VARIATIONAL INTEGRATORS FOR DYNAMICAL SYSTEMS …congress.cimne.com/iacm-eccomas2014/admin/files/filePaper/p1758.pdfduring the time integration. By applying structure preserving variations

11th World Congress on Computational Mechanics (WCCM XI)5th European Conference on Computational Mechanics (ECCM V)

6th European Conference on Computational Fluid Dynamics (ECFD VI)E. Onate, J. Oliver and A. Huerta (Eds)

VARIATIONAL INTEGRATORS FOR DYNAMICALSYSTEMS WITH ROTATIONAL DEGREES OF FREEDOM

THOMAS LEITZ∗, SINA OBER-BLOBAUM†

AND SIGRID LEYENDECKER∗

∗ Chair of Applied DynamicsUniversity of Erlangen-Nuremberg, Germany

e-mail: [email protected]: [email protected]

†Computational Dynamics and Optimal ControlDepartment of Mathematics

University of Paderborn, Germanye-mail: [email protected]

Key words: Variational Integrators, Lie-Group Integrators, Discrete Mechanics.

Abstract. For the elastodynamic simulation of a geometrically exact beam, a variationalintegrator is derived from a PDE viewpoint. Variational integrators are symplectic andconserve discrete momentum maps and since the presented integrator is derived in the Liegroup setting (unit quaternions for the representation of rotational degrees of freedom), itintrinsically preserves the group structure without the need for constraints. The discreteEuler-Lagrange equations are derived in a general manner and then applied to the beam.

1 INTRODUCTION

Instead of deriving and subsequently discretizing the equations of motion for a mechan-ical system, variational integrators are derived from a discrete variational principle. Theresulting integrators inherit the symplectic structure of the continuous Euler-Lagrangeequations and a discrete Noether theorem can be proven. Furthermore, they show verygood energy behaviour, i.e. for conservative systems there is no drift in total energy,though the energy is not exactly preserved. Variational integrators are developed andtheir properties have been studied by Marsden et al. [1].

Rotational degrees of freedom can e.g. be described by elements of the Lie group SO (3)or, alternatively in this case, by unit quaternions. Unit quaternions reduce the numberof unknowns from nine to only four and give rise to the possibility that computing timemight be reduced due to fewer computer operations and smaller memory requirementsduring the time integration. By applying structure preserving variations on the Lie-group

1

Thomas Leitz, Sina Ober-Blobaum and Sigrid Leyendecker

in order to derive the equations of motion, the need for constraints is eliminated and theresulting integrator falls into the class of variational Lie-group integrators.

In geometrically exact beam dynamics, see e.g. Simo [2], the need to represent rotationaldegrees of freedom, computing time concerns, as well as the desire to simulate accuratelycome together. Recent works dedicated to a Lie-group formulation in multibody dynamicsare [3] and [4] and in beam dynamics are [5] and [6]. Quaternions have been used todescribe the rotational degrees of freedom in beam dynamics by Lang et al. [7].

2 QUATERNIONS AND ROTATIONS

The group of quaternions H consists of four dimensional tuples. Their basis is givenby

1, i, j, k i2 = j2 = k2 = ijk = −1

and thus a quaternion can be written as p = p0 + p1i + p2j + p3k where pn ∈ R forn = 0, 1, 2, 3. The group operation is quaternion multiplication which we denote by p r,p, r ∈ H and results in

p r = (p0 + p1i+ p2j + p3k) (r0 + r1i+ r2j + r3k)

= (p0r0 − p1r1 − p2r2 − p3r3)+ (p0r1 + p1r0 + p2r3 − p3r2) i+ (p0r2 + p2r0 + p3r1 − p1r3) j+ (p0r3 + p3r0 + p1r2 − p2r1) k

We write < for the real part and = for the imaginary part of a quaternion. A quaternioncan thus be written as p = <p + =p. The conjugate of a quaternion is denoted byp = <p−=p. The Lie-group of unit quaternions is defined as

H1 := p ∈ H | ‖p‖ = 1

and it can be used to represent rotations. In case of the rotation of a three dimensionalvector s ∈ R3, we consider s as a purely imaginary quaternion and write

Λ (p) s = p s p

where Λ (p) ∈ SO (3) represents the rotation matrix corresponding to the unit quaternionp ∈ H1. The Lie-algebra of H1 is defined as the purely imaginary quaternions

h1 := η ∈ H | <η = 0

with the exponential map exp η = cos

(‖η‖2

)+

η

‖η‖sin

(‖η‖2

)∈ H1 which represents

the rotation around the unit vector n =η

‖η‖with rotation angle Θ = ‖η‖.

