A second order semi-discrete Cosserat rod model suitable fordynamic simulations in real time

Holger Lang and Joachim Linn

Fraunhofer ITWM, Fraunhofer Platz 1, 67663 Kaiserslautern, Germany

Abstract. We present an alternative approach for a semi-discrete viscoelastic Cosserat rod model that allows both fastdynamic computations within milliseconds and accurate results compared to detailed finite element solutions. The model isable to represent extension, shearing, bending and torsion. For inner dissipation, a consistent damping potential from Antmanis chosen. The continuous equations of motion, which consist a system of nonlinear hyperbolic partial differential algebraicequations, are derived from a two dimensional variational principle. The semi-discrete balance equations are obtained byspatial finite difference schemes on a staggered grid and standard index reduction techniques. The right-hand side of the modeland its Jacobian can be chosen free of higher algebraic (e.g.root) or transcendent (e.g. trigonometric or exponential)functionsand is therefore extremely cheap to evaluate numerically. For the time integration of the system, we use well established stiffsolvers. As our model yields computational times within milliseconds, it is suitable for interactive manipulation. Itreflectsstructural mechanics solutions sufficiently correct, as comparison with detailed finite element results shows.

Keywords: Flexible multibody dynamics, Large deformations, Small stresses and strains, Finite rotations, Constrained mechanical systems,Structural dynamics, Partial differential algebraic equations, Finite difference methods.PACS: 40.46.De

INTRODUCTION

The Cosserat rod model [1, 17] is a well known model for the geometrically exact simulation of deformable rods –i. e. slender one dimensional flexible structures – in quasistatics or dynamics. A Cosserat rod can be considered as thegeometrically nonlinear generalisation of a Timoshenko-Reissner beam. In contrast to a Kirchhoff rod, which can beconsidered as a geometrically nonlinear generalisation ofan Euler-Bernoulli beam, a Cosserat rod allows to modelnot only bending and torsion – these are ’soft’ dof –, but as well extension and shearing – these are ’stiff’ dof. For aCosserat rod the overall deformation as response to moderate external loads, i. e. displacements, forces or moments,may become large, although locally the stresses and strainsremain small. This article is concerned with a semi-discretefinite difference model [13] of a Cosserat rod that is firmly based on structural mechanics and applicable to computedynamical deformations very fast at sufficient accuracy. This is in contrast to the usual finite element approach,which is usually favoured in structural mechanics [3, 6, 10]. And this is in contrast to the usual way in multibodydynamics, where flexible structures are usually represented by vibrational modes of Craig and Bampton type [5, 16],which reflect only linear structural response. We derive thecontinuous equations of motion from a two dimensionalvariational principle, where we use unit quaternion parametrisation for the rotatory degrees of freedom. The aim is tohave a model, whose right hand side and Jacobian is cheap to evaluate. We remark that the Cosserat model is only a’skeleton’ model; the reconstruction of the three dimensional displacement, stress and strain distributions throughthecross sections of the rod can be conveniently carried out in apostprocessing by the use of ’warping functions’ [8].

THE MODEL

A continuous Cosserat rod is kinematically characterised by its centerlinex = x(s,t) : U → R3 and its unit quaternion

field p = p(s,t) : U → S3 = q ∈ H : ‖q‖ = 1, the latter inducing its attached frame fieldR p : U → SO(3)

via the Euler mapR : H → RSO(3), p 7→ (2p20 − ‖p‖2)I + 2p ⊗ p + 2p0E (p) with the Levi-Civita pseudotensor

E : R3 = ℑ(H)→ so(3), defined byE (u)v = u× v for u, v ∈ ℑ(H). HereU = [0,L]× [0,T ] is a rectangular domain in

space-time, wheres ∈ [0,L] is the arc length parameter of the centerline of the undeformed rod,t ∈ [0,T ] is the time.The strain vectorΓ, curvature vectorK and angular velocity vectorΩ, each inR3 = ℑ(H) are defined by

