Nonlinear Dyn (2012) 69:705–720DOI 10.1007/s11071-011-0298-6

O R I G I NA L PA P E R

Relaxed representations and improving stabilityin time-stepping analysis of three-dimensional structuralnonlinear dynamics

Salvatore Lopez

Received: 27 June 2011 / Accepted: 6 December 2011 / Published online: 22 December 2011© Springer Science+Business Media B.V. 2011

Abstract In the present paper, we propose a represen-tation of the discrete motion equations in structuralnonlinear dynamics to obtain an improvement in thestability of time numerical integrations. A geometri-cally nonlinear total Lagrangian formulation for three-dimensional beam elements in the hypotheses of largerotations and small strains is presented. In this for-mulation, slopes are used instead of rotation param-eters to compute the nonlinear representations of thestrain measures in the inertial frame of reference. Suchrepresentations of the internal strains—rotations com-patibility are then imposed in their time derivativesversion. The results, related to Newmark approxima-tions for the variations in the displacement and veloc-ity vectors, show a significant increase in the rangeof stability of the time integration process and a re-duction in the number of Newton iterations requiredin the time integration steps. The numerical tests, fur-thermore, show that the variation in the total energy inthe time steps has bounded oscillations about the zerovalue.

Keywords Numerical stability · Time relaxedrepresentation · Structural three-dimensionalnonlinear dynamics

S. Lopez (�)Dipartimento di Modellistica per l’Ingegneria, Universitàdella Calabria, 87030 Rende, Cosenza, Italye-mail: salvatore.lopez@unical.it

1 Introduction

Geometric nonlinearity in elastodynamics is an impor-tant field in structural analysis and in the last 30 yearsthere has been extensive research into time integrationalgorithms. In particular, stability is an important is-sue because while unconditionally stable schemes canbe recovered in linear dynamics, numerical instabil-ity frequently appears in nonlinear cases. In this con-text, to obtain stable solutions, schemes which demandthe conservation or decrease in the total energy of theHamiltonian system within each time step are exten-sively used.

Energy conserving algorithms in elastodynamicshave already appeared in the works of Belytschko andSchoeberle [1] and Hughes et al. [2]. The energy-momentum method introduced by Simo and Tarnow[3, 4] preserves energy as well as linear and an-gular momentum in the time interval. Conservationproperties are enforced in the equation of motion viaLagrange multipliers. Adaptive time stepping proce-dures (Kuhl and Ramm [5]) and controllable numer-ical dissipation (Hoff and Pahl [6], Chung and Hul-bert [7]) can also be introduced to permit larger timesteps and, consequently, to improve computational ef-ficiency. Although finite difference methods appear toprevail in the literature on the numerical treatment ofinitial-value problems, a number of alternative finiteelement methods have been developed for the tempo-ral discretization process. Weighted residual statementand the Galerkin finite element approach for the nu-

706 S. Lopez

merical solution of the equations of motion has beenemployed in Zienkiewicz et al. [8] and in Lasaint andRaviart [9]. More recently, time finite elements, wherethe Newmark family formulas can be recovered by thechoice of representative constants and the algorithmicenergy conservation is implicitly preserved have beencarried out (see Betsch and Steinmann [10] and relatedbibliography).

In this paper, we present an extension of the ap-proach described in [11] for time-stepping nonlinearanalysis to the three-dimensional case. In [11], the in-troduction of strains as additional local variables to therepresentation of the internal energy leads to a signif-icant increase in the range of stability of the classicalNewmark method. In particular, the related additionalequations are governed by a dynamical representa-tion of the internal strains—rotations compatibility. Insuch a dynamical representation, then the expressionsthat define strain measures are imposed in their timederivatives version. Here, the three-dimensional non-linear models are established by the use of Lagrangianmultipliers. Related constraint equations represent thenonlinear definition of the internal strains as a functionof the rotational descriptors. Thus, the imposition ofsuch equations is the strong form of the strain defini-tions while the time derivative of constraint equationsrepresents the related relaxed form.

Of course, the use of the strong internal constraintequations can be replaced by the intrinsic definitionof the related parameters by obtaining, however, sim-ilar computational characteristics in terms of numberof time-steps and approximation properties. The useof the time relaxed description of the compatibilityconditions between deformation and rotational param-eters, instead, leads to an appreciable increase in therange of stability of the time integration scheme. Theincrease in computational effort due to the introduc-tion of new unknowns and the loss of symmetry in theiteration matrix is therefore balanced by the reductionin the number of Newton iterations required in the timeintegration steps.

Note that the described approach, although quitedifferent, has analogies with the procedure of in-dex reduction of the differential-algebraic equations(DAEs); see, e.g., Gear and Petzold [12], Bachmann etal. [13] and Mattsson and Söderlind [14]. Such a pro-cedure is based on a reduction of the DAE problem toan ordinary differential equation (ODE) by a numberof differentiations of the given constraint equations.

Differentiations are carried out until the eliminationof the related Lagrange multipliers typically in termsof velocities and accelerations is possible. The result-ing ODE can then be integrated using effective ex-plicit algorithms, mainly with regard to the stability. Inthe presented approach, one time differentiation of theconstraint equations is performed. Furthermore, suchrelaxed constraints are used to define the Lagrangianextended functional and not directly employed as mo-tion equations. Thus, Lagrange multipliers and theirderivatives are also present in the motion equations.These quantities are coupled with the kinematical un-knowns and are involved in an implicit integration pro-cess. Finally, constraints are not intrinsic equations ofthe mechanical model here but are definitions of me-chanical parameters which can already be eliminateddirectly.

