Efficient MBS-FEM integration for structural dynamics

Transcription

1 Th 2012 World Congrss on Advancs in Civil, Environmntal, and Matrials Rsarch (ACEM 12) Soul, Kora, August 26-30, 2012 Efficint MBS-FEM intgration for structural dynamics *Dragan Z. Marinkovic 1) and Manfrd W. Zhn 2) 1), 2) Dpartmnt of Structural Analysis, TU Brlin, Str. ds 17. Juni 135, Brlin, Grmany 1) ABSTRACT Th MBS softwar packags hav bn originally dvlopd for analysis of nonlinar dynamical bhavior of rigid-bodis. Today, howvr, th rqust for th faturs of MBSsoftwar is much mor dmanding, including considration of flxibl bodis. This fact imposs th nd for dvlopmnts that ar supposd to conciliat th objctivs of high numrical fficincy, which is crucial for MBS systms, and sufficint computational accuracy in cass involving larg and modratly larg dformations, which is intrinsic for FEM systms. Th authors addrss this problm by dvloping adquat FEM formulations capabl of mting both objctivs. Thos solutions opn up opportunitis to coupl th FE and MBS world in a vry fficint mannr. 1. INTRODUCTION Th analysis and dsign of tchnical systms, such as airborn, spac- and road vhicls, wind powr plants or robots, oftn rquir intgration of FEM and MBS modls within th multi-body dynamics applications in ordr to nabl adquat modling of dformabl bodis (Zhn 2005). For this purpos, a possibility of considring flxibl bodis has alrady bcom a fatur of MBS-programs. Th xisting commrcially availabl MBS softwar packags originally offr vry ffctiv and numrically fficint approachs for simulations involving larg (gomtrically nonlinar) rigid-body motions. In ordr to rtain high numrical fficincy charactristic for MBS programs, dformabl body bhavior is typically rsolvd by mans of modal suprposition tchniqu. As a disadvantag, th dformational bhavior considrd in th simulation in this mannr is limitd to small dformations bcaus th abov mntiond tchniqu is intrinsically linar. Extnsions of th approach basd on modal suprposition hav bn dvlopd in ordr to account for modrat gomtric nonlinaritis (Schwrtassk t al 1999, Ditz t al 2003, Marinkovic t al 2010), but thy ar svrly limitd to crtain spcific cass of dformational bhavior. On th othr hand, FEM-programs provid all th possibilitis for simulation of 1) Ph.D. 2) Profssor

