The Contact Dynamics method: A nonsmooth story

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Frédéric Dubois, Vincent Acary, Michel Jean. The Contact Dynamics method: A nonsmooth story . Comptes Rendus Mécanique, Elsevier, 2018, 346 (3), pp.247-262. �10.1016/j.crme.2017.12.009�.

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Contents lists available atScienceDirect

Comptes Rendus Mecanique

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The legacy of Jean-Jacques Moreau in mechanics

The Contact Dynamics method: A nonsmooth story

La méthode de la dynamique des contacts, histoire d’une mécanique non régulière

Frédéric Dubois^a^,^b^,^∗, Vincent Acary^d, Michel Jean^c

aLMGC,Univ.Montpellier,CNRS,Montpellier,France bMIST,Univ.Montpellier,CNRS,IRSN,Montpellier,France cAix-MarseilleUniversité,CNRS,CentraleMarseille,Marseille,France dLJK,INRIA,UniversitédeGrenobleAlpes,Grenoble,France

a r t i c l e i n f o a b s t r a c t

Articlehistory:

Received20May2017

Acceptedafterrevision11October2017 Availableonlinexxxx

Keywords:

Nonsmoothdynamics Shock

Coulomblaw ContactDynamics Discreteelementmethod

Mots-clés :

Dynamiquenonrégulière Chocs

LoideCoulomb Dynamiquedescontacts Méthodeparélémentsdiscrets

Whenvelocityjumpsareoccurring,thedynamicsissaidtobenonsmooth.Forinstance,in collectionsofcontactingrigidbodies,jumpsarecausedbyshocksanddryfriction.Without complianceattheinterface,contactlawsarenotonlynon-differentiableintheusualsense butalsomulti-valued.Modelingcontactingbodiesisofinterestinordertounderstandthe behavior ofnumerousmechanical systemssuchasflexible multi-bodysystems, granular materialsormasonry.Thesegranularmaterialsbehavepuzzlinglyeitherlikeasolidora fluidandadescriptionintheframeofclassicalcontinuousmechanicswouldbewelcome thoughfartobesatisfactorynowadays.Jean-JacquesMoreaugreatlycontributedtoconvex analysis, functions ofbounded variations,differential measure theory,sweeping process theory,definitivemathematicaltoolstodealwithnonsmoothdynamics.Heconvertedall theseunderlyingtheoreticalideasintoanoriginalnonsmooth implicitnumericalmethod calledContactDynamics(CD);arobustandefficientmethodtosimulatelargecollections ofbodies withfrictional contactsand impacts.The CDmethodoffers averyinteresting complementaryalternative tothe familyof smoothedexplicit numericalmethods, often calledDistinctElementsMethod(DEM).Inthispaperdevelopmentsandimprovementsof the CDmethodare presentedtogetherwith acritical comparativereviewofadvantages anddrawbacksofbothapproaches.

©²⁰¹⁷Âcadémie^des^sciences.^Published^byÊlsevier^Masson^SAS.^Thisîsânôpenâccess articleundertheCCBY-NC-NDlicense (http://creativecommons.org/licenses/by-nc-nd/4.0/).

r é s um é

Lorsque des sauts de vitesse se produisent, la dynamique est dite non régulière. Par exemple, dans lescollections de solides supposés rigidesrentrant en contact, lessauts sont causés par les chocs et lefrottement sec. L’absence de déformabilité fait que les lois decontact sont,nonseulementnondifférentiablesausensusuel, maisaussimulti- valuées. Élaborer des modèles de solides en contact est un moyen de comprendre le comportement de nombreux systèmes mécaniques tels que les systèmes multi-corps ﬂexibles, les matériaux granulaires ou les maçonneries. Les matériaux granulaires se comportentde manièreétrange,soit commedessolides,soit commedesﬂuides,etune

* Correspondingauthor.

