TR Use of the Implicit HHT-I3 and the Explicit ADAMS Methods with the Absolute Nodal Coordinate Formulation

Transcription

1 R Use of the Implicit HH-I3 ad the Explicit ADAMS Methods with the Absolute Nodal Coordiate Formulatio Bassam Hussei, Da Negrut, Ahmed A. Shabaa August 007

2 Abstract his ivestigatio is cocered with the use of a implicit itegratio method with adjustable umerical dampig properties i the simulatios of flexible multibody systems. he flexible bodies i the system are modeled usig the fiite elemet absolute odal coordiate formulatio (ANCF), which ca be used i the simulatio of large deformatios ad rotatios of flexible bodies. his formulatio, whe used with the geeral cotiuum mechaics theory, leads to displacemet modes, such as Poisso modes, that couple the cross sectio deformatios, ad bedig ad extesio of structural elemets such as beams. While these modes ca be sigificat i the case of large deformatios, ad they have o sigificat effect o the CPU time for very flexible bodies; i the case of thi ad stiff structures, the ANCF coupled deformatio modes ca be associated with very high frequecies that ca be a source of umerical problems whe explicit itegratio methods are used. he implicit itegratio method used i this ivestigatio is the Hilber-Hughes-aylor method applied i the cotext of Idex 3 differetial-algebraic equatios (HH-I3). he results obtaied usig this itegratio method are compared with the results obtaied usig a explicit Adams-predictorcorrector method, which has o umerical dampig. Numerical examples that iclude bodies with differet degrees of flexibility are solved i order to examie the performace of the HH-I3 implicit itegratio method whe the fiite elemet absolute odal coordiate formulatio is used. he results obtaied i this study show that, for very flexible structures, there is o sigificat differece i accuracy ad CPU time betwee the solutios obtaied usig the implicit ad explicit itegrators. As the stiffess icreases, the effect of some ANCF coupled deformatio modes becomes more sigificat, leadig to a stiff system of equatios. he resultig high frequecies are filtered out whe the HH-I3 itegrator is used due to its umerical dampig properties. he results of this study also show that the CPU time associated with the HH-I3 itegrator does ot chage sigificatly whe the stiffess of the bodies icreases, while i the case of the explicit Adams method the CPU time icreases expoetially. he fudametal differeces betwee the solutio procedures used with the implicit ad explicit itegratios are also discussed i this paper.

3 Cotets. Itroductio Costraied Multibody System Equatios Backgroud HH-I3 method ad Multibody Equatios Itegratio Error ad ime Step Covergece Criterio Explicit ADAMS Method Numerical Examples Summary ad Coclusios Ackowledgemets... 9

4 . Itroductio he itegratio of large deformatio fiite elemet ad multibody system algorithms is ecessary i order to be able to solve may egieerig ad physics problems. Oe approach that bears potetial for the successful itegratio of large deformatio fiite elemet ad multibody system algorithms is based o the absolute odal coordiate formulatio (ANCF) (Dmitrocheko ad Pogorelov, 003; Garcia-Vallejo et al., 003 ad 004; Shabaa, 005; Sopae ad Mikkola, 003; akahashi ad Shimizu, 999; Yoo et al., 004). his formulatio, which results i a costat mass matrix ad zero Coriolis ad cetrifugal forces, leads to displacemet modes, icludig Poisso modes, which couple the cross sectio deformatios, ad bedig ad extesio of structural fiite elemets such as beams ad plates (Hussei et al., 007; Schwab ad Meijaard, 005). hese coupled deformatio modes ca ot be captured usig other existig fiite beam elemets which are based o formulatios that assume that the cross sectio remais rigid. he ANCF coupled deformatio modes ca be sigificat i the case of large deformatios of very flexible structures. I this case, these coupled deformatio modes are ot associated with very high frequecies, ad therefore, havig these modes i the dyamic models does ot lead to umerical problems ad does ot egatively impact the efficiecy of the ANCF solutio algorithm. I this case, little advatage ca be achieved by usig implicit itegratio as compared to explicit itegratio methods. I the case of small deformatios of stiff structures i multibody system applicatios, the fiite elemet floatig frame of referece formulatio, that ca be used to systematically elimiate high frequecy modes, ca be employed. O the other had, i the case of large deformatios of stiff structures, the floatig frame of referece formulatio ca ot be used if the deformatio of the structural compoets assumes a complex shape. Attempts have bee recetly made i several ivestigatios to use the absolute odal coordiate formulatio to solve large deformatio problems i the case of stiff structures. I this case, the ANCF coupled deformatio modes are associated with high frequecies, resultig i a stiff system of dyamic equatios. Explicit itegratio methods ca be very iefficiet or eve fail whe they are used to solve the resultig system of stiff dyamic equatios. For such a stiff system, implicit itegratio methods are kow to be much more efficiet tha explicit methods. As the frequecy icreases, a explicit itegrator selects a smaller step size i order to be able to capture the high frequecy oscillatios i the solutio. he decrease i the step size leads to a icrease i the umber of fuctio evaluatios with a overall icrease i the CPU time. Furthermore, the step size required to capture the high frequecy oscillatios may become less tha the itegrator specified limit value leadig to a termiatio of the simulatio. Implicit itegrators, o the other had, do ot trace high frequecy oscillatios which ca have egligible effect o the solutio accuracy. For this reaso, implicit itegrators ca take a much larger step size i the case of stiff systems, ad ca have better performace i the simulatio of may multibody system applicatios as compared to the explicit itegrators. Furthermore, some implicit itegrators itroduce, i a cotrollable fashio, umerical dampig that results i filterig out of the small amplitude high frequecy oscillatios. Oe of the most popular implicit itegrators is the Hilber-Hughes-aylor (HH) method (Hilber et al., 977). Amog the attractive feature

