IMPACT MODELING BY MANIFOLD APPROACH IN EXPLICIT TRANSIENT DYNAMICS

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1 COMPDYN III ECCOMS Themac Conference on Compuaonal Mehods n Srucural Dynamcs and Earhquake Engneerng M. Papadrakaks, M. Fragadaks, V. Plevrs (eds.) Corfu, Greece, 5 8 May IMPCT MODELING Y MNIFOLD PPROCH IN EXPLICIT TRNSIENT DYNMICS Tmo J. Saksala *, Jar M. Mäknen,, Deparmen of Mechancs and Desgn, Tampere Unversy of Technology P.O. ox 589, FI-33 Tampere, Fnland mo.saksala@u.f, jar.m.maknen@u.f Keywords: Explc dynamcs, Conac mechancs, Lagrange mulpler mehod, Penaly mehod, Consran manfold. bsrac. In hs paper we sudy numercally dfferen mehods o mpose conac consrans n explc dynamcs. The radonal mehods, he penaly mehod and he more recen forward ncremen Lagrange mulpler mehod, are compared, n her performance and accuracy, o a drec elmnaon mehod based on he heory of manfolds. The dea of he mehod s ha when a conac consran s acve he soluon les on he boundary of he conac manfold generaed by conac consrans. Then he conac problem can be solved wh he elmnaon of addonal degrees of freedom. Ths elmnaon echnque can be vewed as a parameerzaon of he conac manfold. We show numercally ha n he case of a rgd mpacor, he elmnaon mehod s hghly more effcen han he forward Lagrange mulpler mehod snce he laer leads o a coupled sysem of equaons for solvng he Lagrange mulplers whle boh mehods produce smlar resuls. In he presen work he elmnaon mehod s also developed for he mpac problem of deformable bodes and he performance of he dfferen mehods are compared n numercal smulaons of he longudnal mpac of hck bars. The modfed Euler mehod s employed n solvng he equaons of moon n me.

2 Tmo J. Saksala, Jar M. Mäknen INTRODUCTION The radonal mehod o mpose conac consrans n explc ransen dynamcs s he penaly funcon mehod. I s, however, an approxmae mehod and has an adverse effec on he numercal sably of explc negraon. For hs reason, Carpener e al. [] nroduced a modfcaon of Lagrange mulpler mehod, called Forward Incremen Lagrange Mulpler mehod (FILM), whch s compable wh explc me negraors and does no affec he sably of negraon. The dea of he mehod s o refer he knemac conac consrans one me sep ahead of he equaons of moon and he Lagrange mulplers. Wh general conac nerfaces, hs mehod leads o a coupled sysem of equaons for he Lagrange mulplers. Therefore, f he number of conac pars s large s no effecve. In hs paper we presen an alernave and effcen mehod for mpac modelng n explc ransen dynamcs. The mehod, called here DME (Drec Manfold Elmnaon mehod) s based on he heory of manfolds. Conac consrans,.e. he dsplacemen nequales, generae a consran manfold wh a boundary []. When a conac consran s acve he soluon les on he boundary of he conac manfold. Ths mehod was employed by Saksala & Mäknen [3] n case of a rgd pson mpacng an elaso-vscoplasc damagng doman. In ha case, he conac consrans,.e. he mpenerably condons beween he pson node and rock surface nodes, were elmnaed. In he presen paper, hs formulaon s also performed n he case of a deformable mpacor and he node-o-node conac nerface. The developed echnque s mplemened n D (axsymmerc) case. In he numercal examples nvolvng mpacs beween lnear elasc bodes and dynamc ndenaon by a rgd ndener, he accuracy and effcency of he manfold mehod s compared o he penaly funcon mehod and he forward ncremen Lagrange mulpler mehod. The modfed Euler explc negraor s used for solvng he FE dscrezed equaons of moon n me. IMPOSITION OF CONTCT CONSTRINTS In hs secon we descrbe he radonal mehods, he penaly mehod and he more recen forward ncremen Lagrange mulpler mehod, as well as he manfold mehod o mpose knemac conac consrans n explc me negraon seng. The background for he manfold mehod s elaboraed only o he level requred for undersandng how he mehod arses. Lnear elascy and small dsplacemens are assumed for smplcy.. Modfed Euler mehod for explc me negraon The explc modfed Euler me negraor s chosen for solvng he equaons of moon. The response of he sysem usng hs mehod s compued by [4]: u u u M u u f ex u u Cu where M s he lumped mass marx, C s he dampng marx (se zero here), f,f n ex are he nernal and exernal force vecors, respecvely, and u s he dsplacemen vecor whle a do above denoes me dervave.. Forward ncremen Lagrange mulpler and penaly mehod f n () The classcal Lagrange mulpler mehod s ncompable wh explc me negraors. For hs reason, Carpener e al. [] nroduced a modfcaon, called here forward ncremen

