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Computational Aspects of Nonlinear

Structural Systems with Large

Rigid Body Motion

Edited by

Jorge A.C. Ambrosio

Michal Kleiber

lOS

Press Ohmsha

NATO Science Series

Series III: Computer and Systems Sciences - Vol. 179

NOllOW AGOR GIOnI 3D"MV'I HIIM. SWtIISAS 'IV"MffiJil"MIS "MVtINI'INON .10 SIJtIdSV 'IVNOIIVlildWOJ

~ATO

Science Series

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Series III: Computer and Systems Sciences - Vol. 179

ISSN: 1387-6694

Arnst Publ

Computational Aspects of

Nonlinear Structural Systems

with Large Rigid Body Motion

rATa Science Programme. conjunction with the NATO

ers

Edited by

ITS

y as the NATO ASI Series.

cience between scientists of cientific meeting generally Torkshops", and the NATO .etings are co-organized by atries - countries of the CIS

pth study of latest advances

j ,I

Jorge A.C. Ambrosio Instituto de Engenharia Mecdnica - Instituto Superior Tecnico,

Lisbon, Portugal

and

Michal Kleiber Institute of Fundamental Technological Research, Polish Academy of

Sciences, Warsaw, Poland

assessment of a field, and

in 1999, the NATO Science

t the following web sites for

IDS

Press ISSN: 1387-6694

A .....

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Amsterdam. Berlin. Oxford. Tokyo. Washington, DC Published in cooperation with NATO Scientific Affairs Division

Proceedings of the NATO Advanced Research Workshop on

Computational Aspects of Nonlinear Structural Systems with Large Rigid Body Motion

2-7 July 2000

Pultusk, Poland

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In the last fe: with computational r body motion and w developments are fir finite elements and I complex industrial communities, such numerical analysis, I each of the scientil scientific progress t potential. Therefore. specific results from research. The need foi large lightweight sn research groups. Sir devising advanced f body motion. At thl dynamics communi mechanical systems system components machinery, flexible developments and cc tools. If on one hat other hand they do contributing to thei capabilities of the fii Conversely, advance multibody formulatii achieve numerical ei take advantage of thr Another aspe for integration of rm of structural compot procedures, develop take advantage of su advanced analysis to The book n Workshop on Com; Body Motion, whicl With the active cor academia, industry presentations were ( this event of some 0

PREFACE In the last few years major developments have been achieved in the fields dealing with computational methods for the analysis of nonlinear structures experiencing large rigid body motion and with flexible multibody systems with nonlinear deformations. These developments are finding their way into the commercial general-purpose software in both finite elements and multibody dynamics packages providing powerful numerical tools for complex industrial applications. Unfortunately, the efforts of different scientific communities, such as those dealing with finite elements, multibody dynamics and numerical analysis, have had very often little interaction. Consequently, major findings in each of the scientific areas are slow to reach other communities preventing that the scientific progress that can be achieved in areas of common interest reaches its full potential. Therefore, it was found desirable to assess both the current state-of-art and specific results from the different schools of thought with a focus on the trends for future research. The need for efficient and accurate analysis tools for the design and analysis of large lightweight structural and mechanical systems has been a driving force for many research groups. Since the mid-seventies, the structural mechanics community has been devising advanced finite element methodologies that enable the treatment of large rigid body motion. At the same time, and very often with little or no contact, the multibody dynamics community has been generalizing methods, originally thought to study mechanical systems made of rigid bodies, to describe the structural deformation of the system components. The design and analysis of deployable structures, high-speed machinery, flexible robotics systems, high-end performing vehicles, etc. motivated such developments and continue to justify the research on more powerful and accurate computer tools. If on one hand the advances obtained in the last decade are very important, on the other hand they do not fully explore the competencies of the different scientific areas contributing to them. Flexible multibody methods can take better advantage of the capabilities of the finite element techniques to deal with nonlinear structural deformations. Conversely, advances in the finite element methods can profit from the suitability of the multibody formulations to describe efficiently large rigid body motion. Finally, in order to achieve numerical efficiency it is necessary to devise and use numerical methodologies that take advantage of the specific form of the equations of motion of these systems. Another aspect that it is now recognized as being of major importance is the need for integration of multibody dynamics and finite element codes for the design and analysis of structural components in multibody systems. The use of well-established optimization procedures, developed for structural analysis and for multibody systems, must be able to take advantage of such integration. In this form, the industrial end-user will be able to apply advanced analysis tools without departing from the numerical software of their choice. The book now published is an outgrowth of the NATO Advanced Research Workshop on Computational Aspects Of Nonlinear Structural Systems With Large Rigid Body Motion, which took place in Pultusk, Poland, during the period of July 2 - 7, 2000. With the active contribution of fifty participants from seventeen countries, representing academia, industry and research institutions, fourteen lectures and twenty other invited presentations were delivered and discussed during the NATO ARW. The participation in this event of some of the most prominent researchers in the scientific areas covered by the workshop led to lively discussions and an in-depth assessment of the state-of-art and the definition of directions for future research and developments. This book contains the lectures delivered at the NATO ARW, reflecting to the large extend the results of the discussions carried out during the meeting.

We are grateful to all the lecturers and participants in the NATO ARW for their excellent contributions to the presentations and the discussions that took place during and after the workshop. We are very much indebted to the members of the Organizing Committee, Prof. Ted Belytschko, Prof. Michael Crisfield, Prof. Michel Geradin and Prof. Edward Haug for their valuable suggestions and advise in the organization of the event. We are also grateful to Prof. Gregory Hulbert for his help and active collaboration in defining the scientific programme of the NATO ARW. Dr. Richard Schwertassek† was deeply involved in the initial discussions that led to the proposal and organization of the ARW and always demonstrated his strong and continued support to the event afterwards. He left forever the company of his friends and colleagues too early, but not before he sent us the manuscript supporting what would be his lecture. He has been greatly missed. Special thanks are due to Dr. Kris Wisniewski for his help to organize and run this ARW. A word of acknowledgement is also due to Miss. Paula Jorge for her competent work and dedication towards the Workshop, which led to its smooth running. The generous support of NATO Scientific Affairs Division, which made the ARW possible, is gratefully acknowledged. We are also thankful to the co-sponsors of the workshop, Fundação para a Ciência e Tecnologia (FCT), Office of Naval Research International Field Office, Europe (ONRIFO), Sociedade Gráfica da Paiã (SOGAPAL), Instituto de Engenharia Mecânica (IDMEC), Institute of Fundamental Technological Research of the Polish Academy of Sciences (IPPT) and the Polish Academy of Sciences (PAN).

Jorge A.C. Ambrósio Michal Kleiber

NATO Advanced Research Workshop COMPUTATIONAL ASPECTS OF NONLINEAR STRUCTURAL SYSTEMS WITH LARGE RIGID BODY MOTION Pultusk, Poland, July 2-7, 2000

MAIN SPONSOR: NATO - North Atlantic Treaty Organization, Scientific Affairs Division

OTHER SPONSORS: FCT – Fundação para a Ciência e Tecnologia NSF – National Science Foundation ONRIFO – Office of Naval Research International Field Office, Europe SOGAPAL – Sociedade Gráfica da Paiã IDMEC – Instituto de Engenharia Mecânica IPPT – Institute of Fundamental Technological Research of the Polish Academy of Sciences PAN – Polish Academy of Sciences

DIRECTORS Prof. Jorge A.C. Ambrósio – IDMEC, Instituto Superior Técnico, Portugal. Prof. Michal Kleiber – IPPT, Polish Academy of Sciences, Poland.

ORGANIZING COMMITTEE Prof. Jorge A.C. Ambrósio, IDMEC, Instituto Superior Técnico, Portugal Prof. Ted Belytschko, Northwestern University U.S.A. Prof. Mike Crisfield, Imperial College, U.K. Prof. Michal Kleiber, IPPT, Polish Academy of Science, Poland Prof. Edward Haug, University of Iowa, U.S.A.

CONTENTS Preface Part I – Flexible Multibody Systems ...................................................................................... 1 Geometric and Material Nonlinear Deformations in Flexible Multibody Systems Jorge Ambrósio ................................................................................................................ 3 Performance of Non-Linear Finite Element Formulations in Flexible Multibody Simulations Ahmed Shabana.............................................................................................................. 29 Modal Representation of Deformation and Stress in Flexible Multibody Simulation Richard Schwertassek..................................................................................................... 41 Part II – Finite Elements Procedures for Structural Systems With Large Rotations .............. 61 Finite Element Analysis of Rigid-Flexible Systems Robert Taylor ................................................................................................................. 63 Dynamics of Complex Flexible Multibody Systems Undergoing Large Overall Motion Adrian Ibrahimbegovic and Said Mamouri ...................................................................... 85 Implicit Kinematical Parameters and Sensitivity of Finite Rotation Shells Kris Wisniewski, Michal Kleiber and Ewa Turska .......................................................... 103 Part III – Numerical Integration Methods for Rigid and Flexible Systems ........................... 119 Energy/Momentum Conserving Time Integration Procedures With Finite Elements and Large Rotations Mike Crisfield and Gordan Jelenić.................................................................................. 121 Efficient and Robust Computational Algorithms for the Solution of Nonlinear Flexible Multibody Systems Gregory Hulbert ............................................................................................................ 141 Sensitivity Analysis and Design Optimization of Differential-Algebraic Equation Systems Linda Petzold, Radu Serban, Shengtai Li, Soumyendu Raha and Yang Cao.................... 153 Multi-Timestep Integration in Computational Dynamics Bill Daniel ..................................................................................................................... 169 Part IV – Advanced Methods in Systems with Large Rigid Body Motion ........................... 191 Nonlinear Structural Multi-Body System Simulation Using Structural and Rigid Body Analysis Software Weidong Pan and Edward Haug .................................................................................... 193 Nonlinear Structural Behavior, Analysis and Design of Deployable Structures Charis Gantes ................................................................................................................ 215 An Embedded Projection Method for Index-3 Constrained Mechanical Systems Marco Borri, Carlo L. Bottasso and Lorenzo Trainelli ................................................... 237 Localized Formulation of Multibody Systems K.C. Park, Carlos Fellipa and Roger Ohayon ................................................................. 253

Part I Flexible Multibody Systems

Geometric And Material Nonlinear Deformations In Flexible Multibody Systems Jorge AMBRÓSIO Instituto de Engenharia Mecânica (IDMEC), Instituto Superior Técnico, Av. Rovisco Pais, 1049-001 Lisboa, Portugal Abstract. Multibody models for which the flexibility of the system components plays a fundamental role in their behavior efficiently describe a large number of mechanical and structural systems of high importance. Vehicles, space satellites, deployable structures, machines operating at high speeds or robot manipulators are just some of the applications where the systems undergo large rigid body motion while their components experience structural deformations. The correct modeling of these systems requires that the formulations used in their description not only account for the motion and deformation of their components but also describe their inertial coupling. The methodology reviewed here uses an updated Lagrangean formulation to obtain the continuum mechanics equations of motion of a general flexible body. The finite element method is then applied to obtain a representation of the flexible bodies suitable to the description of any general flexible component of the multibody system in a computational environment. The equations obtained at this stage are computationally inefficient due to their high nonlinearity. The use of a lumped mass formulation and a change of nodal coordinates lead to much simpler equations of motion, still accounting for the inertial coupling between large rigid body motion and the body flexibility. In the sequel of proposed methodology it is shown that if the flexible body material behavior is linear and the deformations are small the equations obtained can be greatly simplified by using the component mode synthesis. In order for the flexible bodies to be used in the multibody system model it is required that a set of kinematic joints, describing algebraic constraints between the system coordinates, are defined. The concept of virtual bodies provides a general framework to develop general kinematic joints. This only requires that a rigid constraint between a flexible and a rigid body is derived. Then, all kinematic joints are set between rigid bodies only. Taking advantage of the special form of the equations of motion obtained, the explicit use of Lagrange multipliers associated with kinematic joints involving flexible bodies is eliminated from the system equations of motion. The virtues and shortcomings of the formulation proposed are appraised through applications to vehicle structural impact cases and to highly deformable structures. In the process of studying the flexible multibody model impact, a contact model is proposed and discussed also.

1.

Introduction

The design requirements of advanced mechanical and structural systems and the realtime simulation of complex systems exploit the ease of use of the powerful computational resources available today to create virtual prototyping environments. These advanced simulation facilities play a fundamental role in the study of systems that undergo large rigid body motion while their components experience material or geometric nonlinear deformations, such as vehicles, deployable structures, space satellites, machines operating

at high speeds or robot manipulators. If in one hand the nonlinear finite element method is the most powerful and versatile procedure to describe the flexibility of the system components on the other hand the multibody dynamic formulations are the basis for the most efficient computational techniques that deal with large overall motion. Therefore, it is no surprise that many of the most recent formulations on flexible multibody dynamics and on finite element methods with large rotations share some common features. In most of the earlier work dealing with the elastodynamics of mechanical systems the deformations of the system components, assumed elastic and small, is superimposed to their large rigid body motion. In different reviews, Erdman and Sandor [1], Thompson and Sung [2] and Lowen and Chassapis [3] discuss the inertial coupling between the small deformation of the elastic linkages of the system and their gross motion with some contradictory views on its importance. In particular, Thompson and Sung [3], based on experimental work, claim that the inertial coupling is not important in many of the most commonly used mechanical systems. Using reference frames fixed to planar flexible bodies, Song and Haug [4] suggest a finite element based methodology, which yields coupled gross rigid body motion and small elastic deformations. One of the major problems, when the method was proposed, had to do with the computational effort required to use such methodology with complex shaped flexible bodies. The idea behind Song and Haug’s approach is further developed and generalized by Shabana and Wehage [5,6] that use substructuring and the mode component synthesis to reduce the number of generalized coordinates required to represent the flexible components. According to Spanos and Tsuha [7], the selection of modes required for the component mode synthesis vaguely implies the solution of an eigenvalue problem. The methodologies proposed use a limited number of modes to represent the flexible components. Yoo and Haug [8] account for the contribution of the truncated modes by introducing static correction modes. With similar approaches many authors are using this type of formulation to the modeling of complex multibody systems [9-12]. The structure of the equations of motion for flexible multibody systems, obtained with the methods described before, include the mass and stiffness matrices, commonly used for finite element models, the mass and inertia matrices and the gyroscopic and centrifugal force vectors, always present in multibody formulations, and the inertia coupling terms, which are only encountered in these formulations. These coupling terms involve the derivation of matrices, which include the finite element shape functions that are not available in the common finite element literature. Such terms need to be derived if other types of finite elements are to be used in the flexible multibody models, besides beam and eight-node plates/shell elements [13-14]. As a result, most of the flexible multibody models presented in the literature are made of beam elements only. Using a lumped mass formulation for the inertia terms Ambrósio and Gonçalves [15] showed not only that all the terms required in the flexible body equations of motion are readily available from any commercial finite element code but also that any type of linear finite element can be used in the model. The community studying space dynamics was naturally involved in dealing with the dynamics of flexible bodies undergoing large rigid body motion. The problem that attracted their initial attention was the stabilization of spinning spacecrafts with flexible appendages [16]. The orbiting space structural systems, generally involving open chain models, are characterized by the use of very flexible lightweight components. The need to characterize dynamically and control such systems, in particular, motivated valuable investigations on flexible multibody dynamic [17,18]. In the framework of the spinning spacecrafts modeling, Kane, Ryan and Banerjee [19] showed that though most of the flexible multibody methods, at the time, could capture the inertia coupling between the elastodynamics of the

system components and their large motion but they would still produce incorrect results because they neglected the dynamic stiffening effects. This comment motivated a very large amount of research work, in the years that followed, addressing the nature and the solutions of such problem [11,20]. The nature of the observation by Kane and co-workers [19] was in fact well-known by the finite element community where the use of the mode superposition technique has been used mostly for the study of linear elastic structures exhibiting small or moderate rotations about their reference frame. Actually, the floating reference frame methods used in flexible multibody dynamics have the ability to lower the geometric nonlinearities of the flexible bodies, but they do not eliminate them because the moderate rotation assumption about the floating reference frame is still required [21]. The work of researchers in the finite element community, such as that by Belytschko and Hsieh [22], Simo and Vu-Quoc [23] or Bathe and Bolourchi [24] among others, addressing the same type of problems can be easily adapted to the framework of flexible multibody dynamics. Recognizing the problem posed and using some of the approaches well in line with those of the finite element community Cardona and Geradin proposed formulations for the nonlinear flexible bodies using either a geometrically exact model [25] or through substructuring [26]. Defining it as an absolute nodal coordinate formulation, Shabana [27] used finite rotations nodal coordinates enabling the capture of the geometric nonlinear deformations. Another approach taken by Ambrósio and Nikravesh [28] to model geometrically nonlinear flexible bodies is to relax the need for the structures to exhibit small moderate rotations about the floating frame by using an incremental finite element approach within the flexible body description. The approach is further extended to handle material nonlinearities of flexible multibody systems also [29]. Using an updated Lagrangean formulation, the equations of motion obtained for the flexible bodies are general and allow modeling most of the geometric and material nonlinearities. This formulation is reviewed here and applied to cases involving both linear and nonlinear geometric and material deformations. In the sequel of the applications presented proper kinematic constraints between flexible and rigid bodies are developed using the concept of virtual bodies. Also a suitable contact model is presented allowing for the application of this formulation to cases involving contact and impact.

2.

Multibody system equations of motion

A multibody system is a collection of rigid and flexible bodies joined together by kinematic joints and force elements, as shown in figure 1. For the ith body in the system qi denotes a vector of coordinates which contains the Cartesian translation coordinates ri and a set of rotational coordinates pi. A vector of velocities for a rigid body i is defined as vi, which contains the translation velocities r
i and the angular velocities ωi. The accelerations vector for the body is denoted by v
i , which is the time derivative of vi. In a multibody system containing nb bodies, the coordinates, velocities, and accelerations vectors are q, v and v
which contain the elements of qi, vi and v
i , respectively, for i=1,...,nb. The joints between rigid bodies are described by the mr independent constraints

Φ (q ) = 0

(1)

The time derivatives of the constraints result in the velocity and acceleration equations.


= Dv = 0 Φ


v + Dv
= 0

= D Φ

(2) (3)

Current configuration

time t+∆t Z

Co-rotated updated η configuration

ζ ξ

Y

t +∆t


b


h

X

t


b


∆b

Figure 1. General representation of a rigid-flexible multibody system

where D is the Jacobian matrix of the constraints. The constraint equations are included in the equations of motion using the Lagrange multiplier technique [30]

Mv
− DT λ = g

(4)

where M is the inertia matrix, λ is a vector of Lagrange multipliers, and g = g (q, v ) has the gyroscopic terms and the forces and moments that act on the rigid and flexible bodies. If flexible bodies are included, the equations for the constrained multibody system are   Mr M fr  Φ q  r

M rf M ff Φq

f

 ΦTq r  Φ Tq f   0  



r   g r  q u

′      = g f  −  λ   γ 

 sr  s  −  f  0 

 0   K u ′  ff   0 

(5)

In equation (5), the subscript f refers to the flexible bodies quantities. Depending on the formulation used and on the generalized coordinates selected, the matrices and vectors associated to flexible bodies have different structures. These are presented and discussed in the framework of nonlinear geometric and material deformations of nonlinear bodies.

