A VARIATIONAL APPROACH TO MULTIRATE INTEGRATION FOR CONSTRAINED SYSTEMS

Transcription

1 MULTIBODY DYNAMICS, ECCOMAS Themaic Conference J.C. Samin, P. Fisee eds. Brussels, Belgium, 4-7 July A VARIATIONAL APPROACH TO MULTIRATE INTEGRATION FOR CONSTRAINED SYSTEMS Sigrid Leyendecer, Sina Ober-Blöbaum Compuaional Dynamics and Conrol Universiy of Kaiserslauern, PO Box 349, D Kaiserslauern, Germany leyendecer@rhr.uni-l.de, web page: hp:// Compuaional Dynamics and Opimal Conrol Universiy of Paderborn, Warburger Sr., D-3398 Paderborn, Germany s: sinaob@mah.uni-paderborn.de Keywords: mulirae inegraion, srucure preservaion, variaional inegraors, consrained mulibody dynamics Absrac. The simulaion of sysems wih dynamics on srongly varying ime scales is quie challenging and demanding wih regard o possible numerical mehods. A raher naive approach is o use he smalles necessary ime sep o guaranee a sable inegraion of he fas frequencies. However, his ypically leads o unaccepable compuaional loads. Alernaively, mulirae mehods inegrae he slow par of he sysem wih a relaively large sep size while he fas par is inegraed wih a small ime sep. In his wor, a mulirae inegraor for consrained dynamical sysems is derived in closed form via a discree variaional principle on a ime grid consising of macro and micro ime nodes. Being based on a discree version of Hamilon s principle he resuling variaional mulirae inegraor is a symplecic and momenum preserving inegraion scheme and also exhibis good energy behaviour. Depending on he discree approximaions for he Lagrangian funcion, one obains differen inegraors, e.g. purely implici or purely explici schemes, or mehods ha rea he fas and slow pars in differen ways. The performance of he mulirae inegraor is demonsraed by means of he several examples.

2 Sigrid Leyendecer, Sina Ober-Blöbaum INTRODUCTION Mechanical sysems wih dynamics on varying ime scales, in paricular hose including highly oscillaory moion, impose challenging quesions for numerical inegraion schemes. Tiny sep sizes are required o guaranee a sable inegraion of he fas frequencies. However, for he simulaion of he slow dynamics, inegraion wih a larger ime sep is accurae enough. Here, small ime seps increase inegraion imes unnecessarily, especially for cosly funcion evaluaions. Typical examples of sysems exhibiing dynamics on differen ime scales can be found in asrophysics, where depending on he disances beween planes, he resuling graviaional forces can be exremely srong or wea leading o differen ime scales for a fligh rajecory hrough space, or in molecular dynamics, where locally exremely high frequencies superpose global folding processes. In mulibody dynamics, such sysems occur e.g. in combusion engines wih chain drives or in vehicle dynamics, or generally in sysems being composed of rigid and elasic pars wih varying and in paricular wih high siffness. For sysems comprising fas and slow dynamics, mulirae mehods see e.g. Ref. [,, 3] inegrae he slow par of he sysem wih a relaively large sep size while he fas par is inegraed wih a small ime sep. The goal is o save compuaional wor while preserving he accuracy of he simulaion. Thereby, main challenges are he idenificaion of fas and slow pars e.g. eiher by separaing he sysem s energy or by defining disjunc ses of degrees of freedom, he synchronisaion of heir differen dynamics and in paricular he reamen of mixed pars as hey ofen appear when fas and slow dynamics are coupled eiher via poenials or by consrains. Furhermore, resonance phenomena impose resricions on he combinaion of large and small ime seps. Hence, anoher challenge is he sabiliy analysis of mulirae sepping schemes as done for linear problems e.g. in Ref. [4, 5]. Similar o our approach, he laer wor is based on a variaional derivaion, however he resuling mulirae schemes are differen from hose presened here. In his conribuion, a mulirae inegraor is derived in closed form via a discree variaional principle on a ime grid consising of macro and micro ime nodes. Variaional inegraors Ref. [6] are based on a discree version of Hamilon s principle leading o symplecic and momenum preserving inegraion schemes ha also exhibi good energy behaviour. The resuling mulirae variaional inegraor has he same preservaion properies, hus i falls ino he class of srucure preserving inegraors which generally exhibi very good longerm sabiliy. The derivaion of he mulirae inegraor via a discree variaional principle bears grea flexibiliy, since he choice of quadraure in he approximaion of he appearing inegrals heavily influences he degree of coupling in he resuling sysem of discree equaions of moion, he number of necessary funcion evaluaions and he possibiliy o rea fas and slow pars in an implici or an explici way, respecively or vice versa. The performance of he variaional mulirae inegraor is demonsraed by means of a sandard benchmar problem for mulirae inegraion, he Fermi-Pasa-Ulam problem, and for an example from consrained mulibody dynamics. VARIATIONAL MULTIRATE INTEGRATOR. Slow and fas poenial and conrains Consider a mechanical sysem on a manifold Q R n wih he Lagrangian L : T Q R given by Lq, q = T q Uq being he difference beween he ineic energy T and a poenial U. Le he fac ha he Lagrangian conains slow and fas dynamics be characerised by he possibiliy o addiively spli he poenial energy Uq = V q + W q ino a slow