2


3 GROUP STRUCTURE PRESERVING VARIATIONS

When computing the variation of a function f : H1 → R in terms of the variationof the group element, special care has to be taken in order not to violate the groupstructure. The group structure is inherently respected by performing variations in termsof the group operation, namely quaternion multiplication. Consequently, the variation ofp ∈ H1 is given by

δp =d

dε

∣∣∣∣ε=0

p δpε

where δpε can be expressed by the exponential map of a purely imaginary quaternion εηin the Lie algebra h1 where ε ∈ R is a small parameter. Using η = Θn, where Θ ∈ R isthe rotation angle and n ∈ H1 is a purely imaginary unit quaternion, the variation of therotation dependent function is given by

δf (p) =d

dε

∣∣∣∣ε=0

f (p exp (εη)) =∂f

∂p·(

1

2p η

)=

1

2

(p ∂f

∂p

)· η =

1

2=(p ∂f

∂p

)· η

Here we use the fact, that the scalar product of a real number with a purely imaginaryquaternion is zero, and thus only the imaginary part, denoted by =, remains. Using theabove, the variation of functions of the angular velocity ω = 2p p and the angular strainΩ = 2p p′ (to be introduced in Section 4) can be found as

δf (ω) =∂f

∂ω· (ω × η + η) δf (Ω) =

∂f

∂Ω· (Ω× η + η′)

4 CONTINUOUS EULER-LAGRANGE EQUATIONS

Consider a one dimensional reference configuration space variable s ∈ [0, `] ⊂ R, thetime variable t ∈ [0, T ] ⊂ R and the deformation map ϕ : (s, t) 7→ (p, x) ∈ H1 × R3. TheLagrange density L (p, ω,Ω, x, x, x′) is a function of the configuration (p, x) being com-posed by orientation and position in the ambient space, the angular velocity ω = 2pp ∈ h1

and bending and torsional strain Ω = 2p p′ ∈ h1 as well as the translational velocity xand the shear and elongational strain x′, where

˙(·) =d (·)dt

(·)′ = d (·)ds

The action functional is defined as the integral of the Lagrange density over space andtime

S [ϕ] =

T∫

0

`∫

0

Ldsdt

3


Following Hamilton’s principle, the action is stationary with respect to all variations whileholding the boundaries of space and time fixed, i.e. δp = 0, δx = 0 for t = 0, t = T, s = 0and s = `. The resulting Euler-Lagrange equations are

1

2=(p ∂L

∂p

)− ω × ∂L

∂ω− d

dt

(∂L

∂ω

)− Ω× ∂L

∂Ω− d

ds

(∂L

∂Ω

)

∂L

∂x− d

dt

(∂L

∂x

)− d

ds

(∂L

∂x′

)

=

000000

(1)

This coupled system of partial differential equations represents a local balance of angularand linear momentum. The temporal and spatial Legendre transforms for the rotationaland translational part represent the conjugate momenta of time and space

Π =∂L

∂ωΓ =

∂L

∂xΣ =

∂L

∂Ωσ =

∂L

∂x′(2)

In classical mechanics, Π and Γ are angular and linear momenta per unit length whileΣ represents bending and torsional torques and σ are normal and shear forces, see alsoSection 6 on geometrically exact beam dynamics.