5 of this method are the stability of the solutio ad the ability of the method to damp out umerically the small amplitude high frequecy oscillatios. Negrut et al. (007) ivestigated a HH-based algorithm for Idex 3 differetial-algebraic equatios (DAEs) of multibody systems (HH-I3), alog with strategies for error estimatio, itegratio step size cotrol, ad covergece criteria. It is the objective of this paper to examie the use of HH-I3 i the aalysis of flexible bodies modeled usig the fiite elemet absolute odal coordiate formulatio by comparig its performace of the HH-I3 with that of the predictor-corrector explicit Adams itegrator (Shampie ad Gordo, 975). It is importat, however, to poit out that the implicit ad explicit itegratio methods lead to two fudametally differet umerical solutio procedures. Whe the implicit itegrator is used, there is o eed to use the geeralized coordiate partitioig method i order to check o the violatio of the kiematic costraits. he fudametal differeces betwee the two solutio procedures will be discussed i this paper. his paper is orgaized as follows. A brief itroductio of the absolute odal coordiate formulatio ad the DAEs goverig the oliear dyamics of multibody systems is provided i Sectio. I Sectios 3-6, the HH-I3 method, alog with its error estimatio, selectio of the time step size, ad covergece criterio are discussed. I Sectio 7, the predictor-corrector Adams itegratio method ad the associated umerical procedure used i this study are reviewed. Numerical examples are preseted i Sectio 8 to compare betwee the performaces of the implicit ad explicit itegrators. I these examples, the flexible structures are modeled usig the absolute odal coordiate formulatio. he Adams explicit itegrator does ot produce umerical dampig, while the implicit HH-I3 method provides adjustable umerical dampig. A two-dimesioal flexible pedulum example is used i this compariso. he same pedulum example is solved for differet values of the modulus of elasticity i order to show the effect of the stiffess o the solutio accuracy ad itegrator efficiecy. Summary ad mai coclusios of this study are preseted i Sectio 9.. Costraied Multibody System Equatios I the implicit umerical itegratio procedure used i this ivestigatio, the equatios of motio are preseted i the Idex 3 augmeted form (Brea et al., 989). hese equatios are expressed i terms of the kiematic costrait Jacobia matrix ad Lagrage multipliers which ca be used to determie the geeralized costrait forces. he implicit ad explicit methods used i this ivestigatio to solve the resultig dyamics equatios lead to two fudametally differet umerical solutio procedures. For istace, the explicit method requires the use of the geeralized coordiate partitioig, while the implicit method does ot require the idetificatio of the system degrees of freedom. I this study, flexible bodies are modeled usig the fiite elemet absolute odal coordiate formulatio. I this formulatio, the global positio vector r of a arbitrary poit o the fiite elemet j of the flexible body i is defied as ( ) ( t) r ij = S ij x ij e ij ()

6 ij ij where S is the space depedet elemet shape fuctio, x is the vector of the elemet ij spatial coordiates, e is the vector of the time-depedet elemet odal coordiates, ad t is time. he vector of odal coordiates cotais absolute positio ad gradiet coordiates. Usig the priciple of virtual work, the fiite elemet equatios of motio expressed i terms of the odal coordiates ca be writte i the followig matrix form (Shabaa, 005): ij ij ij ij M e = Q + Q () ij ij where Q k is the vector of geeralized elemet elastic forces, Q e is the vector of ij geeralized elemet exteral forces, ad M is the elemet costat symmetric mass matrix which is defied by the followig equatio: = ij ij ij ij ij ij V k e M ρ S S dv (3) ij ij I this equatio ρ ad V are the elemet mass desity ad volume. While the mass matrix is costat, the vector of the elastic forces i the absolute odal coordiate formulatio is a highly oliear fuctio of the odal coordiates. Several methods based o differet models ca be used to evaluate the elastic forces (Schwab ad Meijaard, 005; Hussei et al., 007). If a geeral cotiuum mechaics approach is used to formulate the elastic forces, oe obtais a model that icludes the ANCF coupled deformatio modes. O the other had, if a elastic lie approach is used, the ANCF coupled deformatio modes ca be systematically elimiated. I this ivestigatio, the geeral cotiuum mechaics approach is used to formulate the elastic forces. Usig the fiite elemet equatios, the equatios of motio of the multibody system subject to kiematic costraits ca be writte i the followig form: Mq + C = qλ Q C( q,t ) = 0 I this equatio, M is the system mass matrix, q is the vector of the system coordiates icludig the absolute odal coordiates ad other coordiates used to describe the motio of rigid bodies, λ is the vector of Lagrage multipliers, Q is the vector of all forces icludig exteral, elastic, Coriolis ad cetrifugal forces, C is the vector of the costrait fuctios, ad C is the Jacobia matrix of the kiematic costrait equatios. q he HH-I3 implicit ad the Adams explicit methods used i this ivestigatio to solve the costraied system of Eq. 4 lead to two fudametally differet umerical procedures. I the explicit method, the geeralized coordiate partitioig is used, ad the costrait equatios are imposed at the positio, velocity ad acceleratio levels. he dyamic equatios ad the costrait equatios at the acceleratio levels are solved for the system acceleratios ad Lagrage multipliers. he idepedet acceleratios are idetified ad itegrated forward i time i order to determie the depedet coordiates ad velocities. Depedet variables are determied usig the costrait equatios at the positio ad velocity levels. At the positio level, a iterative Newto-Raphso procedure is used to determie the depedet coordiates. he implicit method, o the other had, does ot require the idetificatio of the idepedet coordiates (degrees of freedom), ad the costrait equatios are imposed at the positio level oly. he extesio of this implicit procedure to satisfy the kiematic costraits at the velocity (4)

