On the role of quadrature rules and system dimensions in variational multirate integrators

Transcription

1 The 3 rd Joint International Conference on Multibody Syste Dynaics The 7 th Asian Conference on Multibody Dynaics June 30-July 3, 04, BEXCO, Busan, Korea On the role of quadrature rules and syste diensions in variational ultirate integrators Tobias Gail, Sigrid Leyendecer, Sina Ober-Blöbau # Chair of Applied Dynaics University of Erlangen-Nureberg Haberstrasse, 9058 Erlangen, Gerany [Tobias.Gail, Sigrid.Leyendecer]@ltd.uni-erlangen.de # Departent of Matheatics University of Paderborn Warburger Str. 00, Paderborn, Gerany sinaob@ath.uni-paderborn.de ABSTRACT Systes lie a chain driven cobustion engine or a satellite in space are echanical systes containing slow and fast dynaics. Multirate schees split such systes into parts which can be siulated using different tie steps. With the fraewor of variational ultirate integration a tie grid with icro and acro nodes is introduced. Siulating the slow variables on the acro grid and the fast variables on the icro grid leads to coputing tie savings. However, increasing the nuber of icro steps per acro step leads to a larger syste of equations that has to be solved iteratively. Therefor, there is a liit to the coputing tie savings. The relation between syste size, nubers of icro steps and coputational effort is investigated by eans of the Feri-Pasta-Ula proble. Therefor, the coputing tie for different nubers of asses for the Feri-Pasta-Ula proble is investigated for an increasing nuber of icro nodes and for different quadrature schees. Relations between the iniu coputing tie and the nubers of variables are derived. Furtherore, another aspect is studied. The convergence rate for different quadrature schees is investigated nuerically for the Feri-Pasta-Ula proble as an unconstrained syste and the triple spherical pendulu as a syste subject to constraints. INTRODUCTION A echanical syste with dynaics on varying tie scales contains slow and fast otions. Such systes are e.g. satellites in space with different gravity of thlanets, chain driven cobustion engines with fast oving valves and slow oving chain drives or the otions inside a olecule where different strong bonds are leading to different frequencies. For such echanical systes, the nuerical integration has to coply with contradicting requireents. On the one hand, to guarantee a stable integration of the fast otions, tiny step sizes are needed. On the other hand, for the slow otions, a larger tie step size is accurate enough and too sall tie steps increase the coputing tie unnecessarily, especially for costly function evaluations. Multirate schees are introduced with the goal to eet those requireents and to save coputing tie. They split the syste into subsystes, as done by e.g. M. Arnold in []. These subsystes can be treated individually and therefor different ethods can be used to solve the as shown by Stern and Grinspun in [9] and Weinan et al. in [7]. Variational integrators are introduced by Marsden and West in [8] and by Lew et al. in [4]. They conservroperties lie syplecticity and oentu aps and have good energy behavior. In the fraewor described as variational ultirate integration introduced in [5], a echanical syste containing slow and fast dynaics is split into two subsystes. Each subsyste is siulated on its own tie grid with its own tie step. With the sae accuracy in the fast dynaics as in the singlerate case, this ethod yields coputing tie savings, because fewer acro steps are needed, see [3]. However, this increases the diension of the coupled syste of equations to be iteratively solved in one acro step. Because of this, there is a liit in the aount of coputing tie that can be saved by using ore icro nodes per acro tie step. The relation between the nuber of variables in the syste and the coputing tie savings is investigated by eans of the Feri-Pasta-Ula proble in this wor.