3 Sigrid Leyendecer, Sina Ober-Blöbaum poenial V and a fas poenial W. Furhermore, le he configuraion be consrained o he n m-dimensional consrain manifold C = {q Q gq = } defined by he holonomic consrain funcion g : Q R m. Then, he consrained Euler-Lagrange equaions of moion on a ime inerval [, N ] R d T d q V q W q T g λ = q gq = can be derived via Hamilon s principle requiring saionariy of he acion inegral s variaion, i.e. δsq, λ = δ N [ Lq, q gqt λ ] d =. Here, λ R m denoes he Lagrange muliplier. See Secion 3 for examples of such addiively spli poenials.. Slow and fas variables We furher assume ha he n-dimensional configuraion variable q can be divided ino n s slow variables q s Q s and n f fas variables q f Q f such ha Q s Q f = Q and q = q s, q f wih n s + n f = n. Le he fas poenial depend of he fas degrees of freedom only, i.e. W = W q f while he slow poenial V = V q depends on he complee configuraion variable as does he consrain funcion g = gq. Wih hese assumpions, he Euler-Lagrange equaions Eq. ae he form d T d q V T g s q λ = s q s d T d q V f q W T g f q λ = f q f gq = Remar. If in addiion, he slow poenial depends on he slow variables only and on op of ha he ineic energy does no conain any enries coupling q s and q f, hen he sysem is compleely decoupled and simulaion can be performed independenly in parallel, wihou any exchange of informaion. This case is rivial and we focus on he scenario described above. Noe ha he inclusion of addiional poenials or consrain funcions depending on he fas or he slow variable only is sraighforward..3 Discree variaional principle In he framewor of variaional inegraors, see Ref. [6], a discree approximaion of he acion funcional is considered. To his end, a discree Lagrangian L d : Q Q R is defined as an approximaion of he acion funcional on a shor ime span T. Raher han choosing Figure : Macro and micro ime grid. 3