3.

Equations of Motion for Nonlinear Flexible Bodies

The motion of a flexible body, depicted by figure 2, is characterized by a continuous change of its shape, due to internal or external forces, and by large displacements and rotations, associated to the gross rigid body motion. Let XYZ denote the inertial reference frame and ξηζ a body fixed coordinate frame. Let the principle of the virtual works be used to express the equilibrium of the flexible body in the current configuration t+∆t and an updated Lagrangean formulation be used to obtain the equations of motion of the flexible body [24]. After linearization, these are written as [28] T T t ′ t′ T t′ t′ t′ ∫ (δ t′e′) t′C t′e′ dv + ∫ (δ t′η′) τ′ dv = − ∫ (δ t′e′) τ′ dv −

t′

t′

V

− ∫

t′ V

t′

ρ (δh )

t′

V

V T t′+ ∆t t′ t′ fb dv +

h dv + ∫ ρ (δh )

T t ′+ ∆t

t′

t′

t′ V

∫ (δh )

t′ A

T t ′ + ∆t t′ t ′ f s da

(6)

t

ζ

t

time t

η

Initial configuration t

ζ

ξ

t


b

time t+∆t t +∆t t +∆t

0 0

Z

η 0

Current configuration

Y

ξ

time 0

X

ζ

t +∆t

ξ t + ∆t

t + ∆t

η


b


h

Co-rotated updated configuration

t ′


b


∆ b

Figure 2 General motion of a flexible body

where δ h is the virtual variation of a material point displacement, t′e′ and t′η′ are the linear and nonlinear terms of the Green-Lagrange strain increments, t′τ is the Cauchy stress tensor, and fb and fs are the body and surface forces respectively. The left superscripts and subscripts refer to the configurations in which a quantity is measured and to the reference configuration respectively. 3.1

Finite element equations of motion

Let the finite element method be used to represent the equations of motion of the flexible body. Referring to figure 1, the assembly of all finite elements used in the discretization of the flexible body results in its equations of motion written as [28]  M rr  Mφr M fr 

M rf Mφφ M fφ

M rf   Mφf  M ff 

0 
r
  g r   s r  0 0 0         ω  
′ ′ ′ 0   =  gφ  −  sφ  − 0 − 0 0   u

′  g′f  s′f  f  0 0 K L + K NL 

0 0   u′

(7)


′ are respectively the translational and angular accelerations of the body fixed where
r
and ω

′ denotes the nodal accelerations measured in body fixed coordinates. reference frame and u The local coordinate frame ξηζ, attached to the flexible body, is used to represent the body’s gross motion and its deformation. Vector u´ denotes the displacements increments from a previous to the current configuration, measured in body fixed coordinates. Equation (7) describes thoroughly the motion of the flexible body. Even if only small elastic deformations occur, this equation is highly nonlinear due to a variant mass matrix, changing external applied forces, gyroscopic and centrifugal forces and non-constant stiffness matrices. The variant mass matrix for the flexible body results from the assembly of the individual contributions of each finite element. The sub-matrices describing the contribution of a single finite element for the mass matrix are given by ~ M rφ j = − A ∫V ρ b′ dv M rr j = I ∫ ρ dv Vj

j

M r f j = A ∫ ρ N dv

~~ Mφ φ j = − ∫V ρ b′b′ dv

~ Mφ f j = ∫ ρ b′ N dv

M ff j = ∫V ρ NT N dv

Vj

Vj

(8)

j

j

Here, A is the transformation matrix from the body fixed coordinate system to the inertial frame, N is the matrix of shape functions of element j and ρ is the mass density. Submatrices Mrr and Mff are constant and represent respectively the mass of the entire

element and the standard finite element mass matrix. Mrf and Mφf are the time variant matrices responsible for the inertia coupling between the gross motion of the body and its deformations. In the numerical implementation of the mass matrix special attention must be paid to the evaluation of Mφφ as large deformation develop. This sub-matrix, representing the inertia tensor of the flexible body is approximately constant if the body deformations are small, otherwise its time variance cannot be neglected. All other submatrices in equation (7) are either null or constant, provided that a proper choice is made for the location and orientation of the body fixed coordinate frame. The right-hand side of equation (7) contains, the vector of gyroscopic and centrifugal forces s and the vector of generalized forces g, which are evaluated for each element j as

  ~′ ~′ sr   A ω ω ∫V j ρ b ′ dv  ~ ~ ~      s φ′  =  ∫V j ρ b ′ ω′ ω′b ′ dv  + 2 s ′f   ~′ω ~ ′b ′ dv    j ∫ ρ NT ω  V   j

  ~  A ω′ ∫V j ρ N dv  ~ ~    ∫V j ρ b ′ ω′ N dv  u
′j   T ~  ∫V j ρ N ω′ N dv   

(9)

The vector of the external generalized applied forces gj for each element is:     ∫A j f s da   ∫V j ρ f b dv  gr   ~ T ~ T        g φ′  =  ∫A j b ′ A f s da  +  ∫V j ρ b ′ A f b dv  g ′f        j  ∫ N T A T f s da   ∫ ρ N T A T f b dv     Vj  Aj

(10)

In equation (10), fb and fs are respectively the body and the surface forces. Matrices KLj and KNLj are the linear and nonlinear stiffness matrices respectively, and fj denotes the vector of equivalent nodal forces due to the state of stress K Lj = ∫

Vj

K NL j = ∫

Vj

fj =∫

Vj

BTL C B L dv

(11)

B TNL τ ′ B NL dv

(12)

B TL τ ′ dv

(13)

In these equations BL and BNL denote the linear and nonlinear strain matrices respectively and τ′ is the Cauchy stress tensor for the updated configuration. Note that the reference to the linearity of the stiffness matrices KL and KNL is concerned to their relation with the displacements. For a constitutive tensor C not constant both KL and KNL are not linear. This is the case when a multibody system experiences elasto-plastic deformations of one or more of its components. For these problems, an elasto-plastic constitutive tensor C must be used in the equation (11). 3.2

Generalized coordinates of the flexible bodies

Equation (7) is not efficient for numerical implementation due to the need to invert the variant mass matrix every time step, during the integration process. A simpler form of the equations of motion for a flexible body is obtained when a lumped mass formulation is

′ are substituted by the nodal accelerations relative to the used and the accelerations u

′f . inertial frame q

Let the nodal accelerations be partitioned into translational and angular accelerations:

[



′ =
δ
′T ,
θ
′T u

]

T

[



′T , α

′f = d

′T q

and

]

T

(14)

with relation to this partition of the flexible coordinates, the mass matrix of the flexible body is now evaluated using lumped masses and inertias at the nodal points [31], leading to ~ M rr j = ∑ m k I M rφ j = −∑ m k Ab′k ~ ~ (15) M r f j = ∑ m k AITk 0 Mφ φ j = −∑ m k b′k T b′k + ∑ µ k I Mφ f j

[ ] ~ = [∑ m b′ I ∑µ I ] T k k k

 m I IT M ff j = ∑ k k k 0 

T k k

  ∑ µ k I k ITk  0

where mk and µk are the lumped mass and inertia of node k respectively, and I k is a Boolean matrix, which associates node k to the finite element node numbering scheme. The lumped mass formulation of the centrifugal and gyroscopic forces leads to  − ∑ m k Aω~′ ω~′ b′k − 2∑ m k Aω~′δ
′k    ~ ~ − ∑ m k b′kω~′ ω~′ b′k − 2∑ m k b′kω~′δ
′k   s= − ∑ m I ω~′ ω~′ b′ − 2∑ m I ω~′ AT δ
′  k k k k k k   0  

(16)

The relation between the relative and absolute nodal accelerations for node k is

′ ~ ′ω ~ ′(x + δ )′ + 2ω ~ ′δ
′  − ~ xk + δ
k  
r
 + ω k k k     ~
′
ω′θ′k  ω   I 

(



′   T d

′f ≡   = u

′k +  A q k

′ k  0 α

)

(17)

where xk is the vector containing the node position in the reference configuration. Equations (15) through (17) are evaluated for all finite elements nodes and substituted into equation (7). The result is simplified yielding [32] n

A

∑ mk d

′k = g r

(18a)

k =1

∑ mk (~x′ + δ ′)k d

′k = gθ′ n

k =1

~

(

(18b)

)



′f = g′f − tt′ f − tt′ K L + tt′ K NL u′ M ff q

(18c)

Equations (18a) and (18b) describe the motion of a system of particles center of mass [33]. If the origin of the body fixed coordinate system is coincident with the center of mass of the flexible body, equations (18a) and (18b) describe the motion of the origin of the ξηζ referential. Equation (18c) describes the flexible body nodal motion, expressed in the body fixed coordinate system. The lumped mass formulation leads to a diagonal mass matrix Mff. 3.3

Partially rigid-flexible body

Let a rigid body with mass and inertia similar to those of the flexible body, as described by equations (18b) and (18c) respectively, be defined. Moreover, assume that the body fixed coordinate system associated to such rigid body has the same location and orientation of the flexible body floating frame. This situation is shown in figure 3, where the flexible body is represented as having rigid and a flexible parts.

Flexible part

Undeformed configuration

Boundary ψ

ζ

Rigid part

Deformed configuration
xk

ξ


r

Z X

Y

η

Node k
uk


dk

Figure 3. Flexible body with a rigid part

The equations of motion of a rigid body i are given by

mi
r
i = f ri ~ ′J′ ω′
′ι = n′i − ω J′i ω ι i i

(19a) (19b)

where fri and n'i are the external forces applied over the center of mass of the rigid part and moments applied over the body respectively. In order for equations (19a) and (19b) to be equivalent to equations (18a) and (18b) it is necessary that the mass and inertia tensor of rigid body i are equal to those of the flexible body, as given in equations (18a) and (18b) respectively. Furthermore, the generalized external forces applied over the rigid body are gr and g′θ given by the first two rows of equation (10) respectively. Equations (19a), (19b) and (18c) represent the equations of motion of the partially flexible body provided that proper reference conditions are set. A suitable set of reference conditions is introduced by enforcing that the flexible and rigid parts are attached by the boundary nodes ψ. This is achieved by imposing kinematic constraints, which ensure that the nodes in the boundary ψ have null displacements, velocities and accelerations with respect to the body fixed coordinate frame. For a given node k this is

′k = 0 u′k = u
′k = u

(20)

Before the constraints implied by equations (20) can be applied they must be expressed in terms of the generalized flexible coordinates q′f. For a constrained node k equations (20) are used in the nodal acceleration equations (17) leading to

′   AT d =   

′ k  0 α

~~ −~ xk′  
r
 ω′ω′x′k  +   
′  0  I  ω

(21)

The acceleration equation of the constrained node is rearranged to obtain a form similar to that of equation (3), which is  − AT   0

~ xk′ I 0  − I 0 I 


r
 ω
′  ~′~′ ′   = ω ω x k 

′   0  d k    

′κ  α

(21)

The nodal constraints are applied to the rigid part equations of motion, equations (19a) and (19b), and to the flexible part equations, represented by equation (18c). This is,

m
r
+ Aλ δ = f r

(22a)

~ ′ J′ ω′
′ι − ~ J ′i ω xk′ λ δ − λθ = n′i − ω ι i i

(22b)

(

)

λ 

′f +  δ  = g′f − tt′ f − tt′ K L + tt′ K NL u′ M ff q  λθ 

(22c)

Using equations (21) and (22), the Lagrange multipliers are eliminated from the equations of motion resulting in the dynamic equations for the rigid-flexible body [28] mI + A M* AT  T  − A M *S  0 

(

0  
r
   f r + AC′δ     * ~ ′J′ω′ − ST C′ − I T C′ 
′  = n′ − ω J ′ + ST M S 0  ω δ θ 

′   ′  ′ ( ) − − + g f K K u 0 M ff  q f   f L NL   − A M*S

)

(23)

where for notation convenience, auxiliary matrices are introduced to represent the existence of more than one constrained node. These are defined as

(

AT = [ A A  A ] ; S =  x
1′ + δ
1′  T

) ( T

x
′2 + δ
′2

)

T

(

 x
′n + δ
′n

)

T

T

 ; 

I = [I I  I ]

T

Vectors Cδ′ and Cθ′ , appearing in equation (23), represent respectively the reaction force and moment of the flexible part of the body over the rigid part, given by

Cδ′ = g′δ − Fδ − (K L + K NL )δ δ δ′ − (K L + K NL )δθ θ′ C′δ = gθ′ − Fθ − (K L + K NL )θδ δ′ − (K L + K NL )θθ θ′

(24) (25)

In these equations, the subscripts δ' and θ' refer to the partition of the vectors and matrices with respect to the translation and rotational nodal degrees of freedom. The underlined subscripts are referred to nodal displacements of the nodes fixed to the rigid part. Note that the equations of motion of the flexible body ensure that the coupling between the rigid body motion and the system deformations is fully preserved. It should also be noticed that the formulation of the body mass matrix, and in particular of its inertia tensor, is valid only when the flexible body experiences small distortions. When that is not the case the inertia tensor must be updated during the dynamic analysis. Finally, the use of the lumped mass formulation implies that the use of higher order finite elements should be avoided in the evaluation of the finite element mass matrix.

4.

Kinematic joints

The use of different sets of generalized coordinates in multibody codes generally implies that new kinematic constraint equations must be derived before the flexible bodies can be used to model general multibody systems. Consequently, either the number of kinematic joints available is duplicated, with the corresponding duplication of development effort, or the full modeling potential of the multibody code is not used. A way of getting around this situation is to use the concept of virtual bodies, which are massless rigid bodies rigidly attached to flexible body nodes. The kinematic joints are defined only between rigid and virtual bodies, allowing the user to include in the model any kinematic joint available in the multibody code [34,35].

ζi ξ

i


r

Z X

Y

η


xk i


δk


dk

ζj ηj


sP ξj

Figure 4. Virtual body attached to a flexible body

Let a point P of virtual body j be rigidly attached to node k of the flexible body j, as shown in figure 4. The corresponding kinematic constraint is represented by the velocity equation A d
′ − r
j + A j ~sP′ j ω′j = 0 d
= r
Pj
≡  i k
≡  k (26) Φ ≡ Φ A i α
′k − A j ω′j = 0 α
k = ω j which is rewritten as

0 0 0 A i
≡ Φ 0 0 0 0

0 Ai

 r
i   ω′   i q
′f  − I A j ~sP′ j  
i  d′k = − A j    0  α
′k     r
j   ω′   j

0  0   

(27)

where the vector of generalized flexible coordinates of the flexible body is partitioned. The time derivative of the velocity equations lead to the acceleration equations, given by

0 0 0 A i
≡ Φ 0 0 0 0

0 Ai


r
i  ω
′   i

′f  q ~ ′d
′ − A ω ~ ~′ ω′  − I A j ~sP′ j  

i   − A i ω i k j ′j sP j j ′ d = k  ~ − A j     0 − A i ω′i α
′k   α

′k   
r
j   ω 
′j 

(28)

Using the Lagrange multiplier technique, the constrained equations of motion of the subsystem formed by the flexible and virtual body are written as

mi I   0  0   0  0   0   0  0   0

0 J ′i

0 0

0 0

0 0

0 0

0 0

0 M ff 0 0 0 0 0 mk I 0 0 0 0 0 µk I 0 0 0 0 0 m jI 0

0

0

0

0

0

Ai

0

0

0

0

Ai

0 0 0 0

J ′j − I A j ~sP′ j 0 0

−Aj

0  
r
i  0   ω
′   i 0 0  q

′    fi  T

′ Ai 0  d k  T   0 Ai

′k  = α   −I 0  
r
j  
′  − ~sP′ j ATj − A Tj   ω  j 0 0   λδ    0 0   λθ  0 0

  f r + AC′δ   T T ~ n′i − ω′i J ′i ω′i − S C′δ − I Cθ′   g′f − f − (K L +K NL )u′    g ′k     n′k   frj     ~ n′j − ω′j J ′j ω′j   ~ ′ d
′ − A ω ~ ~′ ω′   − Ai ω i k j ′j sP j j   ~ ′ α
′ − Aiω i k  

(29)

where g′k and n′k represent all forces applied on node k, including the internal forces of the flexible body. The Lagrange multipliers can now be eliminated from equation (29), adopting a procedure similar to that used to obtain the equations of motion of the rigid flexible body, resulting in the equations of motion of the virtual body, written as    m j + ∑ mk I k    J′j + ∑ µ k I − mk ~sP′ j ~sP′ j  k

(

  
r
j    =
′j   ω 

)

(30)

~ ′d
′ )   f r j + A i ∑ (g′k − mk ω i k   k ~ ′ J′ ω′ + AT A (n′ + µ ω ~ ′ α
′ ) + ~s′ AT A (g′ − m ω ~ ′d
′ ) + ω ~ ′ m ~s′ ~s′ ω ~′  n′ − ω ∑ Pj j i k k i k i ∑ k Pj Pj i  j j j j i∑ k k i k  j k k k 

Note that the accelerations of the constrained node k, attached to the virtual body, are defined by the kinematic acceleration equations (28). Furthermore, it is assumed in equation (39) that the virtual body fixed coordinate frame is attached to the center of mass of the virtual body, including the constrained nodes mass, and its axis orientations are the same as the principal axis of inertia orientations. In order for the formulation of the rigid joint with the virtual bodies to be complete it is necessary to evaluate the reaction force of the virtual body over the fixed nodes. This force is then added to the force vectors gr and g′θ of the rigid part of the partially flexible body. With this purpose in mind, take the 6th and 7th rows of equation (29), and rearrange them as m j I  

frj + λδ   
r
j   =      T T ~ ~
′j  n′j − ω′j J′j ω′j + sP′ A j λδ + A j λθ  J′j  ω j  

(31)

By comparison between equations (30) and (31), the Lagrange multipliers are obtained. For this purpose let it be assumed that the constrained nodes are massless and that, instead, the mass and inertia tensor of the virtual body contain the nodes original mass and inertia. Therefore, the reaction forces of the virtual body on the flexible body center of mass are evaluated as: g i  n′   i

reaction

~ ′d
′ )   A i ∑ (g′k − mk ω i k   k = − ~ ~ ′ α
′ ) + b ~ ′d
′ ) ′ ′ ′ + − ( n µ ( g m ω ω ∑ k k k i k k k i k ∑ k k 

(32)

The equations of motion of the rigid part of the partially flexible body, to which a virtual body is attached, are now written as

mI + A M* AT  * T  − AM S  0 

(

)

 0  
r
  f r + AC′δ + g ireaction      * T T T reaction ~ ′J ′ω′ − S C′ − I C′ + n′
′  = n′ − ω J′ + S M S 0  ω  i δ θ  

′    g′f − f − (K L +K NL )u′ 0 M ff  q f    

− A M*S

(33)

The equations of motion of the flexible part of the body remain unchanged, except for the nodal accelerations of node k, which are now evaluated using equation (28).