Γ = p∂sx p− k, K = 2p∂s p, Ω = 2p∂t p. (1)

Here for a quaternionp = p0+ p1i+ p2 j+ p3k ∈H = R4, p0 = ℜ(p) denotes its real part, ˆp = ℑ(p) = p1i+ p2 j+ p3k

its imaginary part and ¯p = p0− p1i− p2 j− p3k is its conjugate. The potential, dissipation and kinetic energy densitiesare given by

V =12

(

ΓTCΓΓ+ KTCKK)

, D = ΓTCΓΓ+ KTCKK, T =ρ2

(

A‖x‖2+ ΩT IΩ)

. (2)

HereCΓ andCK (CΓ andCK) denote some symmetric and positively definite (visco-)elastic constitutive 3×3 matricesandI is the geometric moment of inertia tensor [1, 11, 12, 17]. With q = (x, p), the constraint density 2g = ‖p‖2−1,exterior forcesF : [0,T ] → ℑ(H), momentsM : [0,T ] → S

3 and the LagrangianL = T −V − gT λ , the variationalprinciple

δ∫∫

UL d(s,t)−

∫∫

U

(

∂D

∂ qδq +

∂D

∂ q′δq′

)

d(s,t) =∫∫

U

(

Fδx +Mδ p)

d(s,t), ˙ =∂∂ t

, ′ =∂∂ s

yields the following system of nonlinear hyperbolic partial differential algebraic equations

ρAx = ∂s(

pF p)

+ pF p

ρ[

µ p− 12

∂p(

pT µ p)

+ ∂p(

µ p)

p

]

= 2x′pF + ∂s(

2pM)

+2p′M +2pM−λ p

0 = ‖p‖2−1

, (3)

which, with given appropriate initial values and boundary conditions, has to be solved [11]. In (3) the internal forcesand moments are given byF = CΓΓ + 2CΓΓ andM = CKK + 2CKK respectively. Hereµ = µ(p) denotes the 4×4quaternion mass matrix [11, 14]. The equivalence of (3) to the classical Cosserat equations [1, 17]

ρAx = ∂s(RF)+ RF

ρR(

IΩ+ Ω× IΩ)

= ∂s(RM)+ ∂sx× (RF)+ RM,

can be readily established [11]. For the method of lines, we use (1) in order to discretise the rod on a staggered gridin the space dimension. The latter means that, given discrete vertex positions 0= s0 < .. . < sN = L, the centroidsx0 ≈ x(s0, ·), . . ., xN ≈ x(sN , ·) are situated here, but the quaternionsp1/2 ≈ p(s1/2, ·), . . ., pN−1/2 ≈ p(sN−1/2, ·) belongto the segment midpointssν = (sν−1/2+sν+1/2)/2 for ν = 1/2, . . . ,N−1/2. Whereas the discretisation of the strainΓon the segment midpoints via finite differences is straightforward, for the discrete curvatureKn on the verticessn, wepropose several choices, depending on how each two neighbouring quaternionspn−1/2 andpn+1/2 are interpolated toa vertex quaternionpn for n = 1, . . . ,N −1, see [12]. Each of these results in a discrete curvature expressionKn that isframe indifferent by construction, i.e. indifferent with respect to superimposed rigid body motions. For the boundaryquaternionsp0 andpN , we use the well known ghost point technique [13].

The resulting semi-discrete model in index one formulationfinally becomes

xn =1

ρA

(

δδ sn

(

p·F· p·)

+ pnFn pn

)

pν =2ρ

µ(pν)−1(

4ρ pνI ˙pν pν +∆x·∆sν

pνFν +∆

∆sν

(

p·M·)

+∆p·∆sν

Mν + pνMν

)

−‖ pν‖2pν

λν = 2

⟨

pν ,4ρ pνI ˙pν pν +∆x·∆sν

pνFν +∆

∆sν

(

p·M·)

+∆p·∆sν

Mν + pνMν

⟩

(4)

with forcesF· = CΓΓ· + 2CΓΓ· and momentsM· = CKK· + 2CKK·. Heren = 0, . . . , N indicates vertex positions andν = 1/2, . . . , N − 1/2 indicates midpoint positions. For the the discrete difference operators in (4), several choicesexist. Table 1 summarises the total operation counts for several model variants based on first order finite differenceoperators. For higher order difference schemes, some care has to be taken. It is clear that (4) yields a consistentdiscretisation of (3).