As regards beam element modeling, the corota-tional approach is one of the classical ways for mod-eling three-dimensional elastic frame structures forsmall strains with large displacements. The motion ofthe continuous medium is decomposed into a rigidbody motion followed by a pure deformation. Thenthe nonlinear motion is obtained by joining the linearkinematic with a rigid body motion that is recoveredby the use of orthogonal transformation matrices. Theevolution of the corotational approach can be traced byreferring to the works: Belytschko and Hsieh [15], Ar-gyris [16], Rankin and Nour-Omid [17], Cardona andGeradin [18], Crisfield [19] and Ibrahimbegovic et al.[20]. These techniques, however, have to be supportedby a robust and economical definition of the rotatedlocal reference system. Basically, attention has to bepaid to avoid singularities in the transformation ma-trices for several angles and complex manipulationsto overcome nonconservative descriptions due to thenoncommutativity of rotations.

Here, we use a small strains—finite rotations for-mulation of a three-dimensional finite element beamwithout the use of rotation parameterizations, whilepreserving the robustness and simplicity of the anal-ysis (see [21]). Based on the Timoshenko beam the-ory, the actual configuration of the element is rigidlytranslated and rotated, and deformed according to theselected linear modes. Rigid and deformation modesare referred to the nodes of the element by a TotalLagrangian formulation. The nonlinear rigid motionis recovered by referring to three unit and mutuallyorthogonal vectors attached to the nodes of the beam

Improving stability in time-stepping nonlinear dynamics 707

element. All nine components of such vectors in theglobal inertial frame of reference are assumed as un-known. As demonstrated in [22], the rotational degree-of-freedom of the element is reduced to only three bysix well-posed constraint conditions.

An approach where the nine components of the ro-tation matrix are used directly has already been pre-sented by Betsch and Steinmann [23, 24] in the fi-nite rotations analysis of rigid bodies and beams. Or-thonormality of the directors is enforced by using sixscalar products. In particular, three unit length andthree orthogonality conditions are imposed on the di-rectors by using the scalar products among them andreciprocally, respectively. Here, instead three scalarproducts and one cross product are used to define theconstraint conditions. In detail, two unit length andone orthogonality conditions are imposed by scalarproducts while the complete definition of a director isobtained by referring to a cross product. In this way,we demonstrate that the present formulation proves tobe well posed for any finite rotations. Furthermore, areduction in the number of unknowns is possible hereusing the cross product between two directors as anexplicit definition of the components of third director.

The remainder of the paper is organized as follows.In Sect. 2, the adopted description of the finite rota-tions in elastic frame structures is given. In Sect. 3, wedefine the kinematics of the beam element while re-lated energetic quantities are evaluated in Sect. 4. Asparticular cases of the given beam element description,the motion equations of prismatic bodies and massparticles are deduced in Sect. 5. Section 6 containsthe semidiscrete motion equations and related time-integration algorithm. Examples of applications of theproposed scheme are given in Sect. 7. Final conclu-sions are drawn in Sect. 8.

2 Description of the finite rotations

In the corotational approach, the motion of the body isdecomposed into a rigid motion followed by a pure de-formation performed in a local corotational frame thatrotates and translates with each element. So, all thenonlinearity of the problem derives essentially fromthe change of reference from the global fixed frameto the local one, the strain energy being governed bytheir relative rotations. In this way, the main difficulty

is the presence of finite rotations in finite kinemat-ics that noticeably complicates the algebra for obtain-ing kinematic expressions. In particular, finite 3D ro-tations must be described through rotation matriceswhich lie in a nonlinear manifold.

In the corotational analysis carried out here, werefer to three unit and mutually orthogonal unknownvectors Eξ , Eη, and Eζ attached to the body and de-fined in the global inertial frame of reference (x, y, z).Classically, these unknown vectors, defining rigid ro-tation in the current configuration, are obtained by ro-tating the corresponding known vectors eξ , eη, and eζ

of the initial configuration. This mapping is realizedby an R rotation matrix. The Rodrigues formula, thenallows R to be expressed in terms of the quantities ly-ing in a vector space and reduces to a three indepen-dent parameter representation:

R(φ) = exp(S(φ)

)

= I + S(φ) + 1

2S(φ)2 + · · · =

∞∑

n

1

n!S(φ)n, (1)

which uses the rotation vector φ = {φξ ,φη,φζ } andthe skew-symmetric matrix

S(φ) =⎡

⎣0 −φζ φη

φζ 0 −φξ

−φη φξ 0

⎤

⎦ . (2)

Here, all nine components of Eξ , Eη, and Eζ vec-tors are assumed as unknown. As demonstrated in[22], by the following six well-posed constraint con-ditions:

gηE = Eη · Eη − 1 = 0,

gζE = Eζ · Eζ − 1 = 0,

gηζE = Eη · Eζ = 0,

gξE = Eη × Eζ − Eξ = 0,

(3)

the rotational degrees of freedom are reduced just tothree. Of course, six conditions being imposed, thedegrees of freedom are at least three. To show alsothat the degrees of freedom are at most three, it wasdemonstrated that nullity(G) is at most three in the so-lution point, where G is the Jacobian matrix related tothe (3) gE = 0 system.

It should be emphasized that the number of arith-metical operations between the two approaches arecomparable if we stop the exponential map expansionat a very low order. In this case, however, a poor ac-curacy of the co-rotational approach results, especially

708 S. Lopez

for appreciably large rotations. On the contrary, in to-tal Lagrange descriptions, the proposed approach is in-sensitive to the rotation values and still effective froma computational point of view.