2 mchanisms bhavior including larg rigid-body motion, but with rathr high numrical costs involvd. For computation of dynamics ovr longr tim-intrvals and with gomtrical nonlinaritis includd, so-calld co-simulations (MBS-FEM) can b usd. This, howvr, rquirs a rathr xpnsiv data transfr in ach tim-stp of numrical intgration as wll as synchronization of diffrnt softwar products (with diffrnt typs of data storag, structur and solvr). Ths solutions ar lss suitabl for th purpos of narly ral-tim simulation (.g. driving simulators, virtual simulation of plant and machinry functions). Th aim of th authors work is dvlopmnt of fundamntals that ar supposd to provid dirct intgration of FE modls for dformabl bodis into MBS programs. Such on objctiv rquirs a vry fficint FEM formulation that can asily cop with larg rigid-body motion of th ntir flxibl body, but also of its sub-domains, in ordr to rtain th advantags of MBS programs and covr computation of rlativly larg dformations. A simplifid co-rotational FEM formulation is dvlopd to mt ths rquirmnts. It yilds rathr promising rsults within an originally dvlopd intractiv tst nvironmnt. Th authors additionally dmonstrat that an implmntation of th dvlopd formulation into an MBS systm is principally possibl, but also that it dmands furthr improvmnts of th collaborativ work of th MBS systm and implmntd FEM formulation. 2. REDUCED VS. FULL FEM MODELS IN MBS DYNAMICS A considrabl amount of work has bn ddicatd to th dvlopmnt of formalisms to simulat flxibl bodis in MBS dynamics. Th common approach is to dscrib th dformation of a flxibl body with rspct to a body-fixd rfrnc fram. In that mannr, th larg rigid-body motion is sparatd from th small dformational motion. This provids considration of nonlinaritis rsulting from larg rigid-body motion, which was actually th original aim of programs for MBS dynamics. It is a common opinion that, vn with this approach, th computation of full FEM modls (nodal approach) within th MBS dynamics is a rathr tim-consuming and dmanding task. Hnc, In ordr to rduc th computational burdn considrably compard to th approach basd on th full FEM modl, a modl rduction is prformd. Th most common approach is th modal approach, which implis that orthogonal mod shaps, calculatd in a stp prior to MBS simulation, dtrmin modal dgrs of frdom, in trms of which th lastic bhaviour of th body is dscribd. Not only is th numbr of dgrs of frdom in this mannr significantly rducd, but th quations for lastic bhaviour ar also dcoupld, i.. th gnralizd mass and stiffnss matrics ar diagonal. Th quality of th rsults obtaind with modal approach strongly dpnds on th quality and numbr of th mod shaps usd in th simulation. Th solution usd by commrcial softwar packag ADAMS is th Componnt Mod Synthsis (CMS) tchniqu, particularly th Craig-Bampton mthod. Th mthod rquirs partition of flxibl body dgrs of frdom (DOFs) into boundary DOFs and intrior DOFs, th formr blonging to th nods of th FE-modl that th usr wants to rtain in th simulation modl mainly for th purpos of dfining (kinmatic or dynamic) boundary conditions. In th nxt stp, th mthod rquirs computation of two sts of mods:

3 1) constraint-mods, which ar static shaps obtaind by giving ach boundary DOF a unit displacmnt, whil all othr boundary DOFs ar fixd; 2) fixd-boundary normal mods, which ar obtaind by fixing all boundary DOFs and computing an ignsolution. Sinc th so-obtaind Craig-Bampton mods ar not an orthogonal st of mods, thy ar orthonormalizd prior to simulation. Th modal approach in its original form is suitabl for small, i.. linar dformations with rspct th body-fixd rfrnc fram. But it should b kpt in mind that th obtaind rsults ar a combination of rigid-body motion and dformabl motion, thus yilding a nonlinar rsult. Obviously, th major rason to avoid using full FEM modls in MBS dynamics is attributd to a rlativly larg numrical burdn rquird to prform th simulation. Additionally, for known points of forc application in th modl, using a full linar FEM modl would not rsult in a significant improvmnt compard to th modl basd on modal rduction with Craig-Bamption mods applid. Howvr, application of a full non-linar FEM modl would provid bttr accuracy (for largr dformations) and significantly gratr simulation flxibility sinc it would not b ncssary to know th points of forc application in advanc. On th othr hand, th numrical burdn with this approach would b in most cass indd prohibitivly larg, particularly if a rigorous non-linar FEM formulation is usd. Th availability of powrful hardwar in th last dcad and its continuous rapid dvlopmnt ncourag, howvr, rconsidration of this aspct. But having mor computational powr at hand is not sufficint to rach th objctiv. Th authors hav, thrfor, invstd fforts in th dvlopmnt of a simplifid co-rotational FEM formulation that is supposd to rspons to th aformntiond challngs by offring a balanc btwn th rquird numrical ffort and achivabl accuracy using full FEM modls. 3. CO-ROTATIONAL FEM FOR IMPLEMENTATION INTO MBS SYSTEMS Th basic ida of th formulation is rathr simpl and actually originats from th xisting principls of incorporating flxibl bodis in Multi-Body-Systm (MBS) dynamics. As alrady discussd, in MBS dynamics th ovrall motion of flxibl bodis is givn as a suprposition of larg rigid-body motion and small dformation with rspct to a local body-fixd c. s. that, of cours, prforms th sam rigid-body motion as th body itslf. In this mannr, th local c. s. xcluds larg rotation from th body motion, thus allowing xtraction of dformabl motion up to a grat xtnt. In th prsnt FEM formulation, th ida is xtndd so that ach lmnt of th finitlmnt assmblag is assignd a local c. s., with rspct to which th bhaviour of th lmnt rmains purly linar. In this mannr, th approach prmits handling dformations in which parts of th flxibl body prform larg rotations with rspct to th rmaining of th body. This concpt allows th computation of linar stiffnss matrics of singl lmnts, K, in a pr-stp prior to intractiv simulation. In ral-tim it is ncssary to us th information about th last dtrmind and th original configuration in ordr to xtract rigid-body rotation for ach singl lmnt, dscribd by th rotational matrix R (Fig. 1). Onc th rotation is known, th last dtrmind configuration of th lmnt is rotatd back, i.. through R -1. Th so-obtaind configuration is compard with th initial configuration to dtrmin th displacmnts fr of rigid-body rotation. Multiplication of