The handling of the rotatory inertia terms is more or less standard [7, 9, 14]. We point out the fact that thecomputation of the spherical tangential inverseµ(p)−1 is exactly as expensive as the computation of the quaternionmass matrixµ(p) itself [11]. Discarding the equations for the Lagrange multipliers in (4) allows to express the modelin fully explicit form q = v, v = f (t, q, v). Stabilisation of the quaternion spherical constraints via the projection orthe Baumgarte method is extremely cheap, see as well the right column in table 1. Usually in finite element literature,the equivalent descriptionsE (K) = RT ∂sR, E (Ω) = RT ∂tR are used instead of (1), but this approach leads to thesophisticated problem to interpolate the rotations in the manifoldSO(3) in a cheap and objective way [2, 3, 4, 15].

TABLE 1. Upper estimates for the total operation counts for the righthand side functionf of(q, v) = f (q,v,t), q = (x, p), q = v, depending on the number of rod segmentsN.

OPS Basic model Variant 1 Variant 2 Variant 3 Variant 4 Stabilisation

+ 174N+34 +10N+10 +11N+11 +37N+37 -26N+00 +8N+00− 111N+36 +15N+15 +27N+27 +03N+03 -10N+00 +1N+00∗ 289N+90 +39N+39 +36N+36 +72N+72 -23N+00 +2N+00/ 3N+03 +30N+30 +31N+31 +03N+03 +03N+00 +0N+002 4N+00 +00N+00 +00N+00 +01N+01 +03N+00 +4N+00√ 0N+00 +00N+00 +01N+01 +01N+01 +01N+00 +0N+00

arccos 0N+00 +00N+00 +00N+00 +01N+01 +00N+00 +0N+00

NUMERICAL EXAMPLES

The total task to evaluatef (q,v,t) for the most robust curvature choice, which corresponds to variant 1, for non-equidistant discretisation and non-symmetric cross sections amounts to only 184N+44 additions, 126N+49 subtrac-tions, 328N+129 multiplications and 33N+33 divisions. The analytical Jacobian∂ f (q,v,t)/∂ (q,v,t), whose non-trivialsubparts∂ f/∂q and∂ f/∂v both have upper and lower bandwidths equal to ten, is about fifteen times as expensive asf if (quaternion skew) symmetry is exploited. Figure 1 showsexcellent agreement of the results of our model, com-puted withonly ten segments, to the 3D finite element solution, computed with the packageABAQUS with 160×12continuum elements, for a quasistatic scenario that displays non trivial coupling between bending and torsion. Otherdynamic and quasistatic examples as well showexcellent agreement both against 1D and 3D finite element solutions[12]. Convergence analysis reveals second order convergence of our finite difference schemes for equidistant discreti-sation. Figure 2 displays the computational times for a swinging rubber rod (left) and a steel string example (right),which are subjected to their own gravity. We used the solversRODAS, SEULEX, RADAU5 – with sparse linear algebra,adapted to second order differential equations –,DASPK = DASSL andDOPRI5 with strong damping on the extensionaland shearing dof.

FIGURE 1. Comparison with ABAQUS three dimensional finite element solution

FIGURE 2. Computational times for a rubber and a steel example withT = 10s

CONCLUSION AND ACKNOWLEDGMENTS

We presented an alternative discrete Cosserat rod model, which yields small computational times at sufficientlycorrect accuracy. The model is thus adequate for multibody dynamics simulations. We wish to thank Ernst Hairerwho provided us with his excellent solvers and who always gave us kind advice.

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