3 Kinematics of the beam element

We refer to the referential coordinate ξ along the el-ement beam centerline −hξ/2 ≤ ξ ≤ +hξ/2. In thefollowing, we denote with i, j and o the nodes re-spectively in ξ = −hξ/2, ξ = +hξ/2 and ξ = 0.Along the beam centerline, we define the displacementvector u(ξ) = {u(ξ), v(ξ),w(ξ)} and three mutuallyorthogonal vectors Eξ (ξ) = {Eξ

x (ξ),Eξy (ξ),E

ξz (ξ)},

Eη(ξ) = {Eηx (ξ),E

ηy (ξ),E

ηz (ξ)}, Eζ (ξ) = {Eζ

x (ξ),

Eζy (ξ),E

ζz (ξ)}, in the global inertial frame of refer-

ence (x, y, z). Director vectors Eη and Eζ are alongthe principal axes of inertia of the cross-section. LetEξ , Eη and Eζ vectors be the columns of the matrixE(ξ):

E(ξ) = [Eξ (ξ)

∣∣ Eη(ξ)∣∣ Eζ (ξ)

]. (4)

The E(ξ) orthonormal matrix is obtained by usingdirector vectors at the i, j, and o nodes and con-strained to the (3) conditions. In particular, we willuse unknown Eo = E(0) to represent the large three-dimensional rotations of the local frame of referenceof the beam element. As mentioned, the initial unitvector in the ξ , η, and ζ element direction, finally, willbe denoted by eξ , eη, and eζ , respectively (see Fig. 1).

In the beam element, global displacement vectoru(ξ) is composed of rigid and deformation compo-nents. In particular, we refer to the u = (u(ξ), v(ξ),

w(ξ)) rigid displacements defined in the initial frameof reference while the deformation u(ξ) = (u(ξ), v(ξ),

w(ξ)) displacements and θ = (θ ξ (ξ), θη(ξ), θ ζ (ξ))

rotations are defined in the local rigidly rotated frameof reference. The deformation kinematics is assumedby the linear interpolations

u = εoξ, v = ϕηo ξ, w = ϕζ

o ξ, (5)

for displacements and the quadratic interpolations

Fig. 1 Total Lagrangian co-rotational formulation: element kinematics and coordinate systems

Improving stability in time-stepping nonlinear dynamics 709

θ η = θηj − θ

ηi

hξ

ξ + 2θ

ηi + θ

ηj

h2ξ

ξ2,

θ ζ = θζj − θ

ζi

hξ

ξ + 2θ

ζi + θ

ζj

h2ξ

ξ2,

(6)

for flexural rotations. The kinematics of the element isthen completed by defining the local θ ξ torque rotationabout the beam centerline. As in (6), we assume

θ ξ = θξj − θ

ξi

hξ

ξ + 2θ

ξi + θ

ξj

h2ξ

ξ2. (7)

Note that zero local rotations at the center of the ele-ment are assumed. Rigid kinematics, then, will be rep-resented by the nodal displacement components andthe degrees of freedom of the vectors Eξ

o attached tothe central node.

Based on the above definitions, local rotations anddirector components are now linked by the field vectoroperations

Eξ (ξ) = Eξo + θ η(ξ)Eη

o + θ ζ (ξ)Eζo,

Eη(ξ) = −θ η(ξ)Eξo + Eη

o + θ ξ (ξ)Eζo,

Eζ (ξ) = −θ ζ (ξ)Eξo − θ ξ (ξ)Eη

o + Eζo .

(8)

As proven, vectors Eξo , Eη

o, and Eζo are unit and mu-

tually orthogonal at the solution points. Then, at thefirst order, Eξ (ξ), Eη(ξ), and Eζ (ξ), there are alsothree unit and mutually orthogonal vectors and theycompletely define the global orientation of the cross-section. We note that the first order accuracy of the(8) representations leads to local evaluations consis-tent with the small strains hypotheses.

By evaluating (8) relations for ξ = −hξ/2 and ξ =hξ/2, respectively, in the i and j nodes, and by usingorthonormality of the directors, we can write

θξj − θ

ξi = 1

2

(Eζ

i · Eηj − Eη

i · Eζj

),

θηj − θ

ηi = 1

2

(Eη

i · Eξj − Eξ

i · Eηj

),

θζj − θ

ζi = 1

2

(Eζ

i · Eξj − Eξ

i · Eζj

),

(9)

and

θξj + θ

ξi = 1

2

[Eζ

o · (Eηi + Eη

j

) − Eηo · (Eζ

i + Eζj

)],

θηj + θ

ηi = 1

2

[Eη

o · (Eξi + Eξ

j

) − Eξo · (Eη

i + Eηj

)],

θζj + θ

ζi = 1

2

[Eζ

o · (Eξi + Eξ

j

) − Eξo · (Eζ

i + Eζj

)].

(10)

Furthermore, by referring to the centerline points, wenow define rigid and deformation components in theinitial frame of reference by

u(ξ) = uo + ξEξox − ξ,

v = vo + ξEηox,

w = wo + ξEζox,

(11)

and

u(ξ) = εoξEξox + ϕ

ηo ξE

ηox + ϕ

ζo ξE

ζox,

v(ξ) = εoξEξoy + ϕ

ηo ξE

ηoy + ϕ

ζo ξE

ζoy,

w(ξ) = εoξEξoz + ϕ

ηo ξE

ηoz + ϕ

ζo ξE

ζoz,

(12)

respectively. Then, in the vectorial notation, the mo-tion of the ξ point is described as

u = uo + ξEξo − ξeξ + εoξEξ

o + ϕηo ξEη

o + ϕζo ξEζ

o .

(13)

Also here, by evaluating (13) relations for nodal coor-dinates ξ = −hξ/2, ξ = hξ/2, and by using orthonor-mality of the directors, we deduce that

uo = 1

2(ui + uj ) (14)

is the central point displacement and

gξI = εo − 1

hξ

[Eξ

o · (uj − ui ) − hξ + hξEξox

] = 0,

gηI = ϕη

o − 1

hξ

[Eη

o · (uj − ui ) + hξEηox

] = 0,

gζI = ϕζ

o − 1

hξ

[Eζ

o · (uj − ui ) + hξEζox

] = 0,

(15)

are the expressions of the axial and shear deformationsas a function of nodal displacement and director com-ponents. In the following, the defined g

ξI , g

ηI , and g

ζI

expressions of internal strain evaluations will be usedin the description of the improving solution scheme.

