www.springer.com/series/76 Peter Betsch Editor Structure-preserving Integrators in Nonlinear Structural Dynamics and Flexible Multibody Dynamics 123 Editor PeterBetsch Institute of Mechanics Karlsruhe Institute of Technology Karlsruhe Germany ISSN 0254-1971 ISSN 2309-3706 (electronic) CISMInternational Centre for MechanicalSciences ISBN978-3-319-31877-6 ISBN978-3-319-31879-0 (eBook) DOI 10.1007/978-3-319-31879-0 LibraryofCongressControlNumber:2016935608 ©CISMInternationalCentreforMechanicalSciences2016 Thisworkissubjecttocopyright.AllrightsarereservedbythePublisher,whetherthewholeorpart of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission orinformationstorageandretrieval,electronicadaptation,computersoftware,orbysimilarordissimilar methodologynowknownorhereafterdeveloped. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publicationdoesnotimply,evenintheabsenceofaspecificstatement,thatsuchnamesareexemptfrom therelevantprotectivelawsandregulationsandthereforefreeforgeneraluse. The publisher, the authors and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authorsortheeditorsgiveawarranty,expressorimplied,withrespecttothematerialcontainedhereinor foranyerrorsoromissionsthatmayhavebeenmade. Printedonacid-freepaper ThisSpringerimprintispublishedbySpringerNature TheregisteredcompanyisSpringerInternationalPublishingAGSwitzerland Preface This volume contains notes based on lectures presented at the advanced course ‘Structure-preserving Integrators in Nonlinear Structural Dynamics and Flexible Multibody Dynamics’ held at the International Centre for Mechanical Sciences (CISM) in Udine, Italy, during October 7–11, 2013. The objective of the five chapters in this volume is to provide insight into state-of-the-art numerical methods for nonlinear structural and flexible multibody dynamics. In the field of structural mechanics, finite element methods are com- monly applied for the discretization in space. Due to the large dimension of the resulting semi-discrete system, one is typically content with second-order accurate schemes for the discretization in time. Based on well-established time-stepping schemes for the linear regime, energy-momentumconsistentschemesandenergydissipatingvariantsthereofhave been developed in the framework of nonlinear structural dynamics during the past 25 years. These schemes are known to possess superior numerical stability and robustness properties when compared to standard methods. ThechapterwrittenbyI.Romeroprovidesageneraloverviewofhigh-frequency dissipative integrators for linear and nonlinear elastodynamics. If the controllable numericaldissipationisswitchedoff,onetypicallygetsbacktoenergy-momentum consistent schemes that are addressed in the chapter authored by P. Betsch. Due to the presence of finite rotations, the configuration space of multibody systemsistypicallynonlinear.InthechapterwrittenbyM.Arnold,A.Cardona,and O.Brüls,LiegroupintegratorsarepresentedwhichpreservetheLiegroupstructure of the underlying nonlinear configuration space by design. Analternativeroutetothedesignofstructure-preservingnumericalmethodsare variational integrators. The newly emerging class of variational integrators is the topic of the chapter authored by A.J. Lew and P. Mata A. Last but not least the Chapter written by J. Gerstmayr, A. Humer, P. Gruber, and K. Nachbagauer pro- vides insight into the absolute nodal coordinate formulation which is increasingly popular in the field offlexible multibody dynamics. v vi Preface The combination of these chapters provides a unique perspective on up-to-date numerical methods for nonlinear structural dynamics and flexible multibody dynamics. Sincere thanks are due to the colleagues for preparing their chapters for this volume. Special thanks to Professors Martin Arnold, Alberto Cardona, JohannesGerstmayr,AdrianLew,andIgnacioRomerofortakingpartatthecourse and presenting their excellent lectures. The course brought together nearly 40 participants from 8 countries. We are grateful to all participants for their interest and the numerous discussions that took place during and after the lectures. We are particularly thankful to the Scientific Council of CISM for supporting this course and recognizing the importance of the topic. We further thank the CISM stafffor the excellent organization, support, and hospitality. Professor Paolo Serafini is gratefully acknowledged for his encour- agementtopublishtheselecturenotesandhispatiencetowaitforthefinalversions. Peter Betsch Contents High Frequency Dissipative Integration Schemes for Linear and Nonlinear Elastodynamics. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 Ignacio Romero Energy-Momentum Integrators for Elastic Cosserat Points, Rigid Bodies, and Multibody Systems . . . . . . . . . . . . . . . . . . . . . . . . . 31 Peter Betsch A Lie Algebra Approach to Lie Group Time Integration of Constrained Systems. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91 Martin Arnold, Alberto Cardona and Olivier Brüls The Absolute Nodal Coordinate Formulation . . . . . . . . . . . . . . . . . . . 