5.

Demonstration examples

Two application examples are commonly found in the literature to demonstrate the coupling between the flexibility of the multibody system components and their large rigid body motion and to show the limitations of the descriptions of the flexible multibody models when only linear elastic deformations are considered. A slider-crank model where the connecting rod is flexible [36] and the spin-up maneuver of a rotating beam [19] are reviewed here to demonstrate the proposed methodology.

5.1

Slider-crank with an elastic connecting rod

Proposed by Chu and Pan [36] and later revisited by many other authors [8,11,35], the study of the dynamic response of a planar slider-crank system composed of a rigid crank, a flexible connecting rod and a massless slider illustrates the importance of using methodologies able to model the inertia coupling between the system elastodynamics and its large gross motion. The slider-crank, pictured in figure 5, is made of a 0.1524 m long rigid crank and a 0.3048 m flexible connecting rod, both having circular cross-sections with 0.00635 m in diameter and a mass density of 7820 Kg/m3. The remaining properties of the connecting rod are a 2.07 1011 N/m2 Young modulus and a Poisson’s ratio of 0.285.
δ

ω = 124.8 rad / s Figure 5 Slider-crank with a flexible connecting rod

The simulation of the system is initiated when the slider and the connecting rod are aligned and no deformation is observed. With a crank angular speed of –124.8 rad/s, the system is simulated for a complete revolution of the crank. The model of the connecting rod is made of 8 beam elements, using the nonlinear beam element proposed by Bathe and Bolourchi [24]. Figure 6 displays the normalized lateral deflection of the connecting rod, i.e, the deflection divided by the connecting rod initial length, versus the crank angle. In two of the models simulated only linear deformations of the flexible component are accounted for, consisting their difference in the amount of structural damping actually used. In a third model the nonlinear geometric deformations of the connecting rod are represented.

Normalized Deflection

0.015 Linear

0.010

Nonlinear

0.005 0.000 -0.005 -0.010 0

1

2

3

4

5

6

7

8

Crank Angle

Figure 6 Normalized lateral deflection of the flexible connecting rod mid-point

The results show that for the model representing only the system linear elastodynamics, the connecting rod mid-point has lateral deformations similar to those reported in the literature [8]. The small differences between the peak displacements are attributed to the three-dimensional characteristics of the beam elements used and to the lumped mass formulation. No significant difference is observed in the response of the linear elastic model that includes structural damping when compared to that of the model not damped. When the model representing the geometric nonlinear characteristics of the connecting-rod is used, the lateral displacements exhibit a similar behavior, but the peak displacements are lower than those displayed by the linear model.

5.2

Spin-up maneuver of a rotating beam

The problem of a cantilever beam attached to a rigid hub, which is spun up from rest to a constant angular speed, was proposed by Kane et al [19] in order to demonstrate the shortcoming of the linear elastic flexible multibody models resulting from the premature linearization of the governing equations. This problem, revisited periodically by different researchers [11,20,32], serves in fact as a benchmark to the capabilities of different codes to handle geometrically nonlinear cases. The spinning beam, represented in figure 7, has a length of 10 m and an annular cross-section with an outside diameter of 0.0652 m and a interior diameter of 0.0612 m. The material has a Young’s modulus of 69 109 N/m2 and a mass density of 3000 Kg/m3. ω

In plane displacement Cross section L

d D

L = 10 m D = 6.52 cm d = 6.12 cm E = 69 10 9 N / m 2 r = 3000 Kg / m 3

Figure 7 Model of a flexible beam attached to a rotating rigid hub

The flexible appendage model, made of 8 beam finite elements, is spun up from rest to a constant angular speed of 6 rad/s according to a prescribed function of time given by  6  15  2π t   t − sin   rad/s 0 ≤ t ≤ 15s ω (t ) = 15  2π  15   6 rad/s t ≥ 15s 

Tip displacement (m)

The results of the simulation, in terms of the displacement of the flexible beam model tip with respect to what would be its undeformed position, are presented in figure 8 for models of the flexible body for which linear and nonlinear deformations are represented. 0 -0.1


δ

-0.3

-0.5

Linear nonlinear

-0.7

-0.9 0

4

8

12

16

20 Time (s)

Figure 8 In-plane displacement of the tip of a flexible beam attached to a rotating rigid hub

The results show that for the linear model the tip displacement becomes unbounded just after 7 seconds of simulation. However, when the nonlinear model is considered, the tip displacement of the appendage, with respect to its undeformed position, increases while the angular velocity increases. When finally the angular acceleration of the hub becomes zero, the tip of the beam simply oscillates about its undeformed position. These results are similar to those obtained by other authors [19,20].

6.

Contact model

Many of the practical applications of flexible multibody systems involving nonlinear deformations also involve contact between different parts of the system or impact with external surfaces. The models for contact-impact require in a first step that kinetic and geometric conditions are setup and in a second step that the contact forces be evaluated.

6.1

Contact detection

Let the flexible body approach a surface during the motion of the multibody system, as represented in figure 9. Without lack of generality, let the impacting surface be described by a mesh of triangle patches. In particular, let the triangular patch, where node k of the flexible body will impact, be defined by points i, j and l. The normal to the outside


surface of the contact patch is defined as n = rij × r jl .





rikn = (rik • n ) n

k


rik


rli


n

l


r jl

k*

i






rikt = rik − (rik • n ) n


rij

j

Figure 9 Contact detection between a finite element node and a surface

Let the position of the structural node k with respect to point i of the surface be

rik = rk − ri

(34)

This vector is decomposed in its tangential component, which locates point k* in the patch surface, and a normal component, given respectively by

( )

rikt = rik − rikT n n

(35)

( )

rikn = rikT n n

(36)

A necessary condition for contact is that node k penetrates the surface of the patch, i.e.

rikT n ≤ 0

(37)

In order to ensure that a node does not penetrate the surface through its ‘interior’ face a thickness e must be associated to the patch. The thickness penetration condition is

− rikT n ≤ e

(38)

The condition described by equation (38) prevents that penetration is detected when the flexible body is far away, behind the contact surface. The remaining necessary conditions for contact results from the need for the node to be inside of the triangular patch. These three extra conditions are

(~r r )

T t ik ij

(

n≥0 ; ~ rikt r jl

)

T

(

n ≥ 0 and ~ rikt rki

)

T

n≥0

(39)

Equations (37) through (39) are necessary conditions for contact. However, depending on the contact force model actually used, they may not be sufficient to ensure effective contact. The contact detection algorithm is also applicable to rigid body contact by using the position of a rigid body point P instead of the position of node k in equation (34). The global position of point P is given by riP = ri + A i s iP ′ , where s iP ′ denotes the point location in body i frame.

6.2

Continuous contact force model

A model for the contact force must consider the material and geometric properties of the surfaces, contribute to a stable integration and account for some level of energy dissipation. Based on a Hertzian description of the contact forces between two solids [37], Lankarani and Nikravesh [38] proposes a continuous force contact model that accounts for energy dissipation during impact. The procedure is used for both rigid body and nodal contact. Let the contact force between two bodies or a system component and an external object be a function of the pseudo-penetration δ and pseudo-velocity of penetration δ


(

)

f s ,i = Kδ n + Dδ
u

(40)

where K is the equivalent stiffness, D is a damping coefficient and u is a unit vector normal to the impacting surfaces. The hysteresis dissipation is introduced in equation (40) by D δ
, being the damping coefficient written as D=

(

)

3K 1 − e 2 n δ 4δ
( −)

(41)

This coefficient is a function of the impact velocity δ
(−) , stiffness of the contacting surfaces and restitution coefficient e. For a fully elastic contact e=1 while for a fully plastic contact e=0. The generalized stiffness coefficient K depends on the geometry material properties of the surfaces in contact. For the contact between a sphere and a flat surface the stiffness is [39]

K=

0.424 r  1 −ν i2 1 − ν 2j    +  πE  π E i j  

(42)

where νl and El are the Poisson’s ratio and the Young’s modulus associated to each surface and r is the radius of the impacting sphere. The nonlinear contact force is obtained by substituting equation (41) into equation (40)

(

)

 3 1 − e 2 δ
 f s ,i = K δ n 1 +  u 4 δ
( −)  

(43)

This equation is valid for impact conditions, in which the contacting velocities are lower than the propagation speed of elastic waves, i.e., δ
( −) ≤ 10 −5 E ρ . 6.3

Example Frictionless impact of an elastic beam

The continuous contact force model is exemplified with the oblique impact of a hyperelastic beam against a rigid wall. The impact scenario, proposed by Orden and Goicolea [40], is described in figure 10. The beam has a mass of 20 kg, length of 1 m and a circular cross-section. Its material has an elastic modulus of E=108 Pa, Poisson’s ration of ν = 0.27 and a mass density of ρ = 7850 kg/m3. For this geometry and material properties the equivalent stiffness coefficient used in equation (43) is K=1.2 108 kg/m2/3. Models of the impacting beam, with 4, 10 and 20 elements respectively, are simulated. The nodal displacements of beam are presented in figure 11. In figure 12, the contact forces developed during nodal impact are presented. The motion of the impacting beam is shown in figure 13 together with that of a rigid bar with similar inertia. The motion predicted for the beam model using 10 elements is similar to that presented by Orden and Goicolea [40] for a

model made of 20 elements and using an energy-momentum formulation to describe contact. Small differences are observed between the response of the beam models made of 10 and 20 elements. node 2 Y

θ=35.2º

v0 = 2 m/s

node 1

d=1m

X

Figure 10 Impact scenario for an oblique elastic beam

X displacement (m)

A predictor-corrector algorithm with variable time-step is used to integrate the beam equations of motion. The size of the integration time-step is controlled, during contact, by the time of travel of the elastic wave across one element, which is Te=4.4 10-4 s for the model with 20 elements [40]. Outside of contact, the integration time-step is controlled by the system response, generally associated to the flexible bodies higher frequencies. Table 1 shows the average time step taken by the algorithm in the various phases of the system motion. 3.00 2.50 2.00 1.50 1.00

(Rigid) node 1

0.50

(10 el) node 2

(Rigid) node 2

0.00

(4 el) node 2

-0.50

(10 el) node 1 (4 el) node 1

-1.00 -1.50 0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

4.0

Time (s)

Figure 11

Force X (N)

10000

Displacement of the beam end nodes for different f.e.m. models

Forces on node 1

Forces on node 2

(c.f.m./u=0) node 1 (c.f.m./u=0) node 2 (Rigid/u=0) node 1

100

(c.f.m./u=.35) node 1 (Rigid/u=0) node 2 (c.f.m./u=.35) node 1

1 0.4

0.6

0.8

1.0

1.2 Time (s)

Figure 12

Forces developed between the (10 elements) beam end nodes and the rigid surface (c.f.m. – continuous force model; k.c. – kinematic constraint model)

Time = 0.0 s

Time = 0.5 s

Time = 0.7 s

Time = 0.8 s

Time = 0.9 s

Time = 1.0 s

Time = 1.6 s

Time = 2.0 s

Figure 13

(a) (b) (c) Frictionless impact of a hyperelastic beam in a rigid surface: (a) Beam model with 4 finite elements; (b) Model using 10 elements; (c) f.e.m. model with 20 elements

Table 1: Average time-step of the integration algorithm Beam model Avg. time-step Avg. time-step before contact during contact Continuous force 4 elements 0.60 10-3 0.20 10-3 -3 Continuous force 10 elements 0.24 10 0.15 10-3 -4 0.34 10-4 Continuous force 20 elements 0.43 10 -3 0.20 10-4 Kinematic constraint 4 elements 0.60 10 -3 0.74 10-5 Kinematic constraint 10 elements 0.24 10

Type of contact model

Avg. time-step after contact 0.55 10-3 0.20 10-3 0.40 10-4 0.55 10-3 0.20 10-3

Observing the contact forces, shown in figure 12, it is clear that the impact

phenomena occurs with multiple contacts. Each of these contacts, for a beam model made of 10 finite elements, lasts for 0.02 s in average, which is similar to the contact duration of 0.018 s estimated by Orden and Goicolea [40] using the elastic wave travel time across the bar length.

7.

Applications to vehicle impact

7.1

Design of a sports vehicle crash-box

The formulation proposed in this paper is applied to the redesign of the front crash-box of the sports car presented in figure 14. The vehicle, a replica of the original Lancia Stratos, was assembled in-house at IDMEC/IST, providing an opportunity to obtain through direct measurements the structural and dynamic characteristics of all car components used.

Figure 14 Prototype of the sports car

The multibody model of the vehicle is composed of 16 rigid bodies and a nonlinear partially flexible body. The system components include a front double A-arm suspension system, a rear McPherson suspension system, wheels and chassis [15]. A partially flexible body where the front part is considered flexible while the remaining structure and shell is considered rigid, as depicted by figure 15 models the chassis of the vehicle. This modeling assumption is valid if plastic deformations are to occur in the parts of the vehicle modeled as flexible regions, such as the front, and no significant deformation takes place in the passenger compartment. This model is suitable to simulate frontal impacts of the sports car. The flexible part of the chassis is composed of 36 nodal points and 38 nonlinear beam elements made of E24 steel and having hollow rectangular cross-sections.

Figure 15 Model of sports car chassis with a nonlinear flexible front crash-box

The front crash-box of the vehicle is modeled and simulated for a frontal impact with an initial velocity of 50 Km/h. It is observed that deformations extend beyond the allowable limit compromising the survivable space. This is due to the mechanism of deformation that forces the crash-box to bend down without absorbing the energy that it is supposed to. A

new design for the crash-box, compatible with the functional geometry of the vehicle front is proposed. The original and new designs for the crash-box are shown in figure 16. The deformation mechanisms resulting from the simulations of the original and improved crash-boxes are presented in figure 17. In the original crash-box the deformation progressed by bending the crash-box down and actually moving the structure out of the way of the remaining of the chassis without forcing the material to reach its limit for energy absorption. Therefore, the deformation progresses into the occupant safety envelope. In the improved design, the deformation of the crash-box is such that all structural components fold, as if they are kept inside the original volume, exploring much better the energy absorption characteristics of the structure.

(a)

(b)

Figure 16 (a) Original and (b) modified crash-box of the sports vehicle.

The new design of the vehicle with the improved crash-box is simulated in different impact situations accounting for several relative orientations of the impact surfaces and various road profiles. For a constant initial velocity of the vehicle of 50 Km/h the impact scenarios are summarized in figure 18.

(a)

(b) Figure 17

Patterns of deformation for the front crash-box: (a) original; (b) improved

Angle 10º no friction

Angle 20º no friction

Angle 20º friction = 0.5

Angle 10º friction = 0.5

10 cm ramp

(a)

Figure 18

(b)

(c)

(d)

(e)

Impact scenarios: a) Frontal impact with surface perpendicular to vehicle heading; b) Impact with a 10º oblique surface, no contact friction; c) Impact with a 20º oblique surface, no friction; d) Impact with a 20º oblique surface, contact friction; e) Same as (d) but the vehicle travels over a ramp with a 10 cm height.

The vehicle motion for the different impact scenarios is presented in figure 19. For the frontal crash scenario it is observed that the deformation of the crash-box progresses with a pattern similar to that predicted for the modified structure shown in figure 10b. At the simulated impact speed the influence of the car suspension elements over the deformation mechanism is minimal.

(a)

(b)

(c)

(d) Figure 19

Motion of the vehicle in different crash scenarios: (a) Frontal impact; (b) 20º Oblique impact without contact friction; (c) 20º Oblique impact with contact friction; (d) Impact with an oblique surface for a vehicle traveling over a ramp

X displacement (m)

In figure 20 the displacement, velocity and acceleration of the center of mass of the vehicle are plotted for the frontal impact scenario. It is observed that all kinetic energy of the vehicle is absorbed, mainly by plastic deformation of the crash-box, leading to the vehicle complete stop 0.09 s after the start of the analysis. Note that the initial conditions of the vehicle are such that the front of the crash-box is 0.570 m from contact surface when the motion starts. The efficiency of the crash-box to dissipate the kinetic energy of the vehicle decreases with the increase of the angle value between surface normal and vehicle heading. It is observed that for the frontal crash the front structure dissipates all kinetic energy. However, for oblique impacts with no surface friction, only part of that energy is absorbed by the crash-box. The vehicle motion is deflected, and would continue if no other component of the car entered in contact with the surface. In figure 21 the forces developed between vehicle and surface are plotted. 1.0 0.8 0.6 0.4 0.2 0.0 0.00

0.05

0.10

0.15

0.20

Velocity (m/s)

Time (s) 16 14 12 10 8 6 4 2 0 -2 0.00

0.05

0.10

0.15

0.20

Acceleration (m/s2)

Time (s) 50 0 -50 -100 -150 -200 -250 -300 -350 -400 0.00

0.05

0.10

0.15

0.20 Time (s)

Figure 20 Displacement, velocity and acceleration of the chassis center of mass in frontal impact

Force (N)

400000 350000 Frontal Oblique (10º) Oblique (20º)

300000 250000 200000 150000 100000 50000 0 0.00

0.05

0.10

0.15

0.20 Time (s)

Figure 21 Forces developed between the front of the vehicle and impact surface

The presence of friction forces between the impacting vehicle and surface is very important for the efficiency of the crash-box, as an energy management component in crash events. For the crash scenarios where friction is modeled, the deflection of the vehicle motion does not occur, enabling the structure to deform with a crushing mechanism similar to that of the frontal impact. Only a slight sideways translation of the vehicle is observed. This result clearly emphasizes the importance of a correct model for the friction forces developed during contact.

6.