E-mailaddresses:frederic.dubois@umontpellier.fr(F. Dubois),vincent.acary@inria.fr(V. Acary),mjean.recherche@wanadoo.fr(M. Jean).

https://doi.org/10.1016/j.crme.2017.12.009

1631-0721/©²⁰¹⁷Âcadémie^des^sciences.^Published^byÊlsevier^Masson^SAS.^Thisîsânôpenâccessârticleûnder^the^CC^BY-NC-ND^license (http://creativecommons.org/licenses/by-nc-nd/4.0/).

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alternative très intéressante à la famille de méthodes usuelles régularisées explicites, commelaméthode deséléments distincts (DEM). Danscetarticle,des développements etdes perfectionnementsde laméthode CD sont présentésainsi qu’uneétude critique comparativedesavantagesetinconvénientsdesdeuxapproches.

©²⁰¹⁷Âcadémie^des^sciences.^Published^byÊlsevier^Masson^SAS.^Thisîsânôpenâccess articleundertheCCBY-NC-NDlicense (http://creativecommons.org/licenses/by-nc-nd/4.0/).

1. Introduction

The genericlabel(DEM)DiscreteElementsMethods,alsocalledDistinctElementsMethods,refers tomethodsthat,op- positely toFinite Elements Methods (FEM)dedicated to thedescription of media inthe frame ofcontinuous mechanics, considera sampleasanassemblyofdistinctbodies.Nowadays,suchmethodsare widelyusedinthenumericalmodeling of divided materials and structures. Naturalapplications concern the simulation ofgranular materials, suspensions,frac- turedmaterials,masonries, rockmass,etc.indomainssuchasgeophysics,mining,chemicalengineering, civilengineering, biomechanics, etc.DEM arealsousedfortheircapabilitytorepresentthevariousstatesofacollectionofsolids(gas,ﬂuid, solid)andtorepresentsomephasechanges(solidtosolidsandviceversa)inthespiritofmeshlessmethods[1]orparticle methods[2,3].Butsuchmethodsarealsoappliedinmulti-bodysystemssuchasmechanismsandrobotics.Usually,insuch methods,oneconsiderscollectionsofrigidbodies,subjecttointeractionlawssuchasfrictionalcontactlawsthataresteep laws.

AnumberofleadingmethodsarederivedfromthepioneeringworkofCundall[4]actuallyreferredtothegenericlabel DEM.ThisworkmaybeconsideredalsoasamodiﬁcationofthegenuineMolecularDynamicsmethodasproposedbyAllen andTidsley[5].Sincesuchmethodsareparticularlypragmatic,steepfrictionalcontactlawsaremodeledasnonlinearlaws using some regularization techniques,andexplicit time integratorsare used to copewiththe nonlinear behavior. At the end, such methodologyleads toa setofuncoupled linearequationsthatcan besolved straightforwardly. ThenameDEM iscommonlyreferringtothosesmoothedandexplicitmethods.Theweaknessesofsuchmethodscomefromtheirprinciple:

small time steps are mandatory due to explicit time integrators, the choice of relevant parameters may be tricky, since they areused to manage severalphenomena:contactcondition, macroscopic mechanicalresponse,dynamical properties, etc. Inparticular, in order to ensurethe stability ofexplicit schemes, it isnecessary to introduce some damping, either generatedbythefrictionalcontactlaws,eitherasanumericaltrick.Thereexistalsoseveral“conﬁdential”methodssuchas DiscontinuousDeformationAnalysis(DDA)[6,7],orDiscreteFractureNetwork(DFN)[8].Forasmallnumberofobjects,the classicalFiniteElementMethod(FEM)canalsomanagecontactproblems[9].

Jean-JacquesMoreauintroducedthe(CD)ContactDynamicsmethodduringtheyear1984.Itisinspiredbyaformulation ofunilateralcontact,shocklaws,Coulombfriction,throughConvexAnalysis.Thecontactinglawsarethusnondifferentiable steep laws.Theyare managed withan implicit method usinga Non-Linear Gauss–Seidel algorithm (NLGS) at each step.

