UNCONDITIONALLY STABLE ALGORITHMS FOR RIGID BODY DYNAMICS THAT EXACTLY PRESERVE ENERGY AND MOMENTUM*

Transcription

1 INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING, VOL. 31, (1991) UNCONDITIONALLY STABLE ALGORITHMS FOR RIGID BODY DYNAMICS THAT EXACTLY PRESERVE ENERGY AND MOMENTUM* J. c. SIMO AND K. K. WONG Division of Applied Mechanics, Deparmen of Mechanical Engineering, Sanford Universiy, Sanford, CA 94304, U.S.A. SUMMARY We show ha, for rigid body dynamics, he mid-poin rule formulaed in body co-ordinaes exacly conserves energy and he norm of he angular momenum for incremenal force-free moions, bu fails o conserve he direcion of he angular momenum vecor. Furher, we show ha he mid-poin rule formulaed in he spaial represenaion is, in general, physically and geomerically meaningless. An alernaive algorihm is developed which exacly preserves energy, and he oal spaial angular momenum in incremenal force-free moions. The implici version of his algorihm is uncondiionally sable and second order accurae. The explici version conserves exacly angular momenum in incremenal force-free moions. Numerical simulaions are presened which illusrae he excellen performance of he proposed procedure, even for incremenal roaions over 65 degrees. The procedure is direcly applicable o ransien dynamic calculaions of geomerically exac rods and shells. 1. INTRODUCTION We examine he formulaion of single ime sepping algorihms for he orhogonal (roaion) group which conserve energy and oal angular momenum. The subjec is of paricular imporance for dynamic problems in non-linear srucural mechanics where finie elemen models for rods and shells resul in roaional degrees of freedom which are described by proper orhogonal marices. The dynamic response associaed wih hese roaional degrees of freedom leads, herefore, o a problem of evoluion in he roaion group. Algorihms ha conserue consans of moion are imporan for wo fundamenal reasons. Firs, consans of moion, such as energy and momenum, are ofen primary physical quaniies of direc engineering ineres. Second, in an algorihmic conex, conservaion properies lead o rigorous noions of non-linear sabiliy; in paricular, he so-called energy mehod (for accouns of his class of echniques see Richmayer and Moron or HughesI3). In his paper we examine wo alernaive algorihms for problems of evoluion in he orhogonal group; in paricular, for classical rigid body dynamics, which are relaed o he classical Newmark family of inegraion algorihms. The firs algorihm, formulaed in Secion 4 and referred o as ALGO-1 in wha follows, differs from ha proposed in Simo and Vu-QuocZ4 * Research suppored by AFOSR under conrac nos. 2-DJA-544 and 2-DJA-771 wih Sanford Universiy Associae Professor of Applied Mechanics *Graduae Research Assisan and Saff Engineer a GE Asro Space Division, San Jose /9 1/01~19-34$ by John Wiley & Sons, Ld. Received 22 Augus 1989

2 20 J. C. SIMO AND K. K. WONG only in one single bu crucial aspec: for a ypical ime inerval [,, n+l]r he rae form of he momenum balance equaion is enforced a he poin,,+,,, 0 < y < 1, and no a n+l. The mid- poin rule version of his modified algorihm has wo remarkable properies no shared by he algorihm in Simo and VU-QUOC:~~ For zero exernal loading in he ime sep [,,, n+l], he algorihm (i) exacly conserves energy and (ii) exacly conserves he norm of he angular momenum. Unforunaely, we show below ha his modified algorihm fails o conserve, in general, he direcion of he oal angular momenum. The second algorihm, formulaed in Secion 5 and referred o as ALGO-C1 in wha follows, is designed so ha balance of angular momenum (boh in magniude and direcion) is auomaically saisjied if no forces are applied in he inerval [,,,+ l]; i.e. for incremenal force-free moions. The crucial design condiion is he enforcemen of balance of angular momenum in conservaion form; i.e. he difference beween he spaial angular momenum a imes, + and, equals he impulse over he ime inerval [n,,+ A similar design condiion was inroduced in Zienkiewicz e al31 in he conex of finie elemens in ime for classical srucural dynamics. For he roaion group, he resuling algorihm has a remarkable propery. We show in Theorem 5.1 ha energy is exacly conserved in incremenal force-free moions for any oalues of he parameers (y, P)E [0, 11 x [O, 31 such ha Ply = 3. The resul is non-rivial since, for rigid body dynamics, he kineic energy is a non-linear funcion of he sae space variables. The proof of discree conservaion of energy (which ensures uncondiional sabiliy) relies crucially on he form aken by he singulariy-free roaional updaes performed via he exponenial map. These updaes exploi he opimal paramerizaion of he roaion group in erms of uni quaernion parameers, and ensure ha he algorihmic procedure is well-condiioned for any magniude of he incremenal roaions. In addiion o he conservaion properies alluded o above, ALGO-C1 has some furher noeworhy feaures. Firs, for he opimal choice B/y = i, he algorihm is compleely independ- en of he angular acceleraion which becomes an uncoupled variable. Thus, as in he class of mehods proposed in Zienkiewicz e ~ l.,~' he acceleraion vecor need no be sored in he acual implemenaion of he algorihm, and is recovered a he end of he ime sep in erms of he final angular velociy. This recovery depends crucially on he choice of he parameer y E [0, 13. We find ha opimal resuls are obained for y = 1. Second, an explici condiionally sable version of he algorihm is also possible. The opimal explici second order accurae mehod, labelled ALGO-C2, is obained for he values p = 0 and y = 1. An ouline of he paper is as follows. Secion 2, included for he convenience and easy reference of he reader, gives a summary accoun of some basic facs on he roaion group and rigid body dynamics which are exploied in he formulaion and analysis of he proposed algorihms. Secion 3 conains a brief accoun of some basic properies of he exponenial map and uni quaernions also used in our algorihmic analysis. Secions 4 and 5 conain he formulaion and analysis of he wo algorihms considered in his paper. Finally, numerical simulaions ha illusrae he excellen performance of he proposed algorihm, ALGO-C1, are presened in Secion 6. A deailed sep-by-sep implemenaion procedure is given in Appendices I and 11. The subjec of conserving algorihms has received considerable aenion in recen years. In paricular, a wo-sep procedure ha enforces conservaion laws for any given algorihm is developed by Bayliss and Isaacson' and Isaacson. l5 A relaed (variaional) procedure which implemens a similar idea via Lagrange mulipliers is considered in Sasaki21 and references herein. See NavonL9 for a comparison of hese echniques in he conex of he shallow-waer equaions. Paricular energy conserving algorihms are considered in LaBudde and Greenspan,l6- and more recenly in Greenspan" where a very ineresing energy-conserving