3 Tmo J. Saksala, Jar M. Mäknen Lagrange mulpler mehod, whch s compable wh explc mehods. The dea of he mehod s o refer he knemac conac consrans one me sep ahead of Lagrange mulplers as: T Mu Cu fn fex G λ () Gu b where b s he vecor conanng he nal dsances beween he conac node pars (whch are known n advance), G s he knemac conac consran marx conssng, usually, of he normal vecors of he conac surfaces, and s he Lagrange mulpler vecor havng he physcal nerpreaon of conac forces. On subsung () no () gves, afer some algebra: ~ T u u M G λ where ~ u u~ u u u u u u M f Cu ex fn ~ u u u T GM G λ Gu~ b λ Ths mehod belongs o he class of predcor-correcor mehods n ha, frs, he acceleraon and dsplacemen of he sysem are predced as f here were no conacs. Then, he conac forces are solved based on he vrual peneraon. Fnally, he acceleraon s correced by subracng he conrbuon of he conac forces and he response of he sysem s updaed. I should be noed ha hs mehod s fully explc only when he node-o-node conac nerface s used. In hs case, wh he lumped mass marx, he equaons for solvng he Lagrange mulplers n (4) are uncoupled,.e. marx GM G s dagonal. Wh more T general conac nerfaces a coupled sysem of equaons mus be solved. Thus, compuaonal effcency of hs mehod consderably depends on he ype of conac nerface. The effcency can be, however, grealy ncreased, a he expense of accuracy and sably however, f he Penaly mehod s employed n he conac force calculaon. ccordngly, he hrd equaon n (4) and Lagrange mulpler n (3) s replaced by f p Gu~ b (3) (4) con (5) where p s he problem dependen penaly coeffcen. Oher equaons go unalered..3 Drec elmnaon mehod based on heory of manfolds Dfferen knemac relaons of consran equaons are shown n Fgure, where geomerc consrans nclude all holonomc and he unlaeral consrans, whch are gven by nequaly equaons wh he funcon of me and a generalzed place vecor only. Unlaeral consrans arse when modelng a knemac relaon beween bodes n conac. Conac consrans,.e. he dsplacemen nequales, generae a conac manfold wh a boundary. When a conac consran s acve he soluon les on he boundary of he conac manfold. Nex, he formal defnon of he conac manfold s gven. 3

4 Tmo J. Saksala, Jar M. Mäknen angen space T x M conac manfold M x geomerc consran manfold (wh boundary) boundary of consran vrual dsplacemen u whou nonholonomc consran Fgure : Geomerc nerpreaon of consrans. Defnon for conac manfold: The geomerc consran equaons nduce a d-manfold ha s defned by: M : n n u R E h(, u) E d (6) The conac manfold s a d-dmensonal smooh manfold wh me as -parameer famly. The conac manfold a fxed me = s denoed by M. In our case, he unlaeral consran equaons are smplyh(, u) Gu b. The Lagrange mulpler mehod and he penaly mehod, descrbed n he prevous secon, are convenonal mehods o mpose he conac consrans. However, he conac problem can also be solved wh he elmnaon of addonal degrees of freedom. Ths elmnaon echnque can be vewed as a parameerzaon of he conac manfold. In he presen paper, we have only node-o-node conacs wh sragh nerfaces (lnes) beween bodes and and boh bodes can be deformable or one of hem can be rgd dependng on he applcaon. In boh cases, he soluon procedure begn wh he deecon of acve conacs by checkng he volaon of he equaon Gu~ b (7) where he predced dsplacemen s as n Equaon (4). Nex, wo dfferen schemes o compue he nodal dsplacemens, veloces and acceleraons are presened. The frs one s for he case n whch he oher body s rgd, hus represenng usually a ool used for ndenaon, and he second s for he mpac of deformable bodes. Case ( s a rgd body, say a punch): If any conac consran s acve, say, hen hs nodal dsplacemenu of he body s elmnaed by solvng he acve consran equaon (7) wh respec ou as: 4