Theselawsaccountroughlyforthemainfeaturesofcontactandfrictionandarerelevantinmulti-bodiescollectionswhere sophisticated lawscannot beexhibitedforsure.Themethoduseslarge timesteps,buteachtime stepistimeconsuming.

So oppositelyto theabove smoothedandexplicitDEMmethod,theCDmethod isanonsmoothandimplicit method.Note that implicitmethods enabletocompute correctlyequilibriumstates,whichisnotalways thecasewithexplicitmethods.

Themethodcanalsoconserve,withasuitablechoiceofparameters,thetotalenergyofthesystemindiscretetime.Inthis paper,weshalldiscusstheprosandconsoftheCDmethodwithrespecttoclassicalsmoothedDEM.

2. MoreaucontributiontoContactDynamicsgenesis

First,it istoberememberedthat Jean-JacquesMoreau wasearlierconcerned withﬂuid mechanics.Heleft anoriginal resultconcerningthehelicityinvariantinperfectﬂuiddynamics(1961)[10].

Jean-Jacques Moreau was introducing himselfas involvedin mechanics, claiming that hewas using mathematicsjust enough forhismechanicalpurpose.Actuallyhewas theauthorofhighlysophisticatedconcepts inmathematicsmainlyin theﬁeldofmeasuretheory.Hedevelopeddeﬁnitelythetheoryoflocallyboundedvariationfunctions[11],thepropermath- ematical settingforthe nonsmoothdynamicsandthesweeping process theory [12–15].Afterthe worksofthepioneers, H. Minkowski andM.W.Fenchel,hesetup ConvexAnalysis,atheoryalsodevelopedatthesametime byR.T.Rockafellar;

see forinstance thenumerousreferencesinthefamous book[16] totheresultsofJean-JacquesMoreau.ConvexAnalysis isthepropertool todealwithnonsmoothmechanicsi.e.mechanicswherethebehaviorlawsarenotdifferentiable inthe

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usual sense [17–19]. Elastoplasticity mechanics is an example. Anotherexample is the mechanics offrictional collisions betweenrigidbodies,aproblemwherevelocityjumpsareexpected. Thefirst-ordersweepingprocess introducedbyJean- JacquesMoreau,motivatedbythequasi-staticevolutionofelastoplasticsystems[20,21],seemstohaveprovidedoneofthe firstoccurrenceofmeasuredifferentialinclusionsintheliterature,withtheworkofM.Schatzman[22].Fortheapplication tononsmooth dynamics,thesecond-ordersweeping process isa fundamental toolforthe mathematicalanalysisandthe designofnumericalschemes.TheformulationoftheSignorinicondition atthevelocityleveliscrucialinthedevelopment oftheCDmethod.Thisconditionmaybeviewedasthetime derivativeoftheunilateralconstraints.Firstly,it allowsone togetsome verygood dissipationpropertiesensuringstability.Thisliesinthefact thatit reducesthe indexofthe asso- ciated switcheddifferential algebraic equation.In other words,theunilateralconstraints iswritten asa relationbetween thevelocityandtheimpulse.Inthisway,themechanicalpowerofthesystemisdescribed ina consistentway.Secondly, thesecond-ordersweepingprocessisadirectextensionoftheNewtonimpactlaw,writtenforthefirsttimeinanumerical tractable way.Some authorscallitthe“Moreau impactlaw”,since itisreallyanoutstandingcontributionofJean-Jacques Moreau.Forthenumericalpractice,italsoensures theNewton impactlawtobesatisfiedindiscretetime.Finally,another major contributionofJean-JacquesMoreau istheformulation offrictionalimpact lawsintermsof ConvexAnalysis, with equivalent formulations, for instance in terms of differential inclusions, pseudo-potentials of dissipation, and variational inequalities.

Atthesametime,otherpeoplescontributedtothedevelopmentofthemethodanditsapplications.Anextensionofthe CDmethodtodeformablebodieshasbeenproposed byM. JeanunderthenameofNonsmoothContactDynamics(NSCD).