7 level, whe the absolute odal coordiate formulatio is used, will be addressed i future ivestigatios. 3. Backgroud he HH-I3 method used i this ivestigatio is based o the Hilber-Hughes-aylor method (also kow to as the α-method) (Hilber et al., 977). his method is widely used i solvig umerically secod order iitial value problems i structural dyamics. he foudatio of the HH method is the Newmark method which was proposed by Newmark (959). For this reaso, the Newmark method is briefly reviewed i this sectio i order to have a uderstadig of the implicit procedure used ad the iheret umerical dampig that characterizes the H method that will be discussed i subsequet sectios. he Newmark method provides a solutio for the structural dyamics equatios of motio which has the followig stadard form: M q + Cq + Kq = F( t) (5) where q, q ad q are the vectors of geeralized positio coordiates, velocities ad acceleratios, respectively, M, C ad K are the symmetric mass, dampig ad stiffess matrices, respectively, ad F is a time depedet exteral force vector. Newmark method itroduces two equatios that relate the positio ad the velocity vectors to the acceleratio vector based o aylor series expasio. he positio vector q + ad velocity vector q + ca be writte at time t + by usig a aylor series expasio as follows. h 3 q q hq q h q + = β m (6) q + = q + hq + γ h qm (7) where h = t + t is the step size, m refers to a state somewhere betwee the states ad +, ad β ad γ are two assumed parameters. Oe ca approximate q m by usig the followig equatio: q m = ( q q + ) h (8) Substitutig Eq. 8 ito Eqs. 6 ad 7 leads to h q+ = q + hq + (( β ) q + βq + ) (9) q = q + h γ q + γ q (0) (( ) ) + + Equatios 9 ad Eq. 0 are the Newmark equatios that defie q + ad q + as fuctios of q, q, q ad q +. By assumig that Eq. 5 is satisfied at ay time, oe ca write the equatios of motio at time t + as follows: Mq Cq + Kq F t () ( ) = + Substitutig Eqs. 9 ad 0 ito Eq. ad assumig that q, q ad q are kow, oe obtais a set of equatios that are fuctios of the acceleratio vector q +. herefore,

8 these equatios ca be used to determie the vector of acceleratios q +, which ca be used i tur to determie q + ad q + usig Eqs. 9 ad 0. he stability of the solutio obtaied usig the Newmark method depeds o the choice of values for the two parameters γ ad β. I order to have a stable solutio these parameters should satisfy the followig relatios (Hilber et al., 977). ( γ + ) γ, β 4 If γ = ad β =, oe obtais the well kow trapezoidal method, which is A-stable 4 ad has secod order covergece property. he mai drawback of the trapezoidal method is that it is ot capable of elimiatig spurious high frequecies. he HH method is a improvemet over the Newmark method i that it has the ability to elimiate udesirable high frequecy oscillatios, while it remais stable ad secod order coverget. he HH method ca be obtaied by replacig Eq. by () ( α ) α ( α ) α ( τ ) Mq + + Cq Cq + + Kq Kq = F (3) where α is a assumed parameter, ad τ + = t + ( + α ) h (4) I order to obtai a stable solutio usig the implicit HH method, the parameters α, γ ad β should be selected to satisfy the followig relatios: 0.3 α 0 γ = α β = ( α ) 4 (5) he smaller the value of α, the more umerical dampig is added to the system. he choice α = 0 leads to the trapezoidal method, which has o umerical dampig. 4. HH-I3 method ad Multibody Equatios he implicit HH-I3 uses the HH method to trasform the differetial equatios of motio of the multibody system ito a set of oliear algebraic equatios. hese oliear algebraic equatios ca be solved simultaeously with the oliear costrait equatios at the positio level to determie the state of the system. he differetial equatios of motio ad the costrait equatios ca be writte i the form of Eq. 4, which is reproduced here for coveiece M q q + C λ = Q t, q, q (6) Note that Eq. 6 ca be writte as ( ) ( ) q C( q,t ) = 0 (7) M q = H, (8) where H = Q C q λ represets the geeralized forces icludig the costrait forces. Recall that whe the absolute odal coordiate formulatio is used, the mass matrix is costat. Furthermore, if Cholesky coordiates are used, the mass matrix becomes the idetity matrix. his importat feature of the fiite elemet absolute odal coordiate formulatio will be exploited whe developig a estimate of the local itegratio error, as discussed i the followig sectio.