Conclusions

A general formulation for the analysis of flexible multibody systems was reviewed here. The main feature of the methodology is its ability to describe the general rigid body motion of the flexible body including its nonlinear deformations and the inertial coupling between these and its gross rigid body motion. Using the floating reference frame approach, the method proposed lowers the geometric nonlinearities of the flexible bodies. By using an incremental finite element approach to the solution of the finite element nonlinear equations the requirement for moderate rotation about the floating reference frame is relaxed. In the sequel of this treatment, the methodology developed allows for the treatment of material nonlinearities also. The formulation developed is to be used in the framework of flexible multibody dynamics. Therefore, the new set of generalized flexible coordinates introduced by the formulation requires that the definition of the kinematic constraints take their nature into account. By using the concept of virtual bodies, a generic procedure to handle kinematic constraints between flexible bodies or with rigid bodies has also been proposed. Contrary to the application of this concept in the case of the flexible bodies experiencing linear elastodynamics, where the mode component synthesis is used, the introduction of virtual bodies does not increase the number of coordinates in the flexible multibody model. This is because the number of coordinates associated to the virtual body equals the number of coordinates for a single node. Through the application to well-understood benchmark problems the merits and drawbacks of the proposed formulation was discussed. If in one hand it is accurate for most types of nonlinear structural behavior of the flexible bodies on the other hand the proposed methodology is computationally expensive when compared with methodologies dealing with linear elastodynamics only. The application of flexible multibody dynamics to the study of complex systems involving contact and impact was also presented. Some attention has been paid to the

contact models used, as they come in play for applications dealing with impact. Through the study of an impacting beam and of a vehicle it was shown that the formulation proposed deals very efficiently with models describing nonlinear material deformations. The quality of the results and the simplicity of the models used indicate that the formulation proposed is suitable for the design and analysis of complex multibody systems.

Acknowledgements The work described here was partially funded by the grant BD/9245/96. The supports by Fundação para a Científica e Tecnologia under the PRAXIS Project 2/2.1/TPAR/2041/95 and by AGARD with the project P-124 are gratefully acknowledged.

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Performance of Non-Linear Finite Element Formulations in Flexible Multibody Simulations Ahmed A. SHABANA Department of Mechanical Engineering, University of Illinois at Chicago 842 West Taylor Street, Chicago, Illinois 60607-7022, U.S.A. Abstract. Because of the nonlinearity and complexity of flexible multibody applications, efficient implementation of multibody formulations becomes necessary. When finite element formulations are used in multibody simulations, several important issues must be considered. Among these issues are the increase in the dimensionality due to the use of the finite element discretization and the accurate representation of the large rotation of the finite element. The dimensionality and the representation of the large rotations are dependent on the finite element formulation used in flexible multibody dynamics. In the floating frame of reference formulation, the most widely used method in flexible multibody dynamics, the dimensionality problem is often solved by using component mode reduction methods. Accurate representation of the large rotation is obtained by using the body floating frame of reference frame. As a consequence, the floating frame of reference formulation leads to zero strain under an arbitrary rigid body motion even when non-isoparametric finite elements are used. Because of the use of component modes and infinitesimal rotations as nodal coordinates, the floating frame of reference formulation has been mainly used to solve small deformation problems. Large deformation problems can be more accurately handled in multibody dynamics by using a full finite element representation instead of modal reduction techniques. A full finite element representation based on the absolute nodal coordinate formulation will be considered in this investigation. This formulation leads to accurate representation of the large rotation, but it also leads to a large dimension as the result of using the finite element nodal coordinates. In this paper, the performance of the absolute nodal coordinate formulation in the large deformation analysis of flexible multibody applications is examined and compared with the floating frame of reference formulation and existing incremental methods. An efficient implementation of the absolute nodal coordinate formulation, based on a Cholesky transformation, that leads to optimum sparse matrix structure of flexible multibody equations is discussed. Numerical results of several structural and multibody applications were presented in previous investigation in order to demonstrate the use of the absolute nodal coordinate formulation in the analysis of small and large deformation problems. The generalization of the absolute nodal coordinate formulation to the three-dimensional analysis as well as the analysis of plates and shells is discussed in this paper.

1.

Introduction

The solution procedures used in flexible multibody simulations as well as the degree and type of nonlinearity of the formulation depend on the choice of the coordinates. Different sets of coordinates may lead to different structures for the dynamic equations. The proper selection of the coordinates, however, must take into consideration the type of application to be investigated. The use of one set of coordinates, as compared to other sets, may prove more efficient in a particular application. For example, in the analysis of small deformation of flexible multibody systems, the use of a mixed set of absolute Cartesian and local elastic coordinates as adopted in the floating frame of reference formulation has a clear advantage as

compared to the use a full finite element representation. The use of elastic coordinates defined in the flexible body coordinate system in small deformation problems leads to a simple and constant stiffness matrix. More significantly, such a choice of coordinates for this type of problems allows the use of modal reduction techniques that significantly reduce the problem dimensionality by eliminating insignificant high frequency modes of vibration. As a result, the numerical integration of the flexible multibody equations becomes much more efficient. There is another important reason for using the mixed absolute Cartesian and local elastic coordinates. This choice of coordinates in the floating frame of reference formulation leads to a consistent formulation when non-isoparametric elements such as beams, plates and shells are considered. The use of this formulation leads to zero strain under an arbitrary rigid body motion, and also leads to exact modeling of the rigid body dynamics. The floating frame of reference formulation has been widely and successfully used in flexible multibody applications, and it has proven to be efficient in handling problems which contain rigid and flexible bodies and in which the small deformation assumptions remain valid. For this reason, the floating frame of reference formulation is implemented in several commercial and research general purpose computer programs. Despite the success and popularity of the floating frame of reference formulation, its use has been, in general, limited to the analysis of small deformations. This limitation is attributed to two main reasons. The first is due to the fact that in the floating frame of reference formulation, linear modes are used in the coordinate reduction, and as such, it is assumed that the deformations are small in order to justify the use of these modes. Second, in multibody applications where beams, plates and shells are used, infinitesimal rotations are used as nodal coordinates. The use of infinitesimal rotations implies that kinematic equations are linearized. Therefore, incremental methods have been more widely used when non-isoparametric finite elements are employed in large deformation problems. Nonetheless, incremental methods do not lead to exact modeling of the rigid body dynamics, and as a consequence, their use in the analysis of small deformation of flexible multibody systems has been much less as compared to the floating frame of reference formulation. In order to obtain accurate representation of the large rotation in flexible multibody dynamics, the absolute nodal coordinate formulation was proposed. In this formulation, global displacement and slope coordinates are used to define the configuration of the finite element. By using global slopes instead of finite and infinitesimal rotations, exact modeling of the rigid body dynamics can be obtained. Furthermore, beams and plates can be considered as isoparametric elements which lead to zero strain under an arbitrary rigid body motion. The absolute nodal coordinate formulation leads to a constant mass matrix and a complex expression for the elastic forces. While the accurate representation of the finite rotation does not pose a problem in the absolute nodal coordinate formulation, the large dimensionality becomes a problem since linear modes can no longer be used. The nodal coordinates are defined in the global system, and as a consequence, the use of modal reduction techniques becomes impractical. Nonetheless, an efficient implementation of the absolute nodal coordinate formulation in general purpose flexible multibody codes can be made. To this end, a Cholesky transformation is obtained and used to define a new set of generalized coordinates. The inertia matrix associated with the new set of coordinates is the identity matrix, thereby defining an optimum sparse matrix structure for the constrained flexible multibody equations. In this paper, the performance of the floating frame of reference formulation and the absolute nodal coordinate formulation in flexible multibody applications is discussed. The results obtained using the two formulations were compared in previous investigations in the case of small deformation problems. The results obtained using the absolute nodal coordinate formulation in the large deformation analysis of structural and multibody systems were compared with the incremental methods which are widely used in structural dynamics. A brief

review of these comparative studies is presented in this paper. The generalization of the absolute nodal coordinate formulation to three-dimensional and shear deformable elements is discussed. It was demonstrated in previous publications that the absolute nodal coordinate formulation can be used to relax many of the assumptions of Euler-Bernoulli and Timoshenko beam theories while the mass matrix of the element remains constant. Because of the isoparametric property obtained using the absolute nodal coordinate formulation, threedimensional plate and shell elements can be systematically developed.

2.

Rotations and Slopes

In the classical finite element literature, the deformation of beams are defined with respect to a beam coordinate system. The axial displacement is interpolated using a linear polynomial while the transverse displacement is interpolated using a cubic polynomial. Such a beam element is not considered as an isoparametric element in the classical finite element literature because infinitesimal rotations are used as nodal coordinates. While it is not justified to use such a beam element in the absolute nodal coordinate formulation because the displacement components are described using different polynomials, it can be demonstrated that such an element can describe an exact rigid body motion if slopes instead of infinitesimal rotations are used as nodal coordinates. In order to demonstrate this fact, we consider a uniform slender beam. The coordinate system of this beam element is assumed to be initially attached to its left end which is defined by point O. The conventional shape function of this element is assumed to be 0 0 ξ 0 0 1-ξ S=  0 1 - 3ξ 2 + 2ξ 3 l( ξ -2ξ 2 + ξ 3 ) 0 3ξ 2 - 2ξ 3 l( ξ 3 - ξ 2 

  )

(1)

where ξ = x/l and l is the length of the beam. The vector of nodal coordinates associated with the shape function of equation (1) is

e = [ e1 e2 e3 e4 e5 e6 ]

T

(2)

where e1 and e2 are the translational coordinates at the node at O, e4, and e5 are the translational coordinates of the second node at A, and e3 and e6 are the slopes at the two nodes. An arbitrary rigid body displacement of the beam is defined by the translation R = [R1 R2]T of the reference point O, and a rigid body rotation θ. As the result of this arbitrary rigid body displacement, the vector of nodal coordinates e can be defined in the global coordinate system as e = [ R1 R 2 sinθ R1 +l cosθ R 2 +l sinθ sinθ ]

T

(3)

Using equations (1) and (3), it follows that

 +x cosθ  S e =  R1  R 2+x sinθ 

(4)

which demonstrates that the element shape function and the nodal coordinates can describe an arbitrary rigid body displacement provided that the coordinates are defined in the global coordinate system and the slopes are defined in terms of trigonometric functions. Therefore, the conventional finite element shape function can be used to obtain an exact modeling of the rigid body displacement [5,6].

3.

Choice of Coordinates

It is clear from the analysis presented in the preceding section that the finite element nodal coordinates can be used to obtain exact modeling of an arbitrary rigid body motion if slopes instead of infinitesimal rotations are used as coordinates. Therefore, a natural question arises with regard to the need for using the floating frame of reference approach in which the large displacement is not described using the nodal coordinates. Obviously, depending on the choice of coordinates, different fundamental problems are encountered in flexible multibody formulations. In the floating frame of reference formulation, a mixed set of absolute and local elastic coordinates are used. The absolute coordinates define the location and orientation of a selected deformable body coordinate system, while the local elastic coordinates define the deformation of the body with respect to its coordinate system. In most applications investigated using the floating frame of reference formulation, the number of elastic coordinates is reduced by using the modal transformation. Therefore, the configuration of the deformable body is described using a mixed set of absolute and modal coordinates. This choice of coordinates leads to fundamental problems related to the selection of the deformable body coordinate system, the boundary conditions, and the appropriate set of mode shapes. This fundamental problem has been addressed in several previous investigations. Despite these fundamental problems, the floating frame of reference formulation remains the most widely used procedure in flexible multibody simulations. It has been implemented in several research and commercial codes and has been successfully used in the analysis of many applications. The floating frame of reference formulation has been widely accepted by the multibody community for several reasons which can be summarized as follows: 1. Many of the multibody applications consist of many rigid bodies and few flexible bodies. Therefore, there is a tendency not to compromise the accuracy of the rigid body analysis. It is also important to use a formulation which is more efficient in the simulation of the rigid bodies. This can be accomplished using the floating frame of reference formulation. 2. The concept of the angular velocity is fundamental in the analysis of multibody systems. This is particularly important when prescribed motion trajectories are considered. In this case, it is much easier to define the angular velocity when a floating frame of reference is used. 3. In multibody simulations, the selection of the degrees of freedom is crucial. Such a selection influence the efficiency of the method as well as the numerical algorithm used for solving the resulting system equations. In many cases, the equations of motion of the multibody system are formulated using the independent joint variables. It is much simpler to develop recursive equations when the floating frame of reference formulation is used. Furthermore, the definition of the driving joint forces and torques becomes simpler. 4. The floating frame of reference formulation is a natural extension of the NewtonEuler formulation which has been extensively used in the analysis of multibody systems consisting of rigid bodies. In the absolute nodal coordinate formulation, on the other hand, absolute displacement and slope coordinates are used. In this case, there is no need for introducing a coordinate system for the deformable body. As a consequence, there is a fundamental difference between the floating frame of reference formulation and the absolute nodal coordinate formulation. This is despite the fact that their equivalence can be demonstrated when slopes instead of infinitesimal rotations are used in the floating frame of reference formulation. Due to the nature of coordinates used in the absolute nodal coordinate formulation, the use of modal reduction techniques is impractical particularly when large deformation problems are considered.

4.

Motion Description

The global position vector of an arbitrary point on the finite element used in the absolute nodal coordinate formulation can be written as follows:

r =Se

(5)

where S is the global shape function, and e is the vector of element nodal coordinates. In the absolute nodal coordinate formulation, no infinitesimal or finite rotations are used as nodal coordinates. The element coordinates are expressed in terms of nodal displacements and slopes which can be determined in the undeformed reference configuration using simple rigid body kinematics. Using this motion description, beams and plates can be treated as isoparametric elements without the need to introduce an orientation coordinate to describe the rigid body rotation of the deformable element. In the case of an arbitrary planar rigid body motion of the beam, the global position vector of an arbitrary point on the beam element can be written as    + x cosθ  r =  r1 =  R1 r 2  R 2 + x sin θ 

(6)

where R1 and R2 are the global coordinates of the endpoint O and θ in this case is the angle that defines the beam orientation. It follows that the slopes in the case of a rigid body motion are defined as ∂r 1 = cos θ , ∂x

∂r 2 = sin θ ∂x

(7)

In this section, we consider cubic polynomials to define the elements of the vector r. It is justified to use the same representation for the elements of this vector since they are both defined in the inertial frame when the absolute nodal coordinate formulation is used. In this case, the global shape function is given by 0 0 1- 3ξ 2 + 2ξ 3 l( ξ - 2ξ 2 + ξ 3 ) S=  0 0 1- 3ξ 2 + 2ξ 3 l( ξ - 2ξ 2 + ξ 3 )  3ξ - 2ξ 2

0  0 l( ξ 3 - ξ 2 )

(8)

0 l( ξ 3 - ξ 2 )

3

0 3ξ 2 - 2ξ 3 and the vector of nodal coordinates is

e=[ e1 e2 e3 e4 e5 e6 e7 e8 ]

T

(9)

where ξ = x/l, l is the length of the element, and e1, e2, and e5, e6 are, respectively, the absolute coordinates of the two nodes of the beam, and e3 =

∂ r 1(x = 0) ∂ r (x = 0) ∂ r (x = l) ∂ r (x = l) , e4 = 2 , e7 = 1 , e8 = 2 ∂x ∂x ∂x ∂x

(10)

Using the simple rigid body kinematic equations previously obtained in equations (6) and (7), one can show that in the case of an arbitrary rigid body motion defined by the translations R1 and R2 of the endpoint O and the rotation defined by the angle θ, the vector of the nodal coordinates e can be written as

e=[ R1 R 2 cos θ sin θ R1 + l cos θ R 2 + l sin θ cos θ sin θ ]

T

(11)

Using this vector of nodal coordinates, and the shape function of equation (8), it is verified that  + x cos θ   r 1 Se =  R1 =  = r  R 2 + x sin θ   r 2 

(12)

which demonstrates that the element shape function of equation (8) and the vector of absolute nodal coordinates of equation (9) can describe an arbitrary rigid body motion.

5.

Absolute Nodal Coordinate Formulation

Assuming that the shape function of the finite element can describe an arbitrary rigid body displacement, the global position vector of an arbitrary point on the element is defined using equation (5). By differentiating this equation with respect to time, the absolute velocity vector can be defined. This velocity vector is used to define the kinetic energy of the element: 1 T = e! T M a e! 2

(13)

where Ma is the constant mass matrix of the element defined as M a = ∫ ρ S T S dV

(14)

V

where ρ and V are the mass density and volume of the finite element. Note that this mass matrix is constant, and it is the same mass matrix that appears in linear structural dynamics. It can be shown that by changing the element shape function used in the absolute nodal coordinate formulation, the effects of the rotary inertia, shear, and torsion in the three dimensional analysis can be taken into account and the mass matrix remains constant [8]. This leads to a zero vector of Coriolis and centrifugal forces. There are two methods that can be used to formulate the elastic forces in the absolute nodal coordinate formulation. In the first method, a local element coordinate system is used to define the element deformation. In this case, a linear strain-displacement relationship can be used. In this case, the element produces zero strain under an arbitrary rigid body motion. By reducing the element size, large deformation problems can still be solved using the linear strain-displacement relationship. The use of this method, however, leads to a complex expression for the elastic forces as reported in the literature. Another method that can be used to formulate a much simpler expression for the elastic forces is to use a continuum mechanics approach in which equation (5) is used to define the displacement Jacobian matrix and the nonlinear strains in terms of the displacement derivatives. In this case, nonlinear straindisplacement relationships must be used in order to obtain zero strains in the case of an arbitrary rigid body motion. It is important to point out that, while this method, leads to a much simpler expression for the elastic forces, the resulting model accounts for the geometric centrifugal stiffening effect. As a consequence, the solution obtained using this method does not wrongly exhibit instabilities at relatively high values of the angular velocities.

6.

Equations Of Motion

Using the results obtained in the preceding two sections, one can show that the matrix equation of motion of the finite element in the case of the absolute nodal coordinate formulation takes the following form:

e + K a e = Qa M a !!

(15)

where Ma is the constant mass matrix, Ka is the nonlinear stiffness matrix, and Qa is the vector of generalized nodal forces. Since the stiffness matrix is a nonlinear function of the element nodal coordinates, the preceding equation can be written as

e=Q M a !!

(16)

Q = Q a -K ae

(17)

where

Since the element mass matrix is constant, the equation of motion of the element can be written as

!! e = M -1 a Q = b( e )

(18)

b = M −a 1Q

(19)

where

7.