5 DISCRETE EULER-LAGRANGE EQUATIONS

In order to derive a variational integrator, space time is discretized on a regular grid asshown in Figure 1. This grid is not necessarily equidistant in space, though the time stepis kept constant in order to allow backward error analysis [8]. Thus, all spatial elementsadvance in time synchronously in contrast to asynchronous integrators developed by Lew[9] and for the beam by Demoures et al. [5]. The discrete Lagrangian is an approximationof the continuous action for one space time element Kj

a, i.e.

Lja ≈

∫ tj+1

tj

∫ sa+1

sa

Ldsdt (3)

and it is generally a function of all four nodes of the space time element Kja

Lja = Lj

a

(pja, p

j+1a , pja+1, p

j+1a+1, x

ja, x

j+1a , xja+1, x

j+1a+1

)(4)

Thus, the discrete action is the sum over all discrete Lagrangians

Sd =N−1∑

j=0

A−1∑

a=0

Lja (5)

4


(pj+1a , xj+1

a

) (pj+1a+1, x

j+1a+1

)

(pja+1, x

ja+1

)(pja, x

ja

)

Kja

sAs0

t0

tN

sa

tj

Kja

s

t

K00

KN−1A−1

Figure 1: Regular discretization of the space time for variational integrators.

The discrete Hamilton’s principle states, that the discrete action is stationary for allvariations vanishing on the boundary of space and time, i.e. ηja = 0 and δxja = 0 for a = 0,a = A, j = 0 and j = N . This leads to the discrete version of (1)

1

2=

(pja

∂Lja

∂pja+ pja

∂Lj−1a

∂pja+ pja

∂Lja−1

∂pja+ pja

∂Lj−1a−1

∂pja

)

∂Lja

∂xja+∂Lj−1

a

∂xja+∂Lj

a−1

∂xja+∂Lj−1

a−1

∂xja

=

000000

(6)

which has to be supplemented by appropriate boundary conditions.

5.1 Solving the discrete Euler-Lagrange equations

For the quadrature rules introduced in Section 5.2, the system decouples to independentsets of six equations for each node a = 0, . . . , A

Rpa

(pj+1a , xj+1

a

)= 0 ∈ R6

In any case, there are seven unknowns and only six discrete Euler-Lagrange equationsfor each node in the grid, if one regards the orientation quaternion naively as a four-dimensional vector. In order to avoid normality constraints to ensure that pj+1

a ∈ H1, weuse the reparametrization pj+1

a = pja Cay f ja where Cay : h1 → H1 is the Cayley map

and f ja ∈ h1 is the increment. The singularity of the Cayley map is avoided by the fact

that the rotation increment Cay f ja is close to the identity for small time steps and thus

far away from rotation about the angle π. The nodal discrete Euler-Lagrange equationsare transformed to

Rfa

(f ja , x

j+1a

)= 0 ∈ R6

5


The equations can be solved for f ja and xj+1

a by a Newton-Raphson scheme and the neworientation is recovered by pj+1

a = pja Cay f ja . The quaternion Cayley map is defined as

Cay f =1 + f

‖1 + f‖

5.2 Quadrature rules

In order to derive the integrator, we have to be more specific in terms of choosingthe discrete Lagrangian (4). Let the generalized coordinate q represent x and p and the

partial derivatives∂ (·)∂q

comprise∂ (·)∂x

and1

2=(p ∂ (·)

∂p

), respectively. Note that we

denote the discrete Lagrangian in the element Kja by Lj

k (instead of Lja) from now on.

Furthermore, time nodes associated with this element are denoted by tjk. As shown in(3) and (4), the discrete Lagrangian Lj

k is an approximation of the action integral overthe space time element, and can generally depend on four nodes. In the following, weintroduce a choice of approximation based on four nodes. To this end, we partition thespace time element Kj

a into four parts denoted by roman numerals as shown in Figure 2.With ∆t = tj+1

k − tjk = const. for all j, k and ∆sk = sa+1 − sa, the time derivatives xjaand space derivatives x′ja are approximated by a forward difference quotient that takesthe form

xja =xj+1a − xja

∆tx′

ja =

xja+1 − xja∆sk

(7)

for the translational degrees of freedom. The temporal and spatial derivatives for therotational degrees of freedom are given by