5 Tmo J. Saksala, Jar M. Mäknen u u u u b u u (8) f where f o, and M =m are he oal force and mass relaed o he acve conac consrans,.e. he summaon s over he degrees of freedom of he acve consrans. o, / M dof dof cve conac par Fgure : Illusraon of node-o-node conac nerface. Case ( and are deformable bodes): When an acve conac par, say, s deeced accordng o Equaon (7), he acceleraon and dsplacemen n he vercal degrees of freedom (dof and dof n Fgure ) of he nodes n conac are calculaed as: uˆ u ˆ uˆ u ˆ f u o, where f o, f o ( dof ) f o ( dof ) and m m M( dof ) M( dof ) are he oal force and mass relaed o he acve conac par wh m and m beng he nodal pon masses of he acve consran for he body and, respecvely. s o he veloces, hey can be solved from he equaons provded by elemenary physcs of colldng rgd bodes: m u e( u m u u ) u ˆ u /( m m u ˆ u ˆ m ) m u ˆ where e s he coeffcen of resuon wh e = for elasc collson and e = for perfecly nelasc collson. y he elmnaon echnque presened, he degrees of freedom of he problem reduce by one for each acve consran. Fnally, he deecon of separaon of he acve conac consran can be examned va he nodal nernal forces. The separaon occurs when he nodal nernal force s a racve force. The racve force convers he acve consran nacve. (9) () 5

6 Tmo J. Saksala, Jar M. Mäknen 3 NUMERICL EXMPLES In hs secon he performance of he dfferen mehods o mpose conac consrans are compared n numercal smulaons nvolvng low-velocy mpac. Frs, he performance of he manfold mehod s demonsraed n he rgd punch ndenaon problem wh varyng number of conac consrans. Then, he problem of longudnal mpac of hck cylndrcal bars s smulaed and he accuracy of he mehods s compared wh each oher and o he analycal soluon as well. The maeral s assumed rgd and/or lnear elasc. ll he smulaons are performed wh self-wren Malab (64-b 9a verson) codes usng a Wndows Vsa pc equpped wh.6 GHz MD Phenom II X3 7 processor and 8. G of RM. 3. Rgd punch ndenaon problem The ndenaon problem wh a rgd punch descrbed n Fgure 3 s solved wh he FILM, DME and Penaly mehods. xsymmery of he problem s exploed and CST elemens are used n he mesh. The maeral of he ndened doman s lnear elasc. The punch s assumed rgd whle s modeled as a sngle node. F ex = punch () Punch node u p,z u p,x R x rse F punch u c,y b xs of roaon Fgure 3: Indenaon wh a rgd punch. The exernal sress pulse, (), s used o smulae he mpac of he rgd punch o he deformable doman. s mulpled by he cross seconal area of he ndener punch, an exernal force, F ex, o be appled o he punch node s obaned. Moreover, he force ha resss he punch peneraon no doman, he conac force F punch, needs o be solved durng he soluon process. Thus, he equaon of moon for he punch node, wh s moonal degrees of freedom u p,x, u p,z, o be added o he FE dscrezed equaon of moon, can be wren as: m u z F F () p p, where m p s a compuaonal mass aached o he punch node. The value of hs mass mus be small enough so ha doesn affec he soluon and bg enough o manan he mass marx well-condoned. ex punch 6