9 Usig Eq. 3 which is the formula for the HH method, Eq. 8 at time step t + ca be writte as Mq H. (9) ( ) ( ) = τ + + where τ + is defied i by Eq. 4. Oe ca use a aylor series expasio to write H ( τ + ) = H( t ) + ( + α ) hḣ ( t ), (0) m where t is a time betwee m t ad Ḣ t m ca be approximated by assumig lier chage i H, that is H ( tm ) = ( H( t+) H( t )) h. () By substitutig this equatio ito Eq. 0, ad the usig the resultig expressio of Ḣ i Eq. 9, oe obtais ( ) τ + Sice τ +, ad ( ) ( Mq ) ( + α ) H( t ) αh( t ). () + = + H = Q C q λ, oe ca write Eq. i the followig form: ( ) ( α )( q ) α ( q ) Mq + + C λ Q C λ Q = 0 (3) + + At time t +, the costraits are assumed to be satisfied. herefore, oe ca use Eq. 7 to write C q, t 0 (4) ( ) + + = While Eqs. 3 ad 4 deped o q +, q +, q + ad λ +, oe ca use Eqs. 9 ad 0 to write these equatios (Eqs. 3 ad 4) as fuctios of q + ad λ + oly. he result is a set of oliear algebraic equatios that ca be solved for q + ad λ + usig a iterative Newto-Raphso method. he iterative Newto-Raphso method requires costructig ad solvig the followig system of equatios at iteratio k : ( k ) F F q λ q F = ( k ) ( k ) F F λ + F q + λ + where idicates Newto differeces, ad F ad F are defied as α ( ) ( q ) ( q ), (5) F = Mq + C λ Q C λ Q + + α + + α F = C ( q+, t+ ) βh Usig these vector fuctios, oe ca show that, istead of usig umerical F q ca be obtaied i aalytical differetiatio, ( F λ ), ( F λ ) ad ( ) + form from the followig equatios: + + (6)

10 F F = C, = 0 q λ + λ + F C q+ = = C q q + βh q + q + he value of F q +, however, is calculated by umerical differetiatio. Sice costructig the Jacobia of Eq. 5 is expesive, it is recommeded agaist updatig this Jacobia at each Newto iteratio. I order to avoid a bad coditio umber of the coefficiet matrix i Eq. 5 whe h 0, Eq. 4 was divided by β h to obtai the secod equatio of Eq. 6. he iterested reader is referred to Negrut et al. (007) for a more complete discussio of this topic. (7) 5. Itegratio Error ad ime Step I order to have a measure of the accuracy of the obtaied solutio, oe must have a accurate estimate of the local itegratio error. I this sectio, a procedure for the error estimatio is discussed. Sice the mass matrix, i geeral, depeds o the coordiates oly, it ca be assumed, i may applicatios, that the chage i the mass matrix with respect to time is small over a certai period of time. I fact, i the case of the absolute odal coordiate formulatio, the mass matrix remais costat, ad if Cholesky coordiates are used this mass matrix becomes the idetity matrix, as previously discussed. I this case, oe ca write the derivative of Eq. 8 with respect to time at t as follows. M q = H (8) ( t ) A equatio similar to Eq. ca be used to approximate the value of Ḣ ( t ). his leads to ( H( t ) H( t )) h M q. (9) = + Assumig that the equatios of motio are satisfied at t, oe has M q = H( t ), (30) Subtractig this equatio from Eq. ad usig Eq. 9, oe obtais q q h( ) q + = + α (3) A aylor series expasio of the exact solutio ( q + ) of the equatio of motio at t e + is give by 3 h h 4 ( q+ ) = q + hq e + q + q + O ( h ) (3) 6 A estimate of the local itegratio error δ + ca be computed by subtractig the exact solutio from the solutio obtaied usig the umerical itegratio. his estimate is give by δ = q q (33) ( ) e

11 Substitutig the values of + 33 yields q ad ( ) q from Eq. 9 ad Eq. 3, respectively, ito Eq. + e 3 h 4 ( q q ) O( h ) δ+ = β h + q (34) 6 By substitutig the value of q from Eq. 3 ad eglectig higher order terms, oe obtais δ h ( q q + β + ) (35) 6( + α ) I order to accept the obtaied solutio, a orm of the error fuctio must be less tha a user specified tolerace. he followig weighted orm for the vector of the local itegratio error was proposed by Negrut et al. (007): where Yi max (, max qi, j j=,..., ) = is the e = p δi, + p i= Yi th i compoet of the weightig vector Y, ad p is the umber of elemets of the positio coordiate vector q. he error fuctio of Eq. 36 ca be writte as h e = β x (37) 6 + α p ( ) p x i where x = ( q + q ), ad represets the weighted orm, that is x =. i= Yi he error must be less tha the user specified tolerace ε, that is, e ε. his is equivalet to Θ (38) where Θ is defied as 4 x h Θ = (39) ψ ad pε ψ = (40) β 6( + α ) he step-size selectio is a importat issue i developig efficiet umerical itegratio methods. A very small step-size may lead to additioal uecessary calculatios for the desired accuracy. O the other had, a very large step-size may lead to a icrease i the umber of Newto-Raphso iteratios required to achieve covergece. Furthermore, a large step size may result i a umerical solutio that is ot accurate eough. he step-size should be selected to accelerate the covergece i the ext time step with a itegratio error withi the limit specified by the user. I order to i (36)

