Table Of ContentCISM 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
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©CISMInternationalCentreforMechanicalSciences2016
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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:ignacio.romero@upm.es
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.