As can be seen, unknown nodal components com-pletely define the (5)–(7) linearized deformation kine-matics of the beam element by (9)–(10) and (15) ex-pressions. Nonlinear rigid kinematics, instead, is de-scribed by the displacement vector in (14) and the un-known director vectors Eo at the central node. The re-maining unknown components of the element are thedisplacements ui , uj and the director vectors Ei , Ej atthe boundary nodes.

710 S. Lopez

4 Evaluation of energetic quantities of the beamelement

We consider the referential coordinates (ξ, η, ζ ) inthe element, where η and ζ are the thickness coor-dinates in the eη and eζ directions, respectively. Bydenoting with uP (ξ, η, ζ ) = {uP (ξ, η, ζ ), vP (ξ, η, ζ ),

wP (ξ, η, ζ )} the displacement of the generic point P

in the element represented in the global referenceframe, we can refer respectively to the expression

uP = uo + ξ(Eξ

o − eξ) + η

(Eη

o − eη) + ζ

(Eζ

o − eζ)

(16)

for the rigid and to expression

uP = uEξo + vEη

o + wEζo − (

θ ζ η + θ ηζ)Eξ

o

+ θ ξ(ηEζ

o − ζEηo

)(17)

for the deformation components of the motion uP =uP + uP .

The principal energetic quantities involved are thekinetic, potential, and external energy:

T = 1

2

∫

V

ρuP · uP dV,

U = 1

2

∫

V

εP : σP dV,

W =∫

V

p · uP dV,

(18)

respectively. In (18), the dot denotes derivatives withrespect to time t , V the volume of beam, p the vectorof external loads and ρ the mass density. Furthermore,εP and σP are the infinitesimal strain and stress ten-sors in the body, respectively.

Kinetic energy is now evaluated by referring to thefollowing expression of the velocity vector:

uP = uo + ξ Eξo + ηEη

o + ζ Eζo + ˙uEξ

o + ˙vEηo + ˙wEζ

o

− ( ˙θ

ζη + ˙

θηζ)Eξ

o + ˙θ

ξ (ηEζ

o − ζEηo

), (19)

obtained by time differentiation of uP and by trunca-tion of the deformation measures to the zero order. Theintegration over the section area A of the square of thevelocity in (19) leads to:∫

A

uP · uP dA

= Auo · uo + ξ2AEξo · Eξ

o + Jζ Eηo · Eη

o

+ JηEζo · Eζ

o + 2ξAuo · Eξo

+ A ˙u · ˙u + Jo˙θ

ξ2

+ Jη˙θ

η2

+ Jζ˙θ

ζ 2

+ 2A( ˙uEξ

o + ˙vEηo + ˙wEζ

o

) · (uo + ξ Eξo

)

+ 2 ˙θ

ξ (Jζ Eη

o · Eζo − JηEζ

o · Eηo

)

− 2(Jη

˙θ

ηEζ

o + Jζ˙θ

ζEη

o

)· Eξ

o, (20)

where Jη and Jζ are the second moments of area aboutthe related principal axes while Jo is the polar mo-ment. By defining the vector γ = (εo,ϕ

ηo ,ϕ

ζo ), kinetic

energy is now computed by further integration over thebeam centerline:

1

2

∫ hξ /2

−hξ /2

∫

A

ρuP · uP dAdξ

= 1

2ρ

{hξAuo · uo + AJξ Eξ

o · Eξo

+ hξJζ Eηo · Eη

o + hξJηEζo · Eζ

o

+ AJξ γ · γ + 2AJξ Eξo · Eo · γ

+ hξ

12

(JoDθ

ξ2 + JηDθη2 + Jζ Dθ

ζ 2)

+ hξ

20

(JoSθ

ξ2 + JηSθη2 + Jζ Sθ

ζ 2)

+ hξ

3

[Sθ

ξ (Jζ Eη

o · Eζo − JηEζ

o · Eηo

)

− (JηSθ

ηEζ

o + Jζ SθζEη

o

) · Eξo

]}, (21)

with the positions Jξ = h3ξ /12, Dθ = (θ

ξj − θ

ξi , θ

ηj −

θηi , θ

ζj − θ

ζi ) and Sθ = (θ

ξi + θ

ξj , θ

ηi + θ

ηj , θ

ζi + θ

ζj ).

Finally, by introducing the geometrical coefficientmatrices JA = diag(Jo, Jη, Jζ ) and JV = diag(AJξ ,

hξJη,hξJζ ) we can write:

T = 1

2ρ

{V uo · uo + Eo · JV · Eo

+ AJξ

(γ + 2Eξ

o · Eo

) · γ+ hξ

12Dθ · JA · Dθ + hξ

20Sθ · JA · Sθ

+ hξ

3

[Sθ

ξ (Jζ Eη

o · Eζo − JηEζ

o · Eηo

)

−(JηSθ

ηEζ

o + Jζ SθζEη

o

) · Eξo

]}. (22)

In (22), the rigid, deformation and mixed kinetic termscan be recognized. In particular, note how the term Eo ·JV · Eo reproduces the rigid inertial components of theangular momentum of the element.

Improving stability in time-stepping nonlinear dynamics 711

The estimation of the potential energy can be car-ried out by extracting the contributions due to the de-formation from the uP motion. Then the projection ofuP in (17) in the Eξ

o , Eηo, and Eζ

o directions gives theinfinitesimal displacements:

u = uP · Eξo = εoξ − θ ζ η − θ ηζ,

v = uP · Eηo = ϕ

ηo ξ − θ ξ ζ,

w = uP · Eζo = ϕ

ζo ξ + θ ξ η.