2 An interesting aspect of every integration ethod is the order of convergence. In this wor, we furtherore study the convergence of variational ulitrate integration nuerically for different quadrature schees by two exaple systes, the Feri-Pasta-Ula proble as a unconstrained syste and the triple spherical pendulu as a syste subject to constraints. In Section, the variational ultirate integrators are derived via a discrete variational principle, i.e. the sae steps arerfored as for the derivation of the constrained ultirate Euler-Lagrange equations in the continuous setting. In Section 3, the three nuerical exaples arresented. By eans of the Feri- Pasta-Ula proble, the liitations of coputing tie savings are investigated in Section 4. An error analysis for the variational ultirate integrator is still wor in progress. Here, we give first insight with nuerical convergence results for two nuerical exaple systes, one with no constraints, the other subject to constraints. DERIVATION OF VARIATIONAL MULTIRATE INTEGRATORS In this section, the variational ultirate integrator is introduced. First, variational ultirate dynaics is deduced fro Hailton s principle and then, perforing the sae steps in the discrete tie setting, the variational ultirate integration schee is derived.. Variational ultirate dynaics Let a echanical syste containing slow and fast dynaics be described by a Lagrange function L(q, q). The configuration vector q(t) Q R n is in the configuration anifold Q. The velocity vector q T q(t) Q R n is in the tangent space of the anifold Q at q. The otion of the echanical syste is constrained by the holonoic, scleronoic vector valued function of constraints requiring g(q) = 0 R c. The echanical syste containing slow and fast dynaics is characterized by a split of the configuration q into slow variables q s R ns and fast variables q f R nf with q = (q s, q f ) and n s +n f = n. Furtherore, we assue that thotential energy can be split into a slow potential V (q) and a fast potential W (q f ). Note that if the slow potential depends only on the slow variables, the fast potential only on the fast variables and the inetic energy and the constraints do not couple slow and fast influences, then the syste is fully decoupled, which is the trivial case and not investigated here. Typically systes are coupled i.e. one of thotentials or the constraints or both depend on slow and fast variables. In this wor, we concentrate on probles where the slow potential and the constraints depend on the coplete configuration and therefor couple the syste. The Lagrange function with the split potential is L(q, q) = T ( q) V (q) W (q f ). In the tie interval [t, t + ], the action integral S(q, λ) = t + t L(q, q) g T (q) λ dt () consists of the Lagrangian, the constraints and the Lagrange ultipliers λ(t) R c. Via Hailton s principle, stating that the trajectory q(t) is a stationary point of the action, δs = 0, the constrained Euler- Lagrange equations are derived as d T dt q s V ( g q s q s d T dt q f V q f W q f ( g q f ) T λ = 0 ) T λ = 0 g(q) = 0. (). Variational ultirate integration In the discrete setting, instead of one tie grid, two tie grids are chosen, one acro and one icro tie grid. Figure shows that the icro tie grid subdivides the acro tie grid into p equal tie steps with

3 Figure. Macro and icro tie grid p the nuber of icro steps. Note that on every acro tie node two icro tie nodes coincide, except for the first and the last acro tie node. The acro tie grid has the acro tie step T and the icro tie grid has the icro tie step t. For the acro and icro tie step, T = p t holds. The acro tie grid provides the doain for the discrete slow variables qd s = {qs }N =0 with qs qs (t ), while the icro tie grid provides the doain for the discrete fast variables q f d = {qf }N =0 wherf = {qf, } p =0 with q f, q f (t ) and the discrete Lagrange ultipliers λ d = {λ } N =0 where λ = {λ }p =0 with λ λ(t ). Instead of discretising the differential equation in Equation (), the variational principle δs = 0 is discretised. The discrete Lagrangian approxiates the action of the Lagrange function with the split potential for one tie step t+ L d (q, s q+, s q f ) T (q, q) V (q) W (q f ) dt. (3) t The discrete constraints are approxiated in the sae way h d (q s, q s +, q f, λ ) t + t g T (q) λ dt. (4) The discrete action is the su over all tie steps and approxiates the action integral in Equation () S d (q s d, q f d, λ d) = N =0 L d (q s, q s +, q f ) h d(q s, q s +, q f, λ ). (5) Via a discrete for of Hailton s principle, requiring stationarity for the discrete action, δs d = 0, the discrete constrained ultirate Euler-Lagrange equations are derived. Assuing that q s, qs, qf,0,..., qf,p are given, solving the following equations deterines q+ s, qf,,..., qf,p and λ 0,..., λp for =,..., N and =,..., p, D q s ( L d (q s, q s +, q f ) + 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 ) ) = 0 D q f,0 ( L d (q, s q+, s q f ) + 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 ) ) = 0 ( ) L d (q, s q+, s q f ) + h d(q, s q+, s q f, λ ) = 0 D q f, D λ h d (q s, q s +, q f, λ ) = 0 D λ +h d (q, s q+, s q f, λ ) + δ,p ( δ,n )D λ 0 + h d (q+, s q+, s q f +, λ +) = 0. (6) The Dirac delta δ is used here, to add the contribution of the constraints at the acro tie nodes. See [5] for further details, in particular on the initialization of throcedure..3 Quadrature rules To approxiate the integral of the Lagrangian and the constraints in Equation (3) and in Equation (4) by discretuantities, quadrature rules are needed. Different quadrature rules can be chosen for the inetic