Heproposedalsoatemplateofnumericalarchitectureinspiredbytheideaofamechanicalschemeastwopairsofspaces in duality(global variables space, local variables space), seethe next section 3.The LMGC90¹ open-source software has beendevelopedaroundthisskeletonbyFrédéricDuboiset al. andenriched byallkindsofcontactingrigidordeformable objectswithavarietyofinteractionlaws.

Currentlythemethodisstill developedby itsprecursors,butsome relevantcontributionsare noticeable.Thetheoryof bi-potentialhasbeenproposed byGeryde Saxcé.ItisinthespiritofConvexAnalysisandusesthesamekindoffrictional contactlawsthan theCDmethod. Itcomes out withanice property,the reactionforce appearing astheprojection ofa linearcombinationofthereactionwiththerelativevelocityontoCoulomb’scone[23].Anotherwayofviewingthismethod is that it may come out asa variational inequality over a cone and not a quasi-variational inequality. Glocker [24] has proposed moresophisticated nonsmoothcontactlaws,alsointhe spiritof ConvexAnalysis, forevolutions whererelative velocities and accelerations have jumps. The results may be quite accurate for multi-bodies systems with few degrees of freedoms. BrogliatoandAcary [25–28] have proposed severalimprovements either concerning impactlaws, enhanced time-steppingschemesortheextensionofthenon-smoothframeworktonewapplicationﬁeldsaselectricalcircuits.There are alsomany methodsandalgorithms dedicatedto quasi-static evolutions. Werestrict the presentpaperto nonsmooth dynamics.

3. Currentappraisal

WeshortlygiveahintofhowtheCDmethodworks.DetailsmaybefoundintheoriginalpapersbyJean-JacquesMoreau [29–31],M. Jean[32],andinthebooks[33,34].

3.1. Arobustframework

Asexplainedby Moreaucontactproblemsindynamicsare non-smooth,dueto1/kinematicunilateralconstraintsthat introducenonsmoothnessinspace,2/collisions thatareexpectedtoinducevelocityjumps,whichleadtononsmoothness intime, 3/multi-valued mappings betweencontactreactionsandthe contactvelocitiesthat introducenonsmoothnessin thelaw.Followingthespiritofthesweepingprocess,thedynamicalproblemiswrittenasameasuredifferentialinclusion;

itallowsustointegrateanddiscretizenon-smoothdynamicalsystems[35,36].

3.1.1. Dynamicsofrigidbodies

Inthefollowingtherigid-bodymotionisgovernedbyaNewton–Eulersystemofnon-smoothequations:

M^dv =^Fext(t)dt+^dI

Jd = −×J+Mext(t)dt+dM (1)

where:

• ^dv^is^thedifferentialmeasureassociatedwithvelocityv,consideredasatimefunctionofBoundedVariations(BV)and dthedifferentialmeasureofspinexpressedinaframeattachedtothebody,

• ^dt ^is^the^Lebesgue^measure,

• M^andJ^representrespectivelythemassandtheinertiamatrices,

1 https://git-xen.lmgc.univ-montp2.fr/lmgc90/lmgc90_user.

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Fig. 1.Contact frame.

• ^Fext(t)andMext(t)aretheforceandtorqueresultantsduetoexternalloads,

• ^dIând^dMâre^theîmpulseândângularîmpulse^measures^due^to^contact.

Remark1.From a practical point ofview, one writestheEuler equation ina body-ﬁxed frame whichmakes J^diagonal.

However, exceptinspecialcases(sphere, cube,etc.),whenJ and arecollinear,theEulerequationremains non-linear anddependsimplicitlyontheorientationofobjects.

