8. THE CONTACT DYNAMICS METHOD
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1 8 THE CONTACT DYNAMICS METHOD 8 Introducton The Contact Dynamcs ( CD ) method was ntroduced at the begnnng of the 990es by M Jean and JJ Moreau (Jean és Moreau, 992; Jean, 999) The method turned out to be extremely fast when smulatng granular flows, rad avalanches, segregaton, vbraton roblems of granular materals etc Most dscrete element methods take nto consderaton the deformablty of the elements eg by concentratng t nto the contacts, lke n the case of BALL-tye models, or by alyng an nternal fnte element mesh lke n UDEC The calculaton of the contact forces between the elements are based on the stffness characterstcs The logc of CD s dfferent The elements are consdered to be erfectly rgd, and the contact forces are not related to any stffness data The contact forces should ensure the eulbrum of the elements, and n addton, they must not volate reurements lke eg Coulomb lmt for frcton, or no tenson n cohesonless contacts, but ther calculaton does not aly any consttutve relatons For statcally hghly ndetermnate systems lke eg granular assembles n general, there exst several vald force systems satsfyng the eulbrum condtons; CD roduces randomly one of them, and several eually vald solutons can be receved f the roblem s calculated agan and agan wth the Contact Dynamcs Method No wonder that ts alcaton s not very wdesread n the engneerng ractce; on the other hand, snce the method s comutatonally very effcent for the smulaton of dynamc roblems, t s rather oular n the granular hyscs lterature The orgnal aers on CD were rather dffcult to decher The aer of Unger and Kertész (2003) was very dfferent from them n ths resect: the authors gave a clear, codewrter-orented ntroducton to the man lne of thought of the method, gvng a sgnfcant hel ths way for those who wanted to wrte ther own code and also for those who just wanted to understand how the method worked whch they were alyng n ther researches The ntroducton below s also based on ths ublcaton 82 The elements and the contacts Unlke n other methods where the basc unt of the analyss s the element or a art of the element around the node, n CD the basc unt s the ar of neghbourng elements The am of the calculatons s to teratvely fnd the sutable contact forces n all ars n the system The elements n the orgnal verson of the CD method were crcular and erfectly rgd Later several ublcatons aeared n the lterature on CD models wth non-crcular and even on deformable elements; but for smlcty, the ntroducton below wll focus only on erfectly rgd, erfectly shercal three-dmensonal elements These elements have sx degrees-of-freedom (3 translatons and 3 rotatons) A reference ont s defned for each element concdng wth ts centre of gravty Focus now on a selected ar of two elements close to (erhas n contact wth) each other We shall collect the state varables and the mechancal characterstcs of ths ar 77
2 Let g denote the dstance between the elements and If ths dstance euals to zero, the two elements get nto a ont-lke contact (denote t by c), and a concentrated contact force can be transmtted between the elements Unlke n most dscrete element technues, now the contact and the elements do not deform, so contactng elements must move n such a way that there would be no overla between them: g cannot become negatve g Fgure 4 The dstance g between the elements and The force actng on by s f c, and the force exressed on by s f c : f c F F F c x c y c z () t () t () t and f c F F F c x c y c z () t () t () t, and these can be reduced to the reference onts of and wth the hel of the B c (t) and B c (t) transton matrces, resectvely: B c c ( t) c c ; B ( t) c c 0 rz ry 0 rz ry c c c c rz 0 r x rz 0 r x c c c c ry rx 0 ry rx 0 The vectors r c and r c ont from the reference onts to the contact The reduced forces are then: c c c c c c f ( t) B ( t) f ( t) ; f ( t) B ( t) f ( t) red Summarze these two forces nto the 2-scalar reduced contact force vector of the ar:: red f f () t f c red c red () t () t Ths vector wll take art n the euatons of moton of the ar 78