3 UNCONDITIONALLY STABLE ALGORlTHMS 21 mehod for one-dimensional sysems is proposed. In conras wih he presen procedure, hese mehods ypically involved a wo-sep predicor-orrecor mehod. Relaed ideas in he conex of he rapezoidal rule for non-linear elasodynamics, in paricular, he variaional mehod, are considered by Hughes e al.,l4 parially moivaed by resuls in 3elyschko and Schoeberle3 and Hughes. In conras wih he presen developmen, his class of mehods relies on he use of Lagrange muliplier echniques o enforce energy conservaion. Momenum conservaion, on he oher hand, does no necessarily hold. In he conex of Hamilonian sysems, a sysemaic developmen of algorihms wih conservaion properies goes back o he pioneering work of De Vogelaere, who coined he denominaion symplecic inegraors. This class of mehods has received recenly considerable aenion; see e.g. Chane4lY4 Chanell and Scovel, Fengs and references herein. In paricular, Zhong and Marsden3O have recenly addressed he developmen of symplecic inegraors for he orhogonal group. Their algorihm, however, appears o be only condiionally sable, firs order accurae and does no conserve energy. In fac, an imporan heorem of Zhong and Marsden3 implies ha symplecic algorihms for he orhogonal group which also conserve energy do no exis. Our algorihm, ALGO-C1, does no conradic his resul since, alhough uncondiionally sable, second order accurae and energy and momenum conserving, is only symplecic up o 0(h3). 2. THE ROTATION GROUP AND RIGID BODY DYNAMICS In wha follows, we summarize some elemenary noions in rigid body dynamics, and a few facs concerning opimal paramerizaion of he special orhogonal group, S0(3), needed for our subsequen algorihmic developmens. Furher deails on rigid body mechanics may be found in sandard reference books; e.g. Wiaker,29 Arnold or Gold~ein.~ Mahemaical and geomeric aspecs no considered here are found in Simo e al Treamens of marix groups in a modern geomeric conex ha include he orhogonal group are given in e.g. Curis6 or Warner.28 A useful engineering perspecive is found in Hughes (Reference 11, Chaper 1) Conjguraion space. Body frame Le c R3 be he reference placemen of a solid body, wih paricles labelled by X. A moion of he body is a one parameer family of maps Recall ha he moion is rigid if a:~-+r3, for E[O, T] c R, (1) [I a( X ) - Qr( X Z ) 11 = I[ X(1 - X Z 11 for all X(l), X( E@ and E[O, TI. This condiion holds if and only if ar is of he form x=q, cp() + N)X where A( ) is an orhogonal ransformaion for all E [0, TI, and I+ cp( ) is a ime dependen vecor funcion. Thus, any mapping ~E[O, T]~(cp(), A()) R3 x SO(3) (4) defines a moion of he rigid body. Consequenly one refers o (2) Q:= R3 x SO(3) (5) (3)

4 22 J. C. SIMO AND K. K. WONG as he absrac conjiguraion manifold of he rigid body. Le pre: a -+ R be he densiy in he placemen B. We choose our co-ordinaes in.4? so ha he cenre of mass is locaed a he origin. We hus have 1, PrefXdX = 0 (6) In wha follows, we le {el, el, e3 I\ denoe he inerzial frame Body frame. We aach a frame {El, E,, E3} a he cenre of mass in.9i? parallel o he inerial frame. By our choice of co-ordinaes, he map H q() defines he cenre of mass a ime (we assume ha q()l,=o = 0). Moreover, he map HA() in (4) defines he orienaion of he frame {El, E,, E3} according o he relaions A():= A(f)EA, A = 1, 2, 3 (7) One calls {A()){A =,,2,3), wih,(o) = E,, he bodyframe. See Figure 1. Noe ha we have he relaion A() E, Velociy jields For a rigid moion (3), he velociy field is given by b, = +() + A()X, (X, ) B x [O, I"] (8) Here, i () is he ranslaional velociy of he cenre of mass. Since A() is in SO(3) for all E [0, I"], we have A()A'() = 1, a condiion which implies ha A() = w()a() = A()W() (9) where G() = A()A'() and W() = AT()A() are skew-symmeric marices. We donoe by w and W he axial vecors associaed wih he skewsymmeric marices & and W, respecively. In componens we have Figure 1. Kinemaics of he rigid body

5 UNCONDITIONALLY STABLE ALGORITHMS 23 Following a sandard noaion, we shall denoe by so(3) he vecor space of skew-symmeric marices; i.e. SO(3) = {+ ~3 -+ rw3p + i~ = 01 Recall ha so(3) is a Lie algebra wih bracke he ordinary marix commuaor. The map A : so(3) + R3 given in co-ordinaes by (1 l), which assigns o a skew-symmeric marix is axial vecor, defines a Lie algebra isomorphism by he relaion (12) i%h = w x h, for any her3 (13) where x is he ordinary cross produc of vecors in R3. The axial vecors w() and W() associaed wih c() and W() are referred o as spaial and conveced angular velociies, respecively Properies of he angular velociy. The following elemenary properies will be used hroughou our subsequen developmens: 1. The spaial angular velociy w is he angular velociy of he body frame, since i, = AEA = AATA 2. In view of he expression = fia = w x,, A = 1,2,3 i, = AWE, = AWA A he following relaion holds (adjoin acion of SO(3) and so(3)): fi = A W A ~ = ~ AW W (16) 3. The componens of he spaial angular velociy w relaive o he body frame equal he componens of he conveced angular velociy relaive o he reference frame {El, E,, E3}. This propery follows by noing ha Consequenly, one has See Figure 2.,*w = AE,.w = E,.ATw = E,* W (17) w = wiei = W,,; W = W,E, Q Figure 2. Spaial and conveced angular velociy fields

6 24 J. C. SIMO AND K. K. WONG 2.3. Angular momenum Using expression (8) for he velociy field along wih relaion (9), he angular momenum vecor, denoed by J(), akes he form s, J():= Prcf%(X) x 4(X)dX = s, P,,fcp() x Ci() + M XldX P,fA()X x C i W + A()Wr()XIdX (19) 5, + In view of relaion (6), equaion (19) reduces o he classical expression where J() = cp() x P() + n() (20) p():= M+(); M:= Ja prerdx Here, p( ) is he oal linear momenum, M is he oal mass of he body, and n( ) is he oal spaial angular momenum relaive o he cenre of mass, which is given by n( ):= D,w( ); I:= A( ) J A *( ) (22) The marix 0, is he ime-dependen spaial ineria dyadic, and 9 is he consan conveced ineria dyadic defined as J:= ~~prn[llxli21 - X@XldX (23) One also defines he conveced angular momenum relaive o he cenre of mass by he expression n():= AT()A() E JW (24) The noion of oal angular momenum is a paricular insance of he concep of momenum map for a mechanical sysem wih symmery; see e.g. Arnold.' This concep plays a crucial role in he exension of he algorihmic ideas discussed in his paper o he more general mechanical sysems suggesed in our concluding remarks (see Secion 7) Equaions of moion Le m() be he applied orque and le n be he applied force a he cenre of mass. In he presen conex, he classical equaions of balance of angular and linear momenum ake he form dn dp - ---m, - d d =n The balance of angular momenum (relaive o he cenre of mass) can be expressed eiher in he spaial or in he convecive (body) represenaions as follows. Le a:=w and A:=W (26) be he rae of change of he angular velociy in he spaial and body descripions, respecively.

7 Observe ha Since W W = 0, (26), and (27) imply UNCONDITIONALLY STABLE ALGORITHMS 25 d a=-[aw] = AW + AW = A[WW + W] d a=aa (28) Relaions (22), (26) and (28) hen lead o he following wo alernaive formulaions of he rae of he angular momenum balance equaion Conveced descripion A=AW J A + W x J W = ATm Spaial descripion A=GA W=A w=a (29) I,a + w x I,w = m The convecive form of equaion (29), is he classical Euler's equaion for he rigid body in body co-ordinaes; see e.g. Arnold' or G~ldsein.~ Observe ha passage from he conveced (body) o he spaial represenaion is performed by ransforming (W, A) o (w, a) according o relaions (16) and (28). In wha follows, wihou loss of generaliy, we assume ha ii = 0 so ha he cenre of mass moves wih consan linear velociy. Thus, he only relevan equaions of moion are (29)(or(25),) which describe he roaional dynamics. Accordingly, he configuraion space is simply Q = SO(3) (30) By choosing, if necessary, an inerial frame wih origin a he cenre of mass we may assume ha p = 0. One speaks of a reducion o he cenre of mass (see e.g. Simo, e az.26). Consequenly he oal angular momenum becomes Finally, we shall consider sandard iniial condiions of he form J=n (31) A()lz=o = 1 and ~()l,=~ = wo (32) Equaions (29) and (32) define an iniial value problem for (A(), W()) in he phase space SO(3) x R3. The disincive feaure of his problem lies in he fac ha he phase space is a differeniable manifold (a Lie group) and no a linear space. This feaure is shared by oher nonlinear models in solid mechanics; such as rods and shells, see Sirno,,, Simo and From an algorihmic poin of view, he spaial discreizaion of hese models by he finie elemen resuls in a sysem of ODE'S wih a srucure idenical o equaions (29) and (32). 3. THE EXPONENTIAL MAP. OPTIMAL PARAMETRIZATIONS The algorihmic reconsrucion procedure of he configuraions from he angular velociy field discussed below reduces o he consrucion of a finie roaion from an incremenal (infiniesimal) roaion via he exponenial mapping. Following Simo and VU-QUOC,~~ he procedure can be inerpreed geomerically as follows; see Figure 3. One views he roaion group as a 'curved surface' whose poins, A ~S0(3), represen configuraions (finie roaions). An infiniesimal roaion is a skew-symmeric marix, i E so(3)