4 Fig. 1 Dcomposition of ttrahdral lmnt motion into rigid-body and dformabl motion th lmnt stiffnss matrix with rotation-fr displacmnts yilds intrnal lastic forcs of th lmnt in th original fram of th lmnt. What rmains is to rotat th forcs to th currnt lmnt fram, i.. through R. Th dscribd oprations ar summarizd in th following xprssion: r F 1 r r ( R v v ) = R K (1) whr v r 0 and v r ar th initial and currnt lmnt configurations, rspctivly. Rarranging Eq. (5) on obtains: r F r r r r = R K R v R K v = K v F (2) whr K R dnots th rotatd lmnt stiffnss matrix. Thus, th ssnc of th concpt consists in computation of intrnal forcs according to Eq. (1) and rotation of ach lmnt linar stiffnss matrix (as don in Eq. (2)), which is furthr usd to rassmbl th complt stiffnss matrix of th dformd structur. It should b notd that th abov dscribd approach is a simplifid on rgarding th spatial rsolution of accounting for matrial rigid-body rotation. Namly, th rotational motion is avragd on th lmnt lvl, maning that a singl rotational matrix dscribs a sort of avrag rotation of th lmnt. Th rotational matrix may b rsolvd ithr by introducing a local lmnt c. s., or, if th lmnt dfinition prmits it, by dfining a transformation matrix that maps original into th last dtrmind lmnt configuration and prforming polar dcomposition of that matrix. In comparison to this approach, th rigorous gomtrically nonlinar FEM approach rquirs a r-computation of ach lmnt tangntial stiffnss matrix basd on its currnt configuration and strsss, thus including also th gomtric stiffnss matrix and chang in th lmnt s gomtry. That would rquir a significantly gratr computational ffort. Anothr important aspct is th solvr. Instad of commonly usd dirct solvrs, th authors us an itrativ solution procdur th prconditiond conjugat gradint mthod (Bath 1996), which bnfits from th systm matrix in th spars form. Th softwar that rprsnts th originally dvlopd tst nvironmnt maks us of implicit tim intgration schm for dynamic computations, which is indd mor xpnsiv R R 0

5 rgarding th ncssary computational ffort for a singl tim-stp, but, on th othr hand, allows significantly largr tim-stps compard to an xplicit solvr. Th fficincy of th itrativ solvr can b noticably improvd by a rasonabl choic of starting vctor of th itrativ procss. In dynamics, th systm of quations is solvd for vlocitis, which do not chang dramatically within a tim-stp. Hnc, taking th vlocitis from th prvious tim-stp as a starting vctor for th itrativ solvr in th nxt tim-stp improvs numrical fficincy of simulation as fwr itrations ar ndd to rach th solution. 4. ACCURACY OF THE PROPOSED CO-ROTATIONAL FEM FORMULATION Th prsntd co-rotational FEM formulation is supposd to combin th advantags of th linar FEM (stability and fficincy of computation) with thos of th gomtrically nonlinar FEM (improvd accuracy). A gratr fficincy compard to th rigorous gomtrically nonlinar FEM is guarantd by th formulation itslf, but th pric to b paid for this advantag is th rducd accuracy. In ordr to gt a glimps into this aspct, a rlativly larg gomtrically (only) nonlinar dformation of a sliding door hing is considrd blow. Th considrd dformation rsmbls a cas whn a car door opns quickly and hits against th bump stop. Th dformation of a part of th sliding door hing that runs ovr th guid rail fixd onto th car chassis is in th focus. Th FEM modl (courtsy of Volkswagn AG) of th considrd part is dpictd in Fig. 2 from two prspctivs. It consists of 3342 ttrahdral lmnts and 1288 nods. Fig. 2 FEM modl and boundary conditions of th sliding door hing Th boundary conditions involv fixd nods ovr th surfac of th hols for th hing pin and th applid xtrnal forc that is larg nough so that gomtrically nonlinar ffcts ar rathr obvious. Th amount of inducd dformation can b visually xplord in Fig. 3 from diffrnt prspctivs (no scaling of dformation). Fig. 3 FEM modl and boundary conditions of th sliding door hing