3 n c Fgure 5 The normal vector of the contact Assgn an (n, t, w) local coordnate frame to the contact: n s the common unt normal of the contact lane, ontng towards element (see Fgure 5); t and w are an arbtrary ar of orthogonal unt vectors n the contact lane The vector v denotes the velocty vector of the (, ) ar: v v () t v () t () t, whch can be exressed as the tme dervatves of the dslacements: v dux ( t) dux( t) duy( t) duy( t) duz ( t) duz( t) ( t) ; v ( t) dx ( t) dx( t) dy ( t) dy( t) dz ( t) dz ( t) Consder now those two materal onts, c and c, whch form the contact Ther veloctes, v c és v c, can be exressed agan wth the hel of the B c (t) and B c (t) transton matrces: c ct v ( t) B ( t) v ( t) c ct v ( t) B ( t) v ( t) The relatve velocty of the contact s the dfference of the veloctes of the two materal onts c and c: 79
4 whch yelds x () t c c μ ( t) y ( t) : v ( t) v ( t) z () t c c ct ct μ ( t) v ( t) v ( t) B ( t) v ( t) B ( t) v ( t) Ths vector shows the relatve translaton of ont c wth resect to ont c, so f the two onts are already n contact, the n-comonent of ths vector must be ostve to exclude overlang Contacts n the CD method are modelled n the followng way: () If the two elements are n contact, e f g = 0, then a contact force can be transmtted If g > 0, then there s no contact, and no contact force exst n the ar The case g < 0 s not ossble (2) The normal comonent of the contact force, N, can only be negatve, e ct ct comressonal, but otherwse ts magntude ( N f ( n) vagy N f n) s arbtrary The (N, g ) relaton s shown n Fgure 6: N g Fgure 6 Normal force versus ga wh n CD (3) The magntude of the tangental T (whch has a t and a w comonent) s lmted by the Coulomb frcton law: T N where s the frcton coeffcent Fgure 7 llustrates ths lmtaton: the vector of the tangental force must ont from the orgn ether to nsde the cone (non-sldng contact) or just to the surface of the cone (sldng contact) 80
5 N arctan T t T w Fgure 7 Columb lmt for the tangental force As long as T s below the frcton lmt, the tangental comonent of the relatve translaton must be zero (elastc deformaton not ossble) When reachng the frcton lmt, the ontact starts to slde: the μ () t relatve velocty has to be just ooste to the drecton of the T force (the tangental relatve translaton must not have a comonent erendcularly to the frctonal force reachng the lmt), and the magntude of T s eual to N The magntude of the relatve translaton s not lmted n the contact model, ths can be determned from knematcal consderatons 83 The euatons of moton CD s a tme-steng method Its fundamental unknowns are the tme-deendent ostons and veloctes of the elements; however, they are not comled nto hyervectors lke n other dscrete element methods, because they are analysed searately ar by ar Assume that at t the state of the system s known: as far as numercally ossble, the ostons and veloctes of the elements are gven: u ( t ) u ; v ( t ) v, and the external, ext forces actng on the elements ( f ) and contact forces for all c ( f c ) are also known The external forces are reduced to the reference onts; the contact forces act n the ont-lke contacts The tme-deendence of the external forces s also known (eg that the gravtatonal force s constant), so the external forces are gven also n t t t (, ext f ) From these data the state of the system at t + (contact forces, the ostons and the veloctes of the elements) s searched for CD ales the mlct verson of the Euler method for ths The basc ste for the and ar can be wrtten as: v v ( M ) f : t v v ( M ) f u u v : t u u v ; 8
6 Here f and f denote the resultants of the external and all contact forces actng on and resectvely, beng reduced to the reference onts (Note that accordng to the mlct scheme, the veloctes and acceleratons belongng to the end of the tme nterval are consdered to be vald along the whole nterval) These reduced forces are, külső k k f B f f ( k ) :, külső k k f f B f ( k ) Summaton over ndex k runs along all contacts of element, ncludng the just analysed contact c as well Smlarly, ndex k runs along all contacts of The transton matrces are assumed to be constant durng the (t, t + ) tme nterval, and eual to ther values at t Indeed, f the dslacement ncrements are small durng the tmeste, the modfcaton of the vectors ontng from the reference onts to the contacts s neglgble Collect the mass and rotatonal nerta of the elements nto the matrces M and M, whch have the followng form n the case of shercal elements: m m m M I I I (Note