If f isarightcontinuouslocallyBVfunction,themeasureof]^tb,te]^by^d^f ^is^given^by

]^tb,te]

df =^f(te)− ^f(t_b) (2)

Recall that every function ofboundedvariations admits left andrightlimits: f⁻ and f⁺. Wechoose to identifythe BV function f asitsrightlimit f⁺.Therefore,velocityandspincanbeobtainedsuchthatv(te)=^v(t_b)+

]^tb,te]dvand(te)= (t_b)+

]^tb,te]d.Thepositioniscomputedusingx(te)=^x(t_b)+_t_e

tb v(t)dt.Theorientationiscomputedbyintegratingwith respecttotheLebesguemeasureR˙ =R˜ where˜x=×^x.

Inthefollowing,V= {^v,}îs^thegeneralizedvelocity, q= {^x,R}îs^thegeneralizedcoordinates,M¯ =^diag(M,J)îs^the generalizedinertiamatrix,F¯ext(t)= {^Fext(t),Mext(t)}^,F¯_quad(t)= {⁰,−×J}ând¯I_]t_b,te]= {

]^tb,te]dI,

]^tb,te]dM}^are^respec- tivelyexternal,quadraticandcontactcontributions.Finally,thesystemtakestheform:

M(¯ ^V(t_e)−^V(t_b))=

te

tb

F¯_ext(t)dt+

te

tb

F¯_quad(t)dt+ ¯^I]t_b,te] (3)

Forsimplicitysake thesymbol ¯ îs ômittedⁱⁿ^the^following.^Note ^that^the ^time^derivative ôf ^q îs^not^directly ^related^to the velocity V since R˙ =R˜.Inthe sequel, we shallwrite q˙=^T(q)V. The expression(3) isno morethan abalance of momentumoverthetimeinterval]^tb,t_e]^.Îtîs^nonlinear^due^to^contactând^to^Fquad(t).

3.1.2. Dynamicsofdeformablebodies

We meanbydeformablebodies,bodiesdescribedintheframeofcontinuousmechanics.Whennumericalcomputation is concerned, those bodies are discretized through some ﬁniteelements techniques, so that the degreesof freedom are commonly the nodes coordinates. For instance, for a viscoelastic body within the small perturbations assumptions, the lineargoverningdynamicalequationis:

M^dv+C^v^dt+K^x^dt=^Fext(t)dt+^dI ⁽⁴⁾

wherexisthecoordinatewithrespecttoareferenceconfiguration,Cîs^the^viscosity^matrix,ândKîs^the^stiffness^matrix.

Inthecaseoflargedeformationsornonlinearbehavior,onecanuseanupdatedLagrangeformulationnestingthelinearized previousapproachinNewton–Raphsonloops.

3.1.3. Contactkinematics

Candidates to contact (shortly contacts) labeled α are selected. By candidates to contact is meant a pair of bodies closeenoughaccordingtosome criteriasothatthereactionbetweenthosebodies(vanishingornot)hastobecomputed.

For anycandidates to contactα betweenbodies i and j,one deﬁnes: the gap gα=^D(qi,qj) (the signed distancefrom

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body i to body j)andthelocalframe {Â,t,n,s}^;^see^{Fig. 1.}^For^the ^sakeôfsimplicity,thecontactlocusisconsidered as punctual. It isthe casewhen strictly convexboundaryare meeting. Knowing thenearest points C andA, it followsthat D(qi,qj)=(xC−^xA)·^n.În^the^caseôf^meshedôbjects,ône^has^to^consider^contactêlements^composedôfâ^node^versusâ faceoftheantagonistdiscretizedboundaries.Therearemoresophisticatedcontactelements.

Foragivencontact α,obviouskinematicrelationsallowonetoexpresstherelativevelocityV^α ât^contactâsâ^function ofthegeneralizedvelocitiesV^{i j} ofcontactingbodiesi and j:

V^α=H^α^,(q_i,q_j)V^{i j} (5)

where V^{i j}= [^Vⁱ^,,V^j^,]^.Ôbviously, ^D(·) ândH^α^,(·) ^depend ôn ^the ^positions ^qi and q_j of bodies i and j,which are not known apriori since they are solutions tothe problem. Thus computingmore accurately H^α^,(q_i,q_j) is anonlinear problem.