(23)

By using this deformation kinematics, we define thefollowing infinitesimal strain components of the εP

tensor:

εξξ = εo − θζ,ξ η − θ

η,ξ ζ,

εξη = 1

2

(ϕη

o − θ ζ − ωζ θξ,ξ

),

εξζ = 1

2

(ϕζ

o − θ η + ωηθξ,ξ

),

(24)

and εηζ = 0. In (24), shearing contributions due tothe torsional mode are modeled by the ωη(η, ζ ) andωζ (η, ζ ) functions. Here, because hη ×hζ rectangularsections were analyzed, we assume the distributions:

ωη = η3

(hη/2)2

(1 − ζ 2

(hζ /2)2

),

ωζ = ζ 3

(hζ /2)2

(1 − η2

(hη/2)2

),

(25)

where εξη = 0 and εξζ = 0 is realized on the bound-aries |ζ | = hζ /2 and |η| = hη/2 of the cross section,respectively.

Extensional components εηη and εζζ are then ob-tained by imposing the statical assumptions σηη =σζζ = 0 on the σP stress tensor. Then we have

εηη = εζζ = − λ

2(λ + μ)εξξ , (26)

where λ and μ are the Lamé coefficients. By using the(26) expressions, the remaining stress components are:

σξξ = 2μεξξ + λ(εξξ + εηη + εζζ ) = 2μ + 3λ

λ + μμεξξ

= Eεξξ ,

σξη = 2μεξη = 2Gεξη,

σξζ = 2μεξζ = 2Gεξζ

(27)

and σηζ = 0. In (27), E and G are the Young and shearmoduli, respectively.

By integrating over the section area the potentialenergy contribution, we have:

∫

A

εP : σP dA

=∫

A

(Eε2

ξξ + 4Gε2ξη + 4Gε2

ξζ

)dA

= E(Aε2

o + Jηθη2

,ξ + Jζ θζ 2

,ξ

)

+ G

[A

(ϕη

o − θ ζo

)2

+ θξ2

,ξ

∫

A

ωζ 2dA − 2

(ϕη

o − θ ζo

)θ

ξ,ξ

∫

A

ωζ dA

]

+ G

[A

(ϕζ

o − θ ηo

)2

+ θξ2

,ξ

∫

A

ωη2dA + 2

(ϕζ

o − θ ηo

)θ

ξ,ξ

∫

A

ωη dA

].

(28)

Note that in (28), to overcome locking effects, the cen-tral value of the θ η(ξ) and θ ζ (ξ) interpolations areused in the shear energy computation. Besides, assum-ing this,

∫A

ωηdA = ∫A

ωζ dA = 0 is obtained. Then,

by defining Jωη = ∫A

ωη2dA and Jωζ = ∫

Aωζ 2

dA,we can write:∫

A

εP : σP dA = E(Aε2

o + Jηθη2

,ξ + Jζ θζ 2

,ξ

)

+ G[A

(ϕη2

o + ϕζ 2

o

)

+ (Jωη + Jωζ )θξ2

,ξ

], (29)

with θηo = θ

ζo = 0. Then the potential energy is com-

puted by further integration over the beam centerline:

U = 1

2

∫ hξ /2

−hξ /2

∫

A

εP : σP dAdξ

= 1

2E

[V ε2

o + 1

hξ

Jη

(Dθη2 + 4

3Sθη2

)

+ 1

hξ

Jζ

(Dθζ 2 + 4

3Sθζ 2

)]

+ 1

2G

[+V

(ϕη2

o + ϕζ 2

o

)

+ 1

hξ

(Jωη + Jωζ )

(Dθξ2 + 4

3Sθξ2

)]. (30)

External work W , finally, is defined in (18) by the(16) and (17) expressions of the displacement vector.Note that kinematics of the element being modeled asa three-dimensional body, only external forces must beassigned. Besides, because finite rotations are replacedby products θEo in the present formulation, we can

712 S. Lopez

see from (17) that the external force vector is a linearfunction of the assumed unknowns.

5 Multibody systems

By referring to the previous beam element model, wecan study the dynamical behavior of multibody sys-tems. In particular, prismatic bodies linked by spheri-cal joints and free mass particles are analyzed.

A prismatic hξ × hη × hζ element is assumed suchthat only the ξ axial deformation ε is present. Springsin the joint point between elements produce, further-more, restraint moments proportional to the relativerotations. In particular, we refer now to the En and Em

directors attached to the central points of the n and m

elements and we denote with Dθξ , Dθη and Dθζ thesmall relative rotations. We have:

Eξm = Eξ

n + DθηEηn + Dθζ Eζ

n,

Eηm = −DθηEξ

n + Eηn + Dθξ Eζ

n,

Eζm = −Dθζ Eξ

n − Dθξ Eηn + Eζ

n.

(31)

Then, by the usual manipulations, we obtain

Dθξ = 1

2

(Eζ

n · Eηm − Eη

n · Eζm

),

Dθη = 1

2

(Eη

n · Eξm − Eξ

n · Eηm

),

Dθζ = 1

2

(Eζ

n · Eξm − Eξ

n · Eζm

).

(32)

With the expression of the rigid component (16) andby zeroing the absent deformation measures in (17),the motion of the body is defined as

uP = u + ξ(Eξ − eξ

) + η(Eη − eη

)

+ ζ(Eζ − eζ

) + ξεEξ , (33)

where u is the central point displacement vector com-puted by the nodal values as in (14).

As before, the integration over the section area ofthe square of the velocity leads to∫

A

uP · uP dA

= Au · u + ξ2AEξ · Eξ + Jζ Eη · Eη

+ JηEζ · Eζ + 2ξAu · Eξ

+ ξ2Aε2 + 2ξAεEξ · (u + ξ Eξ)

(34)

and, then to the kinetic energy evaluation by furtherintegration on the ξ elemental domain

T = 1

2ρ[V u · u + E · JV · E + AJξ

(ε2 + 2εEξ · Eξ

)].