6 Fig. 4 FEM modl and boundary conditions of th sliding door hing Th rsults for displacmnts of th point at which th forc acts in all thr global dirctions yildd by: 1) linar, 2) rigorous gomtrically nonlinar (rsult from ABAQUS) and 3) proposd formulation, ar dpictd in Fig. 4 against th xtrnal forc. Th rsults talk in favor of a vry good agrmnt btwn th rigorous gomtrically nonlinar analysis and th simplifid co-rotational formulation. 5. NUMERICAL EFFICIENCY OF THE FORMULATION Th prsntd FEM formulation is obviously mor fficint than th rigorous gomtrically nonlinar FEM. This is achivd by nglcting crtain aspcts of nonlinar bhavior, such as th influnc of th chang in th lmnt configuration as wll as th strss stiffning ffcts. Th formulation allows th pr-computation of th lmnt stiffnss matrics. This stp is don only onc and prior to th simulation. Ovr th cours of simulation, th rotational lmnt matrics ar xtractd and usd to rotat th lmnt stiffnss matrics and th intrnal forcs. Thos simplifications ar combind with advantags of th itrativ solvr to furthr incras th numrical fficincy. In dynamics, th solvr advantags includ th possibility of an fficint choic of th starting vctor and that would b th vctor of vlocitis from th prvious tim-stp. Additionally, itrativ solvrs offr a quit gnral possibility of limiting th numbr of prformd itrations. Whras this accounts for an advantag with rspct to th objctiv of high numrical fficincy, it should b mphasizd that it coms at th cost of rducd accuracy. To gain an insight into this aspct, a lug structur is considrd in th dvlopd tst nvironmnt. Th structur is discrtizd by 1453 lmnts and 417 nods (Fig. 5). Fig. 5 Lug structur in th simulation tst nvironmnt

7 An Intractiv dynamical simulation is prformd with a tim-stp of 0.01 s. Th boundary conditions can b intractivly rmovd and st again, so that an intractiv forc ithr producs larg dformations (Fig. 5, middl), or larg rigid-body motion (Fig. 5, right). Th fficincy of th prformd simulation strongly dpnds on applid hardwar and it is prsntd hr by th ratio btwn th simulation tim and ral tim (hnc, ratio highr than 1 implis simulation running at a fastr rat than ral tim). Tabl 1 summarizs th rsults for four diffrnt configurations (CPU and graphic card givn as major componnts influncing th rsult). Tabl 1 Numrical fficincy of diffrnt hardwar configurations with th lug modl Hardwar configuration (procssor / graphic card) AMD S140 NVidia 7025 Intl E8500 NVidia 8600GT AMD II X2 250 NVidia 8600GT Intl i7-870 NVidia 550 GTI Ratio : Simulation tim / ral tim IMPLEMENTATION INTO ADAMS Implmntation of th dvlopd formulation into commrcially availabl MBS softwar packag ADAMS provd to b a dmanding task. It has bn don by mans of th currntly availabl usr-dfind subroutins that offr only a limitd flxibility, in th sns what typ of data can b xtractd during a simulation as wll as what can b submittd back to th main program. Th ida of th implmntation is that a subroutin prforms th complt FEM computation, thus avoiding th nd for a cosimulation by using a collaborativ work of commrcially availabl MBS and FEM softwar packags, which would dmand thir synchronization, data transfr, tc, and would b, thrfor, rathr xpnsiv. Gnrally spaking, th approach rquirs from th usr to st up th intrfac nods of th FEM modl. Thos ar th nods at which th FEM modl intracts with th rst of th MBS modl. Th subroutin is supposd to submit to ADAMS ncssary information at intrfac nods (such as intrnal forcs), so that ADAMS may procd with th computation of th MBS modl. Th GFOSUB typ of subroutin has bn usd in th implmntation of th dvlopd formulation. Within th GFOSUB, on may us th SYSARY subroutin to xtract ncssary information about th intrfac nods of th FEM modl and th GFOSUB itslf submits back to th main program a gnral forc (GFOSUB stands for gnral forc subroutin ) containing up to 6 componnts 3 forc and 3 momnt componnts. Th implmntation has bn don in two ways, on of which is mor suitabl for statics, th othr for dynamics.