ωja =

2

∆t=(pja pj+1

a

)Ωj

a =2

∆sk=(pja pj+1

a

)(8)

The discrete Lagrangian is decomposed into

Ljk

(qja, q

j+1a , qja+1, q

j+1a+1

)= ILj

k + IILjk + IIILj

k + IVLjk

where the four parts consist of evaluations of the Lagrangian at different nodes and dif-ference quotients1

IVLjk =

∆t∆sk4

L(qj+1a , qja, q

′j+1a

)IIILj

k =∆t∆sk

4L(qj+1a+1, q

ja+1, q

′j+1a

)

ILjk =

∆t∆sk4

L(qja, q

ja, q′ja

)IILj

k =∆t∆sk

4L(qja+1, q

ja+1, q

′ja

)

1Of course, other evaluations of the Lagrangian depending on a different number or combinations ofnodes are possible, as e.g. an evaluation at midpoints of nodes in the first argument of L (cf. [10]).

6


qj+1a qj+1

a+1

qja+1qja

ILjk

tjk

tj+1k

sa sa+1

IILjk

IIILjk

IV Ljk

Figure 2: Partitioning of the space time element Kja for the integrator and the general case of L =

L (q, q, q′).

The discrete Euler-Lagrange equations (6) for this specific quadrature are

0 =1

4

(∂ ILj

k

∂qja+∂ IILj

k

∂qja+∂ IVLj

k

∂qja

)+

1

4

(∂ ILj−1

k

∂qja+∂ IIILj−1

k

∂qja+∂ IVLj−1

k

∂qja

)

+1

4

(∂ ILj

k−1

∂qja+∂ IILj

k−1

∂qja+∂ IIILj

k−1

∂qja

)+

1

4

(∂ IILj−1

k−1

∂qja+∂ IIILj−1

k−1

∂qja+∂ IVLj−1

k−1

∂qja

)(9)

Quadrature rules of this type lead to an integrator with independent sets of six non linearequations for each spatial node in one time step.

6 GEOMETRICALLY EXACT BEAM DYNAMICS

Modeling geometrically exact beams as a special Cosserat continuum (see e.g., [11]) hasbeen the basis for many discrete formulations starting with [2, 12, 13, 14]. Its Lagrangiandynamics is an example of the formulation described in Section 4.

6.1 Continuous Euler-Lagrange equations

The beam model consists of a central line x(s, t), a curve in the three dimensionalspace, along which rotation matrices Λp = Λ (p (s, t)) represent the orientations of thelocal beam cross sections. Consequently, the cross sections are assumed to stay plane.Hereby, the curve parameter is s ∈ [0, `], where ` denotes the length of the beam in thereference configuration. As mentioned in Section 4, in addition to the configuration, theLagrange density is a function of the angular velocity ω = 2p p ∈ h1 and bending andtorsional strain Ω = 2p p′ ∈ h1 as well as the translational velocity x and the shear and

7


elongational strain x′

L (p, ω,Ω, x, x, x′) =1

2

(ρ ‖x‖2 + ωTJω

)

− 1

2

[(ΛT

p x′ − e3

)TC1

(ΛT

p x′ − e3

)+ ΩTC2Ω

]+ ρ 〈x, g〉 (10)

Here, the kinetic energy density contains the mass density ρ, the mass moment of in-ertia density tensor J , while the gravity potential energy density depends on the grav-ity constant g. In the potential deformation energy density, the diagonal matrix C1 =Diag (GA GA EA) contains the shear and elongation stiffness, while the entries ofC2 = Diag (EI1 EI2 G (I1 + I2)) are the bending and torsional stiffness. Here, A isthe cross section area, I1, I2 are the principal area moments of inertia of the cross section,and the elastic properties are represented by Young’s modulus E and the shear modulusG. Insertion into (1) yields the Euler-Lagrange equations for the beam