(35)

In the potential energy definition. we refer to the EV

axial rigidity and to the like type kη, kζ flexural, andkξ torque stiffness:

U = 1

2

(EV ε2 + kξDθξ2 + kηDθη2 + kζ Dθζ 2)

, (36)

where the relative rotations are defined in (32) whileaxial deformation is defined in the first of (15) expres-sions:

gξI = ε − 1

hξ

[Eξ · (uj − ui ) − hξ + hξE

ξx

] = 0. (37)

Finally, external energy W can be computed by theexpression (33) of the displacement vector of the body.

We note that compatibility in the joint points be-tween elements is realized because nodal displace-ments are assumed as unknowns of the element. How-ever, internal rigidity of the body must be constrainedby the ϕη = 0 and ϕζ = 0 conditions. Then, by thesecond and the third of (15) and the use of λ

ηI and λ

ζI

Lagrange multipliers, in the energetic functional weadd the contribution

ληI g

ηI + λ

ζI g

ζI

= λη[Eη · (uj − ui ) + hξE

ηx

]

+ λζ[Eζ · (uj − ui ) + hξE

ζx

]. (38)

In the three-dimensional mass particles motion, werefer to the ith particle of mi mass. Potential energyof interaction of the particles i and j depends on theinterbody distance hij = (1+ εij )h, where h is the ini-tial distance and εij is the related elongation. Then, bydenoting with ui the displacement vector of the par-ticle and with ka the interaction rigidity, kinetic, andpotential energy are computed by referring to

T = 1

2mi ui · ui , U = 1

2kaε

2ij . (39)

For these models only, the use of the Eij director in thedirection from the ith to the j th particle is necessary.As usual, elongation is defined by

gξI = εij − 1

h

[Eij · (uj − ui ) − h + hE

ijx

] = 0. (40)

The internal connection between the Eij director andthe positions of the related particles is here imposedby the energetic contribution

Improving stability in time-stepping nonlinear dynamics 713

λuI g

uI + λv

I gvI + λw

I gwI

= λu[uj − ui + h − E

ijx (1 + εij )h

]

+ λv[vj − vi − E

ijy (1 + εij )h

]

+ λw[wj − wi − E

ijz (1 + εij )h

]. (41)

As we can see, constraints in (41) define the compo-nents of the Eij vector in the inertial frame of refer-ence.

6 Nonlinear dynamical analysis

We refer to dynamical systems with L(q(t),q(t)) La-grangian function obtained by summing the describedenergetic contributions, where q is the vector of theunknown components of the element. We denote withgE = 0 the constraint conditions in (3) relating the un-known components of the Eξ , Eη, and Eζ vectors andwith gI = 0 the internal compatibility conditions ex-plicitly imposed in the description of the models. Re-lated Lagrange multiplier vectors are denoted with λE

and λI , respectively.Then we obtain the extended functional

LS(q, q,λ) = T (q,q) − U(q) + W(q) + λE · gE(q)

+ λI · gI (q). (42)

In particular, for the beam element model we refer tothe contributions given in (22) and (30) while gI in-ternal compatibility functions are established in (15).Unknown vector q is composed of the three ui, vi,wi ,displacements and the nine components of the direc-tors Eξ

i ,Eηi ,Eζ

i , at the nodes plus the nine components

of the directors Eξo,Eη

o,Eζo , for each element. Six λE

values for each Ei and Eo orthonormal system andthree λI values for each elemental deformation com-plete the group of the unknowns. For the prismatic el-ement model we use the expressions (35)–(38). Theunknown vector is composed of the three ui, vi,wi ,nodal displacements and the nine components of thedirectors Eξ ,Eη,Eζ , at the center of the element. Asmultipliers, then we have six λE and three λI un-known components. Mass particles motion, finally isdescribed by the three ui, vi,wi , displacements of theith mass and the three Eij components for each i–j

connection. Energetic expressions are given in (39)–(41) where related multipliers are the three λu

I , λvI , λ

wI ,

components for the compatibility constraints and theλ

ξI component for the elongation measure definition.

By collecting λE and λI unknowns in the λ vec-tor, the semidiscrete formulation of the motion can bewritten in the form:

∂L

∂q− ∂

∂t

∂L

∂q= 0,

∂L

∂λ= 0,

q(0) = q∗, q(0) = q∗,(43)

where q∗ and q∗ represent the initial values and veloc-ities, respectively.

For the time integration of the semidiscrete initialvalue problem (43) we refer to the constant time stepΔt = tn+1 − tn. Unknown components of the q andλ vectors also collected in the d vector. By assumingthe state variables dn, dn, dn, as known at the time tn

and making the external forces p(t) for all t , the timeintegration is restricted to the subsequent solution ofthe state variables at the end of each step dn+1, dn+1,dn+1. In order to realize this step by step integration,the set of variables is reduced to the unknowns dn+1

only by the Newmark approximations

dn+1 = γ

βΔt(dn+1 − dn) +

(1 − γ

β

)dn

+(

1 − γ

2β

)Δt dn, (44)

dn+1 = 1

βΔt2(dn+1 − dn) − 1

βΔtdn

+(

1 − 1

2β

)dn. (45)

In the following, we use the average accelerationscheme by adopting γ = 1/2 and β = 1/4.