that because of the shercal symmetry, these matrces are constant n tme) In order to determne the velocty of the ar ( v ), the resultants f and f should be known So, n addton to the external forces actng at t +, the contact forces should also be known at the end of the tmeste The Contact Dynamcs models search for these contact forces wth the hel of an teratve solver (whch has to be erformed over and over agan at every tmeste, as the contact forces change wth tme) 84 The teratve solver The solver swees along all ars of neghbourng or nearly contactng elements When consderng a gven ar, an aroxmaton s gven (based on the euatons of moton of the ar), so that the condtons assumed on the mechancal behavour would be satsfed: no overla; Coulomb-frcton etc Then a next ar s consdered After all ars were swet over, the solver starts the ars from the begnnng, and t s reeated over and over agan, untl the next aroxmatons are already suffcently close to the revous ones It means that the contact forces belongng to t + have been found, and the next tme ste can follow 82
7 The aroxmaton of the contact force n the ar (, ) s based on the euatons of moton of that ar Before turnng onto the detals, a few notatons have to be ntroduced: For element, reduce to the reference ont all those forces (external and contact forces) actng at t +, excet from the force exressed by element through contact c:, no _ c, ext k fred, : f f red, k c The f k contact force s only an actual aroxmaton of the force ndeed actng n contact k at t + ; t receves new and new values durng the teratons (At the begnnng of the analyss of the tme ste the contact forces are aroxmated to be the same as ther fnal, just determned values at the end of the revous tme ste, whch s the same as the begnnng of the just analysed tmeste) Smlarly, reduce all the forces actng at t + on excet from that force actng n c to the reference ont of :, no _ c, ext k fred, : f f red,, k c and collect the two vectors nto a hyervector:, no _ c, no _ c f red, fred, :, no _ c fred, Summarze the two transton matrces belongng to c nto a hyermatrx: c B B : c B and the matrces of nerta of and nto a block-dagonal matrx, whose nverse s: M 0 M : 0 M Later the followng two matrces wll also be necessary: and : T M B M B, M : B M B T And now the euatons of moton of the ar (, ) can be comled Frst, the euatons belongng to the end of the tme nterval can searately be wrtten as:, no _ c c v v f red, f red, M, no _ c c t v v fred, fred, Multly both sdes by B T from the left: T, no _ c B M fred, t T c B M B f c c (t was taken nto consderaton that f f ) After some rearrangements: 83
8 , _ t t T no c c B M fred, M f It s easy to notce that on the left sde the vector n the arentheses means that relatve velocty whch would occur n the contact at t + f f c s zero, e f there s no force n the contact Ths vector wll have a secal mortance n the forthcomng dervaton, so a secal notaton s gven to t:, no _ c, _ : T no c t B M fred, The euatons of moton can now be wrtten as: t M f, no _ c c where and f c are the unknowns So the euatons of moton gve the relaton between the unknown contact force and the unknown relatve velocty belongng to the contact Ths wll be the startng ont of the forthcomng calculatons Fnally the normal and tangental comonents of the relatve velocty vector of the contact wll be needed: T n ; n (Remember that n tw n denoted the velocty of the materal ont c relatve to the materal ont c Hence a ostve n means ncreasng ga between the two materal onts) Snce, no _ c the vector belongng to the tme nstant t + can drectly be calculated from the already, no _ c exstng aroxmatons of all other contact forces excet from c, the comonents and, no _ c tw, can also be determned, whle the comonents of the vector n, are unknowns The unknown and f c vectors are determned n three stes: Ste Frst decde whether the two elements wll be n contact at t + : calculate how large wll the ga be between them, assumng zero contact force:, no _ c, no _ c g g t n, A ostve result means that there wll be no contact at t +, and the analyss of another ar can mmedately follow A negatve result, on the other hand, means that wthout a contact force the elements and would overla, so an f c contact force s needed to avod the overla In ths case Ste 2 follows Ste 2 The contact force should modfy the veloctes of the two elements n such a way that nstead of overlang, they would exactly touch each other at the end of the tme ste In Ste 2 the am s to determne f c that satsfes the followng two condtons: () at t + the gawh between and s exactly zero: g t n, 0 () the contact does not slde, so the tangental comonent of the relatve translaton s zero: tw, 0 To satsfy these two condtons, the relatve velocty of the contact should be: 84