4 energy, both potential energies and the constraints, which gives a wide range of possible cobinations. The velocities in the inetic energy are approxiated by finite differences. To ensure that the constraints are fulfilled on every tie node, the trapezoidal rule is used for the approxiation of the integral of the constraints. The integrals of the slow and the fast potential energies are approxiated by either the idpoint rule or an affine cobination. Cobining thesuadrature rules yields the following three schees. Fully iplicit schee Here, the idpoint rule is used in both potential energies. The nonlinear syste of equations which arises fro this quadrature in Equation (6) is solved iteratively with a Newton-Raphson schee. With a linear interpolation between two acro nodes for = 0,..., p, the slow and fast potential read q s, p V d (q, s q+, s q f ) = =0 p W d (q f ) = =0 = p ((p )qs + q s +) tv tw (( q s, (( q f, + q s,+, qf, )) + q f,+ )) + q f,+,. (7) Explicit slow, iplicit fast schee In this variant of the variational ultirate integrator, the slow variables can be coputed in an explicit way while the equations are iplicit in the fast variables, siilar to [9] (see also [3]). The slow potential is therefor approxiated by an affine cobination with α [0, ] ( ) V d (q, s q+, s q f ) = T αv ((q, s q f,0 )) + ( α)v ((qs +, q f,p )). (8) For the fast potential the idpoint rule is used as in Equation (7). Fully explicit schee This version uses the affine cobination in the slow potential as well as in the fast potential with α [0, ], i.e. Equation (8) and p W d (q f ) = =0 ( tα W (q f, ) ) + ( α)w (q f,+. ) 3 NUMERICAL EXAMPLES In this section, the three exaple systes are introduced, naely the Feri-Pasta-Ula proble abbreviated by FPU, a triple spherical pendulu abbreviated by TSP and a siple atoic odel abbreviated as SAM. 3. Feri-Pasta-Ula proble The Feri-Pasta-Ula proble consists of l asses lined with alternating soft and stiff springs as shown in Figure, where it is illustrated that the slow variables are the center of the stiff springs and the fast variables are the lengths of the stiff springs. The FPU is not subject to constraints. For l asses the inetic energy and the slow and fast potential read T ( q) = qt M q [ ] V (q) = l (q s q f 4 )4 + (qi+ s q f i+ qs i q f i )4 + (ql s + q f l )4 W (q f ) = ω l i= (q f i ) i=

5 Figure. Exaple without constraints: Feri-Pasta-Ula Proble with slow and fast variables with M R l l the ass atrix where all asses are equal to one and ω = 500 is the stiffness of the stiff spring. 3.. Triple spherical pendulu The triple spherical pendulu consists of three asses, two rigid lins and one stiff spring shown in the left hand side of Figure 3. The slow variables are thlaceent q R 3 of the first ass slow = 00. The variables q, q 3 R 3 of the two asses fast = fast 3 = are the fast variables. The inetic energy, the slow and fast potential are given by T ( q) = qt M q V (q) = q T M ḡ W (q f ) = ω((q 3 q ) l 3) where M R 9 9 is the ass atrix ḡ R 9 contains the gravity vectors with gravity constant g = 9.8, ω = 5000 is the stiffness of the spring, and l 3 = 3 the unstretched length of the spring. The constraints of constant distance between slow and the origin and fast and slow read g(q) = ( ) (q l) ((q q ) l) with l = 0 and l = Siple atoic odel The siple atoic odel consists of six asses, lined with four rigid lins, two soft and three stiff springs, see right hand side of Figure 3. The slow variables are thlaceent q, q R 3 of the first two asses = 00 and = 40. Thlaceent q 3, q 4, q 5, q 6 R 3 of the asses 3 = 4 = 5 = 6 = are the fast variables. The asses 3, 4 and 5 for a triangle. With, they for a tetrahedron, a very coon structure in olecules, e.g. in carbohydrates. The ass 6 is attached with a rigid lin to 3. With the variables split in this way, the inetic energy and the slow and fast potential read T ( q) = qt M q V (q) = [ ω ((q q ) l ) ] + [ ω ((q q 3 ) l3 ] + qt M ḡ W (q f ) = [ω 3((q 3 q 4 ) l 6) + ω 4 ((q 4 q 5 ) l 7) + ω 5 ((q 5 q 3 ) l 8)] with M R 8 8 ass atrix, ḡ R 8 contains the gravity vectors, ω = 0, ω = the stiffness of the soft springs, ω 3 = ω 4 = ω 5 = 4000 the stiffness of the stiff springs and l = l 3 = l 6 = l 7 = l 8 = 0 the lengths of the unstretched springs. The first constraint of constant distance between and the origin depends only on slow variables. The constraints between and 4 and and 5 depend on slow and fast variables. The last constraint between 3 and 6 depends only on fast variables. They read g(q) = (q l ) (q q 4 ) l 4 (q q 5 ) l 5 (q 3 q 6 ) l 9 where l = l 4 = l 5 = l 9 = 0 are the lengths of the rigid lins.