Computer Implementation for Flexible Multibody Dynamics

Deformable components in multibody systems are subject to kinematic constraints that represent mechanical joints and specified motion trajectories. These constraints can, in general, be described using a set of nonlinear algebraic equations that depend on the system generalized coordinates and time. When the kinematic constraints are augmented to the differential equations of motion of the system, it is desirable to have a formulation that leads to a minimum number of non-zero coefficients for the unknown accelerations and constraint forces in order to be able to exploit efficient sparse matrix algorithms. In this section, a procedure for the computer implementation of the absolute nodal coordinate formulation for flexible multibody applications is described. In this procedure, an optimum sparse matrix structure is obtained for the deformable bodies using a Cholesky decomposition. Using the fact that the element mass matrix is constant, a Cholesky decomposition of the mass matrix is obtained for the deformable body. A constant velocity transformation is used to obtain an identity generalized inertia matrix associated with the second derivatives of the generalized coordinates, thereby minimizing the number of non-zero entries of the coefficient matrix that appears in the augmented Lagrangian formulation of the equations of motion of the flexible multibody systems. The proposed computational procedure can be used for the treatment of large deformation problems in flexible multibody systems. This procedure has also the advantages of the algorithms based on the floating frame of reference formulations since they allow for easy addition of general nonlinear constraint and force functions. In the general computational algorithms developed for the nonlinear dynamic analysis of multibody systems, nonlinear constraint equations that describe mechanical joints and specified motion trajectories can be systematically introduced to the dynamic formulation. This provides the flexibility of building computer models for mechanical systems with complex topological structure. By utilizing sparse matrix algebra, these models can be efficiently simulated and modified. In order to maintain the generality of the dynamic formulation, several of the general purpose computer algorithms developed for the simulation of multibody applications are based on solving the following system of equations [5]:  M CTq  q !!   Q      =  e  Cq 0   λ  Q d 

(20)

where M is the mass matrix of the system, Cq is the Jacobian matrix of the kinematic constraints, q is the vector of the system generalized coordinates, λ is the vector of Lagrange multipliers, Qe is the vector of forces that include external, gravity, Coriolis, centrifugal, and elastic forces, and Qd is the vector resulting from the differentiation of the constraint equations twice with respect to time. For instance, the constraint equations can be written as: C (q, t ) = 0

(21)

where C is the vector of constraint functions, and t is time. Upon differentiating the constraint equations twice with respect to time, one obtains q = Qd Cq !!

(22)

where Qd is the vector defined as

(

Q d = -Ctt - Cq q!

)q q! - 2 Cqt q!

(23)

The coefficient matrix of equation (20) has a sparse matrix structure. An efficient solution of this system of equations can be obtained if the number of non-zero entries of the nonlinear coefficient matrix is minimized [8]. To this end, a procedure based on Cholesky decomposition of the inertia matrix of the deformable body that leads to a constant, square and triangular velocity transformation matrix is employed in this investigation. If a standard finite element assembly procedure for the deformable body is used, the element connectivity constraint forces do not appear in the equations of motion. In this case, the equations of motion of the unconstrained deformable body i can be written as: i M i !!ei = Q

(24)

where Mi is the mass matrix of the deformable body, ei is the vector of nodal coordinates of the deformable body i, and Qi is the vector of nodal forces. In this case, the inertia matrix Mi of a deformable body i in the system is constant, symmetric and positive definite. Therefore, a Cholesky decomposition of this matrix can be obtained as T Mi = L L

(25)

where L is a non-singular lower triangular matrix. In order to define a new set of generalized coordinates that have an identity inertia matrix, we introduce the following coordinate transformation: i e! i = B q! r

(26)

where qri is the new set of generalized coordinates, and B is the velocity transformation matrix chosen in this case such that B=

( LT )

−1

(27)

Substituting equation (26) into equation (24), pre-multiplying by the transpose of B, and keeping in mind that the inverse of the transpose is equal to the transpose of the inverse; one obtains the following equation of motion for the deformable body:

!!ir = BT Q q

(28)

This is the equation of motion of the deformable body expressed in terms of the generalized coordinates qri. The generalized inertia matrix associated with these coordinates is the identity matrix, and as a consequence, the procedure discussed in this section leads to a minimum number of inertia coefficients.

8.

Performance in Static Problems

One of the main advantages of using the absolute nodal coordinate formulation is the simplicity of the inertia forces. Such an advantage can only be utilized in dynamics problems. However, before the dynamic problems are presented, two classical large deformation static examples with known analytical solutions were considered in order to shed light on some of the issues associated with large element rotations. The solutions of the problems obtained using the absolute nodal coordinate formulation were compared with the solutions obtained using the general purpose finite element code ANSYS that employs the co-rotational procedure proposed by [3]. The first problem is a cantilever beam subjected to an end moment that bends the beam into a full circle [1,2,7]. The beam was divided into ten two dimensional beam elements. The results obtained using the ANSYS code were almost identical to the exact solution. The total CPU time of the solution obtained using the ANSYS code on an HP-Convex SPP1200/XA-16 was found to be 7.5 sec. The same results were also reported by [3] who emphasized that the co-rotational procedure gives good results in this kind of problems because the large deflections of the beam are converted into much smaller deformation increments at each load step. They also demonstrated that the results obtained using the conventional incremental approach were not accurate and that no solution could be obtained when the free end rotation reaches the vicinity of 90°. The same problem was solved using the absolute nodal coordinate formulation. The solution obtained using the absolute nodal coordinate formulation was found to be quite accurate and the error was found to be less than 6% as compared to the exact solution. The total CPU time used to obtain the absolute nodal coordinate formulation was found to be 7.2 sec on a PC Pentium 90 MHz. The second large deformation static example considered was the elastica problem in which a cantilever beam is subjected to an overcritical compressive load applied at the free end [2,4,7]. The analytical solution of the problem is available in the literature. The results obtained for this problem showed that the ANSYS solution using twenty elements agrees well with the exact solution in the range of rotation 40° - 120° of the free end. Rankin and Brogan [3] affirmed that no solution could be obtained for the elastica problem using the conventional incremental approach. When the load approaches the critical value (for free end rotations ≈ 40°), the ANSYS results are different from the exact solution. This is due to the fact that, in this range of critical load, very small change of the load causes very large change in the solution. By changing the ANSYS tolerance parameters in order to obtain better refinement, a better solution is obtained when the free end rotation is approximately 40°. However, no solution for the ten and twenty element models could be obtained using Rankin and Brogan’s incremental procedure when the rotation of the free end exceeds 140° [2]. A linear elastic model was used in the absolute nodal coordinate formulation to obtain a solution for the elastica problem. The results presented in this figure show that the solution obtained using the absolute nodal coordinate formulation is very close to the exact solution. Unlike the ANSYS case, a very accurate solution was also obtained for a free end rotation that exceeds 140°. 9.

Performance in Dynamics Problems

In the preceding section, the performance of the absolute nodal coordinate formulation in static applications was discussed. In this section, the dynamic multibody problems are considered. The results obtained using the absolute nodal coordinate formulation were compared in previous investigations with the results obtained using the floating frame of reference formulation and the incremental methods. A summary of the results of this

comparison is presented in this section. Comparison with the floating frame of reference formulation was made only in the case of small deformation problems. A four-bar mechanism was used as one of the study models. The data for this mechanism are presented by [2]. The interest was focused on the midpoint deformation of the connecting rod of the mechanism. The results obtained for this model showed a good agreement between the three different formulations (absolute nodal coordinate, floating frame of reference, and incremental method) when small deformation problems are considered. In order to examine the performance of different formulations in large deformation multibody applications, a lower value of the modulus of elasticity of the connecting rod was considered. A comparison of the results of the midpoint deformation of the connecting rod obtained using the absolute nodal coordinate formulation and the floating frame of reference formulation was presented. It shown that the solutions obtained using the two formulations agree well in the case of large deformation problems when a low value of the angular velocity is used [2]. As the angular velocity increases, significant differences between the results of the two formulations were observed. At the high value of the angular velocity, the effect of the linearization used in the incremental method becomes significant. Another example which was considered to study the performance of the absolute nodal coordinate formulation is a simple flexible pendulum [1]. The results obtained using this model also showed that the absolute nodal coordinate formulation has excellent convergence characteristics as demonstrated in the study presented by Berzeri et al [1].

10.

Generalization of the Method

The absolute nodal coordinate formulation can be generalized to the case of threedimensional analysis, while the mass matrix remains constant. In this case, the vectors of centrifugal and Coriolis forces are identically equal to zero. The following assumed displacement field can be used in the case of three-dimensional analysis of beams: n

r = a0 + x a1 + y a2 + z a3 + x y a 4 + x z a5 +

∑ x k -4 ak

k =6

where ai are three-dimensional vectors that represent the coefficients of the interpolating polynomials, and (n - 4) is an integer that defines the order of the polynomials. It can be shown that the preceding assumed displacement field can be used to obtain exact representation of the rigid body motion, and at the same time accounts for the effects of rotary inertia, shear, and torsion. Such an assumed displacement field when used with a general continuum mechanics approach for formulating the expression of the elastic forces relaxes all the assumptions of the Euler-Bernoulli and Timoshenko beam theories. In this case, one does not need to assume a value for the shear coefficient. The use of the continuum mechanics approach leads to a relatively simple expression for the elastic forces that captures all the nonlinearities and accounts for the effect of the geometric centrifugal stiffening. By developing isoparametric elements using the absolute nodal coordinate formulation, curved structures can be systematically modeled. The absolute nodal coordinate formulation can be used to develop plate and shell elements which can be efficiently used in the small and large deformation analysis of mechanical and structural systems.

11.

Summary and Discussion

In this paper, the performance of three different finite element formulations in flexible multibody applications was discussed. The results obtained using the floating frame of reference formulation, the incremental method and the absolute nodal coordinate formulation were compared in previous investigations. Because of the nature of the coordinates selected for the floating frame of reference formulation, this formulation has been widely and efficiently used in small deformation problems. The incremental method and the absolute nodal coordinate formulation can be used in the small and large deformation problems. The implementation of the absolute nodal coordinate formulation for flexible multibody dynamics was discussed. A Cholesky transformation was used to define a constant velocity transformation matrix. This matrix can be used to obtain an optimum sparse matrix structure for the multibody equations in which the absolute nodal coordinate formulation is used to model the deformable bodies. The performance of the absolute nodal coordinate formulation in the large deformation static problems was examined [1,2]. Two problems with known analytical solution were analyzed using the absolute nodal coordinate formulation and the incremental method implemented in the general purpose finite element code ANSYS. It is shown using these two problems that the absolute nodal coordinate formulation is more efficient as compared to the incremental methods. Furthermore, the ANSYS solution does not converge in the elastica problem when rotations exceed a certain limit. On the other hand, the absolute nodal coordinate formulation converges for all ranges of rotations despite the fact that a linear strain displacement relationship is used with the absolute nodal coordinate formulation. The performance of the floating frame of reference formulation, the incremental method and the absolute nodal coordinate formulation in the dynamic case was also investigated in previous investigations [1,2]. A flexible multibody four-bar mechanism and a flexible pendulum were used as the study models. It was shown that the results obtained using the three multibody formulations considered agree well in the case of small deformations and low angular velocities. In the case of large deformation, the results obtained using the absolute nodal coordinate formulation and the incremental method were compared. The result of this comparative study showed that in the case of large deformation and low angular velocity, the solutions obtained using the absolute nodal coordinate formulation and the incremental method agree well. The effect of the linearization used in the incremental method becomes significant as the angular velocities increase. In this case, significant differences between the solutions obtained using the two formulations were observed. The absolute nodal coordinate formulation can be generalized to the three-dimensional case. The effects of rotary inertia, shear and torsion can be systematically accounted for while the mass matrix remains constant. A simple expression for the elastic forces that captures all the nonlinearities can be obtained using a continuum mechanics approach. Isoparametric plate and shell elements can systematically developed using the absolute nodal coordinate formulation.

Acknowledgments This research was supported by the Army Research Office, Research Triangle Park, NC.

References [1]

[2] [3]. [4] [5] [6] [7] [8]

M. Berzeri, M. Campanelli, and A.A. Shabana, Incremental Finite Element Formulations and Flexible Multibody Dynamics, In: Proceedings of 1999 ASME Design Engineering Technical Conferences, September 12-15, Las Vegas, Nevada, 1999. M. Campanelli, Computer Methods for the Dynamics and Stress Analysis of Track Chains, Ph.D. Thesis, Department of Mechanical Engineering, University of Illinois at Chicago, 1998. C.C. Rankin, and F.A. Brogan, An Element Independent Corotational Procedure for the treatment of Large Rotations, ASME Journal of Pressure Vessel Technology, 108 (1986), 165-174. R. Schwertassek, and O. Wallrapp, Dynamik flexibler Mehrköper Systeme, Vieweg Publishers, 1999. A.A. Shabana, Dynamics of Multibody Systems, Second edition, Cambridge University Press, 1998. A.A. Shabana, Finite Element Incremental Approach and Exact Rigid Body Inertia, ASME Journal of Mechanical Design, 118(1996), 171-178. S. Von Dombrowski, Modellierung von Balken bei grossen Verformungen für ein kraftreflektierendes Eingabegerät, Diploma Thesis, German Aerospace Research Center and University of Stuttgart, 1997. R.Y. Yakoub, and A.A. Shabana, Use of Cholesky Coordinates and the Absolute Nodal Coordinate Formulation in the Computer Simulation of Flexible Multibody Systems, Nonlinear Dynamics, 20 (1999), 267-282.

Modal Representation of Deformation and Stress in Flexible Multibody Simulation Richard SCHWERTASSEK Institute of Robotics and Mechatronics, German Aerospace Center (DLR) D-82234 Wessling, Germany. Abstract. The approach most widely used for the modelling of flexible bodies in multibody systems has been called the floating frame of reference formulation. In this methodology the flexible body motion is subdivided into reference motion and deformation. The displacement field due to deformation is approximated by the Ritz method as a product of known shape functions and unknown variables depending on time only. The shape functions may be obtained using entire finite-element-models of flexible bodies in a multibody simulation. Such a nodal approach results in a detailed system representation but also in a high number of system equations. The number of system equations can be reduced considerably using a modal representation of deformation. A modal approach, however, leads to the fundamental problem of selecting the proper shape functions. Two examples are chosen to demonstrate the effects of various choices of shape functions and associated body reference frames. The focus of the discussion is on the representation of the body deformation and on the computation of the corresponding stresses, as well as on the range of validity of equations of motion, which have been linearized assuming the deformations to be small. An acceptable representation of stresses is achieved by appropriate selection of quasi-comparison functions. These can be selected also in ways to increase the range of validity of the linearized equations of motion. The latter goal is achieved as well by so called substructuring techniques. Combining both of the methodologies one obtains efficient models for flexible multibody simulation.

1.

Introduction: The floating frame of reference formulation

A general multibody system, as considered here, is shown in figure 1. The elements of the model are bodies, force elements, joints and a global reference frame. The bodies may be rigid or flexible, and they are the only system components, which are assumed to have inertia. On the surface of the bodies there are parts, called nodes, at which the joints and force elements are attached. The force elements are used to model applied forces and torques. As shown in figure 1, they may represent external action (e.g. due to gravity) or interaction between the bodies, resulting from dampers, springs, actuators or contact. All of these forces and torques are functions of the position and velocity of the system bodies, the constraint forces (e.g. in case of dry friction) and the state of a control system in case of the actuators. The joints are any devices that constrain the relative motion of the nodes on the bodies, and they result in unknown constraint forces and torques. Joint deformations as a result of the interaction between the system bodies are not considered. The global reference frame is used to model a known global system motion in inertial space. Multibody formalisms are computer oriented procedures to generate the equations of motion for systems of the general form shown in Figure 1, based on data, which describe the system elements and the system topology, i.e. the way the nodes on the system bodies are interconnected by force elements and joints [1, 7, 10, 11, 17, 19, 21, 23, 34]. Two groups of

formalisms may be distinguished. The first group yields the Lagrangian equations of type one, which contain the unknown generalized constraint forces in terms of Lagrangian multipliers [2]. These differential equations are accompanied by a set of algebraic constraint equations. The resulting representation of the system motion is sometimes called the descriptor form of the equations of motion. It is simple to generate, but it requires the numerical solution of differential-algebraic equations [6]. By contrast, the second group of formalisms provides the state space representation of the system motion, i.e. a minimal set of first order (kinematical and dynamical) differential equations, in which the constraint forces have been eliminated. Numerical methods for solving the latter equations are often considered to be more mature with respect to computational efficiency.

controller input signal actuator external force

damper

sensor

body (rigid)

coupler spring

body (flexible)

universal joint

rubber bearing

joint with kinematic excitation global reference frame (inertial or accelerated)

body (rigid)

sensor

prismatic joint friction force

external force

Figure 1. General multibody system model and its elements.

The starting point for the development of both types of formalisms are the equations describing the motion of a representative system body i, acted upon by the applied external and internal forces and torques due to the force elements and the unknown internal joint forces and torques between the system bodies. Methods of modelling flexible bodies in a multibody system have been reviewed in [26]. The most widely used approach has been called the floating frame of reference formulation. It is described in [21, 22] using the general body model shown in Figure 2. The model is based on the modelling assumptions of continuum mechanics. By introducing so-called internal constraint equations for the relative motion of the points of the body – see e.g. [30] – it may be specialized to obtain the body models, such as beams, plates, shells or finite element structures, used in multibody simulation. The floating frame of reference formulation is based on a separation of the flexible body motion into a reference motion and a deformation. The former is the motion of the body in its reference configuration. It may be described as the motion of a body reference i i frame {O , e } . Deformation is the motion of the points of the body with respect to its reference frame. Such a separation of the body motion allows to linearize the equations of motion assuming the deformation to be small.

coordinate system {O k,i, e k,i } in reference configuration

body i (actual configuration)

node k on body i

body i (reference configuration) P ui

{O i, e i }

r k,i

R ri

P

u k,i

R k,i

coordinate system {O k,i, e k,i } in actual configuration

k,i i

inertial coordinate system {O I, e I }

Figure 2. General model of a representative body i of a multibody system.

Applying notation described in [22] and [21] the location and velocity of an arbitrary point P of a system body i are given by I

T

ρ(R , t ) = A i (t ) ρ(R, t ) where ρ(R , t ) = ρi (t ) + R + ui (R , t ) ,

(

)

v (R, t ) = v i (t ) + u! i (R , t ) + ω" i (t ) R + ui (R, t ) , where v i = ρ! i + ω" i ρ i .

(1) (2)

i i In these relations, ρ , R and u are the coordinates of vectors ρ , R and u – see figure 2 – i i in the body frame {O , e } . The elements of the matrix R are the material coordinates of the I I I body. Matrix ρ contains the coordinates of ρ in the inertial frame {O , e } and the matrii i i i ces A and ω describe orientation and angular velocity of {O , e } with respect to I {O I , e } . For various flexible body models, e.g. beams, there exist coordinate systems {P, e } at the points P of the body, which remain undeformed during body deformation [22]. To obtain a meaningful multibody system model, the existence of such coordinate systems is mandatory at the attachment points of joints and force elements, i.e. at the nodes k on body i denoted k, i in figure 2: The modelling of joint deformation has been excluded in the multibody system model and as a consequence a rigid surface element is required on a flexible body to attach a rigid joint. More general, rigid surface elements and the associated coordinate systems k ,i {P, e } = {O k ,i , e } are required, because the joints and the force elements are considered to transmit single forces and torques. These interactions are the resultants of the system of distributed forces at the nodes on the bodies. Only when the nodes remain undeformed, the resultants have the same physical effects as the system of distributed forces. When using flexible body models without coordinate systems {P, e } , their existence at the attachment points of joints and force elements has to be assured by boundary conditions. They state that the strains are zero on parts of the body surface, at which the elements are fixed. The rotation of a basis e = e(R,t) at a point P is described by

(

)

A (R, t ) = E − ϑ" i (R, t ) Γi (R ) A i (t ) ,

ω (R, t ) = ω i (t ) + ϑ! i (R , t ) .