9 g n t (the ngatve sgn means that f the gawh was larger than zero, then should get closer to to touch t) The f c has to be such a force that f contnuously actng between and durng (t, t + ), at t + the relatve velocty would be just eual to μ From the euatons of moton, ths force turns out to be eual to: c, no _ c f M g n t t Now the ueston s whether ths force volates the consttutve condtons There were two condtons on the comonents of the contact forces The frst one reured the normal force a comresson Ths s automatcally satsfed because of Ste The second one was the Coulomb-condton: T c N c If ths holds for the calculated f c, then the analyss of the (, ) ar s ready, and a next ar can follow However, f the tangental comonent exceeds the frcton lmt, then the calculated contact force cannot be transmtted n the contact: the contact sldes, whch means that the tangental comonent of s not zero, and the calculaton based on zero tangental comonent should be corrected Ths correcton s done n Ste 3 Ste 3 In a sldng contact the tangental force comonent has to satsfy the followng to condtons, and as the thrd condton the euatons of moton: () The contact s sldng, so the magntude of the tangental force comonent s eual to the Coulomb-lmt: T c N c () The drecton of the tangental relatve velocty s just ooste to the drecton of the tangental force comonent: c T tw, c T tw, () the euatons of moton: c, no _ c f M g n tw, t t From these condtons the unknowns f c és can be calculated, and the analyss of the (, ) ar s ready The next ar can follow These calculatons ntroduced above gve an aroxmaton for the contact force n a ar, assumng that all other contact forces are unchanged and kee ther values last aroxmated When turnng to the next ar, the latest aroxmatons n other ars are aled Proceedng from ar to ar ths way, an aroxmaton s receved for the whole system of contact forces By sweeng through the comlete set of contacts over and over agan, the results (at least, hoefully) get closer and closer to what should really exst at t + The modfcatons caused by the consecutve teraton cycles cause smaller and smaller modfcatons n the contact forces; and the teraton can be termnated as the modfcatons 85
10 decrease under a rescrbed threshold Now the state belongng to t + has been found, and a new tme ste can be analysed The convergence of the method s an oen ueston n the lterature The order accordng to whch the ars are consdered wthn an teraton ste s random; the only reurement s that every ar should be consdered once In the next teraton stes the orderng s dfferent, rescrbed also by a random number generator If the same roblem s analysed twce, by startng the random number generator from two dfferent laces, the two resultng contact force systems wll be dfferent Ths s evdent for an engneer, artcularly f eulbrum s searched for: for a statcally hghly ntedermnate system several eulbrated force systems can be found, and wthout flexblty data the correct one cannot be selected There are several ublcatons dealng wth the ndetermnacy of the results Exerences show that though the order of the ars greatly affect the ndvdual contact forces and even the toology of the system, the overall, macro characterstcs lke average stress tensor or freuency dagram of contact force magntudes reman the same, aart from slght statstcal devatons 85 Alcatons The Contact Dynamcs Method s rather oular among hyscsts studyng granular dynamcs roblems (eg Daudon et al, 997; Radja et al, 998; Unger et al, 2004), but not aled yet n the engneerng ractce, artly because of the doubts on ts relablty and the random nature of the results Questons 8 Wrte the euatons of moton of ar n CD, and exlan the meanng of the uanttes n t! 82 Descrbe the mechancs of the contacts n CD! 83 Exlan the analyss of a tme ste n CD! 86
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