4 Sigrid Leyendecer, Sina Ober-Blöbaum one ime grid for he approximaion as for sandard variaional inegraors, for he mulirae inegraor, wo differen ime grids are inroduced, see Fig.. Wih he ime seps T and where T, a macro ime grid { = T =,..., N} and a micro ime grid { m = T + m =,..., N, m =,..., p} are defined. Noe ha excep for he boundary nodes, N, wo micro ime nodes coincide wih a macro ime node, i.e. p = = for =,..., N, see Fig.. The micro and macro ime grids provide he domains for he discree macro rajecory of he slow variables q s d = {q s } N = wih q s q s and he discree micro rajecory of he fas variables q f d = {qf }N = = {{qf,m } p m=} N = wih q f,m q f m Since he consrains depend on he complee configuraion variables, he Lagrange mulipliers can no be separaed in a fas and a slow par and mus be compued on he fine ime grid. Thus, he discree rajecory of Lagrange mulipliers aes he form Noe ha p = λ d = {λ } N = = {{λm } p m=} N = wih λ m λ m and herefore also qf,p = qf, and λ p = λ hold. As an approximaion o S, he discree acion is defined as S d qd, s q f d, λ d = N = [ ] L d q, s q+, s q f h dq, s q+, s q f, λ The discree Lagrangian L d = T d V d W d approximaes + Lq, q d and reads L d q s, q s +, q f = T dq s, q s +, q f V dq s, q s +, q f W dq f 4 while h d is approximaing + gq T λ d. Omiing he argumens of L d and h d, saionariy of he discree acion N { δs d = Dq s L d + h d δq s + D q s + L d + h d δq++ s = p m= [ Dq f,m L d + h d δq f,m + D λ m h d δλ m ]} = wih independen variaions δq s, δqf,m, δλ m, =,..., N, m =,..., p yields he discree Euler-Lagrange equaions. Le = and assume ha an iniial configuraion q, s q f, being consisen wih he consrains, i.e. gq, s q f, =, and an iniial conjugae momenum p s, p f, is given. Then, q, s q f,,..., q f,p and λ,..., λ p are deermined by solving he following se of equaions for m =,..., p. IC s D q s L d q, s q, s q f + h d q, s q, s q f, λ IC f D q f, L d q, s q, s q f + h d q, s q, s q f, λ = p s = p f, D λ h d q, s q, s q f, λ = = DEL f,m D q f,m L d q, s q, s q f + h d q, s q, s q f, λ h d q, s q, s q f, λ + δ m,p D λ h d q, s q, s q f, λ = D λ m

5 Sigrid Leyendecer, Sina Ober-Blöbaum These equaions can be considered as iniial condiions, since hey deermine he unnowns in he firs macro ime inerval from given iniial daa. Noe ha variaion wih respec o λ is unnecessary, since he iniial configuraion does fulfil he consrains a priori. Analog o he variaional inegraors for consrained sysems on a single ime grid, see e.g. Ref. [6, 8], here variaion wih respec o λ m yields he condiion gq, s q f,m =. Therefore, he las condiion gq, s q f,p = is composed by conribuions from variaion wih respec o he mulipliers λ p and λ which are equal which is ensured using he Dirac dela in he las equaion. To proceed furher in ime for =,..., N assuming ha q s, qs, qf,,..., qf,p are given, solving he following discree Euler-Lagrange equaions for m =,..., p deermines q+ s, qf,,..., qf,p and λ,..., λp. DEL s D q s L d q s, qs +, qf + L dq s, qs, qf + h dq s, qs +, qf, λ + h d q s, qs, qf, λ = DEL f, D q f, L d q s, qs +, qf + L dq s, qs, qf + h dq s, qs +, qf, λ + h d q s, qs, qf, λ = D λ h d q s, qs +, qf, λ = DEL f,m D q f,m L d q s, qs +, qf + h dq s, qs +, qf, λ = D λ m+h d q s, qs +, qf, λ + δ m,p δ,n D λ + h d q+ s, qs +, qf +, λ + = 6 Again, a he macro nodes he consrain equaions include an addiional erm which is added using he Dirac dela. Noe however, ha his erm does no exis a he very end node N. Remar. Due o he variaional derivaion of he mulirae inegraor, we can sae ha i has he following wo properies which classify i as being srucure preserving. The discree symplecic form is preserved along he discree soluion rajecory. If he discree Lagrangian is invarian under a group acion on [, + ], hen he corresponding momenum map is preserved. This follows from he discree Noeher Theorem, see Ref. [6]..4 Discree acion influence of quadraure The quadraure rules in use for he discree Lagrangian Eq. 4 and he discree consrain erm in Eq. 3 deermine he degree of coupling beween he discree equaions Eq. 5 and Eq.6, respecively. This can range from a fully implici scheme over varians being explici in he macro and implici in he micro quaniies o fully explici schemes. We consider Lagrangians of he form Lq, q = T q V q W q f.4. Kineic energy Assume ha he ineic energy can be decomposed in a conribuion from he fas and he slow variables, i.e. T q = qt M q = qs T M s q s + q f T M f q f, where M s and M f are he mass marices for he slow and fas variables, respecively, yielding he oal mass marix as M = diagm s, M f. In he following he velociies q s and q f are 5