8 f.sinz 26 J. C. SIMO AND K. K. WONG 7 IS0 (3) f so (3) f Figure 3. Graphical illusraion of he exponenial map wih axial vecor XE R3, which is inerpreed as defining a angen vecor o he surface (SO(3)). Now consider a one-parameer family of infiniesimal roaions E H fe:= cf E so(3) inerpreed geomerically as a line angen o SO(3). This sraigh line is angen a he ideniy 1 o he curve EH A, in SO(3) which is he one-parameer subgroup uniquely defined via he exponenial map as m 1 For an arbirary marix group, he exponenial map is defined by he infinie series (33). The crucial fac exploied in our reconsrucion procedure (configuraion updae) is ha, for S0(3), he series (33) has a closed-form expression given by he following classical formula of Euler and Rodrigues; (see e.g. Wiaker29). sin II x II + A = exp[f] = 1 + I1 x II e II x II 1 2 C3IlXlll' (34) By seing E = 1, i follows ha formula (34) uniquely defines he proper orhogonal marix A:= exp [f] associaed wih a given skew-symmeric marix f E so(3). Furhermore, since i x = 0, we conclude from (34) ha exp Cflx = x (35) hence, x is he eigenvecor of exp [ x] associaed wih he uni eigenvalue. This observaion leads o he sandard inerpreaion of (34) as afinie roaion, wih roaion vecor x E R3, and roaion angle 1) x 1) ; see Figure Opimal paramerizaion. Quaernion parameers Several alernaive paramerizaion of he roaion group SO(3) are possible; see e.g. Curis6 or Hughes." I is now well esablished ha he opimal singulariyfree paramerizaion is defined in

9 UNCONDITIONALLY STABLE ALGORITHMS 27 Figure 4. Mechanical inerpreaion of he exponenial maps in erms of roaions erms of he (four) uni quaernion parameers, denoed by (qo, q). Uni quaernion parameers are elemens of he uni sphere S3 in R4 subjec, herefore, o he consrain qi + 11q1I2 = 1; see Curis (Reference 6. Chapers 1 and 2). The paramerizaion ( go, q) E S3 H A E SO(3) is defined via he sandard formula Noe ha he inverse paramerizaion A E SO(3) H (qo, q)e S3, defined hrough he relaions is no singulariy free. The algorihm of Spurrier3 provides an opimal quaernion exracion procedure ha avoids he singulariy inheren in formulae (37). However, quaernion exracion procedures, in paricular Spurrier s algorihm, play no role in our implemenaion of he algorihms described below. The paramerizaion of SO(3) in erms of uni quaernion parameers plays a crucial role in our implemenaion of he algorihm proposed in Secion 5 and described in deail in Appendices I and 11. In paricular, recall ha, given quaernion parameers (po, p), (go, q), (ro, r) associaed wih orhogonal marices P, Q, RE S0(3), marix muliplicaion and quaernion muliplicaion are in one-o-one correspondence Algorihmically, numerical round-off leads o evenual loss of he orhogonaliy propery when marix muliplicaion is employed. By conras, he orhogonaliy propery can be exacly preserved using quaernion muliplicaion along wih he following simple renormalizaion procedure: (ro, r)-(ro/l, r/l), where I:=,/= (40) Of course, in exac arihmeic 1 = 1. This renormalizaion procedure is imporan in pracical compuaions.

10 28 J. C. SIMO AND K. K. WONG 4. A GENERALIZED NEWMARK ALGORITHM FOR SO(3) In his secion we consider a ime sepping algorihm for he orhogonal group based on he classical Newmark algorihm; see e.g. Hughes (Reference 13, Chaper 8). We show ha he midpoin rule version of his exended Newmark's algorihm exacly preserves energy and he norm of he angular momenum, bu violaes in general conservaion of angular momenum. Our version of Newmark's algorihm for SO(3) is closely relaed o ha proposed by Simo and Vu- Quoc.~* A crucial difference, however, concerns he way in which balance of angular momenum is enforced (see Remarks 4.1 below). In conras wih he presen procedure, he mid-poin version of he algorihm in Simo and VU-QUOC~~ neiher conserves energy nor conserves he norm of he angular momenum The general algorihm: ALGO-1 Le [,,,,+l] c [0, T] be a ypical ime inerval, where N LO, TI = (J Cn,n+lI n=o Le h = A:= n+l -, be he ime sep inerval. Assume ha a, he following iniial daa are known: (A,,, W,, A,,) S0(3) x R3 x R3 (given) (42) The objecive is o obain an algorihmic approximaion o he acual soluion (A(,,+ W(,+ convecive represenaion. Consider he following algorihm: (An+~,Wn+l,An+l)~S0(3) x R3 x R3 (43) A(n+ 1)) of he evoluion equaions (29) in he ALGO-1: Exended Newmark's Algorihm Sep 1. Define he updaed configuraion via he exponenial map as: A,,+ = A,, exp [6] (44) where 0 E R3 is he conveced relaive (incremenal) roaion vecor. Sep 2. Define he conveced relaive (incremenal) R3 in erms of( W,, A,,, A,+ 1) by he formula 0 = hw, + h2[(f - P)A, + PA,+1] (45) where BE [O, 31 is a parameer wih idenical significance as in he classical Newmark algorihm. Sep 3. Define he updaed conveced angular velociy by he formula Wn+l =Wn+hC(1-~)An+~An+11 (46) where y E [0, 11 is a parameer wih idenical significance as in he classical Newmark algorihm. Sep 4. Enforce rae of momenum balance a, + yh: JAn+y + W n+y x JW,+, = A;~+,I~I,,+~ (47)

11 UNCONDITIONALLY STABLE ALGORITHMS 29 Remarks The above algorihm differes from ha of Simo and Vu-QuocZ4 only (bu crucially) in ha momenum balance is enforced a n+,, insead a, The algorihm is convergen, and second order accurae for y = The algorihm is uncondiionally sable for 72% and b2$ (48) 4. Observe ha velociies (and acceleraions) a differen ime seps are added in he conueciue (or body) represenaion, and no in he spaial represenaion. Tha his procedure furnishes he physically correc and geomerically consisen updae procedure for he velociy field is demonsraed in he example given below. Example 4.1. Le ~(A(), W()) be a given moion of he rigid body. Consider he moion defined by he expression A+() = QA() (49) obained by superposiion of a consan roaion Q E SO(3) on he given moion. The velociy field associaed wih he final moion (a +(), A '()) is hen obained via ime differeniaion of (49) as so ha A'() = QA() (50) This relaion gives he resul ~;+:=A+(A+)T=Q~;QT W++:=(A+)TA+ =w (51) w+=qw and W+=W (52) In view of (52), we conclude ha direc addiion of velociy vecors in body co-ordinaes make physical sense; in paricular, for he presen example we have W + - W = 0. On he oher hand, a direc addiion of spaial velociies is meaningless unless he spaial velociy is ransformed according o he classical rule (52),. rn 4.2. The conveced mid-poin rule: Conservaion properies The mid-poin rule is recovered from he preceding algorihm merely by seing y = 4 and 0 = $. This choice of parameers resuls in he expressions h 0= hw,++:=-(w, + W,+i) A,++ := 5 (An+ 1 + A,) = Wn = [W,+ 1 - W,,] h2 h Subsiuing (53) ino (47) yields he following expression for he (rae of) momenum balance equaion expressed in erms of he relaive roaion vecor 0: 1 h2 1 J x J8 = hj W, + - Az++m,++ (54) 2 2 where A,,++:= A,,exp[f&] and An+l = A,exp[&] (55) (53)