ϑi = ϑαi  ,

(3) (4)

i i i In Equations (3) and (4) A and A = A (α ) represent the absolute orientation of e(R,t) and i i i ei(t). Matrix Γ = Γ ( γ ) describes the orientation of e(R,t) with respect to ei(t) in the i i i reference configuration of the body. As indicated, the matrices A = A (t ) and Γ are parameterized by the angles α i (t ) = ααi (t )  and γ i = γ αi  , respectively. The rotation of e(R,t) from the reference configuration into the actual configuration is given by the angles ϑi . When the deformations are small, the linearized form of the rotation matrix, as used in Equation (3), is sufficient to describe the motion. Matrix ω represents the absolute angular i ! i (R , t ) of the angular velocity of e(R,t) as a sum of ω and the linear approximation ϑ velocity due to deformation. i i i i To summarize, the variables ρ (t ) , v (t ) , A (t ) and ω (t ) represent the reference moi tion of a representative body i and its deformation is described by the functions u (R , t ) and ϑi (R , t ) and their time derivatives. Selecting a set of admissible shape functions Φi(R), Ψ i(R) – see [15] – and introducing a Ritz approximation of the form

ui (R, t ) = Φi (R ) qi (t ), ϑi (R , t ) = Ψ i (R ) qi (t ),

Φi =  Φαi k  Ψ i =  Ψαi k 

  

qi (t ) =  qki (t )  , k = 1, 2, … nqi ,

(5)

the variables describing position and velocity of a body i are found to be  ρi (t )   v i (t )  i i i z (t ) =  z (t )  = α (t )  , z II (t ) =  z IIj (t )  = ω i (t )  , j = 1, 2, … nzi , nzi = 6 + nqi .  q i (t )   q! i (t )      i I

i Ij

(6)

The definition of the two parts of the motion requires specifying the body frame i {O i , e } . Popular choices are coordinate systems, which have been called tangent-, chordand mean-axes- or Tisserand- or Buckens-frame in [22]. The definition of these frames i i results in requirements to be satisfied by the variables u (R , t ) and ϑ (R , t ) , which apply either for all of the points R of the body or for specific points only. The latter requirements yield geometrical boundary conditions. All of the requirements affect the selection of the i i shape functions, which implies that the choices of {O , e } and of Φi(R), Ψ i(R) are interrelated closely. When modelling flexible bodies in multibody systems as finite element structures, the so-called nodal and modal approaches are applied to describe body deformation [21,22]. In the nodal approach the shape functions are provided by the finite element model. The reduction of the number of variables in the modal approach is highly desirable to increase computational efficiency, but this method results in the fundamental problem of selecting shape functions that represent body deformation with appropriate accuracy. The problem is not limited to a modal representation of the deformation of finite element structures: when using other models, such as beams or plates, a modal approach is applied frequently, [20]. The selection of shape functions in the modal approach is of particular importance for the representation of the stresses in the flexible system bodies. Often, multibody system codes yield good results to describe the body deformation, whereas the representation of stresses is poor. To overcome such deficiencies, it has been proposed to determine the stresses by post-processors, using finite element codes [4, 12]. But an acceptable representation of internal forces in the flexible system bodies can be obtained with multibody models as well when using instead of admissible functions so-called quasi-comparison functions [8, 9, 14-16] as shape functions Φi(R) and Ψ i(R). A set of quasi-comparison functions may be obtained by combining eigenfunctions and static deformation modes. The reference motion of a flexible body in a multibody system is large and fast in general, but its deformation remains small in many applications. In such cases computational efficiency of a simulation can be increased by linearizing the dynamical equations of motion

i i of a system body i, assuming the variables u (R , t ) and ϑ (R , t ) to be small. The linearization procedure may require the consideration of geometric stiffening, applying the general rules given in [20]. In terms of the variables used in equation (6) the kinematical and dynamical equations of motion of a representative system body i, as required for the development of multibody formalisms based on the floating frame of reference formulation, have the general forms

z! iI = Zi (z iI ) z iII ,

M i z! iII = hia + h ic ,

i = 1, 2 , … n

.

(7)

In these equations Zi is the coefficient matrix as required to relate the velocity variables z iII i to the time derivatives of the position variables z I . The generalized masses and the generalized applied and constraint forces, which correspond to the generalized velocities z iII , are collected in matrices M i , hia and hic , respectively. The generalized constraint forces hic result from the joints between the nodes on the system bodies.

data of beam model

data of FE-model

static and dynamic analysis static modes and eigenmodes computation of standard data for beam model

choice of shape functions

data of FE-model in MBS

FE-code

standard FE-output data

computation of standard data for FE-model

standard data for bodies in multibody systems

MBS-code I

MBS-code II

other multibody system data (joints, force elements, system topology and global reference frame)

Figure 3.

Multibody system data and pre-processors for the computation of body data as required for multibody system simulation. The pre-processors to compute the body data and the generality of the interface data allow to combine any finite element code with any multibody program.

Any type of multibody formalism may be developed from equations (7) by introducing the joint constraint equations and applying one of the principles of dynamics. The resulting procedure for the generation of the system equations provides the definition of the data to describe a multibody system. They may be subdivided into data describing bodies, joints, force elements, the motion of the global reference frame and the system topology. All of these have been well discussed in the literature on dynamics of systems of rigid bodies. In [18], an object oriented data model has been proposed for such systems. It has been augmented by a set of standard input data describing flexible bodies in multibody systems [31]. The definition of these data is based on equations (7), which represent the motion of a general flexible body i as shown in figure 2. The specific equations for the body models used in multibody analysis are found by formulating the internal constraint equations for the relative motion of the points of the body. The body data for the various models may be computed by the pre-processors shown in figure 3, [32, 33]. The selection of shape functions required in this context cannot be made in a formal way. It depends on the analyst's judgement and it is emphasized in figure 3 by the circle in the flowchart for the computation of the standard body data.

The summarized description of the floating frame of reference formulation for the simulation of systems of flexible bodies suggests two possibilities to increase computational efficiency: application of the modal approach and linearization of the system equations assuming the deformations to be small. These topics will be discussed here, using two simple examples. The first one gives an analysis of the effects of various shape functions on the representation of deformation and stress. An application of the linearized equations requires the deformations to be as small as possible. It will be demonstrated by the second example, that a wide range of applicability of the linearized equations may be achieved by selection of specific shape functions and by subdividing the flexible bodies as proposed in [35]. A combination of the two methodologies provides efficient system equations for an analysis of flexible multibody systems.

2.

Effects of various body reference frames and shape functions

2.1. Study Model For a discussion of the choice of shape functions the structure shown in figure 4 is considered. It consists of a flexible beam AB, which is connected to a rigid body BC by a bracket joint, allowing no relative motion. The beam is attached at points A and D to a fixed, supporting structure. At A, there is a revolute joint together with a rotational damper. Point D is connected with C by a spring. External forces acting on the structure are the result of gravitational acceleration, defined by the vector g . The influence of shape functions is studied by analyzing the • equilibrium configuration I, in which the structure is supported by the spring, • motion and the equilibrium configuration II, obtained after cutting the spring at C, • internal forces in the beam AB in the equilibrium configuration II. e 2I O

$0 D

I

A revolute joint including damper

supporting structure

e 1I

spring attached at revolute joints

beam structure C

intersection

g B

bracket joint

Figure 4. Rotating beam structure supported by a spring.

The problems are solved using the multibody model shown in figure 5. Origin and basis of an inertial frame (identical with the global reference frame i = 0) are shown in figure 4. In this frame the coordinates of the gravitational acceleration are [0, –g, 0]T, g = 9.81 m/s2. The distance of the points A and D is $0 = 2 m. To represent the motion of body i = 1 by the variables given by Equation (6), a body reference frame has to be defined. Three types of frames, the tangent-, chord- and Buckensframe will be compared [21]. Figure 6 shows a chord-frame, the variables of Equation (6) 1 1 representing its motion and the variables w1 ( x, t ) and w2 ( x, t ) , x ≡ R1, describing the deformation. The data of the beam model are: length $1 = 2 m, height h1 = 0.003 m, area of cross section A1 = 0.0006 m2, area moment of inertia of the cross section J1 = 4.5·10-10 m4, density ρ1 = 8400 kg/m2 and modulus of elasticity (Young's modulus) E1 = 7·1010 N/m2.

6

7

joint s = 1 force element r = 4 force element r = 3 global reference frame i = 0 5 1 4 body i = 2 3 node k =2 joint s = 2 body i = 1 Figure 5.

Multibody model of the beam structure. General node-, joint- and force element-labels are k, s and r, respectively. Node labels are encircled. The arrows at the joints and force elements show the way, in which the relative motion across those interconnections is defined [21].

Body 2 is rigid, and figure 7 shows location and orientation of its reference frame 2 {O 2 , e } . In this frame the data of the body are: location of center of mass c2 = [0.2, 0, 0]T [m] and moment of inertia I2 = 0.32·10-10 kg m2, given with respect to the origin O2 of the body frame. The length of the body is $2 = 0.4 m. 3 h

ϑ e1

2

1,1 3

w

2

2

O

I

e1

1 2

$1

1

1 1 = O

eI

w 1 1

1

α

ρ1

1 3

x = R1

eI

1

Figure 6. Notation used to represent the motion of body i = 1.

Location and orientation of coordinate systems {Ok,i, ek,i}, shown in figure 2 at the nodes k,i, are specified by matrices Rk,i and Γk,i in the reference configuration. For most of the nodes, Rk,i = 0 and Γk,i = E. The only matrices having different values are 1  2 0.4  2  0 1 0 R 2,1 = 0  , R 3,1 =  0  , R 5,2 =  0  , R 6,0 =  0  , G3,1 =  −1 0 0  0   0   0   0   0 0 0 

(8)

with the Rk,i given in m. Node k = 2 on body i = 1 is not required for attachment of a system element – it is used to define a point for evaluation of deformation and internal forces. eI

e2

2

OI

e 1I

ρ2

2

5

e 12

α 23

$2

2 4= O

Figure 7. Notation used to represent the motion of body i = 2.

The axis of the rotational joint at A is given in the basis e1 by [0, 0, 1]T. The force element r = 4 at this joint results in a torque about e13 , depending linearly on the relative angular velocity across the interconnection – the damping factor is 40 N m s/rad. The spring, connecting the nodes k = 5 and k = 6, has an undeformed length of 0.1 m, and its stiffness is 300 N/m. 2.2. Definition Of The Deformation Models To solve the problems mentioned above, three families of models are used to represent the deformation of body 1. They correspond to a chord-, tangent- and Buckens-frame – see figure 8 – and they are distinguished by labels C, T and B, respectively. Buckens-frame chordframe R1 = 0

CMi

1

w2C

Euler-Bernoulli beam, cross section R1 0

1

2

2 2

w1T

1

w2B

w1B

tangent-frame

R1 = $

w1C

w2T

Figure 8. Deformations wk = wkC , wk = wkT and wk = wkB , k = 1, 2 of an Euler-Bernoulli beam when using a chord- , tangent- and Buckens-frame, respectively. i

i

i

The displacements of the beam axis result from stretching and bending and they are i represented by nq = nq1 + nq 2 shape functions in the form nq 1

nq 2

w ( x, t ) = ∑ W ( x) q (t ) , w ( x, t ) = ∑ W21j ( x) qk1 (t ) where k = nq1 + j . 1 1

j =1

1 1j

1 j

1 2

j =1

(9)

The shape functions have to satisfy the conditions resulting from the choice of the body reference frame. In case of a chord- and tangent-frame these are geometrical boundary conditions of a simply supported and a cantilever beam, respectively. In case of a Buckensframe the linearized conditions for a mean-axis-frame must be satisfied. They state that the frame translates and rotates as to guarantee that the linear and angular momentum due to deformation is zero, i.e. OJi = 0 and OHi = 0 – see [22]. These conditions are satisfied when using free-free modes after deleting the rigid body modes [29]. A variety of models, all of them using the chord-frame as the body reference frame, i will be considered first. In these models the deformations wk = wkC , k = 1, 2 of body i = 1 are described as suggested by Equation (9), using various numbers and types of shape functions. The models are characterized by identifiers, which are combinations of C and 1 1 consecutive numbers. In addition to the identifiers, the numbers nq of the variables q j (t ) , which are used to represent deformation, are given in parentheses. The models are: 1 1 C1(2): The first eigenfunctions W11 ( x) and W21 ( x) for longitudinal and lateral vibrations of a simply supported beam are selected to represent deformation. 1 1 C2(6): Same as C1, but 3 functions W1 j ( x ) and 3 functions W2 j ( x ) . 1 1 C3(15): Same as C1, but 5 functions W1 j ( x ) and 10 functions W2 j ( x ) . C4(6): Same as C2, but the shape functions used here are based on eigenfunctions of a beam, which is supported by a rotational joint at x = 0. Therefore, a (rotational) rigid body mode appears among the eigenfunctions. It is deleted and

to obtain the shape functions, the remainder of the eigenfunctions is transformed, to satisfy the chord-frame boundary conditions [25]. 1 1 C5(2): In this case W11 ( x) and W21 ( x) are the static deformations of a simply supported beam due to a longitudinal force and to a torque at node k = 3, respectively. C6(4): Combination of C1 and C5. C7(8): Combination of C2 and C5. The shape functions, used in these models, are shown in figure 9 for C2, C4, C5 and C6. The functions used in C1 are a subset of C2, and in case of C3 additional, higher modes of the type shown for C2 are considered. C5

C2

C4

T2

T7

Figure 9.

B2

C6

B8

Mode shapes used to represent the deformation of body i = 1. Dotted and solid curves correspond to longitudinal and lateral motion, respectively. In all of the diagrams the horizontal axes represent values of x and in vertical direction the function values W11j ( x ) and W21j ( x ) are given.

In the second group of models a tangent-frame is used as a body reference frame. Its origin coincides with the origin of the chord-frame shown in figure 6. Using notation similar to the one introduced for the C-models, the tangent-frame models are: T2(6): Same as C2, but eigenfunctions of a cantilever in place of those of a simply supported beam. T7(8): Same as C7, but eigenfunctions and static forms of a cantilever beam, i.e. 3 vibration modes for both, longitudinal and lateral motion, and static forms due to a force and a torque at node k = 3. The shape functions, used in the two T-models, are shown also in figure 9. 1 1 1 Finally, a Buckens-frame is chosen for {O , e } . Its origin O coincides with the center 1 of mass CM of the deformed beam, which means that the linear momentum OJi due to 1 body deformations disappears. Similarly, its basis e rotates as to guarantee that the angular momentum OHi due to body deformation is zero [22]. In this case eigenfunctions of the free-free beam are used, with the rigid body modes deleted. The models considered are: B2(6): Same as C2, but with eigenmodes of the free-free beam. B7(8): B2 combined with C5 – by analogy with C7. B8(9): B7 with an additional static deformation mode of a simply supported beam, due to a linearly distributed load in the longitudinal direction. The shape functions, used in these models, are shown in Figure 9. The diagram for B8 shows the longitudinal modes only. The models B7(8) and B8(9) require a comment. The quasi-comparison functions, used in Equation (5) to approximate the solution of the partial differential equations describing body deformation, have to be elements of a complete set of functions, they have to satisfy the geometrical boundary conditions and a linear combination of them has to be capable to satisfy the dynamical boundary conditions [8, 9, 14-16]. In case of a Buckens frame the body deformation is described by differential equations satisfying the constraints i i OJ = 0 and OH = 0. The constraints guarantee the deformations to become a minimum – see [22], Equation (38) – and they are fulfilled automatically when using free-free modes after deleting the rigid body modes [29]. When augmenting the free-free modes by static

modes as in case of the models B7(8) and B8(9) one has two choices: One considers the constraints OJi = 0 and OHi = 0 when solving the multibody system equations or one renounces exploiting any properties of the Buckens frame resulting from these constraints. In the latter case the body deformation is no longer the smallest possible. In this study the second option has been used for the models B7(8) and B8(9). In all models, mentioned heretofore, the modal approach has been used to describe the deformation of the beam i=1 shown in figure 6. A reference solution (ref) will be generated using the nodal approach. In this reference model, the body i=1 is subdivided into 10 finite beam elements, in which a tangent-frame is used to represent the element deformation. The displacement field of the elements is interpolated by linear polynomials for the longitudinal motion and by cubic ones for bending [36]. The results will be compared also with those obtained by the absolute nodal coordinate formulation (anc), which has been proposed recently [24]. A large reference motion of a flexible body is described by the nodal coordinates of a finite element model in this methodology [27]. The beam is modelled in this formulation by 20 elements and their displacements are interpolated by third order polynomials [5].

2.3. Deformation Results The three problems, described at the beginning of this section, have been solved using the C1-model first. As an example, figure 10 shows the equilibrium configuration I. All of the results on the motion and deformation of the structure are in good correspondence with the reference solution and they are not modified significantly, when using the other models defined above. This statement is detailed in table 1. It contains results for the two equiI k ,i I k ,i librium configurations I and II as obtained with various models. The results ρ1 and ρ 2 describe the location of the nodes k , i = 2,1 and k , i = 3,1 in the inertial frame and the angles α 3k ,i with k , i = 1,1 and k , i = 3,1 represent the orientation of e k ,i with respect to e I – see figure 10. For the configuration I the results are given as obtained by the models C1 and C7. Their comparison demonstrates that the complex model C7 does not modify the results obtained with the simple model C1 very much. Both of them are found to be in good correspondence with the reference solution and the results provided by the absolute nodal coordinate formulation. For the configuration II results are presented in table 1 as obtained with the C-, T- and B-models together with the reference results ref and anc. They clearly I k ,i demonstrate that all the models are capable to describe the displacements ρα , excluding k ,i the model C5, which uses static modes only. In case of the angles α 3 the deviations between the results obtained with the various models are more pronounced. e 2I e 1,1 2

global reference frame − Iρ

7 1

− α

2,1 1

1,1 3

e 3,1 1 body

2 i =1

e 1I

6 − I ρ 2,1 2

e 1,1 1

body

i =0

i =2

5

α

e 3,1 2

4 3

Figure 10. Equilibrium configuration I of the beam structure.