12 describe the procedure used i this ivestigatio to select the step size, the defiitio x = ( q q + ) ad Eq. 3 are used to write x as a fuctio of q as follows: x = h + α q (4) ( ) By assumig that there is o sigificat chage i q ad weightig vector Y i betwee two cosecutive time steps, oe ca defie a costat a as follows: p ( ) qi, a = + α (4) i= Yi By usig the precedig two equatios ad the defiitio of the orm, oe obtais By substitutig x from this equatio ito Eq. 39, oe obtais x = ah (43) 6 ah Θ = (44) ψ he ew step-size h ew should be selected such that the itegratio error is withi the user specified tolerace, or equivaletly, Θ ew =. herefore, 6 ah ew = (45) ψ I this ivestigatio, a safety factor s = 0. 9 is itroduced. he precedig two equatios ca the be used to obtai a estimate of the ew step-size as follows: h = sh ew (46) 6 Θ his equatio is used i this study to determie the ew step-size. 6. Covergece Criterio Havig developed a procedure for error estimatio ad step size selectio, oe eeds to establish a criterio for stoppig the iteratio process. his criterio determies how accurately Eq. 3 should be satisfied, or i other words, how accurately should the vector be calculated before the result are accepted by the user. I order to develop the q + covergece criterio, assume that the exact value of the vector q + is approximated ( k ) after k iteratios by q +. I this case, oe ca write the local error after k iteratios by usig Eq. 37 as follows: ( k ) h ( k ) e = β x (47) 6 + α p where ( ) (k ) ( k ) ( k ) x is the approximated value of x, that is, x = ( q q + ) (k ) Covergece is assumed if the relative error betwee e ad e is less tha a ( k ) e e e. Sice the value of e is ot available, it certai specified value c, that is, ( ) c.

13 is approximated i the deomiator by ε, ad the covergece test ca be writte as follows: ( e e ε By usig Eq. 37 ad Eq. 47, oe ca write e e ( k ) I order to obtai a approximatio for k ) c h ( k ) ( + α ) (48) β x x (49) 6 p (k ) x x, oe ca use the liear covergece property of Newto method, which assumes that there is a costat ξ, 0 ξ <, such that ( k ) ( k ) x + ξ x (50) (k ) ( k ) ( k ) ( k ) where x is the correctio at the iteratio k, x = x x. he iequality of Eq. 50 ca be used to write the followig: ( k ) ( k ) x + ξ x (5) ( k + m) ( k ) ( k + m) ( k + m ) ( k + m ) ( k + ) ( k + ) ( k ) Sice x x = ( x x ) + x + x + ( x x ) = oe ca write... x i= Usig the iequalities of Eqs. 5 ad 5, oe ca write m i= ( k + i) m ( k + m) ( k ) ( k + i) x x x (5) m ( k + m) ( k ) ( k ) i x x x ξ (53) i= (k ) x coverges to x after a large umber of iteratios, that is, Assumig that ( ) lim k + ξ x = x m, ad sice for 0 ξ <, i ξ =, the m i= ξ ( k ) ( k ) ξ x x x (54) ξ Usig the iequalities of Eqs. 49 ad 54, oe has ξ β ( k ) h ( k e e x 6( + α ) ) (55) p ξ From Eqs. 40 ad 55, the covergece test of Eq. 48 is satisfied if the followig coditio is met: ξ ψ ξ h his is the covergece test used i the HH-I3 method employed i this ivestigatio that is focused o the absolute odal coordiate formulatio models. I summary, the procedure used i the implicit HH-I3 method described i this ( k ) x c (56) 4 ad λ which ca be made usig Newto study requires a iitial guess of q + + differeces or values obtaied at the previous time step. Equatio 5 is the solved,

14 iteratively to fid more accurate solutio for q + ad λ +. After at least two iteratios, the parameter ξ ca be calculated, ad the covergece coditio of Eq. 56 is tested. If this coditio is ot satisfied, the more iteratios are required. If the coditio is satisfied, the the coditio of Eq. 38 is tested. If this coditio is satisfied, the the ew step-size is updated usig Eq. 46, ad the itegrator proceeds to the ext time step. If the coditio of Eq. 38 is ot satisfied, the the step-size has to be updated usig Eq. 46 ad the same time step solutio has to be reevaluated. 7. Explicit ADAMS Method he Adams method is a explicit predictor-corrector method for the umerical solutio of first order ordiary differetial equatios. his method ca ot be used to directly solve a system of DAEs. he algebraic equatios must be first elimiated usig the geeralized coordiate partitioig techique, which is i priciple equivalet to the embeddig techique (Shabaa, 005). he embeddig techique, that elimiates the reactio forces ad the depedet coordiates, leads to a miimum set of differetial equatios expressed i terms of the idepedet coordiates (degrees of freedom) oly. Usig the geeralized coordiate partitioig, oe ca write the system coordiates as follows: = i d q q q (57) where q i is the vector of idepedet coordiates or degrees of freedom, ad q d is the vector of depedet coordiates. Oe ca defie the velocity trasformatio matrix Bdi that relates the system velocities to the idepedet velocities ( q = Bdiq i ) as follows. I Bdi = - Cq C d qi By pre-multiplyig Eq. 6 by B, oe obtais di It ca be show that (Shabaa, 005) di di q di (58) B Mq + B C λ = B Q (59) B C 0 (60) di q = By takig the secod derivative of the oliear algebraic costrait equatios of Eq. 7 with respect to time, oe obtais C q = Q (6) where d = tt qt ( q ) q q Q C C C q q is a vector that is quadratic i the velocities. Oe ca rewrite Eq. 6 i terms of q i ad q d as follows: C q + C q = Q (6) Usig Eqs. 57, 58 ad 6, oe has = d where - d ( q d ) qi i qd d d d q = B q + Q (63) di i d Q 0 C Q. Substitutig Eq. 63 ito Eq. 59 ad usig Eq. 60 yields B MB q + B MQ = B Q (64) di di i di d di