6 Sigrid Leyendecer, Sina Ober-Blöbaum approximaed using bacward difference operaors on he macro and micro grid. Then he discree ineic energy is defined on he ime inerval [, + ] as T d = T q s + q s T M s q s + q s p + T T.4. Consrains m= q f,m+ q f,m T M f q f,m+ q f,m The discree funcion h d is approximaing he inegral + gq T λ d. Similar o he approximaion on a sandard ime grid in Ref. [8], a rapezoidal rule is used here. h d q s, q s +, q f, λ = p m= [ gt d q, s q+, s q f,m λ m + ] gt d q, s q+, s q f,m+ λ m+ For he discree consrain funcion g d, an inuiive example is he following. Example. The slow variables q s can be linearly inerpolaed beween q s and qs + on he micro ime grid as q s,m = p p mq s + mq s + Then, he discree consrain funcion reads.4.3 Poenial energy g d q s, q s +, q f,m for m =,..., p 7 = gq s,m, q f,m When sandard variaional inegraors are used for problems wih very siff poenials, heir discree counerpars are ofen based on midpoin evaluaions of he coninuous poenials such ha he corresponding inegraion scheme is implici. On he oher hand, sofer poenials can be approximaed by evaluaions of he coninuous poenial on he lef or righ node yielding explici schemes a leas as long as here are no consrains presen, which are of course much cheaper regarding he compuaional coss. For he mulirae inegraor, a large variey of combinaions is possible. Example Implici fas and explici slow forces. Le s firs consider a special case where he dynamics is no subjec o any consrains. Then, choosing an affine combinaion as approximaion in he slow poenial ha involves only macro nodes V d q, s q+, s q f = T αv q, s q f, + αv qs +, q f,p 8. wih α and a micro node based midpoin rule in he fas poenial p W d q f = q f,m + q f,m+ W m= 9 leads o discree conservaive forces in he discree Euler-Lagrange equaions which are explici for he slow poenial and implici for he fas one. Thus, only few evaluaions of he gradien 6

7 Sigrid Leyendecer, Sina Ober-Blöbaum of V are necessary which is advanageous when he slow poenial s evaluaion is very cosly compared o he fas one. The resuling scheme can be inerpreed as a variaional spliing mehod which is symmeric and symplecic, since i is a symmeric composiion of symmeric and symplecic mehods. When his mehod is formulaed wih α = on only one ime grid wih a consan ime sep i.e. = T and p = and wihou spliing he configuraion variable ino fas and slow variables, one obains he IMEX mehod in Ref. [7] which is an example of so called impulse mehods, see Ref. [] and references herein. Example 3 Fully implici scheme. In his example, he slow variables are inerpolaed according o Eq. 7 and hen midpoins are insered ino he slow poenial p V d q, s q+, s q f = q s,m V m= + q s,m+, qf,m + q f,m+ and ino he fas poenial as in Eq. 9. As a resul, he discree Euler-Lagrange equaions are fully coupled and have o be solved simulaneously using an ieraion mehod. Anoher quadraure yielding a fully implici scheme is given by p V d q, s q+, s q f = m= αv q s,m, q f,m + αv q s,m+, q f,m+ Example 4 Fully explici scheme. In he absence of consrains, using he affine combinaion of he slow poenial evaluaed a he macro nodes in Eq. 8 and he affine combinaion of micro node evaluaions of he fas poenial p W d q f = m= αw q f,m + αw q f,m+ leads o discree Euler-Lagrange equaions Eq. 6 ha can subsequenly be solved wihou ieraion, i.e. firs q f, is obained from DEL f,, hen DELf,m yields q f,m+ for m =,..., p. A any ime, q+ s can be compued from DELs. For α =, his choice of quadraure leads o he scheme in Ref. [5] for he special case ha a synchronised ime grid is used here. More general mulirae schemes are obained for differen choices and combinaions of quadraure. Depending on he complexiy of he evaluaion of he poenial funcions and heir gradiens, he compuaional coss of he overall simulaion is heavily influenced by he choice of quadraure. 3 NUMERICAL EXAMPLES 3. Fermi-Pasa-Ulam problem The performance of he presened mulirae approach is firs demonsraed by means of he Fermi-Pasa-Ulam FPU problem see e.g. Ref. []. Consider l uni poin masses ha are chained ogeher by sof and siff springs as shown in Fig.. Wih an appropriae choice of he coordinaes i is possible o separae he slow and he fas variables of he mulirae sysem. The slow variables qi s, i =,..., l, correspond o he locaion of i-h siff spring s cener, while he lengh of i-h siff spring is a fas variable q f i, i =,..., l. The Lagrangian is composed by he ineic energy of slow and fas variables and he spring poenials L = l i= q s i + q f i 4 [ ] l q s q f 4 + qi+ s q f i+ qs i q f i 4 + ql s + qf l 4 i= 7 ω l i= q f i