By inserting relations (44) and (45) in (43), we ar-rive at the nonlinear equation of the form:

F(dn+1) = 0. (46)

This represents the nonlinear system of algebraicequations defined at the tn+1 time with the dn+1 un-known vector. The velocities and accelerations at theend of the time step can then be obtained by relations(44) and (45), respectively. Newton like iterative meth-ods can be used to solve system (46) by linearization

F(d(k+1)

n+1

)

= F(d(k)

n+1

) + ∂F(d(k)n+1)

∂dn+1

(d(k+1)

n+1 − d(k)n+1

)

+ O(Δt2) = 0. (47)

714 S. Lopez

The iterative process is here initialized by choosingd(0)

n+1 as the linear extrapolation of the previously com-puted dn and dn−1 vectors when n > 0, while the for-mula d(0)

1 = d∗ +Δt d∗ is used when n = 0. By choos-ing the fixed tolerance η = 10−8, the formula∥∥d(k+1)

n+1 − d(k)n+1

∥∥/∥∥d(k+1)

n+1 − dn

∥∥ ≤ η (48)

is adopted as convergence criterion.We note that the use of the gE = 0 constraint equa-

tions are necessary for the description of the elementmodel. The use of the gI = 0 explicit internal con-straint equations instead can be avoided by the intrin-sic definition of the related parameters with resultingdecrease in the number of unknowns. Such a benefit,nevertheless, is balanced by an increase in the com-plexity in the evaluation of the coefficients in the al-gebraic system and a larger band in the iteration ma-trix. Similar computational characteristics in terms ofcomputation time and approximation, however, can beverified in both cases.

Here, the use of the constraint equations is suitablefor improving stability in the time integration scheme.As shown in [11], the use of a time relaxed descrip-tion of the connection between deformation and rota-tional parameters leads to an appreciable increase inthe range of stability of the time integration schemeand a reduction in the number of Newton iterations re-quired in the time integration step. Such a relaxed de-scription is obtained by referring to the extended func-tional

LR(q, q,λ)

= T (q,q) − U(q) + W(q) + λE · gE(q)

+ λI · gI (q), (49)

where the constraint equations are replaced by theirtime differentiations. In such a representation, we notethat consistent initial conditions for directors and de-formations must be provided to preserve gE(q) = 0and gI (q) = 0.

In the following, we denote the (42) and (49) func-tionals as the strong and relaxed form of the La-grangian formulation respectively. Similarly, relatednonlinear Newmark integration algorithms are de-noted by LS and LR schemes, respectively.

7 Numerical tests

A set of numerical examples shows the application ofthe proposed method for the time integration of the

motion equations. Stable behavior of the time integra-tion method, in the absence of numerical dissipation,is verified by referring to the condition

ΔE = Un+1 − Un + Tn+1 − Tn − ΔW = 0, (50)

as a sufficient stability condition in the nonlinear dy-namical schemes (see, e.g., [1], [3]). Equation (50) ex-presses the conservation of the total energy ΔE, whereUn+1 and Un represent the internal energy at the endand the beginning of the time step, Tn+1 and Tn beingthe corresponding kinetic energy and ΔW symboliz-ing the work done by external forces within the timestep. We use the trapezoidal rule to calculate the workof the external forces:

ΔW = 1

2(un+1 − un) · (pn+1 + pn). (51)

As regards consistency note that as in the numericaltests, for such a Δt in which both LS and LR represen-tations have a stable behavior, the difference betweenrespective kinematic quantities of interest proves tobe negligible. In this way, the methods prove to besecond-order accurate.

Numerical tests were performed for increasing val-ues of the Δt time step. Let steps be the number oftime steps effected by the integration process to ana-lyze the behavior for t = 0 . . . T . In the following, werefer to the mean value of the Nwi Newton iterationsin the ith step

Nwm =step∑

i=1

Nwi/steps, (52)

carried out, unless it becomes unstable (div), in theprocess.

7.1 Motion of a dumbbell

We investigate the motion of a dumbbell with initialinterbody distance h = 1 modeled as a two-particleproblem and defined in the three-dimensional am-bient space. We assume m1 = m2 = 1 with u ={u1 v1 w1 u2 v2 w2}. The initial conditions are givenby u∗ = 0, u∗ = {0 0 2 5 5 10} and Eij = {1 0 0} is thedirector in the x direction. Accordingly, εij = 0 and,by time differentiation of expressions (40) and (41),εij = 0 and Eij = {0 3 5} is obtained.

The interaction of the two bodies is assumed tobe governed by a Lennard–Jones potential U(rij ) =A[(σ/rij )

5 − (σ/rij )3] which is often employed in

Improving stability in time-stepping nonlinear dynamics 715

Fig. 2 DumbbellΔt = 0.02 for T = 2:sequence of configurationsobtained by the (a) LS and(b) LR scheme, respectively

molecular dynamic simulations. The distance betweenthe centers of the two particles is rij = h + εij . Wemake σ = (3/5)1/2, such that εij = 0 characterizesthe internal force free configuration. Here, we considerthe quasi-rigid connection A = 106 that classically hassevere instability restrictions. In such a case, by lin-earization of U(rij ) for rij = 1, ka = 6Aσ 3 can be in-terpreted as the spring stiffness. We can refer to Gon-zales and Simo [27] and Crisfield and Shi [28] for thenumerical instabilities which are introduced in suchdynamic systems. Additional background material onthe motion of a several particle system in a potentialfield can be found in standard books on classical me-chanics; see, e.g., Goldstein [25] and Arnold [26].

To illustrate the motion, Fig. 2 contains a sequenceof configurations calculated in the LS and LR rep-resentation with Δt = 0.02 and t = 0 . . .2. At aboutt = 1.4 an unphysical motion indicates the unstablebehavior of the Newmark scheme in the LS represen-tation. Related evolutions of the (50) increment of totalenergy are plotted in Fig. 3. The incipient non stablebehavior is highlighted because the total energy os-cillates and increases up until unbounded values. LR

representation always shows oscillations contained inthe neighborhood of zero. As p = 0 is problem withnonzero initial values, in (50) the value of ΔW is re-placed by the initial kinetic energy. Table 1 finally re-ports the mean value (52) of Newton iterations carriedout in the process for T = 10 in both schemes.