7 Tmo J. Saksala, Jar M. Mäknen The geomery of he punch (ndener) s defned as b,.e. he dsances beween he punch node and he conac nodes on he ndened doman boundary. Thus, usng he noaon n Fgure 3, he knemac conac consrans can be wren n form: u c, z u p, z b () Equaons of hs form are wren for all conac nodes and, consequenly, he consran equaon of form Gu = b s obaned. s here s only one DOF (frcon s no aken no accoun) assocaed o he punch node, he sysem of consran equaons s coupled. Ths sysem s solved (when usng he FILM mehod) usng he Malab backslash operaon. The conac compuaons are performed as explaned n Secon. Nex, he problem descrbed n Fgure 3 s smulaed wh varyng number of conac consrans (acheved easly by dfferen values of radus R z ) usng he mesh conssng of 6 CST elemens shown n Fgure Fgure 4: CST mesh and a deal The maeral parameers are: E = GPa, =.3 and = 785 kg/m 3. compuaonal mass of. kg s used for he punch node mass m p. me sep = s s chosen o keep he compuaons sable. Three dfferen smulaons are carred ou n wha follows, correspondng o hree dfferen number of conac consrans: Sm wh 7 (R z = m), Sm wh 38 (R z = 3 m) and Sm3 wh 6 (R z = 6 m) conac consrans. The analyss me s 4 = s. The sress pulse amplude s MPa whle s duraon, rse and descen mes are -4 s, -5 s, respecvely. The resuls of he smulaons for Sm are presened n Fgure 5. 7

8 Tmo J. Saksala, Jar M. Mäknen u p,z /m 5 x -3 FILM 4 DME Penaly, p = 6e Tme /s x -4 F punch /N 7 x FILM DME p = 6e9.5.5 Tme /s x p = 6e9 FILMLag FILMToal DMEToal CPU Tme /s (c) Tme sep Fgure 5: Punch peneraon, conac force and CPU mes for conac soluon (c) n Sm. ccordng o Fgure 5a he FILM and DME (Case ) mehods produce equal punch dsplacemen durng loadng phase bu slgh, barely noceable dfference occurs durng unloadng. In conras, he vercal dsplacemen by Penaly mehod devaes consderably from he oher mehods. The value p = 6 9, found by ral and error, gave he bes resuls n hs problem. The force ressng peneraon, F punch, s also que smlar o all mehods besdes he oscllaons seen n he DME curve n Fgure 5b. s for he CPU mes (measured by Malab c funcon), s seen ha he me spend for conac compuaons by he DME and Penaly mehods (DMEToal and p = 6e9) s abou half of he lowes mes by he FILM mehod. Moreover, mos of he oal me used by he FILM mehod s consumed n solvng for he Lagrange mulplers from Equaon (4), (compare he FILMLag and FILMToal curves n Fgure 5c). Fnally, he FILM mehod dsplays heavy varaons n CPU mes whch are aesed by he hgh peaks n Fgure 5c. Nex, he resuls for Sm and Sm3 are shown n Fgure 6. 8

