Version April 30, 2004.Submied o CTU Repors. EXPLICIT TIME INTEGRATORS FOR NONLINEAR DYNAMICS DERIVED FROM THE MIDPOINT RULE Per Krysl Universiy of California, San Diego La Jolla, California 92093-0085, USA pkrysl@ucsd.edu Keywords: Time inegraion, rigid body moion, midpoin rule, symplecic Euler, Verle, Newmark, midpoin Lie algorihm We address he design of ime inegraors for mechanical sysems ha are explici in he forcing evaluaions. Our saring poin is he midpoin rule, eiher in he classical form for he vecor space seing, or he Lie form for he roaion group. By inroducing discree, concenraed impulses we can approximae he forcing impressed upon he sysem over he ime sep, and hus arrive a firs-order inegraors. These can be hen composed o yield a second order inegraor wih very desirable properies: sympleciciy and momenum conservaion. 1. Inroducion I is well-known ha he Verle algorihm (explici Newmark for a cerain choice of is parameers) may be wrien as a composiion of wo firs order algorihms, he symplecic Euler and is adjoin [HLG (2003)]. Wha is perhaps less known is ha here is an inerpreaion of his composiion in erms of an approximaion of he midpoin rule, which is of course implici in he evaluaion of he forcing impulse. Our goal in his brief noe is o poin ou ha his mechanically inspired derivaion yields he well-known second order explici Verle algorihm in he vecor space seing, bu also an exremely accurae inegraor for rigid body roaions when he midpoin rule is inerpreed in he Lie sense.
2. Vecor space midpoin rule approximaion Le us wrie he iniial value problem for a mechanical sysem wih configuraion u fairly general form as p=f &, p(0) = p 0 u=m & p, u(0) = u0 where p& is he rae of linear momenum, andf = f ( u, ) is he applied force. For simpliciy we shall assume p= Mv, where M is a ime-independen mass marix, and v is he velociy. The midpoin approximaion of he second equaion is u( +) u( ) =M p( +/ 2) where p( + /2) = p( u( + /2), +/ 2) makes his formula implici. To approximae he midpoin momenum, we may use he equaion of moion in inegral form, + h p( + h) = p( ) + f ( τ ) dτ, and numerically evaluae he impulse inegral. This we propose o rea by recourse o he concep of concenraed, discree impulses delivered a pre-seleced ime insans. In paricular, he impulse may be delivered a he end of he ime sep, in which case +/2 f ( τ ) dτ 0. On he oher hand, he impulse may be imposed a he beginning of he ime sep, and +/2 f ( τ) dτ f( ). Therefore, we obain wo algorihms. The firs one, Euler mehod, and he second, *, as is adjoin. n in a, may be recognized as he symplecic p p+ p +f (symplecic Euler) u u+ u +M p+ * p p+ p +f+ (symplecic Euler adjoin) u u+ u +M p These algorihms are symplecic [HLG (2003)], and momenum conserving. Their accuracy is only linear in he ime sep, bu heir composiion preserves boh sympleciciy and momenum conservaion and yields a second order accurae algorihm. Tha algorihm may be recognized as he well-known Verle (explici Newmark wih γ = 1/ 2 ). Ξ = o * /2 /2 (Verle)
3. Midpoin rule approximaion on he roaion group Now we shall discuss dynamics of rigid bodies roaing abou a fixed poin. (More deail is available in Reference [K(2004)].) The iniial value problem may be wrien in he conveced descripion (body frame) as Π & = skew[ I ΠΠ ] + T, Π(0) = Π0 R=R & skew[ I Π], R(0) = R0 where Π & is he rae of body frame angular momenum, R is he roaion ensor, T is he applied orque in he body frame, skew[ ] is defined by skew[ h] h = 0, and I is he ime-independen ensor of ineria in he body frame. The second equaion is no in a form suiable for midpoin discreizaion, because he roaion ensor consiues poins of he Lie group SO(3), which is no a vecor space and linear combinaions are no legal operaions on he roaion ensors. To ransform he iniial value problem o a form suiable for our purposes, we shall inroduce he roaion vecor represenaion of he roaion ensor. The equaion of moion is wrien in he spaial frame as π & =RT, where π& is he rae of he spaial angular momenum, and consequenly he equaion of moion may be wrien in inegral form as T Π ( ) = exp[ skew[ Ψ]] Π( 0 ) + R ( ) ( τ ) ( τ) dτ R T 0 where exp[ skew[ Ψ]] = R T ( ) R( 0) is he incremenal roaion hrough vecor Ψ. Upon ime differeniaion and idenificaion wih he original differenial equaion of moion, we obain Ψ& ( ) 1 d exp skew[ ] = Ψ I Π where d exp ( ) is he differenial of he exponenial map. The iniial value problem may skew[ Ψ] be herefore rewrien as Π & = skew[ I ΠΠ ] + T, Π(0) = Π ( skew[ Ψ] ) Ψ& = d exp I Π, Ψ(0) = 0 The midpoin approximaion applied o he second equaion yields ( Ψ = 0 ) Ψ+ = dexp 1 + skew[ ] I Π /2 Ψ 2 which may be simplified by noing d exp 1 Ψ + = Ψ + o give skew[ Ψ+] 2 0 1 + = +/2 IΨ Π. This equaion needs o be solved for he roaion vecor. Therefore, as for he vecor space seing we ge wo differen algorihms, depending on he chosen approximaion of Π + /2. For he impulse applied a he beginning of he ime sep we obain he SO(3) counerpar of he symplecic Euler inegraor:
( ) Π Π+ exp[ skew( Ψ+ )] Π + T R R+ R exp[skew( Ψ+ )] where Ψ solves + 1 1 IΨ+ = exp[ skew( Ψ+)] ( Π +T) 2 On he oher hand, he oal orque impulse applied a he end of he ime sep yields he adjoin mehod * Π Π+ exp[ skew( Ψ+ )]( Π +T+) R R+ R exp[skew( Ψ+ )] where Ψ solves + 1 1 IΨ+ = exp[ skew( Ψ+)] Π 2 Boh algorihms are firs-order, symplecic, and momenum conserving. As before, he composiion of hese wo algorihms in one ime sep provides us wih a second order accurae algorihm, which is an analogy of he Verle (explici Newmark wih γ = 1/ 2 ) algorihm for he vecor space dynamics Ξ = o * /2 /2 The above algorihms have been called he explici midpoin Lie varian 2 and 1 respecively, and he alernaing midpoin Lie algorihm in Reference [K(2004)]. I bears emphasis ha hese algorihms are no simply he symplecic Euler and is adjoin. They all reduce o he full midpoin Lie algorihm for orque-free moion, and i is only he approximaion of he midpoin momenum evaluaion ha disinguishes hem from he implici midpoin Lie rule. 4. Example The accuracy of he explici midpoin Lie algorihms is raher remarkable as may be seen in Figure 1. We show convergence graphs for he fas spinning heavy op (he kineic energy is dominan, and a numerical mehod has o effecively deal wih precession and nuaion which are moions of disinc frequencies). Even he firs-order mehods perform very well for larger ime seps, and he presen alernaing midpoin Lie algorihm is he sronges performer ou of a selecion of he bes currenly available implici and explici algorihms, including he implici midpoin Lie mehod.
Figure 1. Fas Lagrangian op; on he lef hand side convergence in he norm of he error in body-frame angular momenum; on he righ hand side convergence in he norm of he error in he aiude marix: AKW: implici midpoin rule of Ausin, Krishnaprasad, Wang [AKW (1993)]: SW: Simo and Wong [SW (1991)]; NMB: Krysl, Endres Newmark algorihm [KE (2004)]; LIEMID[I]: implici midpoin Lie: LIEMID[E1]: adjoin of he symplecic Euler (explici midpoin Lie varian 1); LIEMID[E2]: symplecic Euler (explici midpoin Lie varian 2); LIEMID[EA]: alernaing explici midpoin Lie mehod. 5. Conclusions We have presened an approach o he consrucion of rigid body inegraors, in paricular for general 3-D roaions, ha are explici in he evaluaion of he forcing, momenum-conserving, and symplecic. The saring poin is he midpoin (implici) rule, which is hen reaed wih numerical inegraion of he forcing wih concenraed impulses. The resuling algorihms conserve momenum, are symplecic, firs-order, and hey are adjoin. Consequenly, heir composiion leads o a second order algorihm, which may be readily inerpreed as a Verle (explici Newmark) inegraor, boh in he vecor space seing, and in he seing of he special orhogonal group of roaions. (We would like o sugges as an appropriae name explici midpoin Lie algorihm for he laer.) For roaional dynamics, his inegraor is a new addiion o he lineup of curren high performance algorihms, and in fac numerical evidence suggess ha i is he bes second-order inegraor o dae. Applicaions abound: molecular dynamics, micro magneics, rigid body dynamics, finie elemen dynamics of deformable solids. Acknowledgemens: Suppor for his research by a Hughes-Chrisensen award is graefully acknowledged. 6. References [AKW (1993)] M. Ausin, P. S. Krishnaprasad, and L. S. Wang. Almos Lie-Poisson inegraors for he rigid body. Journal of Compuaional Physics, 107:105-17, 1993.
[HLG (2003)] E. Hairer, C. Lubich, and G. Wanner. Geomeric Numerical Inegraion. Srucure-Preserving Algorihms for Ordinary Differenial Equaions., Springer Series in Compu. Mahemaics, Springer-Verlag, volume 31, 2002. [K (2004)] P. Krysl, Explici Momenum-conserving Inegraor for Dynamics of Rigid Bodies Approximaing he Midpoin Lie Algorihm, Inernaional Journal for Numerical Mehods in Engineering, 2004. Submied. [KE (2004)] P. Krysl and L. Endres. Explici Newmark/Verle algorihm for ime inegraion of he roaional dynamics of rigid bodies. Inernaional Journal for Numerical Mehods in Engineering, 2004. Submied. [SW (1991)] J. C. Simo and K. K. Wong. Uncondiionally sable algorihms for rigid body dynamics ha exacly preserve energy and momenum, Inernaional Journal for Numerical Mehods in Engineering, 31, 19-52, 1991.