15 his system of equatios is expressed i terms of the idepedet coordiates, leadig to the elimiatio of the costrait forces. I order to use the explicit Adams method, oe has to trasform the secod order differetial equatios ito a system of first order ordiary differetial equatios (ODEs). his is accomplished by itroducig a ew vector y, defied as = i i y q q (65) akig the time derivative of this vector ad substitutig the value of q i from the solutio of Eq. 64 yields q i y = ( (66) B ) ( dimbdi BdiQ BdiMQd ) Note that the right had side of this equatio is a fuctio of q i, q i ad t. herefore, the origial problem is reformulated as y = f ( y,t), (67) which is the stadard form of a first order ODE that ca be solved usig the Adams method. he explicit Adams method used i this study is the Adams-Predictor-Corrector method discussed by Shampie ad Gordo (975). I this method, a predicted value p y + at time step t + based o the solutio at time step t ad several previous values of f is obtaied by usig the Adams-Bashforth formula: Here m,( t) values of f ( y,t), that is, With t+ p = + + m, t ( t) y y P dt. (68) P is a Lagrage iterpolatig polyomial that iterpolates the previous m p + m m t t + k P m, ( t) = f+ j (69) j = k = t+ j t+ k k j p p f y t. he corrector Adams- y available, oe ca the fid the value of ( ) Moulto formula is the used to fid a corrected value of * Here ( t) m, t + t + +, + p y +, which is deoted by + ( t) y : * y = y + P dt. (70) + P is a iterpolatig polyomial that iterpolates the previous m poits i additio to the predicted value f p +, that is m m m * p t t + k t t+ k P ( ) + m, t = f + f+ j (7) k = t+ t+ k j= k = 0 t+ j t+ k k j Oe ca the use the corrected value y + to evaluate the fuctio f+ which is used i the ext time step. A full discussio of the procedures used i this explicit method for m,

16 error cotrol ad selectio of the order ad time step ca be foud i the literature (Shampie, ad Gordo, 975). here are two poits that eed to be emphasized whe the explicit Adams method is used. First, the itegratio i Eqs. 68 ad 70 ca be easily performed aalytically, ad i fact Shampie ad Gordo (975) provide closed form solutios for those itegrals. Secod, at every time step the umber of poits used i the iterpolatio should be less tha or equal to ; thus, at the begiig of the simulatio, there is o sufficiet umber of poits to start the itegratio with a higher order formula. For this reaso, a lower order formula is used util iformatio at a sufficiet umber of poits becomes available. he formal derivatio of the embeddig formulatio preseted i this sectio shows that the mass matrix associated with the idepedet coordiates may ot be, i geeral, a sparse matrix. I the actual implemetatio of the embeddig formulatio, full advatage is take of sparse matrix techiques at the positio, velocity ad acceleratio levels. For example, the costrait equatios at the velocity level ca be writte as C q = C (7) q If the idepedet velocities are assumed to be kow from the umerical itegratio, a elemetary matrix I d that has zeros ad oes ca be easily costructed. his matrix ca be used to extract the idepedet coordiates or velocities from the total vector of the system coordiates or velocities. herefore, this costat sparse matrix satisfies the followig relatio: I dq = q i. (73) By combiig the precedig two equatios, oe has (Shabaa, 00) Cq Ct q = (74) Id q i he solutio of this system of algebraic equatios defies the depedet coordiates. Note that the coefficiet matrix i this equatio is also sparse, ad therefore, efficiet sparse matrix techiques ca be used. A similar procedure ca be used at the positio level whe the iterative Newto-Raphso method is used to solve the oliear algebraic costrait equatios for the depedet coordiates. his ca be achieved by replacig i the precedig equatio q, q i ad C t by q, 0 ad C, respectively. Similarly, the total vector of acceleratios ca be determied by solvig the augmeted form of the equatios of motio obtaied by combiig the first equatio i Eq. 6 with Eq. 6, leadig to M C q q Q = (75) Cq 0 λ Qd he coefficiet matrix i this equatio is agai a sparse matrix. herefore, by usig sparse matrix techiques, the precedig equatio ca be solved efficietly for the acceleratios ad Lagrage multipliers. he idepedet acceleratios ca be idetified ad itegrated forward i time usig the explicit itegratio method. herefore, i the actual implemetatio of the embeddig techique, full advatage is take of sparse matrix techiques. Furthermore, there is o eed to idetify the costrait subjacobia matrices C q ad C i q or fid the iverse of C d q. d t