(ΛT

p x′)×

(C1

(ΛT

p x′ − e3

))− ω × (Jω)− Jω + Ω× (C2Ω) + C2Ω

′ = 0

−ρg + ρx+ Λp

(Ω× C1

(ΛT

p x′ − e3

))+ ΛpC1

((Λ′p)Tx′ + x′′

)= 0

The conjugate momenta (2) are given by

Π =∂L

∂ω= Jω Γ =

∂L

∂x= ρx

Σ =∂L

∂Ω= −C2Ω σ =

∂L

∂x′= −ΛpC1

(ΛT

p x′ − e3

)

They can be identified as angular momentum per length Π, linear momentum per lengthΓ, and Σ represents the bending and torsional momenta while σ comprises shear andelongation forces2. In case of the beam, in addition to prescribing the position or orienta-tion at specific nodes, further boundary conditions in space are the momenta Σ0,ΣA andforces σ0, σA on the end nodes of the beam. Further boundary conditions in time are theangular momenta Π0,ΠN in the beginning and the end of time, as well as linear momentaΓ0,ΓN .

7 NUMERICAL RESULTS

The integrator is implemented in Matlab R© using the discretization discussed above.The discrete Euler-Lagrange equations are derived using Matlab R©’s symbolic toolbox andautomatic code generation.

2The unit of the Lagrangian density isJ

m= N , i.e. energy per length. Therefore, the units for the

conjugate momenta are

[Π] =Ns

m=

kgm2

sm

[Γ] =Ns

m=

kgm

sm

[Σ] = Nm [σ] = N

8


In order to illustrate the performance of the integrator, the dynamics of a beam withthe physical parameters shown in Table 1 is simulated according to Section 5. The firstnode of the beam is translationally fixed and the initial configuration consists of thebeam hanging downward (in the negative e3-direction). During the first two seconds ofthe simulation time, a torque is applied to the first node. During that timeframe, theenergy and the linear and angular momenta of the beam are expected to change, whileafter two seconds, we expect the total energy to be nearly constant and the e3-componentof the angular momentum Πj

3 to be exactly preserved, since the Lagrangian is invariantwith respect to time and rotations around the e3-axis. Figure 3 illustrates the expectedbehavior in the plots for the energy and the linear and angular momenta, while Figure 4presents snapshots of the configuration of the beam at different times. Figure 5 shows theevolution of the conjugate momenta at the nodes a = 0, a = 5 and a = 11. The torque Σat node a = 0 is the actuation that decays after 2 seconds. Afterwards, the orientation atthis node is free and thus the reaction momentum is zero. However, the position of nodea = 0 is fixed, thus, the reaction force is non zero. This is in contrast to node a = 11,since at the free end of the beam, reaction forces and momenta are always zero.

Comparing this implementation to a similar implementation using orthonormal matri-ces, i.e. elements of SO (3), for the representation of rotational degrees of freedom, thecomputation time is reduced by approximatly 20% for the same model and simulationparameters.

length 2m

cross section 0.01m× 0.01m

mass density 1000kg

m3

Young’s modulus 5 · 106 N

m2

Poisson ratio 0.35

number of elements 11

spatial element ∆sk =2

11m

time step size ∆t = 4 · 10−4s

simulation time 10s

Table 1: Simulation parameters.

8 CONCLUSIONS

Group structure preserving variations based on the representation of rotational degreesof freedom by unit quaternions is derived in the first part of this paper. It is then usedto derive continuous and discrete Euler-Lagrange equations from a PDE point of view.The discrete Euler-Lagrange equation are applied to geometrically exact beam dynamicsand the resulting variational integrator shows all of the expected properties of thesetypes of integrators. The implementation using quaternions shows a significantly reduced

9


0 2 4 6 8 10−2

−1

0

1

2

time in s

ener

gyin

J

TVE

0 2 4 6 8 10−0.5

0

0.5

1

time in s

linea

rm

om.

inkgm

2

s Γ1

Γ2

Γ3

0 2 4 6 8 10

−0.5

0

0.5

1

time in s

angu

lar

mom

.in

kgm

2

s Π1

Π2

Π3

Figure 3: Energy, linear and angular momenta for the simulation of the beam.

t = 0.0s t = 0.3s t = 0.6s t = 0.9s

Figure 4: Snapshots of the beam configuration. The color gradient indicates the absolute torque.

computation time compared to a similar implementation using orthonormal matrices,i.e. elements of SO (3), for the representation of rotational degrees of freedom.