Fig. 3 Dumbbell Δt = 0.02 for T = 2: total energy incrementΔE versus time t ; dotted line for LS and continuous line for LR

representations, respectively

7.2 Two body system

Here, we study the behavior of a pair of equal pin-jointed elements as described in Sect. 5. Starting fromrest, the dynamics of the two prismatic hξ = 0.5, hη =hζ = 0.02, bodies is stimulated by an impulsive forcep(t) acting on the system as illustrated in Fig. 4. Weassume ρ = 1.5 × 104 and E = 5 × 109, kξ = 103,kη = 5 × 102, kζ = 2 × 102, for the evaluation of the(35) kinetic and (36) potential energy. Time-steppingschemes applied to N-body problems can be found inBetsch and Steinmann [10].

The motion of the two body system, computed inboth representations with Δt = 0.0002 and T = 0.3,

716 S. Lopez

Table 1 Dumbbell T = 10:mean value Nwm of theNewton iterations

Δt 0.002 0.005 0.01 0.02 0.05 0.1

LS -scheme 3.000 3.185 3.486 div

LR-scheme 2.665 2.986 3.050 3.575 4.100 5.000

Fig. 4 Two body system:initial configuration andproblem definition

Fig. 5 Two body systemΔt = 0.0002 for T = 0.3:sequence of configurationsobtained by the (a) LS and(b) LR scheme, respectively

is shown in Fig. 5. The unstable motion of the sys-tem in the LS representation can be observed at aboutt = 0.21 where chaotic relative rotations in the jointpoint occur. The reason for this behavior can be iden-tified by means of the time histories of energy givenin Fig. 6. A stable and conservative behavior of theLR scheme can be verified by the time histories re-ported in Fig. 7. Applications of LS and LR rep-resentations for increasing values of the time stepΔt are then investigated. Table 2 reports the Nwm

values performed in the time integration process forT = 1.

7.3 Toss rule in space

The characteristics of the time-integration schemeapplied to the different Lagrangian representationswill be shown here for the example of the three-dimensional movement of a toss rule (see Kuhl andRamm [29] for a solution to such a dynamical prob-lem). The beam, with zero initial displacements andvelocities, is discretized by nine described finite el-ements. The geometry, position of loads, and loadfunction of the rule are described in Fig. 8. The ma-terial is characterized by the values ρ = 7.8 × 103,E = 2.06 × 1011 and G = E/2.

Improving stability in time-stepping nonlinear dynamics 717

Table 2 Two body systemT = 1: mean value Nwm ofthe Newton iterations

Δt 0.00002 0.00005 0.0001 0.0002 0.0005 0.001 0.002

LS -scheme 3.010 3.312 3.665 div

LR-scheme 2.045 2.518 2.965 3.004 3.777 3.965 4.892

Fig. 6 Two body systemΔt = 0.0002 for T = 0.3:energy versus time for LS

representation

Fig. 7 Two body systemΔt = 0.0002 for T = 0.3:energy versus time for LR

representation

Fig. 8 Toss rule: geometry, loads and load function

718 S. Lopez

Fig. 9 Toss ruleΔt = 0.00005 for T = 0.04:sequence of configurationsobtained by the LR scheme

Fig. 10 Toss ruleΔt = 0.00005 for T = 0.04:total energy increment ΔE

versus time t ; dotted linefor LS and continuous linefor LR representations,respectively

Fig. 11 Toss ruleΔt = 0.0001 for T = 0.04:energy versus time for LR

representation

Selected deformed configurations computed within

the t = 0 . . .0.04 time by the LR representation are

shown in Fig. 9. A stable behavior of the LR scheme

is obtained. Figure 10 shows total energy increment

versus time with Δt = 0.00005 for both representa-

tions. An incipient nonstable behavior of the integra-

tion scheme in the LS representation at about t =0.026 can be observed, while bounded oscillations in

the neighborhood of zero value can be verified in theLR representation.

For the LR scheme with Δt = 0.0001, the time his-tories of energy are given in Fig. 11. Finally, Table 3shows the behavior of the two different schemes withrespect to the time integration step for T = 0.1. Wenote that, although a long time stable behavior is ob-served also for large Δt , weak disagreements betweenthe W and U + T values are recorded.

Improving stability in time-stepping nonlinear dynamics 719

Table 3 Toss rule T = 0.1:mean value Nwm of theNewton iterations

Δt 0.00001 0.00002 0.00005 0.0001 0.0002 0.0005

LS -scheme 3.202 3.635 div

LR-scheme 2.104 2.478 3.000 3.225 4.360 4.970

8 Conclusions

In the hypothesis of large rotations and small strains, atechnique to analyze the dynamical behavior of three-dimensional finite element beam frames has been pre-sented. A vectorial approach for the rotation parame-terizations and linear strain definitions based on slopeunknowns has been used. The analyzed models do notuse angle measures and a total Lagrangian and implic-itly conservative mechanical description of the motionis obtained.

The use of relaxed representations in the inter-nal strains—rotations compatibility appears to signif-icantly improve the behavior of the time integrationscheme with regard to stability. In the numerical ap-plications of the approach, it can be seen that the vari-ation in the total energy in the time steps has boundedoscillations about the zero value. Arbitrary implicitone-step time integration schemes with Newmark ap-proximations are suitable for use as basic algorithmsfor the proposed procedure. Finite element assemblingprocedures, moreover, are unchanged by maintainingtheir typical local nature.

The tests show a good convergence of the internalNewton iteration. The increase in computational effortdue to the introduction of new unknowns and the lossof symmetry in the iteration matrix is balanced by thisreduction in the number of Newton iterations requiredin the time integration steps. The computed equilib-rium paths are in agreement with the results reportedin the literature for similar models and the stability inthe solution process has been shown to be insensitiveto large incremental steps, at least for acceptable timeincrements in regard to the approximation error.

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