6 g e e e 3 l slow g slow l 4 l 5 l 3 4 l 7 fast 5 5 l 8 l 9 l l 6 3 fast 3 fast 4 fast 6 Figure 3. Exaples subject to constraints: triple spherical pendulu (left) siple atoic odel (right) with slow and fast variables fully iplicit FPU SAM tie in sec p Figure 4. Coputing ties for p {, 0,..., 00} and t = const for the SAM (green) and the FPU (blue), the red circles illustrate inial coputing tie 4 LIMITATIONS OF COMPUTING TIME SAVINGS FOR FPU In general it can be expected that the coputing tie decreases for an increase in p since less acro nodes arresent where the slow variables need to be deterined. In fact, this is the reason for using variational ultirate integration. However, in the coputing tie easureents for variational ultirate schees, above a certain threshold value of p the coputing tie does not decrease with the increase of icro steps per acro steps, see [3]. Figure 4 shows the coputing tie for the FPU and the SAM for the fully iplicit schee versus p. With an increase in p and a constant icro step size t = 0.0 for the FPU and t = 0.00 for the SAM the acro step size T = p t is increasing and at first the coputing tie is decreasing. At p = 0 for the SAM and at p = 60 for the FPU, the coputing tie starts to increase again. Thesoints are illustrated by the red circles in the figure and are referred to as inial coputing tie at optial p = p opt. At thesoints, the tie to solve the linear syste of equations in the Newton iteration is starting to doinate the coputing tie, see the explanation in [3]. This behavior is dependent on the nuber of variables X = n s + pn f of the syste. The nuber of variables X of the inial coputing ties for the SAM is X = 307 and for the FPU it is X = 83. This otivates to investigate thuestion whether the inial coputing tie is at approxiately the sae X for different echanical probles. The nuber of degrees of freedo in the FPU can be varied with the nuber of asses l. Measuring the coputing tie for increasing p, we loo at which nuber of icro steps p opt and nuber of variables