159 Johannes Gerstmayr, Alexander Humer, Peter Gruber and Karin Nachbagauer A Brief Introduction to Variational Integrators . . . . . . . . . . . . . . . . . . 201 Adrián J. Lew and Pablo Mata A vii High Frequency Dissipative Integration Schemes for Linear and Nonlinear Elastodynamics IgnacioRomero Abstract Timeintegrationschemeswithcontrollable,artificial,highfrequencydis- sipationareextremelycommoninpracticalengineeringanalysesforintegratingin timeinitialboundaryvalueproblemspreviouslydiscretizedinspacewithfiniteele- ments or similar techniques. In this chapter, we describe the structure of the most commonly employed integration schemes of this type and focus in their numeri- calanalysisforlinearandnonlinearproblems.Theseincludespectral,energy,and backwarderroranalyses.Forthenonlinearcase,additionally,westudythepreserva- tionofconservationlawsandtheapproximationofrelativeequilibria.Thechapter shouldprovideageneraloverviewofdissipativemethods,theirissues,andthetools availablefortheirformulationandanalysis. 1 Introduction Stiffordinarydifferentialequations,suchastheonescommonlyappearinginsolid dynamicsandmanyotherareasofappliedmathematics,havesolutionsthatinvolve characteristic times of very different orders of magnitude. Whereas one is often interestedonlyintheslowresponse,integratingthefastesttimescalesissometimes necessary and always dictated by the time step choice (Wood 1990; Hairer and Wanner1991). In systems of ordinary differential equations resulting from the spatial dis- cretizationofpartialdifferentialequations,themodeswithhighestfrequenciesare inevitablyresolvedverypoorlybythemesh.Thisisthecase,forexample,insolid mechanics,whereafiniteelementmesh—orasimilardiscretizationtechnique—is B I.Romero ( ) DepartmentofMechanicalEngineering,TechnicalUniversityofMadrid, Madrid,Spain e-mail:[email protected] I.Romero IMDEAMaterialsInstitute,Getafe,Madrid,Spain ©CISMInternationalCentreforMechanicalSciences2016 1 P.Betsch(ed.),Structure-preservingIntegratorsinNonlinearStructural DynamicsandFlexibleMultibodyDynamics,CISMInternationalCentre forMechanicalSciences565,DOI10.1007/978-3-319-31879-0_1 2 I.Romero employed toapproximate theinitialboundary valueproblemofcontinuum elasto- dynamicsbyasemidiscreteinitialvalueproblem,governedbythesameequations astheproblemofstructuraldynamics,whichcanthenbeintegratedintimenumer- ically.Insuchaprocess,theoneofinterestinthischapter,thehighestfrequencies modes are completely spurious, and so poorly resolved that their precise value is oftenirrelevantfortheanalyst. The mathematical analysis and the numerical experience accumulated during decades indicates that, in nonlinear problems, the poorly resolved, high frequency modes of the solution are ultimately responsible for many instabilities observed inthenumericalsolutionofstiffevolutionproblems.Since,asalreadymentioned, thosesamemodesarepoorlyresolvedwhenderivingfromaspatialdiscretization, time integration algorithms that possess some kind of high frequency controllable dissipationarefrequentlyfavoredinresearch andcommercial codes.Itisthegoal ofthischaptertodiscussinwhichsensethisisavalidapproachandhowitshould beaddressedfromthestandpointoftheuserandthealgorithmicdesigner. Tounderstandthestrengthsandlimitationsofhighfrequencydissipativeintegra- tors for solid mechanics, it is convenient to start by studying them in the context oflinearelastodynamics.Theequationsthatdescribethisproblemareamenableto acompletemathematicalanalysisandguidethechoiceofalgorithmsthatcanlater be applied to more complex nonlinear problems. Most of the efforts in this regard have been addressed toward the development of direct integration schemes, simi- lar to Newmark’s classical method (Newmark 1956), but with optimal dissipation propertiesandmaximumaccuracy.Infact,somemembersoftheNewmarkfamily of methods, possibly the most commonly used integrators in solid and structural dynamics, have controllable high frequency dissipation, although all of these are only first-order accurate (Hughes 1983). The design of Newmark-like integrators forsolidandstructuraldynamicswithsecond-orderaccuracyandcontrollablehigh frequency dissipation motivated a large amount of works since the 1960s (Wilson 1968;BatheandWilson1973;Hilberetal.1977;Woodetal.1981;BazziandAnder- heggen1982;Zienkiewiczetal.1984;ChungandHulbert1993;ModakandSotelino 2002;ZhouandTamma2004).Manycommercialfiniteelementandmultibodycodes haveadoptedoneofthesemethodsasthedefaultintegratorforimplicitdynamical problems. Thestabilityofatimeintegrationmethod,whenemployedinthesolutionoflinear elastodynamics,isbestunderstoodwhenacompletespectralanalysisisperformed (Hughes1983, 1987;Wood1990;Bathe1996).Suchananalysischaracterizesthe evolution in time of each of the independent modes that contribute to the global solution,identifyingtheirgrowthordecay,phaseerror,overshoot,etc.Alternatively, amoredirectmethodofanalysisbasedontheenergyofthesolutioncanbeemployed toassessthepropertiesoftheintegrators(Hughes1976, 1983;Romero2002, 2004). The latter approach, although less systematic than the spectral analysis, furnishes globalinformationthatcompletestheinformationobtainedfromtheformer.
The list of books you might like