8 Sigrid Leyendecer, Sina Ober-Blöbaum Figure : Fermi-Pasa-Ulam problem: l poin masses ha are chained ogeher by sof and siff springs. where he second erm is he sof spring poenial V q s, q f depending on he complee configuraion variable, while hird erm is he siff poenial W q f ha depends on he spring lenghs only and includes he siffness ω R which is supposed o be large. For his sysem, no consrains are presen. The Fermi-Pasa-Ulam problem is a mulirae sysem, i.e. i shows differen behaviour on differen ime scales confirm Ref. []. The vibraion of he siff linear springs aes place on he ime scale ω, while ω is he ime scale of he sof nonlinear springs moion. Furhermore, on he ime scale ω, energy exchanges among he siff springs. For he simulaions, we consider 6 poin masses i.e. l = 3 wih mass m = and he siffness of he siff springs is ω = 5. The sysem has an iniial displacemen q s = and an iniial exension q f = ω, iniial velociies are q s = and q f =. All remaining iniial values are zero. In his simulaion, he quadraure Eq. wih α = is used for he slow poenial and he midpoin rule Eq. 9 for he fas one. As a reference soluion, a sandard variaional inegraor p = wih he ime sep T =. is used. This ime sep is small enough o resolve he fas oscillaions of he siff springs exensions. In he lef hand side plo in Fig. 3, he configuraion and momenum of he firs slow and he firs fas variable i.e. he firs siff spring s cener and he lengh of he firs siff spring are shown. Using a bigger ime sep T =.3, he fas moion can no be capured anymore as can be seen on he righ hand side of Figure 3. Keeping a macro ime sep of T =.3, he mulirae variaional inegraor is used for a differen number of inermediae micro seps. In Fig. 4, micro red solid and macro blue dashed soluions for configuraion and momenum of he firs slow and he firs fas variable are shown for p = micro seps on he lef and for p = 3 micro seps on he righ. For an increasing number of micro seps, he approximaion of he fas variables becomes beer. For p = 3, he micro sep size =. is equal o he sep size of he sandard variaional inegraor in he reference soluion. As a resul, he discree soluion of he fas variable nicely coincides wih he reference soluion alhough he macro soluion alone red solid does no resolve he fas dynamics. In Fig. 5, he exchange of energy beween he siff springs blue, blac, cyan is shown. The oal oscillaory energy, i.e. he sum of he siff springs energy red remains close o a consan value his is called an adiabaic invarian of he Hamilonian sysem, see Ref. [] which is nicely visible in Fig. 5 a for he reference soluion. Using he macro ime sep T =.3 and p = Fig. 5 b, he oal energy oscillaes much more and can no be considered a consan value anymore. However, for p = Fig. 5 c and p = 3 Fig. 5 d micro seps he oscillaions become smaller, and for p = 3 he same qualiaive long erm energy behaviour as for he reference soluion is obained. 8