12 30 J. C. SIMO AND K. K. WONG In he absence of exernal forces; i.e. under he assumpion of zero applied orque, Euler s equaions of rigid body dynamics possess wo firs inegrals of he moion corresponding o: (i) conservaion of energy, and (ii) conservaion of oal (spaial) angular momenum. We examine wheher hese inegrals of he moion are also presen in he discree algorihmic problem defined by he mid-poin rule version of ALGO-1 (obained for y = i, p = a). Recall ha he angular momenum in he convecive represenaion (body frame) a imes, and,+ is given by he expressions n,+ = 5 W,+ and n, = 5 W, (56) To examine he conservaion properies of he conveced mid-poin rule, we assume zero orque in [,,,+ 1], so ha m,++ = 0. Combining (53) and (54), he momenum balance equaion a, + $12 becomes Wih his relaion a hand, we have he following resuls.* (i) Conservaion of he norm of angular momenum. By aking he do produc of (57) wih (n,+ + 11,) and using he elemenary fac ha a, x a2 -a2 = 0, we obain Therefore (nn+ 1 - fin) * (nn+ 1 + nn) = II nn+ 1 II - I/ n n II z= 0 (58) and ALGO-1 conserves he norm of he angular momenum for y = i, p = a. (ii) Conservaion of(kineic) energy. By aking he do produc of (57) wih (W,+ + W,) and using once more he fac a1 x a,-a, = 0, we obain (nn+, - n n ). ( w n + l + Wn) =Wn+1.5Wn -Wn.JWn+ = 0 (60) By expressing he kineic energy of he sysem as H =$w.o,w _$W.JW (61) we conclude from (60) and (61) ha i.e. he conveced mid-poin rule also conserves energy. Remarks Alhough in he absence of applied orque he norm of he spaial angular momenum is conserved by he coveced mid-poin rule; in general he spaial angular momenum is no conserved; i.e. in general J,+, = # J, = A,. *Resuls (59) and (62) were known o P. S. Krishnaprasad

13 2. 3. UNCONDITIONALLY STABLE ALGORITHMS 31 For more general mechanical sysems he oal angular momenum J (i.e. he momenum map) does no, of course, coincide wih he spaial momena. For ALGO-1, he only choice of parameers leading o boh conservaion of energy and conservaion of he norm of angular momenum is in fac y = 3 and fi = $. By conras, he conservaion properies of he algorihm discussed below in Secion 5 hold for any (y, P)E [0, 13 x [0, ] such ha fi/y =. 5. A MODIFIED ENERGY AND MOMENTUM CONSERVING ALGORITHM In his secion, we develop a modified ime sepping algorihm which exacly conserves energy and, in addiion, enforces by consrucion conservaion of oal angular momenum. Remarkably, he resuling algorihm is considerably simpler and easier o implemen. We remark ha hese conservaion properies are achieved wihou resor o Lagrange mulipliers (compare wih Hughes e all4) Reconsrucion based on he momenum map: ALGO-C1 The basic idea underlying he mehod is raher simple. Essenially, one replaces he rae form of he momenum balance equaion, as given by (47), by is firs inegral (conservaion form) over he ime sep [,,,, A similar idea was employed by Zienkiewicz e wihin he framework of classical srucural dynamics. In he presen conex, recall ha he ime derivaive of he oal angular momenum is given by he expression d J = ic = -[l,~] d = I,w + I,W = w x 0,w + I,W = m (63) Inegraion of his expression over [,, n+l] and using he relaion II = All = AJW, yields he discree conservaion law II,+~ -A, = An+lJWn+l - A,JW, = l+' m()d (64) Evaluaion of he inegral on he righ-hand side of (64) (he impulse) by a generalized ype of midpoin rule leads o he algorihm summarized below. ALGO-CI: Modijied Energy-Momenum Conserving Algorihm Sep 1. Define he configuraion updae exacly as in ALGO-1: A,+ = A, exp ~631 (65) Sep 2. Define he conveced angular velociy in erms of he relaive roaion vecor 0 as (by subsiuion of (45) ino (46))

14 32 J. C. SIMO AND K. K. WONG Sep 3. Formulae he momenum balance equaion in conseroaion form: where a possible choice for ae(o,1] is An+1JW,+1 - A,JW, - hm,,, = 0 Sep 4. Updae he conveced angular acceleraion as (using (46)) 1 ( 3 A,+1 =-[Wn+l - W,] A, Yh Remarks By consrucion, for zero applied exernal orque in he inerval [,,,+ he algorihm ALGO-C1 conserves he oal angular momenum in he inerval I,,,,,] since from (67); J,+, = J, when m,+, = 0. One speaks of an incremenal force-free moion. 2. The choice of he evaluaion poin,+, for he exernal orque in expression (68) is no essenial, bu is moivaed by he proof of Theorem 5.1 below. There, we show ha energy conservaion in incremenal force-free moions holds if and only if /?/y = $, a choice which reduces (67) o he mid-poin rule. 3. I should be noed ha, for a choice of parameers such ha o! = P/y = i, he configuraion, velociy updae and momenum balance equaion, defined by (65),(66) and (67) respecively, are independen of he conveced angular acceleraions A, and A,, The algorihm hen becomes defined solely in erms of (0, W}, and formula (69) merely gives a decoupled updae procedure for he conveced angular acceleraion. 4. For he opimal choice /?/y = 3, our numerical experimens indicae ha he value y = 1 (which implies /? = 3) yields he bes resuls for he conveced acceleraion. This choice gives A, + = (l/h)( W,, - W,), making he acceleraion updae insensiive o error propagaion via A,. Alernaive updae procedures for he acceleraion oher han (69) are also possible. In paricular, A,, +, could be compued (for = f) by enforcingforce balance a, +. For all pracical purposes, his approach appears o yield resuls idenical o hose obained wih he choice y = A sandard argumen shows ha ALGO-C1 is second order accurae (in configuraion and velociies) for ply = 3. (See also Secion below.) 6. In general he orque m is configuraion dependen. This is ypically he case in space-craf dynamics where he loading is consan in he body frame, bu configuraion dependen relaive o he inerial frame. For rods and shells, he configuraion variable A also depend on he spaial variables. Afer a spaial finie elemen discreizaion, however, he nodal equaions are of he form (67) where m(a) is now configuraion dependen. The evaluaion of he impulse inegral in (64) can hen be accomplished by a second order accurae algorihm, eiher he mid-poin I:+' or he rapezoidal rule. In general we hus have m(a())d x hm(a,+,), wih A,+,:= A, exp[ao] (70) (69)

15 UNCONDITIONALLY STABLE ALGORITHMS 33 where 0 is he relaive roaion vecor defined by (65). A analogous expression holds for he generalized rapezoidal rule, see (A10) of Appendix I Relaion beween ALGO-1 and ALGO-CI. The relaionship beween he algorihms ALGO-1 for B = 2 and y = +, and ALGO-C1 for ply = 3 can be esablished by means of he following argumen. Subsiuion of (66) ino he momenum balance equaion (67) yields From (55) we hen have An+1= A ~++~XPC+~I (72) which yields h h2 exp[i@]jo = -Ai+&(A,,+l 2 + A,)JW, + TA:++m,++ (73) Nex, by expanding in power series he exponenial, we have e~p[fq]j@=[l+~+~~+ 6 lo 1. - and 1 = JO + -0 x JO + U (03) 2 (74) By subsiuion of (74) and (75) ino he momenum balance equaion (73) we obain JO + 1 h x JO = hjw, + -A:++m,++ + hco(0 ) + +(a3) 2 which differs from he momenum balance equaion (54) only by erms of hco(02) and 0(03); or, equivalenly, by erms of order O(h3), since 0 = Co(h). Noe ha his argumen also implies ha ALGO-C1 is second order accurae for ply =, since ALGO-1 is second order accurae for /3 = and y = Conservaion of energy We have shown above ha ALGOX1 enforces by design conservaion of angular momenum in force-free moions. Furhermore, we claim ha algorihm ALGO-C1 also conserves energy in force-free moions for he choice of parameers Ply = 4. The proof of his resul is given in he following. Theorem 5.1. Le he applied orque in he inerval [,,,, conserved by ALGOX1 if and only if ply = 3; i.e. be zero. Then he oal energy is