7 nuber of asses fully iplicit nuber of asses explicit slow, iplicit fast nuber of asses fully explicit tie in sec tie in sec tie in sec p p p Figure 5. Coputing ties for the FPU for p {, 4, 8,..., 00} for the fully iplicit schee schee (left) with l {0,... 80}, the explicit slow, iplicit fast schee (iddle) and the fully explicit (right) schee with l {0,..., 00} X the inial coputing tie is reached. Figure 5 shows the coputing tie easureents for p {, 4, 8,... 00} for the fully iplicit schee with l {0,..., 80} and for the explicit slow, iplicit fast and the fully explicit schee with l {0,..., 00}. The different lines represent the nuber of asses and l increases fro botto curve to top curve. The red circles illustrate inial coputing ties. The figure shows that with an increase in l the inial coputing tie is at a lower p opt (red circles drift to the left while going upwards through the curves) which leads to the conjecture that the inial coputing tie could be at the sae X. However, starting at a certain nuber of asses l, the inial coputing tie stays at the sa opt for different l. This can be seen for all threuadrature schees in Figure 5: for the fully iplicit schee at p opt = 8 in the range of l = {90, 00} and p opt = in the range of l = {70, 80}, for the explicit slow, iplicit fast at p opt = 8 in the range of l = {0,..., 80} and p opt = in the range of l = {70,..., 0}, and for the fully explicit schee for p opt = 4 in the range of l = {40,..., 00} and p opt = 8 in the range of l = {70,..., 30}. For the fully iplicit schee with l 0 and for the explicit slow, iplicit fast schee with l 90, we hav opt =, i.e. the ultirate schee does not yield coputing tie savings anyore copared to the singlerate variational integrator. We draw the conclusion that the inial coputing tie is not only depending on the nuber of variables X but is highly proble dependent. 5 NUMERICAL CONVERGENCE ANALYSIS Next, the results of nuerical convergence analysis arresented. For this, a reference solution is coputed with singlerate integration on a fine tie grid for all quadrature schees. For the investigation of convergence, two scenarios arossible. Here, we loo at the scenario where the nuber of icro steps is constant and the two tie step sizes decrease i.e. p = const, T 0, t 0. The error of the configuration variable is then given by = sup = 0,..., N =,..., p { q q(t ), q f, q(t ) } (9) and for the conjugate oentu the error is defined analogously. First, we consider the convergence of the FPU being an unconstrained syste, then we investigate the TSP as an exaple of a syste subject to constraints. 5. Feri-Pasta-Ula The reference solutions is coputed with a tie step T = 0 6 for the configuration, and the conjugate oenta are coputed in post processing via the discrete Legendre transfor, see [5]. For the FPU, siulations arerfored with two constant nubers of icro steps p = 5 and p = 0 and both tie step sizes getting saller as T {0., 0.05, 0.0, 0.0, 0.005, 0.00, 0.00, , 0.000, 0.000} and t = T p.

8 0 qrls: expl qrlf: expl p= 5, Hhdec pfix 0 qrls: expl qrlf: expl p= 0, Hhdec pfix error q, p, icac 0 0 error q, p, icac Figure 6. Nuerical convergence of configuration (blue) and conjugate oentu (red) for FPU with fully explicit schee using p = 5 (left) and p = 0 (right) 0 qrls: expl qrlf: idp p= 5, Hhdec pfix 0 qrls: expl qrlf: idp p= 0, Hhdec pfix error q, p, icac 0 0 error q, p, icac Figure 7. Nuerical convergence of configuration (blue) and conjugate oentu (red) for FPU with explicit slow, iplicit fast quadrature using p = 5 (left) and p = 0 (right) Fully explicit schee Figure 6 shows the global error of the configuration and the conjugate oentu versus the acro tie steps size T. In both plots, the convergence of the configuration and the conjugate oentu is of order one. Explicit slow, iplicit fast schee In Figure 7, the global error of the configuration and the conjugate oentu versus the acro tie step size T is presented. Again, for p = 5 and p = 0 the figure shows a convergence of order one for the configuration and the conjugate oentu. Fully iplicit schee For the fully iplicit quadrature schee, Figure 8 illustrates the global error of the configuration and the conjugate oentu versus the acro tie step size T again for p = 5 and p = 0. Both plots show a convergence of order two for the configuration and the conjugate oentu. Rear. These are the expected convergence rates for all quadrature schees. 5. Triple spherical pendulu For the TSP, the sarocedure is perfored as for the FPU. First, a reference solution is coputed with a tie step size T = 0 5. Again, we loo at the scenario where the nuber of icro steps is constant and the acro and icro tie step size decrease, i.e. p = const, T 0, t 0. Then, siulations are perfored for two nubers of icro steps, for p = 5 and p = 0. The global error of the configuration and the conjugate oentu is coputed for the siulations with Equation (9).