Usingdualityconsiderations(equalityofpowerexpressedintermsofglobalorlocalvariables), q,V ← Equations of motion → ^I

H↓ ↑H

g,V ← Interaction laws → I

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thelocalcontactimpulsemaybemappedintoageneralizedimpulse:

I^{i j}=H^α(q_i,q_j)I^α ⁽⁷⁾

where H^α^,(q_i,q_j) is the transpose of H^α(q_i,q_j) and I^{i j}= [Îⁱ^,,I^j^,]^. Ânother împortant ^relation ^relates ^the ^normal componentoftherelativevelocitytothegap,

˙

g^α=V_N^α ⁽⁸⁾

whichmeansthatatthecontact αthetimederivativeofthegapisthenormalcomponentoftherelativevelocity.

3.1.4. Frictionalcontactlaws

We presentheretwo basic nonsmoothlawsthat are used inthe kernel ofthe CDmethod.Other more sophisticated frictional contactlawsmaybe derived fromtheseseed lawsasmentionedlater on (section4.1). Consider foramoment that thereaction exerted betweentwocontacting bodiesatthe contactα isa force R^α ^andV^α ^is ^the^relative ^velocity.

R^α ^has^componentsôn^the^local^frame R^α_T,R^αs (thecomponentsofthefriction force)andR^α_N.Therelativevelocity V^α hasforcomponentsV_T^α^,Vs^α (thecomponentsoftheslidingvelocity)andV_N^α^.Âctually^the^relative^velocityînvolvedⁱⁿ^these^laws istherightvelocityV⁺^α^,î.e.^the^velocity^justâfter^the^consideredînstantôf^contact^(whereâ^possible^velocitydiscontinuity occurs,forinstancecausedbyashock).Inordertosimplifythewriting,weconsiderthetwo-dimensionalcase(wedisregard thes-components,andweomitthesymbolα).Thesigneddistancebetweenthebodiesisg (thegap).

Theinelasticshocklawreadsas,

ifg>0 thenRN=⁰ ⁽⁹⁾

ifg^{0 then}VN⁰ RN⁰ VNRN=⁰ ⁽¹⁰⁾

Thelastrelation,acomplementaryrelation,impliesthat,whenbodiesaretouchingeachother,therightcomponentofthe velocityisseparatingthebodies(sincethereisnoadhesion),butifthenormalcomponentofthereactionispositive(some pressureisexerted),thenormalcomponentoftherelativevelocityvanishes,whichmeansthatthecontactingbodieskeep stickingorslidingoneachother.Jean-JacquesMoreauhasproventhattheselawsensuresimpenetrability betweenbodies;

seeJean-JacquesMoreauviabilitylemma[35,37,36].

TheCoulomblawreadsas,

|Rt|μRn ∀S^{such as}|S|μRnthenVt·(S−Rt)⁰ ⁽¹¹⁾

where μ 0 isthefrictioncoeﬃcient.

Both laws may be written in several equivalent manners in terms of Convex Analysis, for instance using the sub- differential of the indicatrix function of a proper convex set. The graphs of these laws are shown in Fig. 2. Those are graphs ofmultimappings.Those lawsbeingpositively homogeneous,one isinclined to admitthat they are stillrelevant whenR^α îsânîmpulseI^α însteadôfâ^force.^Sometimes,^this^may^beâ^strong^modelingassumption.

Inthe caseofdeformablebodies,a restitutionshocklawmaybe irrelevant. Nevertheless,whena deformablebody is numericallymodeled,forinstanceusingafiniteelementmethod,theconfigurationisdescribedbynodecoordinatesacting asmaterials pointsyielding afinite-dimensionalmechanicalsystem.The inelasticshocklawisusually adopted.Coulomb’s lawisalsorelevantforflexiblesystems.