16 34 J. C. SIMO AND K. K. WONG Proof: Firs, for convenience, we rewrie (66) in erms of he following noaion: W,+,=--0-Wn+ Y 2-- ph ;>w,* W,* = W, + iha, Nex, wih his expression in hand, using he fac ha exp[6]0 = -0 (see (35)), we obain he following relaion: Wn + 1 = An+ 1Wn + I Wn+l + W, - exp[&]w, + (exp[6] - 1) = An Wn A,+ 1 W, + [An+ 1 - An] (79) where we have used he updae formulae (65) and (78). Hence Now recall ha wn+1 -Wn=AnWn+l -An+,Wn+ CAn+1 -An1 2Hn+1-2H,:= nn+l*wn+i -ffn wn R,+ = n, Thus, making use of (80) and (81) we obain 2(Hn+ 1 - Hn) = nn.anwn+ (by consrucion) I herefore follows ha =nn.wn+l which vanishes if and only if y = 2p. ( 5) =W,*JWn+l-JWn+l.Wn+ 2-- ( zyg) [nn+l -Hn].W,* (82) Hn+l - H, = 1 -- [nn+l -nn]*w,* (83) I should be noed ha algorihm ALGO-C1 conserves momenum regardless of he value of YE [0, 13 and BE [0,9]. and conserves energy for any combinaion of he parameers such ha y = 2p. In addiion, he preceding heorem shows ha he algorihm is uncondiionally sable according o he energy mehod; see Richmayer and Moron. w

17 5.3. Ouline of he linearizaion of he algorihm UNCONDITIONALLY STABLE ALGORITHMS 35 The momenum balance equaions (47) or (67), along wih he corresponding updae formulae for he conveced angular velociy and conveced angular acceleraion, for he wo algorihms described above, define a non-linear equaion for he relaive roaion vecor 0 of he form Gdy,(@):= Gdyn(An exp [&I) = 0 (84) where A,,+l = A, exp[&]es0(3) is clearly a non-linear funcion of 0. For he proposed algorihm ALGO-C1, for insance, Gdyn akes he following form: Gdyn(-O) = hmn+= - A, The ieraive soluion of (84) by Newon's mehod requires performing he linearizaion of (84) abou a sequence of approximae soluions (Ai),-O(i))~S0(3) x R3. The exac linearizaion of (84) akes he form L[cdyn(@(i))] = Gdyn(@(i') -I- DG~,,(-O~i))'A8'i' (86) where D( - )- 80"' denoes he direcional derivaive of (-) in he direcion A@. The direcional derivaives of he roaion marix A(i) and he relaive roaion vecor 0") (in body co-ordinaes) abou an inermediae soluion (Ac0, 0")) are given by (see Simo and Vu-Quocz4) DA(i). = A(i)AG(i) and DO"). A@(') = T(@(i))A@(i) where he linear map T: so(3) + so(3) is defined by he expression T(-O):= e + 4 II Q II 0 [l - e@e] + 46, wih e:= - an CiII -O II 1 II 6 II Wih hese relaions in hand, a Newon ype of ieraive soluion procedure can be easily consruced. Concepually, a each sep of he ieraion process, he relaive roaion vecor in he conveced (body) represenaion is updaed according o he expression exp (87) + 1'1 = exp [&(')I exp [@)I (89) Deails of a sep-by-sep implemenaion of his soluion procedure are given in Appendix I. One can also express he preceding algorihms and he linearizaion procedure in erms of he spaial incremenal vecor Wi) = A,O(i). See Simo and VU-QUOC~~ for deails Explici momenum conserving algorihm: ALGOL2 I is of ineres o examine in some deail he form aken by he conserving algorihm ALGO-Cl for explici ransien calculaions. Formally, he explici version of ALGO-C1 is obained by seing y = 1 and p = 0. Observe ha his choice is a variance wih he explici version of he sandard Newmark algorihm which enforces y = 4. In he conex of he conserving algorihm ALGO-C1, he imporance of selecing y = 1 (and no y = 4) will be illusraed numerically in he following secion. Deails peraining o his explici implemenaion are given below. ALGO-C2: Explici Momenum Conserving Algorihm Sep 1, Compue relaive roaion vecor OER' in he convecive descripion in erms of (Wny An), hz 0 = hw, +-A, 2 (90)

18 36 J. C. SIMO AND K. K. WONG Sep 2. Updae he configuraion via he exponenial map An+l = A,exp[6] Sep 3. Updae he convecive angular velociy Wn+l = J-'exp[ - 6 ][JW, + hazm,,++] Sep 4. Updae he convecive angular acceleraion Observe ha, again by consrucion, he preceding algorihm conserves he oal angular momenum in incremenal force-free moions. Furhermore, as in he implici case, we remark ha opimal resuls are obained for he choice y = 1, see Secion NUMERICAL SIMULATIONS In his secion, we presen represenaive numerical simulaions ha illusrae he performance of he algorihms discussed and developed in he preceding secions. The firs algorihm, denoed as ALGO-0, is he mid-poin rule version (y = 4,b = $) of ha proposed by Simo and Vu-Q~oc~~ in which momenum balance is enforced a,+ 1. The second algorihm considered is he mid-poin rule version (y = +,/.I = $) of ALGO-1 in which momenum balance is enforced a n+y. The performance of hese algorihms is hen compared wih he energy and momenum conserving algorihm (ALGO-C1) for he opimal choice b/y = *, and he wo values y = $ and y = Unsable roaion abou he axis of inermediae momen of ineria I is well known ha permanen rigid body roaions are sable only abou he axis of maximum and minimum momen of ineria; e.g. see Thomson (Reference 27, Chaper 5.12). Roaion abou he axis of inermediae momen of ineria is unsable in he sense ha small disurbance will resul in unsable oscillaion. Our numerical experimen is precisely concerned wih he simulaion of his unsable moion, and is performed according o he following seps. 1. A ime = 0, wih he rigid body iniial a res, a consan orque (relaive o he inerial frame) is applied a he cenre of mass in he direcion parallel o he axis of inermediae momen of ineria. 2. A ime = *, he consan orque is removed and a consan disurbance orque, acing normal o he iniial orque, is applied o he body for a duraion of h, he ime sep incremen. 3. Thereafer, he rigid body undergoes an exernal orque-free unsable moion. The ime hisory of he applied exernal orque is hus given by he expression C,e,; O<<* m=[ C,e,; * < < * + h (94) 0; >*+h where C1 and C2 are consans. The numerical consans and iniial condiions used in he