8 2.1. Implmntation with just a fw intrfac nods Th first vrsion of implmntation allows th us of just a fw intrfac nods, which is actually th original aim of th subroutin. Thos ar for th simulation ncssary nods at which th intraction with th rst of th MBS modl is achivd. Th GFOSUB is providd with th currnt position of only thos nods with rspct to a prdfind rfrnc fram. Th usr can mak an arbitrary choic of th rfrnc fram and it dos not hav to b fixd in spac. It is typically a coordinat systm dfind at on of th intrfac nods, which offrs an asy way to includ larg rigidbody motion in th simulation, though th proposd FEM formulation alrady incorporats this ffct implicitly. Within this vrsion, th subroutin contains a complt FEM solvr that rsolvs th currnt positions of all nods of th FEM modl and computs th intrnal forcs, which is th information submittd back to ADAMS. Th subroutin is suitabl for statics as th position of th FEM nods is rsolvd assuming static dformation. Using this approach for dynamics rmains at this point an opn qustion. What nds to b rsolvd is th inclusion of inrtial and damping ffcts, which rquirs furthr information at FEM nods, i.. th information about currnt nodal positions alon is not sufficint. This vrsion has bn tstd with th xampl dscribd in Sction 4 and th obtaind rsults ar congrunt with thos from th originally dvlopd program (dnotd as Prsnt in Fig. 4). Howvr, th computation is not as ffctiv as with th authors program, bcaus th MBS program s solvr tsts various nodal displacmnts in ordr to stimat th right dirction and amount of displacmnt that yild th balanc btwn th intrnal and xtrnal forcs. This implis that th GFOSUB has to rsolv a rlativly larg numbr of dformation stats bfor th convrgd solution has bn rachd Implmntation suitabl for dynamics Th authors hav dvlopd anothr vrsion of subroutin that uss somwhat diffrnt logic and is, as a consqunc, actually mor suitabl for dynamic computations. As a mattr of fact, th prvious vrsion, in th xisting form, can also b usd for th purpos by using ADAMS dynamics, but th problm rsids in th fact that th mass proprtis would hav to b condnsd to th fw intrfac nods only, sinc ADAMS ss only thos points. This, howvr, is an unaccptably rough simplification of lastic body inrtial proprtis. Th ida of altrnativ implmntation is rathr simpl, but thrfor not without crtain drawbacks. ADAMS modl of th flxibl body nds to contain concntratd masss at ach of th FEM modl nods. This mans that ADAMS modl nds to includ all th nods from th FEM modl of th involvd flxibl body, which is th advantag and th main drawback, at th sam tim. Th concntratd masss in th ADAMS modl corrspond to th ntris of th lumpd mass matrix of th FEM modl. Th GFOSUB submits back th currnt valu of th intrnal forc at a nod. Compard to th prviously discussd GFOSUB vrsion, th diffrnc is significant. Th GFOSUB for statics nds to rsolv th static dformation of th whol FEM modl du to th prdfind positions (displacmnts) of th intrfac nods in ordr to giv back th information on th intrnal forcs at th intrfac nods. On th othr hand,