3,1 3

Table 1.

model

ρ12,1 ρ 2,1 2 I 3,1 ρ1 I 3,1 ρ2 α 1,1 3 α 3,1 3 I I

Results describing the deformation of body i = 1 in the equilibrium configurations I and II. Displacements I ραk ,i are given in mm and angles α 3k ,i in 103 rad. configuration I configuration II C1 C7 C2 C5 C7 T2 T7 B2 B7 ref anc 840.6 837.0 –20.5 –12.7 –19.8 –18.9 –20.0 –21.0 –19.7 –19.9 –19.8 -569.8 -577.9 –1001 –1002 –1001 –1000 –1000 –1001 –1001 –1000 –999.8 1825.5 1825.8 –137.0 –141.1 –136.1 –135.9 –136.3 –136.4 –136.1 –136.0 –136.0 –0.817 –816.4 –1995 –1995 –1995 –1999 –1999 –1995 –1995 –1999 –1991 –698.4 –721.4 –1576 –1564 –1581 –1580 –1581 –1567 –1580 –1581 –1581 1427.7 1410.7 –197.5 –225.2 –237.8 –161.7 –237.1 –178.3 –237.8 –236.9 –238.3

To summarize: Using eigenfunctions as shape functions in the modal approach one obtains a good representation of the system motion and of the displacements due to body deformation. Angle variables describing the deformation of the beam investigated here are still acceptable but the stresses due to body deformation, as obtained with the various models, differ significantly. This will be demonstrated now.

2.4. Stress Results The forces and torques, required for the discussion of the internal forces due to body k ,i k ,i deformation, are defined in figure 11. Symbols F and L denote forces and torques applied at the nodes k on body i – see figure 2. They result from force elements and joints at k ,i k ,i i the nodes on the bodies. Both, F and L , are resolved in the bases e , shown in figures k ,i k ,i 6 and 7 for i = 1 and i = 2, to obtain the coordinates Fα and Lα . These coordinates are determined in an analysis of the multibody system using the laws for the forces and torques across the force elements and the generalized constraint forces across the joints. F

1,1

e 1,1 2 e 4,2 = e 3,1 1

L

1,1

5

rigid body i = 2

1

− α 1,1 3 e 4,2 = 2

1

e 1,1 1

Lb (x)

1

− F 1 (x)

− Lb (x)

F

n

2 Fn1 (x)

flexible body i = 1

L

= −L

= −F

L

e 3,1 2

4,2

4

4,2

F

3 3,1

3,1

1

4,2

4,2

Figure 11. Interaction-forces and -torques upon the bodies of the beam structure together with internal force and moment due to deformation of the beam.

Vectors F n ( x ) = Fn1 ( x) e1 and Lb ( x ) = L1b ( x ) e3 represent the normal force and the bending moment applied at a cross section x of the beam. The functions Fn1 ( x ) and L1b ( x) are found from the variables describing deformation by using the material law 1

1

1

1

S111 (R ) = E 1 G111 (R ) .

(10)

1 1 Here S11 (R ) and G11 (R ) represent stress and strain at a point R of the beam. Stresses and strains are identical for any values R3 in case of the planar problem shown in figure 6. 1 1 1 Therefore, it is sufficient to consider S11 ( R1 , R2 ) ≡ S11 ( R1 , R2 , 0) and G11 ( R1 , R2 ) ≡ 1 G11 ( R1 , R2 , 0) . From the stress distribution S111 ( R1 , R2 ) one obtains the normal stress S n1 (due 1 to stretching) and the shear stress Sb (due to bending) as

S n1 ( x) = S111 ( x, 0) and Sb1 ( x, R2 ) = S111 ( x, R2 ) − S n1 ( x) , x ≡ R1 .

(11)

1 The maximum values of shear stress are obtained for R2 = ± h / 2 . The effects of the 1 stresses S11 ( R1 , R2 ) distributed over the rigid cross section at the point x = R1 of the beam's 1 1 axis can be described by a single force F n ( x) at R2 = 0 and a moment Lb ( x ) . These resultants of the system of forces acting upon the rigid cross section x are found to be

Fn1 ( x ) = A1 S111 ( x, 0) and L1b ( x) =

(

)

2 J1 1 S11 ( x, h1 / 2) − S n1 ( x) . h1

(12)

The strain distribution appearing in Equation (10) is given for an Euler-Bernoulli-beam by

(

G111 ( R1 , R2 ) = w1′1 ( x) − R2 w2′′1 ( x) + 12 w2′1 ( x )

)

2

, x ≡ R1 .

(13)

1 The deformations w1 and w2 in Equation (13) are obtained from the variables q , appearing 1 2 in equation (6), by equation (9). The nonlinear expression ( w2′ ) in Equation (13) serves to develop the geometric stiffening terms due to longitudinal forces as described in [20]. The 1 1 forces Fn ( x ) and moments Lb ( x) are obtained using the linearized strain-displacement relations of Equation (13) in Equations (10) to (12). 1 1 The two functions Fn ( x ) and Lb ( x) have to satisfy dynamical boundary conditions at x = 0 and x = $1:

. Fn1 (0) = − F11,1 , L1b (0) = − L1,1 and Fn1 ($1 ) = F13,1 , L1b ($1 ) = L3,1 3 3

(14)

Some of the shape functions shown in figure 9 violate these boundary conditions. In such 1 1 cases the representation of stress described by the functions Fn ( x ) and Lb ( x) is poor. 2.4.1.Representation of Stress Using the Chord-Frame-Models A static analysis, based on the model C1, yields for the equilibrium configuration I:  Fn1 (0) = 44.3 N , Fn1 ($1 ) = 0 N , F11,1 = − 49.4 N , F13,1 = 9.0 N ,  whereas   1 1 1 L1,1 L3,1 3 = 0 Nm , 3 = − 6.9 Nm ,   Lb (0) = 0 Nm , Lb ($ ) = 0 Nm.

(15)

It is obvious that the shape functions used in the C1-model cannot satisfy the dynamical boundary conditions of Equation (14). A similar result is obtained by an analysis of the equilibrium configuration II. In this case the model C1 results in  Fn1 (0) = 154.7 N , Fn1 ($1 ) = 0 N , F11,1 = −157.4 N , F13,1 = 58.7 N ,  whereas   1 1 1 L1,1 L3,1 3 = 0 Nm , 3 = − 11.6 Nm ,   Lb (0) = 0 Nm , Lb ($ ) = 0 Nm.

(16)

Because of the violation of the dynamical boundary conditions, the representation of the 1 1 internal forces Fn ( x ) and moments Lb ( x) is not satisfactory. Such behaviour, i.e. good motion and deformation results and poor stresses, suggests that one may determine the deformations by multibody simulation programs and compute the stresses, using post-processor finite element codes [4, 12]. However, an acceptable representation of stresses can also be obtained with multibody models using quasi-comparison functions. This will be demonstrated now, considering all of the chord-frame models introduced above.

0

150

[Nm]

[N]

-2

125 100

C2 = C4

L b(x )

-4

75

C1

1

1

Fn (x )

C4

C2

ref

C3

-6

ref 50

C1

C3

-8

25

-10

x

0 0

0.5

150

1

1.5

[m]

[m]

2

L -6

ref

[m]

2

ref

C5 -4

1 Fn (x )

75

1.5

-2

C5

100

1

[Nm]

C6

125

0.5

0

C7

[N]

x 0

2

1 (x ) b

C6

50

C7

C6

-8

25

-10

x

x

0 0

0.5

1

1.5

[m]

2

0

0.5

1

1.5

Figure 12. Normal force Fn1 (x) and bending moment L1b (x) in body 1 as obtained with the chord-framemodels for the equilibrium configuration II. 1 1 Figure 12 shows the functions Fn ( x ) and Lb ( x) for the equilibrium configuration II, when using the chord-frame models. The reference solution obtained with the finite element method is denoted by ref in the diagrams. In the reference model the displacement field of the finite elements is interpolated by linear polynomials for the longitudinal motion. This explains, why the element normal forces are constant, resulting in the discontinuities of the 1 reference solution for Fn ( x ) in figure 12. The points (•), marked in the diagrams at x = 0 k ,i k ,i and x = $1, represent the values of the forces F1 and the torques L3 at the nodes k = 1 and k = 3, as obtained with the reference solution from the constraint forces in the joints. These results are: 3,1 . F11,1 = −157.7 N , L1,1 = 58.8 N , L3,1 3 = 0 Nm , F1 3 = − 11.44 Nm

(17)

The internal forces at x = 0 and x = $1 found with the reference solution have slightly 1 1 different values. They are, together Fn and Lb at the node k = 2 on body i = 1: Fn1 (0) = 152.8 N, L1b (0) = 0 N m ,

Fn1 ($1 / 2) = 108.3 N, L1b ($1 / 2) = − 2.47 N m,

Fn1 ($1 ) = 63.8 N, L1b ( $1 ) = − 11.37 N m.

(18)

The results presented in figure 12 demonstrate that the eigenfunctions used in models C1 to C3 cannot satisfy the dynamical boundary conditions. The functions minimize a mean value of the errors, as suggested by the Ritz method. By adding more functions, the error becomes 1 smaller, but as demonstrated clearly by the results of the C3-model for Lb ( x) , the solution starts to oscillate heavily in the neighbourhood of x = $1 when using higher numbers of 1 1 3,1 eigenfunctions, which are unable to satisfy the dynamical boundary condition Lb ($ ) = L3 . Such a behaviour is similar to Gibb's phenomenon, as known from the representation of discontinuous functions by Fourier series, e.g. [13], p. 392. Selecting the transformed eigenfunctions of a beam supported by a rotational joint at x = 0 does not improve the solution, as demonstrated by the results obtained with the C4-model. The results of the model C5 1 1 demonstrate that the representation of Fn ( x ) and Lb ( x) by the static modes only is poor.

But combining their capability to satisfy the dynamical boundary conditions with the ability 1 1 of the eigenfunctions to represent Fn ( x ) and Lb ( x) , convergence is significantly improved, as demonstrated by the results of the C6- and C7-models. 0

[Nm] -2 -4

L 1b( x )

T2 T7

T2

C2

-6

C7

-8

ref

T7 -10

x 0

0.5

1

[m]

1.5

2

Figure 13. Normal force Fn1 ( x) and bending moment L1b ( x ) in body 1 as obtained with the tangent-framemodels. Results determined with the corresponding C-models are presented here, as well.

2.4.2.Representation of Stress Using the Tangent-Frame-Models Figure 13 shows the results, when using a tangent-frame (the T-models, defined previously). Eigenfunctions and static deformation modes for the longitudinal motion are identical for chord- and tangent-frames, which implies that the force representations, 1 obtained by the C- and T-models, do not differ. This explains why figure 13 shows Lb ( x) only. The chord-frame results to be compared with the tangent-frame results are shown in figure 13 again. The T-models yield larger errors than the corresponding C-models. The errors result from violation of the dynamical boundary condition at x = 0, which is enforced by the eigenfunctions satisfying the chord-frame conditions – see diagrams for T2 and C2. 1 As in case of the C-models, the representation of Lb ( x) is improved, when adding static modes, but the errors of the T-models remain larger than those obtained with the corresponding C-models. 2.4.3.Representation of Stress Using the Buckens-Frame-Models 1 1 Finally, figure 14 shows the functions Fn ( x ) and Lb ( x) when using free-free-modes. The conditions on the location and orientation a Buckens-frame are satisfied automatically, when using these eigenfunctions after deleting the rigid body modes. By contrast with the 1 eigenfunctions of a simply supported beam, satisfying Fn (0) ≠ 0 , those of a free-free beam 1 1 yield Fn (0) = 0 . As a consequence, the errors in the representation of Fn ( x ) are increased in the model B2 as compared to C2. When adding the static modes used in the C5-model to 1 obtain the B7-model, the representation of the moment Lb ( x) is improved, providing an approximation similar to the one obtained by the model C7. 0

150

1

[N]

F n (x )

125

-2

1

B7

B7

100

B7

[Nm]

ref

C7 = B8

L b (x )

-4

B2

C7 = B8

C2

75

-6

ref

50

-8

B2

C2

25

B2

0

0.5

1

1.5

B7

-10

x

0

[m]

2

x 0

0.5

1

1.5

[m]

2

Figure 14. Normal force Fn1 ( x) and bending moment L1b ( x ) in body 1 as obtained with the B-models together with the results provided by the corresponding C-models.

1 The representation of the force Fn ( x ) with the B7-model is still insufficient as compared to the C7-model. The reason is simple: By adding the static mode for longitudinal deformation of the C5-model to the eigenfunctions of the free-free beam, one has the freedom to represent non zero forces at x = 0 and x = $1, but these have to be identical – see diagram 1 B7 for Fn ( x ) . Only after adding another static mode to obtain the B8-model, different forces at x = 0 and x = $1 can be represented by the system of static forms and eigen1 functions. As a consequence, the B8-model yields an approximation of Fn ( x ) of roughly the same accuracy as C7. The curves for the two models C7 and B8 cannot be distinguished in a diagram as shown in Figure 14.

2.5. Concluding Remarks The results demonstrate that admissible functions (here eigenfunctions of various types) yield an acceptable representation of body deformation, as suggested by the expansion theorem detailed in [3], vol. I, p. 311 or [28], p. 111. When the admissible functions violate the dynamical boundary conditions an acceptable representation of the stresses is obtained by high numbers of shape functions only. In particular, at points where the admissible functions violate the dynamical boundary conditions a behavior similar to Gibb's phenomenon is observed. Quasi-comparison functions as proposed in [8, 9, 14-16] (here a combination of eigenmodes and static modes) result in a significant improvement of the representation of stress.

3.

Range of validity of linearized equations

A second example, the slider/crank mechanism shown in figure 15, is used to demonstrate the influence of the choice of the body reference frame on the range of validity of the linearized equations of motion. It is studied by analysing • the equilibrium configurations I and II of the mechanism, obtained when fixing the 1 1 1 angle β to the values β = 0° and β = 57.3° , respectively; 1 • its motion due to a driving torque, sustaining a constant angular velocity Ω . To define problems with unique solutions, the friction force in the prismatic joint at D is assumed to be zero for the determination of the equilibrium configurations. Ω

B

1 = const.

revolute joint

g

slider

coupler

revolute joint crank

β

1

(t)

A motor frame

C D

prismatic joint and dry friction E

Figure 15. Slider/crank mechanism with elastic elements.

3.1. Description Of The Models The multibody model of the mechanism is shown in figure 16. The models of crank and coupler are Euler-Bernoulli beams and the slider is assumed to be rigid. The data of bodies i = 1 and 2 are with reference to the labelling scheme shown in figure 16 and using notation by analogy with the previous example: $1 = 0.6 m, $2 = 2.0 m, Ai = 0.0006 m2, Ji =

body i = 1 e 21

joint s = 2

e 22

2

3 =O2

e11

e2I

1=O1

body i = 2 e 12 5

node 4 joint s = 3

e3 2

6 =O 3

joint s = 1 e1I

9= O I

e13 n4

8

global reference framei = 0

7

body i = 3

joint s = 4 force elementr = 5

Figure 16. Multibody model of slider/crank mechanism. Node k,i = 8,0 is at E from figure 15, where D = E for β 1 = 0 ° and bodies are not deformed. Vector n 4 represents the axis of the joint s = 4.

4.5·10-10 m4, ρi = 8400 kg/m3 and Ei = 7·1010 Nm2. The data of body 3 are: m3 = 8.4 kg, I3 = 0.028 kg m2, where I3 is given with respect to O3. The latter point coincides with the center of mass CM3. The axes of the joints are clear from the figures, and the dry friction coefficient is 0.05. The driving angular velocity is Ω 1 = 0.2 rad/s and the coordinates of the gravitational acceleration in the inertial frame are [0, –g, 0]T, g = 9.81 m/s2. 1 In a preliminary analysis the deformation w2 of the crank, as given by Equation (9), is described using the tangent-frame shown in figure 16 and the first eigenmode of a cantilever 2 beam. The deformation w2 of the coupler is represented using a chord-frame and the first i mode of a simply supported beam. Longitudinal deformations w1 , i = 1, 2 are neglected for both of the bodies. This model is denoted as model A in table 2. Table 2. Models used to study the effects of substructuring and definition of body reference frames. model

crank, body i = 1

coupler, body i = 2

A

One body, tangent-frame, one bending mode One body, chord-frame, one bending mode of a cantilever beam. of a simply supported beam.

B

As in model A.

Two bodies, chord-frame and one bending mode of a simply supported beam each.

C

As in model A.

Two bodies, tangent-frames and one bending mode of a cantilever beam each.

D

One body, chord-frame and one bending mode As in model B. of a simply supported beam.

ref

Two bodies, chord-frame and one bending mode of a simply supported beam each, together with two static modes, due to a force F2 and a torque L3 at x = $1/2.

anc

Absolute nodal coordinate formulation using 8 As for body i = 1 but 20 elements. elements – deformation of the elements is described using a chord-frame. The linearized strain-displacement relations (13) are used.

Three bodies, chord-frame and one bending mode of a simply supported beam each, together with two static modes, due to torques L3 at x = 0 and x = $2/3.

The equilibrium configurations I and II and the motion of the mechanism have been obtained using the model A. The corresponding shape functions do not satisfy the dynamical boundary conditions and as a consequence the representation of internal forces turns out to be poor. Moreover, the results demonstrate that the deformations, especially those of the coupler, are quite large. In case of the equilibrium configuration I (β1 = 0˚) one observes that the rotational displacement of the beam cross section at the node k, i = 5, 2 becomes ϑ35,2 = 0.51 rad = 29.5˚. This is far beyond the values acceptable for an application of linearized equations assuming the deformations to be small. The magnitude of the deformations may be reduced, considering two possibilities to improve modelling: One may subdivide the flexible bodies, as proposed in [35], and one may modify the body reference frame to minimize the magnitude of deformations. Various options will be analyzed now for the case of the equilibrium configuration I, using the models described in table 2. Only chord- and tangent-frames are considered, because the orientation of a Buckens-frame is not modified as compared to a chord-frame in case of the beam models used here. Moreover, a shift of the origin of the body reference frame to the center of mass, resulting from a choice of a Buckens-frame, does not affect the magnitude of the angles leading to the problem mentioned above. In case of model B the coupler is subdivided into two bodies, interconnected by a bracket joint. Deformations of the subbodies are represented by the first eigenmode of a simply supported beam, as suggested by the boundary conditions of a chord-frame. In model C this chord-frame is replaced by a tangent-frame, using the first eigenmode of a cantilever beam. In case of D the coupler is modelled as in B and to reduce the magnitude of deformations of the crank, a chord-frame is used instead of the tangent-frame.