9 Tmo J. Saksala, Jar M. Mäknen u p,z /m p = e9 FILM DME CPU me /s DME FILM p = e Tme /s x Tme sep.. FILM DME CPU me /s (c) 3 4 Tme sep Fgure 6: Punch peneraon and CPU mes n Sm, and CPU mes for conac soluon (c) n Sm3 s he number of conac consrans ncrease, he Penaly mehod becomes more llbehavng as seen n Fgure 6a. In fac, no value of penaly coeffcen gvng reasonable resuls for punch dsplacemen, u p,z, was found n Sm3,.e. when he number of conac consrans s 6. Ths defcency of he penaly mehod s relaed o he relavely long conac nerface n hs problem: a value of he penaly coeffcen gvng good resuls close o he axs of symmery s far oo small, by many orders of magnude, furher away from. s o he CPU mes spen n he conac compuaons, he DME mehod s clearly he mos effcen one. Indeed, n Sm3, see Fgure 6c, s a leas mes faser han he FILM mehod. I s also wce as fas as he Penaly mehod, as seen n Fgure 6b. In hs specfc problem, havng 336 DOFs, he second mos me consumng operaon durng a me sep s he compuaon of he nernal force vecor by f n = Ku where K s he global lnear elasc sffness (sparse) marx. In he presen smulaons hs me vared beween o ms whch s approxmaely of he same magnude as he me needed o perform he conac compuaons wh he DME mehod. Therefore, he effcency of conac compuaons s crucal n hs knd of explc ransen elaso-dynamcs nvolvng conac-mpac. 3. Longudnal mpac of cylndrcal bars The problem of longudnal mpac of hck cylndrcal bars descrbed n Fgure 7a s solved wh he FILM, DME and Penaly mehods n order o compare her performance. xsymmery of he problem s exploed and CST mesh shown for he lef bar n Fgure 7b s used. Relavely coarse mesh, wh 46 for he lef and 46 elemens for he rgh bar s used. 9

10 Tmo J. Saksala, Jar M. Mäknen r v v z a =.m (radus of he bar) = 78 kg/m 3 =.3 E = GPa v = m/s (nal velocy) z... r (c) xs of roaonal symmery Fgure 7: Longudnal mpac of hck bars, he CST mesh and a deal (c). The node-o-node conac nerface s employed here and he conac compuaons are performed as explaned n Secon. The number of conac pars s very small, 6 pars, whch s nsgnfcan from he compuaonal pon of vew. me sep = -6 s s chosen o T keep he compuaons sable. s he marx GM G s dagonal for he node-o-node conac nerface, he average of s enres,.9, s aken for he penaly parameer. In he resuls n Fgures 8-, he gap compued as Gu b and conac forces are ploed a he conac nodes as a funcon of me for dfferen mehods. Gap /m x -5 r = r =. r =. r =.4 r =.6 r =.8 Conacs open 5 x 6 - Tme /s x -4 Tme /s Fgure 8: Smulaon resuls wh FILM mehod: gap, and conac forces. F con /N 4 3 r = r =. r =. r =.4 r =.6 r =.8 x -4 ccordng o he resuls n Fgure 8, he mpenerably condon s accuraely sasfed,.e. he gap s zero all he me bars are n conac, wh he FILM mehod. s he sress waves refleced a he free ends of he bars reach he conac surfaces, he conacs open and bars sar o rerea. Ths even s ndcaed n Fgure 8a. Nex, he same resuls are presened for he Penaly mehod n Fgure 9.

11 Tmo J. Saksala, Jar M. Mäknen Gap /m x -4 r = r =. r =. r =.4 r =.6 r =.8 F con /N 3.5 x r = r =. r =. r =.4 r =.6 r = Tme /s x -4 Tme /s x -4 Fgure 9: Smulaon resuls wh Penaly mehod (p =.9 ): gap, and conac forces. s expeced, he Penaly mehod does no sasfy he mpenerably condon. Indeed, peneraon occurs and he conac force s based on hs peneraon. Fnally, he same resuls are presened for he DME mehod wh he perfecly nelasc collson scheme (e = ) n Fgure. Gap /m x -6 r = r =. r =. r =.4 r =.6 r = Tme /s x -4 Tme /s x -4 Fgure : Smulaon resuls wh DME mehod: gap, and conac forces. F con /N 6 x r = r =. r =. r =.4 r =.6 r =.8 The resuls for he DME mehod wh he perfecly nelasc collson assumpon sasfy he mpenerably condon accuraely, as aesed by he zero gap values (see Fgure a) unl he arrval of he sress wave refleced from he free end of he bar. The conac forces n Fgure b are somewha smlar o hose produced wh he FILM mehod. Nex, he mehods are compared o he analycal soluon of he problem by Valeš e al. [5]. The dmensonless axal sress a he locaon r =. m and z =. m, a locaon close enough o he conac surfaces so ha he dfferences n he resuls beween he mehods are sll vsble (and no smoohed oo much by he San-Venan prncple). In addon, he dfferen schemes for calculang he velocy of conac nodes wh he DME mehod, Equaon (), are compared. The resuls are shown n Fgure.