10


0 2 4 6 8 10−2

0

2

·10−3

time in s

forc

ein

N

a = 0

σ1σ2σ3

0 2 4 6 8 10−2

02468·10−4

time in s

torq

ue

inN

m

a = 0

Σ1

Σ2

Σ3

0 2 4 6 8 10

−1

0

1

2·10−3

time in s

forc

ein

N

a = 5

σ1σ2σ3

0 2 4 6 8 10−1

0

1

2·10−4

time in s

torq

ue

inN

m

a = 5

Σ1

Σ2

Σ3

0 2 4 6 8 10−1−0.5

00.5

1

time in s

forc

ein

N

a = 11

σ1σ2σ3

0 2 4 6 8 10−1−0.5

00.5

1

time in s

torq

ue

inN

m

a = 11

Σ1

Σ2

Σ3

Figure 5: Forces and torques as stress resultants at nodes 0, 5 and 11.

REFERENCES

[1] J.E. Marsden and M. West. Discrete mechanics and variational integrators. ActaNumerica, 10:357–514, 2001.

[2] J.C. Simo. A finite strain beam formulation. the three-dimensional dynamic problem.part I. Computer Methods in Applied Mechanics and Engineering, 49(1):55 – 70, 1985.

[3] O. Bruls and A. Cardona. On the Use of Lie Group Time Integrators in MultibodyDynamics. Journal of Computational and Nonlinear Dynamics, 5, 2010.

[4] O. Bruls, A. Cardona, and M. Arnold. Lie group generalized-α time integration ofconstrained flexible multibody systems. Mechanism and Machine Theory, 48(0):121– 137, 2012.

11


[5] F. Demoures. Lie Group and Lie Algebra Variational Integrators for Flexible Beamand Plate in R3. PhD thesis, Ecole Polytechnique Federale de Lausanne, Lausanne,2012.

[6] F. Demoures, F. Gay-Balmaz, T. Leitz, S. Leyendecker, S. Ober-Blobaum, and T. S.Ratiu. Asynchronous variational lie group integration for geometrically exact beamdynamics. In ECCOMAS Thematic Conference on Mutlibody Dynamics, 1-4 July2013.

[7] H. Lang, J. Linn, and M. Arnold. Multi-body dynamics simulation of geometricallyexact cosserat rods. Multibody System Dynamics, 25(3):285–312, March 2011.

[8] E. Hairer, C. Lubich, and G. Wanner. Geometric numerical integration, volume 31of Springer Series in Computational Mathematics. Springer, 2002.

[9] A. Lew, J.E. Marsden, M. Ortiz, and M. West. Asynchronous variational integrators.Archive for Rational Mechanics and Analysis, 167(2):85–146, April 2003.

[10] J.E. Marsden, G.W. Patrick, and S. Shkoller. Multisymplectic geometry, variationalintegrators, and nonlinear PDEs. Communication in Mathematical Physics, 199:351–395, 1998.

[11] S.S. Antmann. Nonlinear Problems in Elasticity. Springer, 1995.

[12] J.C. Simo and L. Vu-Quoc. A three-dimensional finite-strain rod model. Part II:Computational aspects. Comput. Methods Appl. Mech. Engrg., 58:79–116, 1986.

[13] J.C. Simo and L. Vu-Quoc. On the dynamics in space of rods undergoing largemotions – A geometrically exact approach. Comput. Methods Appl. Mech. Engrg.,66:125–161, 1988.

[14] J.C. Simo, J.E. Marsden, and P.S. Krishnaprasad. The Hamiltonian structure ofnonlinear elasticity: The material and convective representations of solids, rods, andplates. Archive for Rational Mechanics and Analysis, 104:125–183, 1987.

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VARIATIONAL INTEGRATORS FOR DYNAMICAL SYSTEMS …congress.cimne.com/iacm-eccomas2014/admin/files/filePaper/p1758.pdfduring the time integration. By applying structure preserving variations