17 8. Numerical Examples I order to compare the performace of the implicit HH-I3 ad the explicit Adams methods, a dyamic simulatio of a flexible pedulum is performed by usig the two itegratio methods. he pedulum used i this example is assumed to be iitially horizotal, as show i Fig.. he pedulum is assumed to fall uder the effect of the gravity force. he pedulum model is developed usig the fiite elemet absolute odal coordiate formulatio. he beam i this pedulum example is divided ito two twodimesioal fiite beam elemets alog its legth (Omar ad Shabaa, 00). he pedulum beam is assumed to have a udeformed legth of 0.4 m, cross sectioal area of m, a desity of 7800 kg/m 3, ad a Poisso ratio of 0.3. I order to examie the effect of the stiffess of the beam o the performace of the implicit ad explicit itegratio methods, several values of the beam modulus of elasticity are used. 5 he pedulum example is solved by assumig the followig moduli of elasticity: 0, 6 0, 7 0, 8 0, 9 0, 0 0 ad 0 N/m. he beam elastic forces are formulated usig the geeral cotiuum mechaics approach. his approach leads to the ANCF coupled deformatio modes that also iclude Poisso modes. For all stiffess levels, these modes couple the deformatio of the cross sectio with bedig ad extesio of the beam. For very stiff structures, the ANCF coupled deformatio modes, as previously metioed, ca be associated with very high frequecies. A eigevalue study of these frequecies was preseted i the literature (Schwab ad Meijaard, 005). Explicit itegratio methods have show to perform poorly i the presece of these modes i the case of very stiff structures. Figure shows the vertical positio of the pedulum tip whe the modulus of 5 elasticity is equal to 0 N/m. he results preseted i this figure show the solutios obtaied usig the two itegratio methods. Figure 3 shows the mid-poit trasverse deformatio, which is defied as show i Fig. 4, for the same value of the modulus of elasticity. It is clear from the results preseted i these figures that there is o sigificat differece betwee the solutios obtaied usig the two itegrators. Figure 5 shows the 8 mid-poit trasverse deformatio whe the modulus of elasticity is 0 N/m. he results preseted i this figure show that, for this moderate stiffess, the results of the two itegrators are almost the same, ad that there is still a good agreemet betwee the solutios obtaied usig the implicit ad explicit itegratio methods. As a example of a highly stiff structure, the simulatio was performed assumig a modulus of elasticity of 0 N/m. Figure 6 shows the vertical positio of the pedulum tip for the stiff structure case. he results preseted i this figure do ot show sigificat differece betwee the two referece motio solutios, ad therefore, the umerical dampig of the HH-I3 method does ot have a sigificat effect o the rigid body motio of the beam. Figure 7 shows the mid-poit trasverse deformatio for this relatively high modulus of elasticity. It is clear from the results preseted i this figure that the implicit HH-I3 method damped out some high frequecy oscillatios ad it produces a smoother solutio tha the oe obtaied by the explicit Adams method.

18 Figure 8 shows the performace, i terms of CPU time, of the two itegrators as fuctio of the modulus of elasticity. he results suggest that there is o sigificat differece i CPU simulatio time betwee the implicit HH-I3 method ad the explicit Adams method i the case of very flexible structure. As the stiffess of the structure icreases with the icrease i the modulus of elasticity, the CPU time of the explicit Adams method grows expoetially. I the case of such stiff structures, the implicit HH-I3 method is much more efficiet, ad it is capable of dampig out the ANCF coupled deformatio modes, which for the stiff structure do ot have a sigificat effect o the accuracy of the solutio. 9. Summary ad Coclusios Ulike existig fiite elemet structural elemet formulatios, the absolute odal coordiate formulatio allows for the use of the geeral cotiuum mechaics theory to formulate the structural elemet elastic forces. he use of the geeral cotiuum mechaics approach to formulate the elastic forces leads to the ANCF coupled deformatio modes, icludig Poisso modes. hese modes couple the cross sectio deformatios ad bedig ad extesio of the structural elemets, as discussed i the literature (Schwab ad Meijaard, 005; Hussei et al., 007). I the case of very flexible structures, the effect of the ANCF coupled deformatio modes ca be sigificat as they capture dyamic effects that ca ot be captured usig existig fiite elemet beam models. I this case of very flexible structures, the ANCF coupled deformatio modes are ot a source of umerical problems, ad explicit itegratio methods have prove to be efficiet. For very stiff structures, o the other had, the ANCF coupled deformatio modes ca be associated with very high frequecies. I these special cases, if the deformatio shape is simple, istead of the absolute odal coordiate formulatio, it is recommeded to use the floatig frame of referece formulatio, which systematically elimiates high frequecy modes. If the deformatio shape is complex, such that there is o clear advatage of usig the floatig frame of referece formulatio, the absolute odal coordiate formulatio ca still be used with implicit umerical itegratio methods, as demostrated i this study. I order to examie the performace of the implicit ad explicit itegratio methods whe the absolute odal coordiate formulatio is used, two itegratio methods are examied i this ivestigatio. he first method used (HH-I3) is the implicit Hilber- Hughes-aylor method modified to hadle Idex 3 differetial-algebraic equatios (Negrut et al., 007). he secod method is the explicit Adams-predictor-corrector method, which has a variable order ad variable step size (Shampie ad Gordo, 975). hese two differet itegratio methods require the use of two fudametally differet solutio procedures. Ulike HH-I3, the explicit itegratio method requires the idetificatio of the system degrees of freedom ad the associated state equatios. he formulatios ad solutio procedures used with the two itegratio methods employed i this ivestigatio are discussed. Numerical examples are preseted i order to compare the performace of the implicit ad explicit itegratio methods. he results obtaied i this study show that i the case of very flexible structures, where the effect of the ANCF coupled deformatio modes ca be sigificat, there is o sigificat differece betwee the CPU times of the implicit ad explicit itegratio methods. I this case explicit