9 0 0 qrls: idp qrlf: idp p= 5, Hhdec pfix 0 0 qrls: idp qrlf: idp p= 0, Hhdec pfix 0 ï 0 ï 0 ï 0 ï error q, p, ic ac 0 ï 3 0 ï 4 0 ï 5 error q, p, ic ac 0 ï 3 0 ï 4 0 ï 5 0 ï 6 0 ï 6 0 ï 7 0 ï 7 0 ï 8 0 ï 4 0 ï 3 0 ï 0 ï 0 ï 8 0 ï 4 0 ï 3 0 ï 0 ï Figure 8. Nuerical convergence of configuration (blue) and conjugate oentu (red) for FPU with fully iplicit quadrature using p = 5 (left) and p = 0 (right) 0 3 qrls: idp qrlf: idp p= 5, Hhdec pfix 0 3 qrls: idp qrlf: idp p= 0, Hhdec pfix error q, p, icac error q, p, icac Figure 9. Nuerical convergence of configuration (blue) and conjugate oentu (red) for TSP with fully iplicit schee using p = 5 (left) and p = 0 (right) Fully iplicit schee Figure 9 shows the global error for the TSP using the fully iplicit quadrature versus acro tie step T. It illustrates that the constrained syste has a convergence of order two. Fully explicit schee In Figure 0, the global error for configuration and conjugate oentu versus tie step size T is illustrated. Thlot shows convergence of order one for p = 5 and p = 0. 6 CONCLUSION AND OUTLOOK It is shown that the iniu coputing tie and the optial nuber of icro nodes per acro tie step is not only dependent on the nuber of variables X. That the inial coputing tie for different nubers of asses in the FPU proble is at the sae optial nuber of icro steps indicates that the inial coputing tie is depending on the specific echanical proble. The convergence is presented with nuerical convergence results for the FPU and the TSP. For the fully explicit and the explicit slow, iplicit fast quadrature schees for the FPU the convergence is of oder one. The convergence of order two is shown for the fully iplicit schee for the FPU. Constraints in the TSP are ipleented with the Lagrange ultipliers ethod. To siulate the TSP, an index three differential algebraic equation is discretised. For this, the convergence for the fully iplicit schee is of order two and for the fully explicit schee of order one. In the future, it would be interesting to use the null space ethod [, 6] in cobination with the variational ultirate integrator, because it reduces the nuber of variables to a inial set. This could yield ore

10 0 qrls: expl qrlf: expl p= 5, Hhdec pfix 0 qrls: expl qrlf: expl p= 0, Hhdec pfix error q, p, icac 0 0 error q, p, icac Figure 0. Nuerical convergence of configuration (blue) and conjugate oentu (red) for TSP with fully explicit schee using p = 5 (left) and p = 0 (right) coputing tie savings, especially for ultibody systes. REFERENCES [] Arnold M.: Multi-rate tie integration for large scale ultibodysyste odels. In P. Eberhard (Ed.) IUTAM Syposiu on Multiscale Probles in Multibody Syste Contacts, pp. 0, Springer, 007. [] Betsch P.; Leyendecer S.: The discrete null space ethod for the energy consistent integration of constrained echanical systes. Part II: Multibody dynaics. International Journal for Nuerical Methods in Engineering, Vol. 67, pp , 006. [3] Gail T.; Leyendecer S.; Ober-Blöbau S.: Coputing tie investigations of variational ultirate integrators. In Z. Terze, M. Vrdolja (Ed.) Mulitbody Dynaics ECCOMAS Theatic Conference, pp , 03. [4] Lew A.; Marsden J. E.; Ortiz M.; West M.: Variational tie integrators. International Journal for Nuerical Methods in Engineering, Vol. 60 pp. 53, 004 [5] Leyendecer S.; Ober-Blöbau S.: A Variational approach to ultirate integration for constrained systes. In: J.C. Sain, P. Fisette (Ed.) Multibody Dynaics, Coputational Methods in Applied Sciences, Vol. 8, pp. 97, Springer, 03. [6] Leyendecer S.; Marsden J.E.; Ortiz M.: Variational integrators for constrained dynaical systes. Journal of Applied Matheatics and Mechanics, Vol. 88, pp , 008. [7] Weinan E.; Engquist B.; Li X.; Ren W.; Vanden-Eijnden E.: Heterogeneous ultiscale ethods: a review. Counications in Coputational Physics, Vol., No. 3, pp , 007. [8] Marsden J. E.; West M.: Discrete echanics and variational integrators. Acta Nuerica, pp , 00 [9] Stern A.; Grinspun E.: Iplicit-explicit variational integration of highly oscillatory probles. Multiscale Model. Siul., Vol. 7, No. 4, pp , 009.