9 Sigrid Leyendecer, Sina Ober-Blöbaum. qrls: expl qrlf: midp dt=. p=.4 qrls: expl qrlf: midp dt=.3 p= q s. q s.5. q s..5 q s q f.5 q f.. q f q f a configuraion T =., p = b configuraion T =.3, p = qrls: expl qrlf: midp dt=. p= qrls: expl qrlf: midp dt=.3 p= p s.8 p s.5.6 p s p s p f p f.5.5 p f p f c momenum T =., p = d momenum T =.3, p = Figure 3: FPU problem. Simulaion resuls using a sandard variaional inegraor p = wih ime sep T =. lef and T =.3 righ. Configuraion a,b and momenum c,d of firs slow op and firs fas boom variable. 9

10 Sigrid Leyendecer, Sina Ober-Blöbaum. qrls: expl qrlf: midp dt=.3 p=. qrls: expl qrlf: midp dt=.3 p= 3 q s q s.5.5 q s. q s q f, macro q f, micro. q f, macro q f, micro.. q f q f a configuraion T =.3, p = b configuraion T =.3, p = 3 qrls: expl qrlf: midp dt=.3 p= qrls: expl qrlf: midp dt=.3 p= 3 p s p s.5.5 p s p s p f, macro p f, micro p f, macro p f, micro.5.5 p f p f c momenum T =.3, p = d momenum T =.3, p = 3 Figure 4: FPU problem. Simulaion resuls using a mulirae variaional inegraor wih macro ime sep T =.3 and p = lef and p = 3 righ micro seps. Configuraion a,b and momenum c,d of firs slow op and firs fas boom variable.

11 Sigrid Leyendecer, Sina Ober-Blöbaum.4 qrls: expl qrlf: midp dt=. p=.6 qrls: expl qrlf: midp dt=.3 p= I I. I I 3.4 I I 3 I=I +I +I 3 I=I +I +I 3..8 I p I p a T =., p = b T =.3, p =.4 qrls: expl qrlf: midp dt=.3 p=.4 qrls: expl qrlf: midp dt=.3 p= 3 I I I I. I 3. I 3 I=I +I +I 3 I=I +I +I I p I p c T =.3, p = d T =.3, p = 3 Figure 5: FPU problem. Energy of he hree siff springs blue, blac, cyan and he oal oscillaory energy red.

12 Sigrid Leyendecer, Sina Ober-Blöbaum Figure 6: Triple pendulum consising of one large slow and wo small fas masses. 3. Triple spherical pendulum For he riple spherical pendulum in Fig. 6, he slow variable q s = q R 3 is he placemen of he large mass m slow =, while q f = q, q 3 R 6 conains he placemens of he wo smaller masses m fas = m fas 3 =. The slow poenial energy reads V q = q T M ḡ wih he consan mass marix M R 9 9 and he graviy vecor ḡ R 9 acing in he negaive e 3 - direcion wih acceleraion 9.8. Massless rigid lins of lenghs l = and l = 3 connec he large mass o he origin and he firs small mass o he large one, respecively. They give rise o a purely slow consrain g s q s = q l and a consrain g sf q = q q l coupling he slow and he firs fas mass. Boh consrains are combined ino he vecor valued consrains funcion g = g s, g sf. The second small mass is conneced o he firs one by a linear spring wih he siffness ω = 5, hus he fas poenial aes he form W q f = ωq 3 q l3 where l 3 = 3 is he lengh of he unsreched spring. Iniially, he riple pendulum is aligned wih he e -axis and he spring is presreched by. The slow mass has an iniial velociy of q s =,, 3 and he fas masses iniial velociy is q f =, 3l, l, l +l 3, 5l +l 3, l +l 3. In he simulaion of he riple pendulum s dynamics, he midpoin evaluaion Eq. 9 is used in he fas poenial. Since he graviy poenial is a linear funcion, i yields a consan force vecor, which is independen of he choice of quadraure. The wo lef hand side plos in Fig. 7 show he evoluion of he configuraion and conjugae momenum of he second fas mass, being compued via a sandard variaional inegraor p = wih T =. as a reference soluion. The righ hand side plos show he resuls from he variaional mulirae scheme wih T =.8 and p = 5, while he corresponding resuls for p = and p = are depiced in Fig. 8. The lines connec he values a he marco nodes and he inermediae micro node values are indicaed by lile crosses. One can see clearly, ha he macro grid wih T =.8 is oo coarse o resolve he fas moion. For an increasing number of micro nodes, he fas oscillaions of he second small mass become more and more visible. Finally, a numerical indicaor for he variaional characer of he proposed mehod is given in Fig. 9. The riple pendulum s Lagrangian is invarian wih respec o roaion abou he graviaional axis, hus he corresponding angular momenum componen L 3 is conserved exacly along he rajecory. The algorihm does conserve L 3 o numerical accuracy, independen of he macro or micro ime sep size.