19 UNCONDITIONALLY STABLE ALGORITHMS 37 simulaions are Comparison of he hree implici algorihms. The numerical resuls obained wih he implici algorihms ALGO-0, ALGO-1 and ALGO-C are compared for wo differen ime sep inervals: h = 0-01 and h = 0.1. For he smaller ime sep (h = 0-01) he hree algorihms give essenially idenical resuls and exhibi conservaion of energy and momenum. A sample of hese resuls is given in Figure 5, where he ime hisory of he kineic energy, angular momenum, norm of he incremenal roaion vecor, roaion vecor and conveced velociy are shown. On he oher hand, for he larger ime sep (h = 0-l), he differences beween he hree algorihms, as shown in Figures 6, 7 and 8 become eviden. In paricular, ALGO-0 neiher conserves energy, nor angular momenum (boh in direcion and magniude) for > 2.0 when he body is free of exernal orque. ALGO-1 exhibis conservaion of energy and conservaion of he norm of he angular momenum, bu fails o conserve he direcion of he angular momenum vecor (noe he drif of he spaial angular momenum vecor II from is iniial orienaion along el). ALGO-C1, on he oher hand, conserves boh he energy and vecor of angular momenum for he orque-free moion corresponding o > 2. These resuls are in agreemen wih our analysis in Secions 4 and 5. I should be noed ha for his choice of ime sep he magniude of he incremenal roaion vecor 0 is over 65" (resuls ploed are in radians), he perfec we& condiioning exhibied by he hree algorihms illusraes he singulariy free naure of our roaional updaes employing paramerizaion of roaion marices by quaernion parameers. Finally, observe for > 2.0, he rigid body undergoes a spinning/umbling moion which periodically flips i up-side-down (wih respec o he el) and is well-capured, even for he 'unreasonably large' ime seps by ALGO-C1. The sep-by-sep implemenaion o ALGO-C1 for his problem is given in Appendix I Conservaion properies for ALGO-CI. Choice of y. Theorem 5.1 shows ha ALGO-C1 conserves energy for any (b, y )[0, ~ 31 x [0,1] such ha Ply = $. In addiion, second order accuracy is also obained for ply = 3. Thus, he choice of ~E[O, 13 auomaically dicaes he value of BE [0, )] (so ha energy conservaion and second order accuracy are obained) and affecs only he acceleraion updae given by (69). In wha follows, we consider wo possible choices (i) y=$andb=$. (ii) y = 1 and = i. The conveced angular acceleraions obained wih hese wo values are given in Figure 9. Also given are he acceleraions calculaed using ALGO-0 and ALGO-1 for comparison. The resuls from ALGO-Cl for he angular acceleraion show a marked improvemen for he value y = 1 over ha for y = Explici version: ALGO-CL Choice of y. Finally, we examine he performance of he ALGOX2 for wo possible choices of y E [0,11: (i) y=)andb=o. (ii) y = 1 and j3 = 0.

20 ALCO-Cl (B/y = = 1.0) h = 0.01 ALGO-Cl (p/y = 0.5, y = 1.0) h = 0.01 I Normalized Spaial Angular Momenum ALGO-C1 (@/y = 0.5, y = 1.0) h = 0.01 I -154.,.,.,.,. 1.,,.,., Om 0nc.b (e) Norm of Angular Momenum ALGO-Cl (@/r = = 10) h = 0.01 Conveced Angular Velociy ALGO-C1 (p/y = 0 5, y = 10) h = 0.01 (0 Norm of Incremenal Roaion Roaion Vecor Figure 5. ALGO-Cl (y = 1, fi = 4). Free spinning rigid body: Resuls for ime sep h =

21 . I ALGO-0 (p/y = 0.5. y = 0.5) h = 0.1 (4 Kineic Energy -01.,,., (4 Normalized Spaial Angular Momenum I h = 0.1 ALGO-0 (@/y = 0.5, y = 0.5) h = 0.1 '" I I (b) Norm of Angular Momenum -15.,.,.,.,., I (e) Conveced Angular Velociy ,. I I (c) Norm of Incremenal Roaion Roaion Vecor Figure 6. ALGO-0 ( y = f. /? = a). Free spinning rigid body: Resuls for ime sep h = 0-1

22 ALCO-I = = 0.5) h = (1 I 1 =':U (4 Kineic Energy c 3 - l= \. 03- Ol- -a 1, ,.,,,.,,,,, ALGO-I (P/r = 0 5. y = 0 5) h = 0.1 I I (b) Norm of Angular Momenum ALGO-I (@/y = 05. y = 05) h = ,.,.,....,.,.,,,.,, I Conveced Angular Velociy ALGO-I (Pi7 = 0.5. y = 05) h = 0.1 Norm of Incremenal Roaion (f) -41.,,,.,.,.,.,.,,, Roaion Vecor om oncns I Figure 7. ALGO-1 (y = i, = 4 ). Free spinning rigid body: Resuls for ime sep h = 0.1

23 , ~~ ALCO-Cl (P/y = = 1.0) h = 0.1 ALGO-C1 (P/y = 0 5, y = 1.0) h = 0.1 I I? iw ~ *o o ~..,,.,.,.,.,.,.,.,. (4 Kineic Energy -01.,.,.,.,.,.,.,.,. I, (4 Normalized Spaial Angular Momenum 50 ALCO-C1 (P/7 = 05, y = 10) h = LO I (b) Norm of Angular Momenum -154., 1 (e).,.,.,.,. I 1 I ' > ' I Conveced Angular Velociy p-fi ALGO-C1 (P/y = 0 5. y = 10) h = 0.1 I, Norm of Incremenal Roaion - 4,.,.,,,.,.,.,.,,,., (0 Roaion Vecor Figure 8. ALGO-Cl (y = 1, p = )). Free spinning rigid body: Resuls for ime sep h = 0.1

24 ALCO-CI = = 1.0) h = 0.01 pt= -100.,.,.,,.,.,.,.,., _.-3 (a) Conveced Angular Acceleraion 100 IW w w u) m a 0 a w W -IW (b) Conveced Angular Acceleraion (d) Conveced Angular Acceleraion s IW W ALGO-0 (@/7 = 0.5. Y = 0.5) h = 0.1 IW W ALCO-I (B/y = 0.5. y = 0.5) h = 0.1 M M 40 u1 m m d o m -4a w -M -m -m -lm -IW (d Conveced Angular Acceleraion 6 Conveced Angular Acceleraion Figure 9. Free spinning rigid body: Comparison of conveced angular acceleraion for differen algorihms and ime seps

25 (4 Kineic Energy Conveced Angular Acceleraion ALGO-c'? (p = 00. y = 05) h = % - i ALCO-C2 (/3 = 0.0, y = 0.5) h = I 1 0 s I5 Norm of Angular Momenum (e) rn Conveced Angular Velociy.2 c3 ALGO-C2 (@ = = 0.5) h = ALCO-C2 (/3 = 0.0, y = 0.5) h = ILL--- 0 w2 Ow0 (4 Norm of Incremenal Roaion (0 Roaion Vecor Figure 10. ALGOK2 (Explici). Free spinning rigid body: Resuls for a = 0 and y = f. Time sep h = O W1

26 ALGO-C2 = 0.0, y = 1.0) h = I I. I. r. I. I.,. 0 2 I 6 8 (4 Kineic Energy h = Conveced Angular Acceleraion ALGO-C2 (p = 0.0. y = 1.0) h = Norm of Angular Momenum -15.,.,.,.,.,.,.,,,., (e) r* Conveced Angular Velociy ALGOKC2 (p = 00. y = 10) h = ALGOLC2 (p = 0 0, y = 10) h = Norm of Incremenal Roaion Roaion Vecor Figure 11. ALGO-C2 (Explici). Free spinning rigid body: Resuls for p = 0 and y = 1. Time sep h = 0-001