19 itegratio methods ca be used. O the other had, i the case of very stiff structures i which the ANCF coupled deformatio modes are associated with very high frequecies, there is a sigificat differece betwee the CPU times of the two methods. he implicit HH-I3 method, because of its umerical dampig properties, ca be much more efficiet as compared to the explicit ADAMS method. 0. Ackowledgemets his research was supported by the Army Research Office, Research riagle Park, ad the Iteratioal Program of the Natioal Sciece Foudatio. his fiacial support is gratefully ackowledged. Refereces. Brea, K.E., Campbell, S.L., ad Petzold, L.R., 989, Numerical Solutio of Iitial-Value Problems i Differetial-Algebraic Equatios, North-Hollad, New York.. Dmitrocheko, O. N. ad Pogorelov, D. Y., 003, Geeralizatio of Plate Fiite Elemets for Absolute Nodal Coordiate Formulatio, Multibody System Dyamics, 0, o. : García-Vallejo D., Mayo J., Escaloa J. L., Domíguez J., 004, Efficiet Evaluatio of the Elastic Forces ad the Jacobia i the Absolute Nodal Coordiate Formulatio, Joural of Noliear Dyamics, Vol. 35 (4), pp Garcia-Vallejo, D., Escaloa, J. L., Mayo, J. ad Domiguez, J., 003, Describig Rigid-Flexible Multibody Systems Usig Absolute Coordiates, Noliear Dyamics, 34, o. -: Hilber, H. M., Hughes,. J. R., aylor, R. L., 977, Improved Numerical Dissipatio for ime Itegratio Algorithms i Structural Dyamics, Earthquake Egieerig ad Structural Dyamics, Vol. 5, pp Hussei, B., Sugiyama, H., ad Shabaa, A.A., 007, Coupled Deformatio Modes i the Large Deformatio Fiite Elemet Aalysis: Problem Defiitio, ASME Joural of Computatioal ad Noliear Dyamics, Vol. (), pp Negrut, D., Rampalli, R., Ottarsso, G., Sajdak, A., 007, O a Implemetatio of the Hilber-Hughes-aylor Method i the Cotext of Idex 3 Differetial- Algebraic Equatios of Multibody Dyamics, Joural of Computatioal ad Noliear Dyamics, Vol. (), Ja. 007, pp Newmark, N.M., 959, A Method of Computatio for Structural Dyamics, Joural of Egieerig Mechaics Divisio, ASCE, pp Omar, M. A., Shabaa, A. A., 00, A wo-dimesioal Shear Deformatio Beam for Large Rotatio ad Deformatio, Joural of Soud ad Vibratio, Vol. 43 (3),

20 0. Schwab A. L., Meijaard J. P, 005, Compariso of hree-dimesioal Flexible Beam Elemets for Dyamic Aalysis: Fiite Elemet Method ad Absolute Nodal Coordiate Formulatio, Proceedigs of ASME Iteratioal Desig Egieerig echical Cofereces ad Computers ad Iformatio i Egieerig Coferece - DEC005, September 4-8, 005, Vol. 6 (B).. Shabaa, A.A., 00, Computatioal Dyamics, Secod Editio, Joh Wiley & Sos.. Shabaa A. A., 005, Dyamics of Multibody Systems, hird Editio, Cambridge Uiversity Press. 3. Shampie L. F., Gordo M. K., 975, Computer Solutio of Ordiary Differetial Equatios, W.H. Freema, Sa Fracisco. 4. Sopae, J.. ad Mikkola, A. M., 003, Descriptio of Elastic Forces i Absolute Nodal Coordiate Formulatio, Noliear Dyamics, 34, o. -: akahashi, Y., ad Shimizu, N., 999, Study o Elastic Forces of the Absolute Nodal Coordiate Formulatio for Deformable Beams, Proceedigs of ASME Iteratioal Desig Egieerig echical Cofereces ad Computer ad Iformatio i Egieerig Coferece, Las Vegas, NV. 6. Yoo, W.S., Lee, J.H., Park, S.J., Soh, J.H.. Pogorelov, D., ad Dimitrocheko, O., 004, Large Deflectio Aalysis of a hi Plate: Computer Simulatio ad Experimet, Multibody System Dyamics, Vol., No., pp

21 Y Gravity force Cross sectio X Figure. Flexible pedulum iitial cofiguratio ip vertical positio (m) ime (s) 5 Figure. Vertical positio of the beam tip (E= 0 N/m ) (, ADAMS method;, HH-I3 method).

22 0.0 Mid-poit trasverse deformatio (m) ime (s) 5 Figure 3. Mid-poit trasverse deformatio (E= 0 N/m ) (, ADAMS method;, HH-I3 method). Beam local coordiates Y Beam deformed shape Mid-poit X rasverse deformatio Figure 4. Mid-poit trasverse deformatio.

23 Mid-poit trasverse deformatio (m) ime (s) 8 Figure 5. Mid-poit trasverse deformatio (E= 0 N/m ) (, ADAMS method;, HH-I3 method) ip vertical positio (m) ime (s) Figure 6. Vertical positio of the beam tip (E= 0 N/m ) (, ADAMS method;, HH-I3 method).

24 4.0x x0-7 Mid-poit trasverse deformatio (m).0x0-7.0x x x x x ime (s) Figure 7. Mid-poit trasverse deformatio (E= 0 N/m ) (, ADAMS method;, HH-I3 method) CPU time (s) Modulus of elasticity (N/m^) Figure 8. CPU time versus modulus of elasticity (, ADAMS method; I3 method)., HH-