12 Tmo J. Saksala, Jar M. Mäknen FILM p =.9e p = 5e DME, e = DME, e = Fgure : nalycal soluon for dmensonless axal sress [5], smulaon resuls wh dfferen mehods and resuls wh dfferen velocy schemes of DME mehod (c). s seen n Fgure b, he axal sress predced wh he FILM and DME (e = ) mehods are dencal. The resul wh he elasc collson scheme for velocy (DME, e = ) devaes slghly from hese. These mehods predc he analycal soluon wh a good accuracy. For example, he arrvals of he dfferen wave ypes are clearly denfable n he numercal resuls as well. In conras, he Penaly mehod wh he value p =.9 s que poor when compared o he analycal soluon. Wh he hgher value, p = 5, he Penaly mehod seems o perform slghly beer frs bu hen, afer me saon c /a = 5, overshoong oscllaons develop n he response. Moreover, hs value provded poor resuls a he conac nodes. 4 CONCLUSIONS Dfferen mehods, ncludng he forward ncremen Lagrange mulpler mehod (FILM), he Penaly mehod and he drec manfold elmnaon mehod (DME), o mpose conac consrans n ransen explc dynamcs were numercally compared n hs paper. mehod based on drec elmnaon of he conac consrans was developed and esed for he node-o-node conac nerface. The heorecal background of he mehod based on he heory of manfolds was elaboraed wh some deal. In he numercal smulaon of he ndenaon problem wh a rgd punch, he DME mehod was found o be overwhelmngly more effcen han he FILM mehod whle boh mehods provded smlar resuls. s o he Penaly mehod, whle effcen, faled o produce reasonable resuls when he conac nerface was relavely wde. I s more suable for pon-lke conac problems. Moreover, he Penaly mehod has an adverse effec on he numercal sably of he explc negraon and he problem specfc pan o fnd he correc value of he parameer

13 Tmo J. Saksala, Jar M. Mäknen In he mpac problem of longudnal bars, he FILM and DME mehods predced he analycal soluon for axal sress wh a good accuracy. These mehods, he DME wh he perfecly nelasc collson assumpon, sasfed he mpenerably condon accuraely and predced dencal resuls for axal sress. In conras, he resuls wh he Penaly mehod devaed from he analycal soluon and from he oher mehods as well. Due o s effcency and accuracy, he DME mehod can be regarded as superor o he FILM mehod n problems where he FILM mehod requres a soluon of coupled sysem of equaons for he Lagrange mulpler. These problems nclude he rgd punch ndenaon smulaed n hs paper and problems wh a more general conac nerface. In he fuure developmens of he DME mehod, wll be formulaed for more general node-o-segmen conac nerfaces. REFERENCES [] N.J. Carpener, R.L. Taylor M.G. Kaona, Lagrange Consrans for Transen Fne Elemen Surface Conac. Inernaonal Journal for Numercal Mehods n Engneerng, 3, 3-8, 99. [] J.M. Mäknen, Manfolds on Connuum Mechancs. In:. Koppel & J. Oja Eds. Connuum Mechancs, Nova Scence Publshers. pp. -5,. [3] T.J. Saksala, J.M. Mäknen, Numercal Modellng of -Rock Ineracon n Percussve Drllng by Manfold pproach, In:. Erksson and G. Tber Eds. Proceedngs of he 3rd Nordc Semnar on Compuaonal Mechancs, KTH Sockholm, Sweden, Ocober -,. [4] G. D. Hahn, Modfed Euler Mehod for dynamcal analyses. Inernaonal Journal for Numercal Mehods n Engneerng, 3, , 99. [5] F, Valeš e al., Wave Propagaon n a Thck Cylndrcal ar Due o Longudnal Impac. JSME Inernaonal Journal, Seres : Mechancs and Maeral Eng, 39, 6 7,