27 UNCONDITIONALLY STABLE ALGORITHMS 45 el Figure 12. The heavy symmerical op wih one poin fixed Numerical resuls for hese wo se of values are presened in Figures 10 and 11. A ime sep of h = is chosen. I is apparen ha he values in (i) lead o unaccepable resuls. By conras, choice (ii), he one advocaed here, leads o excellen resuls comparable o hose obained wih he implici version of he algorihm, ALGO-C The heavy symmerical op wih one poin $xed In his numerical example we consider he moion of a symmerical op wih oal mass M and axis of symmery E, in a uniform graviaional field - ge,. Conservaion of angular momenum a he op s cenre of mass yields he following equaion of moion: k=m= -M gr x e,; r=ar (95) where r is he spaial posiion of he op s cenre of mass relaive o he fixed conac poin (origin of inerial frame) and R = 1E, is he corresponding conveced (body) descripion, see Figure 12. In Secion we showed ha he mid-poin versions of ALGO-1 and ALGO-C1 differ only in erms of hq(0 ) and 0(0O3). Thus, since he superoriy of ALGO-Cl over ALGO-1 for large ime seps was clearly esablished in he preceding example, we shall omi furher reference o ALGO-1 and focus our aenion in his example on ALGO-C1. Using (64) we obain he following difference form of he momenum balance equaion: An+lJWn+l - A,JW, = hi&++ = - Mglh(A,++E, x e,) (96) The numerical consans and iniial condiions used in he simulaions are as follows: Since he applied orque is posiion dependen, small ime seps are required o obain accurae soluions. Two differen ime seps are considered. Figures 13 and 14 show he numerical resuls obained for ime seps h = 0.04 and h = 0.001, respecively. Figure 14 demonsraes he op s

28 Energy Normalized Spaial Angular Momenum ALGO-C1 (p/7 = 0 5. y = 10) h = i.,.,.,.,.,., -,.,. I, I (b) Norm of Angular Momenum ALGO-Cl (B/y = 0 5. Y = 10) h = 0.04 Conveced Angular Velociy ALGO-Cl (p/y = 0 5, y = 10) h = Oil (C) Norm of Incremenal Roaion -04.,.,.,.,.,., Cener of Mass Figure 13. ALGO-C1 ( y = 1, p = i). Spinning (fas) op: Resuls for large ime sep h = 004.,.,.,.

29 ALGO-CI = 0.5, y = 1.0) h = ALCO-C1 (B/y = 0.5, 7 = 1.0) h = Energy ALCO-CI (P/y = 0 5. y = 10) h = I I Normalized Spaial Angular Momenum ALCO-Cl (@/7 = 0 5. y = 10) h = Norm of Angular Momenum ALGO-C1 (@/y = 0 5, 7 = 10) h = Conveced Angular Velociy ALGO-C1 (@/y = 0.5, y = 10) h = P :j,, 006-,,,,,,, j.,,,,,., 04- O M om oncna Norm of Incremenal Roaion Cener of Mass Figure 14. ALGO-C1 ( y = 1, p = ). Spinning (fas) op: Resuls for small ime sep h = OQ01

30 48 J. C. SlMO AND K. K. WONG nuaion and procession abou is fixed conac poin, which becomes eviden form he posiion of is cenre of mass defined by r. For a fas op (ha is, a op wih kineic energy 9 poenial energy), Goldsein (Reference 9, Chaper 5.7) gives he following approximae relaions for he angular frequency of nuaion on and precession w,: on=-w J33 Jll Mill and up=- J33 w3 ; (Jll = Jzz) For our choice of numerical parameers, hese relaions lead o values on = 10.0 and op = 0.4. The simulaion resuls wih he smaller ime sep h = yields on = 9.2 and up = Noe ha, since he exernal orque is posiion dependen, he ime sep h = 0.04 resuls in raher large values of he incremenal roaions and leads o inaccurae resuls. Neverheless, he e3 componen of R is conserved (since m,++ -e3 = 0). 7. CONCLUDING REMARKS We have developed a class of single sep ime sepping inegraion algorihms for he roaion group ha is uncondiionally sable, second order accurae, and exacly preserve energy and oal angular momenum in incremenally forced-free moions. The crucial idea underlying his developmen is he enforcemen of balance of angular momenum in conservaion form over a ime sep [,,,+ We have shown ha an exension of he classical Newmark algorihm o he roaion group, which enforces he rae of balance of angular momenum a he mid-poin, conserves energy and he magniude of he oal angular momenum, bu fails o conserve he direcion of he oal angular momenum. We have demonsraed he good performance of he proposed algorihm, even for very large incremenal roaions, in a number of numerical simulaions. Concepually, he exension of he proposed algorihms o non-linear srucural models, such as rods shells, involves he evaluaion of a conjguruion dependen impulse over a ypical ime sep. Such an exension can be accommodaed easily wihin he proposed framework, alhough preservaion of he conservaion properies is no longer ensured in his more general conex. Our fuure research will address hese and relaed issues. (97) ACKNOWLEDGEMENTS We hank P. S. Krishnaprasad and J. E. Marsden for helpful discussions. In paricular, conservaion properies (59) and (62) were known o P. S. Krishnaprasad. Suppor was provided by A.F.O.S.R. gran nos. 2-DJA-544 and 2-DJA-771 wih Sanford Universiy. This suppor is graefully acknowledged. Sep-by-sep implemenaion of ALGOL1 APPENDIX I Sep 0. Daa available from converged soluions of previous ime sep,: R, := spaial angular momenum, W, := conveced angular velociy, A, := conveced angular acceleraion, (x,,, x), := quaernion parameers associaed wih roaion vecor 1,.

31 UNCONDITIONALLY STABLE ALGORITHMS 49 Remark. The quaniies x and (xo, x) are differen possible paramerizaions of he roaion marix A. We have he relaions x = &-'C(Xo, x)l A = W (xo, x)i = exp Cil (xo, x) = &[XI - where he following operaors are defined in Appendix 11: &[.] := operaor which compues he uni quaernions from he roaion vecor &- C. ] := operaor which ransforms he uni quaernions ino he roaion vecor B[ '1 := operaor which ransforms he uni quaernions ino he roaion marix In he implemenaion ha follows, i is crucial o employ he quaernion parameers as he basic paramerizaions of he roaion marix. Sep 1. Iniializaion for ime sep in [,,,,,I. 0 Conveced angular velociy (leing A!,') = 0): 0 Conveced incremenal roaion: WLo), = W, + (1 - y)ha, (see (46)) 0"' = hwn + (1 - p)h2a, (see (45)) 0 Quaernion parameers associaed wih 0"): (40, q)'o' = &[O'O'] 0 Quaernion parameers associaed wih roaion marix A:? (xo,x)lp!1 = (~o,x)no(qo,q)'~' where he quaernion produc operaor 0 is defined by (39). 0 Roaion marix: 0 Spaial angular momenum: 1 = BC(X0, x)lp! 1 1 (0) - A(0) JW(0) Rn+1- n+l n+l Remark. Daa available a beginning of ieraion i + 1 a ime sep,+,: { R:! 1, (xo, x):! A:i,, W:!,, O"), (qo, q)(i)} Sep 2. Compue residual R!,?+, o non-linear (angular momenum) equaion and check for convergence. 0 Compuaion of R:! : (Al) ('42) R:\, = hm,,, + R, - n:!+, (A91 where m,,, is he loading evaluaed a,+,. Alernaively, if he exernal loading is specified as a funcion of ime, one can use a linear inerpolaion and se: m,,, = (1 - a)m, + am,,,

32 50 J. C. SIMO AND K. K. WONG 0 Check for convergence: IF II R:\ II < Tolerance, THEN: 1. Compue angular acceleraion a n+l: 2. Begin new ime sep; le n +- n + 1; go o Sep 0. ELSE: Coninue o Sep 4. Sep 3. Compue angen marix K:L 0 Compuaion of KC\ 1: and solve for ieraive roaion incremen. [See Appendix I1 for he compuaion of he linearizaion marix T(@(i)).] 0 Solve for he conveced incremenal roaion vecor A@(') from $i o :::): 0 Compue quaernion parameers associaed wih A@(i): A@(" = [K$)+l]-lR:)+l (A 13) (Aq,, Aq)(i) = &[A@("] Sep 4. Updae configuraion and velociy for given incremen A@"): 0 Quaernion parameers associaed wih roaion marix A:::): (x0, x)$:' = (x0, x)$i o(aqo, Aq)@) (A 15) 0 Quaernion parameers associaed wih incremenal roaion vecor 'I: 0 Conveced incremenal roaion vecor: 0 Conveced angular velociy: (40, q)(i+') = (40, q)(""(aqo, 1) = ('416) &- lc(40, qyi+ "I (A 17) 0 Roaion marix: A::) = W[(X,, x)$:;)] 0 Spaial angular momenum: n$:;)=~::;)~~p;;) Sep 5. Begin new ieraion; se i i + 1; go o Sep 2.