9 th vrsion of th GFOSUB discussd in this subsction xtracts th currnt position of all FEM nods from ADAMS, sinc thy ar now availabl in ADAMS modl, and uss that information to comput th intrnal forcs. This practically mans that th GFOSUB for dynamics only computs th intrnal forcs and th whol solution procdur is lft to ADAMS, i.. no solvr is ndd within th subroutin. That is th advantag. Th abov mntiond srious drawback is in th fact that all FEM nods nd to b includd in th ADAMS modl, which significantly incrass th numbr of dgrs of frdom of th MBS modl and maks th numrical burdn prohibitivly larg alrady for rlativly modrat FEM modls. And th sam issu as in th cas of th first discussd GFOSUB vrsion is also prsnt hr although intrnal forcs at all FEM nods ar computd in a subroutin call, th intrnal forc at only on of th nods can b submittd back to th main program pr subroutin call. In Fig. 6, 16 nods of an FEM modl containing 15 ttrahdral lmnts ar shown. Th bottom nods ar fixd. Fig. 5, lft, dpicts th original, undformd configuration togthr with constant xcitation forcs, whil th othr figurs dpict dformd configurations during th transint simulation. On may notic larg (gomtrically nonlinar) dformations without artificial modl nlargmnts, which ar charactristic for linar modls and rprsnt th consqunc of inability of linar FEM modls to handl larg local rotations. Fig. 6 Nods of th FEM modl in ADAMS and snapshots of dformd configurations during a transint simulation 6. CONCLUSIONS Th papr discusss th simplifid co-rotational FEM formulation dvlopd for flxibl body implmntation into MBS softwar packags. Implmntation of full FEM modls into MBS modls brings th advantag of gratr flxibility during th simulation, as boundary conditions and points of forc application do not nd to b known prior to simulation. Highr accuracy of simulation is anothr advantag. Th discussd FEM formulation also covrs gomtrically nonlinar ffcts up to a grat xtnd. Howvr, rgarding numrical ffort, it is of cours significantly gratr compard to modls basd on modal rduction tchniqu, but it is kpt rlativly low whn compard to th rigorous gomtrically nonlinar FEM formulations. With th modrn hardwar tchniqu, on may rconsidr th possibility of using th proposd formulation within th framwork of MBS for th parts that xprinc rlativly larg gomtrically nonlinar dformations and whos FEM modls ar of rlativly small or

10 modrat siz. It has bn dmonstratd that th implmntation of th approach in MBS systms is principally possibl and th obtaind rsults ar ncouraging, but also that th implmntation, basd on th xisting possibilitis to achiv it (in ADAMS), dos not allow making full us of th advantags th formulation offrs. Th issu of mor fficint implmntation is to b addrssd in th futur work. REFERENCES Bath, K.J. (1996). Finit lmnt procdurs, Prntic Hall, Nw Jrsy Ditz, S., Wallrapp, O., Widmann, S. (2003). Nodal vs. modal rprsntation in flxibl multibody systm dynamics, Procdings of Multibody Dynamics 2003, IDMEC/IST, Lisbon Marinkovic, D. and Zhn M. (2010). Gomtric stiffnss matrix in modal spac for multibody analysis of flxibl bodis with modrat dformations, Procdings of Intrnational Confrnc on Nois and Vibration Enginring, ISMA 2010, Luvn, Blgium, 2010 Schwrtassk, R. and Wallrapp, O. (1999). Dynamik flxiblr Mhrkörprsystm, Viwg Vrlagsgsllschaft, Braunschwig/Wisbadn Zhn, M. (2005). MBS and FEM: A Marriag-of-Convninc or a Lov Story?, BENCHmark Int. Magazin for Eng. Dsign&Analysis, 12-15