13 Sigrid Leyendecer, Sina Ober-Blöbaum a configuraion T =., p = b configuraion T =.8, p = 5 c momenum T =., p = d momenum T =.8, p = 5 Figure 7: Triple pendulum. Simulaion resuls using a sandard variaional inegraor p = wih ime sep T =. lef and a mulirae variaional inegraor wih macro ime sep T =.8 and p = 5 righ micro seps. Configuraion a,b and momenum c,d of second fas mass m fas 3. 3

14 Sigrid Leyendecer, Sina Ober-Blöbaum a configuraion T =.8, p = b configuraion T =.8, p = c momenum T =.8, p = d momenum T =.8, p = Figure 8: Triple pendulum. Simulaion resuls using a mulirae variaional inegraorwih macro ime sep T =.8 and p = lef and wih p = righ micro seps. Configuraion a,b and momenum c,d of second fas mass m fas 3. x 4 qrls: midp qrlf: midp dt=. p= x 4 qrls: midp qrlf: midp dt=.8 p= L L L L L 3 L 3 angular momenum 3 angular momenum a configuraion T =., p = b configuraion T =.8, p = Figure 9: Triple pendulum. Evoluion of angular momenum using a sandard variaional inegraor p = wih ime sep T =. lef and a mulirae variaional inegraor wih macro ime sep T =.8 and p = righ micro seps. 4

15 Sigrid Leyendecer, Sina Ober-Blöbaum 4 CONCLUSION A unified framewor for he derivaion of differen mulirae inegraors for consrained dynamical sysems is presened. All schemes are derived in closed form via a discree variaional principle on a ime grid consising of macro and micro ime nodes. Being based on a discree version of Hamilon s principle, he resuling variaional mulirae inegraors are symplecic and momenum preserving inegraion schemes and also exhibi good energy behaviour. The choice of quadraure in he slow and fas poenials of he sysem can be adaped o he simulaion goal lie e.g. a low number of funcion evaluaions of a cosly poenial or obaining a parly of fully explici scheme. In paricular, if he number of micro nodes is large enough, fas oscillaions can be resolved wihou solving for he slow variables on he micro grid. This unified variaional framewor allows he analysis of a large class of mulirae schemes, which has o be done in fuure wor wih paricular focus on sabiliy problems caused by resonance phenomena. REFERENCES [] C.W. Gear and R.R. Wells. Mulirae linear mulisep mehods. BIT, 4,484 5, 984. [] E. Hairer and G. Wanner and C. Lubich. Geomeric Numerical Inegraion: Srucure- Preserving Algorihms for Ordinary Differenial Equaions. Springer, 4. [3] M. Arnold. Muli-rae ime inegraion for large scale mulibody sysem models. Proceedings of he Proceedings of he IUTAM Symposium on Muliscale Problems in Mulibody Sysem Conacs, Sugar, Germany, February 3, 6. [4] E. Barh and T. Schlic. Exrapolaion versus impulse in muliple-imesepping schemes. II. Linear analysis and applicaions o Newonian and Langevin dynamics. Journal of Chemical Physics, 9, , 998. [5] W. Fong, E. Darve and A. Lew. Sabiliy of Asynchronous Variaional Inegraors. Journal of Compuaional Physics, 7, , 8. [6] J.E. Marsden and M. Wes. Discree mechanics and variaional inegraors. Aca Numerica,, ,. [7] A. Sern and E. Grinspun. Implici-explici inegraion of highly oscillaory problems. SIAM Muliscale Modeling and Simulaion, 7, , 9. [8] S. Leyendecer, J.E. Marsden and M. Oriz. Variaional inegraors for consrained dynamical sysems. ZAMM, 88, , 8. 5