33 Summary of operaions wih uni quaernions UNCONDITIONALLY STABLE ALGORITHMS 51 APPENDIX I1 In his appendix, we summarize he basic operaions involving quaernions, roaion vecors and orhogonal (roaion) marices needed for he acual implemenaion of he sep-by-sep algorihm oulined in Appendix I. II.1. Marix T(0) associaed wih he linearizaion of 0. The linearizaion of he relaive roaion vecor is given by he expression DO.AO = T(O)AO, where T(0) is compued as follows : IF IJ0[I < 1, THEN: ELSE: T=l The erm x/an(x) is compued using he series expansion an (x) X x2 + -x4 + -x6 + -x* +... wih x = /2 o avoid numerical difficulies when x Definiion ofhe operaor &[.]. The uni quaernion parameers (xo, x) associaed wih he roaion vecor x, denoed by (xo, x) = &(x), are given by For small )I x , one employs he Taylor series expansion sin (x) x2 x4 x6 -= X X which circumvens numerical ill-condiioning as 1) x 1) Recall ha uni quaernions have he propery 1 = xi + x'x, which is ensured by normalizing he resuls compued using (A24) hrough division by d m Definiion of he operaor I-' parameers (xo, x), denoed by x = &-'[(x,,, [*I. Le x be he roaion vecor wih associaed quaernion x)]. Making use of he resuls in 11.2 we have X x = II x II wih II x II = 2 sin-- ( I1x II 1 (A261 II x II For 11 x 1, se x = 0. As noed in Simo and VU-QUOC,~~ i is numerically more accurae o compue 11 x 11 as given above han using he alernae formula 11 x I( = 2 cos-' (xo), since he sine funcion is more sensiive for small I( x 11 han he cosine funcion.

34 52 J. C. SlMO AND K. K. WONG II. 4. Dejiniion of operaor 9C.1. The orhogonal marix A associaed wih a se of uni quaernion parameers (xo, x), denoed by A = 9[(xo, x)], is given by x$ + x: - f XlX2 - x3xo x1x3 + x2xo [ 1 [A]=2 x2x1 +x,xo x$+x$-* x2x3-xlx0 wih {x}= x2 (A27) ~3x1 + ~2x0 ~3x2 - ~ 1x0 X$ + X: - f I Observe ha he inverse operor W- [.I, which defines he quaernion exracion from an orhogonal marix, is no needed in our implemenaion oulined in Appendix I. Consequenly, he need for Spurrier s algorihm for quaernion exracion is enirely by-passed. REFERENCES 1. V. I. Arnold, Mahemaical Mehods of Classical Mechanics, Springer-Verlag, Berlin, A. Bayliss and E. Isaacson, How o make your algorihm conservaive, No. Amer. Mah. Soc., A.594-AS95 (1975). 3. T. Belyschko and D. F. Schoeberle, On he uncondiional sabiliy of an implici algorihm for nonlinear srucural dynamics, J. Appl. Mech. ASME, 42, No. 4, (1975). 4. P. J. Chanell, Sympleic inegraion algorihms, Repor AT-6: ATN Los Alamos Naional Laboraory, P. J. Chanell and J. C. Scovel, Symplecic inegraion of Hamilonian sysems, Preprin, M. L. Curis, Marix Groups, 2nd edn, Springer-Verlag, Berlin, R. De Vogelaere, Mehods of inegraion which preserve he conac ransformaion propery of Hamilonian equaions, Repor 4, Deparmen of Mahemaics, Universiy of Nore Dame, Feng Kang, Difference schemes for Hamilonian formalism and symplecic geomery, J. Comp. Mah., 4, (1986). 9. H. Goldsein, Classical Mechanics, 2nd edn, Addison-Wesley, Reading, MA, D. Greenspan, Conservaive numerical mehods for x =f(x), J. Comp. Phys. 56, 2841 (1984). 11. P. C. Hughes, Spacecraf Aiude Dynamics, Wiley, New York, T. J. R. Hughes, Sabiliy, convergence, and growh and decay of energy of he average acceleraion mehod in nonlinear srucural dynamics, Comp. Sruc., 6, (1976). 13. T. J. R. Hughes, The Finie Elemen Mehod, Prenice-Hall, Englewood Cliffs, N.J., T. J. R. Hughes, W. K. Liu and P. Caughy, Transien finie elemen form ulaions ha preserve energy, J. Appl. Mech. ASME, 45, 36G370 (1978). 15. E. Isaacson, Inegraion schemes for long erm calculaions, in R. Vichnevesky (ed.), Advances in Compuer Mehods for Parial Differenial Equaions 11, 1.M.A.C.S (AICA) R. A. LaBudde and D. Greenspan, Energy and momenum conserving mehods of arbirary order for he numerical inegraion of equaions of moion. Par 1, Numer. Mah., 25, (1976). 17. R. A. LaBudde and D. Greenspan, Energy and momenum conserving mehods of arbirary order for he numerical inegraion of equaions of moion. Par 11, Numer. Mah., 26, 1-16 (1976). 18. J. E. Marsden, Lie-Poisson Hamilon Jacobi heory and Lie-Poisson inegraors, Phys. Le. A, o appear. 19. I. M. Navon, Implemenaion of a poseriori mehods for enforcing conservaion of poenial ensrophy and mass in discree shallow-waer equaions models, Monhly Weaher Rev., 109, (1981). 20. R. D. Richmayer and K. W. Moron, Difference Mehodsfor Iniial Value Problems, 2nd edn, Inerscience, New York, Y. K. Sasaki, Variaional design of finie-difference schemes for iniial value problems wih an inegral invarian, J. Comp. Phys., 21, 27G278 (1976). 22. J. C. Simo, On a one dimensional finie srain beam heory: The 3-dimensional dynamic problem, Comp. Mehods Appl. Mech. Eng., 49, (1985). 23. J. C. Simo and D. D. Fox, On a sress resulan geomerically exac shell model. Par I: Formulaion and opimal paramerizaion, Comp. Mehods Appl. Mech. Eng., 72, (1989). 24. J. C. Simo and L. Vu-Quoc, On he dynamics in space of rods undergoing large moions-a geomerically exac spproach, Comp. Mehods Appl. Mech. Eng., 66, (1988). 25. J. C. Simo, J. E. Marsden and P. S. Krishnaprasad, The Hamilonian srucure of nonlinear elasiciy: The maerial and convecive represenaion of solids, rods and plaes, Arch. Raional Mech. Anal., 104, (1989). 26. J. C. Simo, T. Posbergh and J. E. Marsden, Sabiliy analysis of geomerically exac rods and rigid bodies: The energy-momenum mehod, Phys. Repors, o appear. 27. W. T. Thomson, Inroducion o Space Dynamics, Dover Publicaions, New York, F. W. Warner, Foundaions of Differeniable Manifolds and Lie Groups, Springer-Verlag, Berlin, E. T. Wiaker, A Treaise on Analyical Dynamics, Dover Publicaions, New York, Zhong Ge and J. E. Marsden, Lie Poisson Hamilon-Jacobi heory and Poisson inegraors, Le. Phys., o appear C. Zienkiewicz, W. L. Wood and R. L. Taylor, An alernaive single-sep algorihm for dynamic problems, Earhquake eng. sruc. dyn., 8, 3140 (1980). 32. R. A. Spurrier, Commen on Singulariy-free Exracion of a Quaernion from a DirecorCosine Marix, J. Spacecraf, IS, 255 (1978).