0.4

e2I

0.2 0

2,1 –Iρ2

1 9 body 1

-0.2

23

-0.4 -0.5

4,2 –Iρ2

D

C

0.5

1

6

B

A body 2 0

5

4 1.5

e1I

ref 2

2.5

Figure 17. Equilibrium configurations I of slider/crank mechanism as obtained by the models A to D and ref. Models A and C result in deformations too large for application of linearized equations.

The models A to D allow an analysis of the size of deformations due to substructuring and to the choice of body reference frames. The corresponding results for the equilibrium configuration I are given in figure 17 and table 3. Deformations of the coupler are significantly smaller when using model B instead of A: Angle ϑ35,2 is reduced from 0.514 rad to 0.209 rad. Model C demonstrates that this is true only when using a chord-frame. The choice of a tangent-frame does not pay off, when trying to reduce the magnitude of deformation. Angle ϑ35,2 is reduced by subdividing the coupler but, as demonstrated by the values of ϑ 32,1 for A, B and C in table 3, the deformation of the crank is still too large. It is reduced in model D by selecting a chord-frame and the corresponding eigenmode. This

Table 3. Values characterising the equilibrium configuration I as obtained with the models defined in table 2. model

A

location

I

ρ 2,1 2

location

I

ρ 4,2 2

location

I

ρ16,3

I

I

I

I

I

I

of node 2,1 in {O , e } [m] of node 4,2 in {O , e } [m] of node 6,3 in {O , e } [m]

B

C

D

ref

anc

–0.134 –0.134 –0.136 –0.079 –0.132

–0.130

–0.394 –0.277 –0.253 –0.250 –0.359

–0.355

2.596

2.481

2.470

2.551

2.621

2.548

angle

α 2,1 3

at node 2,1 [rad]

–0.307 –0.308 –0.312 –0.264 –0.326

–0.322

angle

α 3,2 3

at node 3,2 [rad]

–0.447 –0.356 –0.265 –0.385 –0.412

–0.409

angle

α 34,2

at node 4,2 [rad]

0.067

0.071

0.059

0.042

0.072

0.071

angle

α 5,2 3

at node 5,2 [rad]

0.581

0.490

0.371

0.463

0.547

0.542

deformation

ϑ 2,1 3

at node 2,1 [rad]

–0.307 –0.308 –0.312 –0.132 –0.074 –0.0014

deformation

ϑ 3,2 3

at node 3,2 [rad]

–0.514 –0.214

deformation

ϑ 5,2 3

at node 5,2 [rad]

0.514

0.209

0.430

0.211

0.088

–0.0023

= − µ3 at node 1,1 [Nm]

–38.1

–38.3

–38.7

–38.1

–37.4

–37.18

at node 4,2 [Nm]

25.4

0.0

34.6

0.0

22.9

22.76

bending torque

1,1 Lb

bending torque

4, 2 Lb

0.0

–0.213 –0.090 –0.0023

yields a value of ϑ 32,1 = − 0.132 rad. If this is still considered too large for linearization, the bodies may be subdivided further, as suggested for the ref-model defined in table 2. The analysis demonstrates that the deformations may be kept small enough as to justify an application of equations, which are linearized assuming small deformations. The goal may be achieved by subdividing the bodies and using chord-frames. But the previous example demonstrates that one cannot expect to represent body deformation, and especially internal forces, properly, when using one mode only. Sources of the deficiencies of the models A to D may be found by looking at the values of table 3: Whereas models A to C 2,1 I 2,1 yield coordinates ρ2 of vector ρ in the inertial frame, i.e. of the location of node k , i = 2,1 , of ca. − 0.13m , the same value is lowered to − 0.079m in model D. The problems become even more apparent when comparing with the solution anc, obtained with a finite element model based on the absolute nodal coordinate formulation as described in the last row of table 2. The results found with this model (last column of table 3) can be approximated closely by the ref-model from table 2. In this case static modes are used to satisfy the dynamical boundary conditions. The resulting equilibrium configuration I is marked in figure 17 by ref. The curve cannot be distinguished from the one obtained with the anc-model. Based on the results of this analysis of the equilibrium configuration I, one can recompute the equilibrium configuration II and the system motion, using the ref-model to obtain more reliable results. Because of the preliminary analysis, these computations can be done with an efficient model based on the modal approach. For the analysis of the mechanism motion one may want to add static modes to satisfy dynamical boundary conditions due to the driving torque. All of the results have been obtained using Mathematica, Version 2.2. The corresponding notebook files may be found at http://www.vieweg.de/downloads.

4.

Conclusions

The floating frame of reference formulation is based on a separation of the flexible body motion into a reference motion and a deformation. The definition of the two motions requires the specification of a body reference frame, which in turn is tied to the choice of

shape functions in the Ritz method. Two statements, which follow from the derivation of the system equations given in [21, 22] have been verified by discussion of examples: 1. The modal approach to represent body deformation may reduce the number of system variables considerably and thus increases computational efficiency, but it raises the problem of how to select the shape functions. Quasi-comparison functions, obtained by combining eigenfunctions (satisfying equations resulting from the definition of the body reference frame) with static modes to satisfy the dynamical boundary conditions, significantly improve the representation of internal forces, i.e. stresses. This statement applies for any choice of the body frame and the corresponding shape functions, but specific choices may require less variables than others. The effort required for the selection of suitable quasi-comparison functions pays off: in addition to an increased efficiency of multibody simulation, it often eliminates the need for finite element postprocessing the simulation results to obtain the stresses. 2. The linearization of the system equations requires the deformations to be as small as possible. This is guaranteed by selecting a so-called Buckens-frame as a body reference frame. If the orientation of the Buckens-frame is not significantly affected by deformation, a properly chosen chord-frame serves the purpose as well. In addition, deformations can be kept small by subdividing flexible bodies. A preliminary analysis of simple problems may provide the numbers of subbodies and shape functions required for the representation of deformation and internal forces in more complicated problems. Efficient models, found by following such routes, increase the efficiency of multibody system analysis. To summarize: The results improve the capabilities of multibody codes and they increase the range of validity of the linearized system equations, often used in the floating frame of reference formulation.

References [1]

H. Bremer and F. Pfeiffer, Elastische Mehrkörpersysteme. Teubner Studienbücher, Mechanik, B. G. Teubner, Stuttgart, 1992. [2] A. Budó, Theoretische Mechanik. 4. Auflage, Deutscher Verlag der Wissenschaften, Berlin, 1967. [3] R. Courant and D. Hilbert, Methoden der mathematischen Physik. Heidelberger Taschenbücher, Band 30, Springer-Verlag, Berlin, 1968. [4] S. Dietz, H. Netter and D. Sachau, Fatigue Life Prediction by Coupling FEM and MBS Calculations, In: First Symposium on Multibody Dynamics and Vibration at 16th Annual Conference of ASME on Mechanical Vibration and Noise, Sacramento, CA., 1997. [5] S. v. Dombrowski and R. Schwertassek, Analysis of Large Flexible Body Deformation in Multibody Systems Using Absolute Coordinates, In: J.A.C. Ambrosio and W. Schiehlen (eds.), Euromech Colloquium 404, Advances in Computational Multibody Dynamics, Instituto Superior Tecnico, Lisboa, Portugal, (1999), pp. 359-378. [6] E. Eich-Soellner and C. Führer, Numerical Methods in Multibody Dynamics. Teubner-Verlag, Stuttgart, 1998. [7] J. Garcia de Jalon and E. Bayo, Kinematic and Dynamic Simulation of Multibody Systems, The Real Time Challenge. Mechanical Engineering Series, ed. F. F. Ling, Springer-Verlag, New York, 1994. [8] P. Hagedorn, Quasi-Comparison Functions in the Dynamics of Elastic Multibody Systems, In: Proc. of 8th VPI&SU Symposium on Dynamics and Control of Large Structures, Blacksburg, VA, Virginia Polytechnic Institute and State University, (1991), pp. 97-108. [9] P. Hagedorn, The Rayleigh-Ritz Method With Quasi-Comparison Functions in Nonself-Adjoint Problems, Journal of Vibration and Acoustics, 115, (1993), 280-284. [10] E. J. Haug, Computer-Aided Kinematics and Dynamics of Mechanical Systems, Volume I: Basic Methods. Allyn and Bacon, Boston, 1989. [11] R. L. Huston, Multibody Dynamics. Butterworth-Heinemann, Boston, 1990.

[12] M. Jahnke, Ein Beitrag zur Untersuchung elastischer Mehrkörpersysteme unter Nutzung von FiniteElemente-Software. Fortschritt-Berichte der VDI-Zeitschriften, Reihe 11: Schwingungstechnik, Nr. 214, VDI-Verlag, Düsseldorf, 1994. [13] K. Knopp, Theorie und Anwendung der unendlichen Reihen. 4. Aufl., Die Grundlehren der mathematischen Wissenschaften, eds. W. Blaschke, et al., Vol. II, Springer-Verlag, Berlin, 1947. [14] L. Meirovitch and P. Hagedorn, A New Approach to the Modelling of Distributed Non-Self-Adjoined Systems, Journal of Sound and Vibration, 178(2), (1994), 227-241. [15] L. Meirovitch and M. K. Kwak, Convergence of the Classical Rayleigh-Ritz Method and the Finite Element Method, AIAA Journal, 28(8), (1990), 1509-1516. [16] L. Meirovitch and M. K. Kwak, On the Modeling of Flexible Multi-Body Systems by the Rayleigh-Ritz Method, In: Proc. of AIAA Dynamics Specialists Conference, Long Beach, CA, 1990. [17] P. E. Nikravesh, Computer-Aided Analysis of Mechanical Systems. Prentice Hall, Englewood Cliffs, New Jersey, 1988. [18] M. Otter, et al., An Object Oriented Data Model for Multibody Systems, In: W. Schiehlen (ed.) Proc. Int. Symp. on Advanced Multibody System Dynamics, Kluwer Acad. Publ., 1993, pp. 19-48. [19] R. E. Roberson and R. Schwertassek, Dynamics of Multibody Systems. Springer-Verlag, Berlin, 1988. [20] R. Schwertassek, Flexible Bodies in Multibody Systems, In J. Angeles and E. Zakhariev (eds.), Computational Methods in Mechanical Systems, Springer-Verlag, Berlin, 1998, pp. 329-363. [21] R. Schwertassek and O. Wallrapp, Dynamik flexibler Mehrkörpersysteme. Friedr. Vieweg Verlag, Braunschweig, 1999. [22] R. Schwertassek, O. Wallrapp and A. Shabana, Flexible Multibody Simulation and Choice of Shape Functions, Nonlinear Dynamics, 20(4), (1999), 361-380. [23] A. A. Shabana, Dynamics of Multibody Systems. J. Wiley & Sons, New York, 1989. [24] A. A. Shabana, An Absolute Nodal Coordinate Formulation for the Large Rotation and Deformation Analysis of Flexible Bodies, Department of Mechanical Engineering, University of Illinois at Chicago, Techn. Report MBS96-1-UIC, 1996. [25] A. A. Shabana, Resonance Conditions and Deformable Body Coordinate Systems. Journal Sound and Vibration, 192(1), (1996), 389-398. [26] A. A. Shabana, Flexible Multibody Dynamics: Review of Past and Recent Developments, Multibody System Dynamics, 1, (1997), 189-222. [27] A. A. Shabana and R. Schwertassek, Equivalence of the Floating Frame of Reference Approach and Finite Element Formulations, International Journal of Non-Linear Mechanics, 33(3), (1998), 417-432. [28] A. N. Tychonoff and A. A. Samarski, Differentialgleichungen der mathematischen Physik. Deutscher Verlag der Wissenschaften, Berlin, 1959. [29] B. F. Veubeke, The Dynamics of Flexible Bodies, International Journal on Engineering Sciences, 14, (1976), 895-913. [30] E. Volterra, The Equations of Motion for Curved and Twisted Elastic Bars Deduced by the Use of the "Method of Internal Constraints". Ingenieur-Archiv, XXIV, (1956), 392-400. [31] O. Wallrapp, Standard Input Data of Flexible Members for Multibody System Codes, In: W. Schiehlen, (ed.), Advanced Multibody System Dynamics — Simulation and Software Tools, Kluwer Academic Publishers, Dordrecht, 1993, pp. 445-450. [32] O. Wallrapp, Beam — A Pre-Processor for Mode Shape Analysis of Straight Beam Structures and Generation of the SID File for MBS Codes, User's Manual, Deutsche Forschungsanstalt für Luft- und Raumfahrt (DLR), Inst. Robotik und Systemdynamik, Oberpfaffenhofen, Report Version 3.0, 1994. [33] O. Wallrapp, A. Eichberger and J. Gerl, FEMBS - An Interface Between FEM Codes and MBS Codes, User Manual for ANSYS, NASTRAN, and ABAQUS, INTEC GmbH, Wessling, Report Version 3.0, January 1997. [34] J. Wittenburg, Dynamics of Systems of Rigid Bodies. Leitfäden der angewandten Mathematik und Mechanik, ed. H. Görtler, Vol. 33, B. G. Teubner, Stuttgart, 1977. [35] S. C. Wu and E. J. Haug, Geometric Nonlinear Substructuring for Dynamics of Flexible Mechanical Systems, International Journal of Numerical Methods in Engineering, 26, (1988), 2211-2226. [36] O. C. Zienkiewicz and R. L. Taylor, The Finite Element Method: Solid and Fluid Mechanics Dynamics and Non-linearity. Vol. 2, McGraw-Hill Book Company, London, 1991.

Part II Finite Elements Procedures for Structural Systems With Large Rotations

Finite Element Analysis of Rigid-Flexible Systems Robert L. TAYLOR Department of Civil and Environmental Engineering, University of California at Berkeley, Berkeley, CA 94710-1710, USA Abstract. This paper addresses methods for analysis of coupled flexible and rigid body solids undergoing large motions and deformations. The formulation is based on a finite element model for the entire system and is presented using a natural coordinate approach which is free of rotation parameters. This form avoids the need to find ‘compatible’ integration formulae for translations and rotations. The motion of each rigid body is represented in terms of natural coordinates which are expressed in terms of the nodal parameters of a simplex element subject to constraints which ensure rigid motions. Flexible structural members for rods and shells are then expressed in terms of displacement and relative displacement parameters leading to total system of equations involving only translation degrees of freedom. The motion are integrated using classical energy and momentum conserving schemes, thus leading to systems which are unconditionally stable for Hamiltonian (elastic-rigid) systems. Sample problems illustrate the performance and conservation properties.

1. Introduction Many situations are encountered where treatment of the entire system as deformable bodies is neither necessary nor practical. For example, the frontal impact of a vehicle against a barrier requires a detailed modeling of the front part of the vehicle but the primary function of the engine and the rear part is to provide inertia – deformation being negligible for purposes of modeling the frontal impact. In such situations it is efficient to model the problem with fully rigid parts that are included only to preserve the inertial properties. The literature on rigid body analysis is extensive with many books devoted to the subject (e.g., see references [1] and [2]) In addition there are many papers devoted exclusively to this subject and here, for example, we refer the reader to the published papers for additional details on numerical methods and formulations beyond those covered here [3]- [4]. In this work we illustrate how rigid-body behavior can be described and combined in a finite element system. Two approaches are introduced, the first being one in which the motion of the rigid body is described by a translation and an orthogonal tensor representing rigid rotations. In the second approach we describe a form which can be expressed entirely in terms of translation parameters. In the work presented here it is assumed that a complete description of the system is available in the form of a finite element model. The construction of a rigid-flexible system is then defined by suitable constraints on parts of the system. The constraints to be introduced naturally group onto three distinct parts: (1) Constraint on a group of elements to impose a rigid body motion only; (2) constraints between rigid and flexible elements along a surface between the parts; (3) joint constraints to impose limitations on the motions which can occur between two or more rigid or flexible bodies.

φ(X,t)

Ω

Ω

0

x X X2,x2

X ,x 1

1

Figure 1. Reference and deformed configurations for finite deformation problems

The final form for a rigid-flexible system defined from a finite element model is a set of ordinary differential equations subject to a set of algebraic constraints. In the present work these are solved using an energy-momentum conserving scheme originally proposed by Simo and co-workers [5]- [6] and extended to treat rigid-flexible bodies by Chen and Taylor [7],[8].

2. Equations for rigid bodies In this paper points in the deformed position of a body are denoted by x and are defined in terms of points in the a fixed inertial reference system X through the mapping (see Fig. 1)

x = (X; t)

(1)

The material velocity is given as the time derivative of the motion

where

V = x_ = _ (X; t)

(2)

x_ = @@tx = @@t

(3)

x=X+U

(4)

denotes partial differentiation with respect to time with X held fixed. The position in the deformed configuration may be written in terms of a displacement 1 U as The change in shape of a solid body undergoing the motion  may be described in terms of the deformation gradient

 @U F = @@X =1+ @X

(5)

E = 12 FT F 1

(6)

in which 1 is a second order identity tensor. For a deformable body the Green-Lagrange strain tensor is then expressed as


and enters constitutive equations to define the second Piola-Kirchhoff stress, S. 1

A slight abuse of notation is used since normally a shifter is introduced to transform components between reference and current states. For simplicity, however, the shifter is omitted here.

2.1. Weak form for rigid body motion The weak form of the balance of momentum equation for a solid body undergoing large motions and deformations may be written in a reference configuration form as

@ Æ = @t

Z



Æ U 0 V d + T



Z

Æ ET S d Æ ext = 0

(7)

in which 0 is the mass density at the reference state, Æ (
) denotes a variation (or virtual) quantity, and ext is the effect of external loading. For conservative loading the external term may be written as

Æ ext =

Z

ÆU

T

bv d +

Z t

 d

Æ UT T

(8)

 a specified boundary traction on the in which bv is a body force per unit of volume and T part t . Points on u ( u [ t  ) are assumed to have specified position or displacement  or x = x . given by U = U For a rigid body deformations are ignored and the balance of momentum becomes

@ Æ = @t

Z



Æ U 0 V d

T



Æ ext = 0

(9)

subject to the constraint

E=0

(10)

2.2. Rigid bodies using translation and rotation parameters Traditionally, the constraint given by Eq. (10) is satisfied by representing points in the current configuration by the mapping [9]

x = r(t) + (t) (X R)

(11)

in which R is a position of a point in the reference body with respect to the fixed inertial frame and r is the position of the same point in the current configuration. In addition,  is an orthogonal second order tensor which describes the orientation of the rigid body relative to the fixed inertial frame (see Fig. 2). Upon noting Eq. (4) we have Æ x = Æ U. Using Eq. (11) we obtain the velocity

[ [

V = r_ + !b  (X R) = r_ (x r) ! (12) and, upon noting Eq. (4) we have Æ x = Æ U, the variation Æ U = Æ r + Æ b (X R) = Æ r (x r) Æ  (13) in which (b
) denotes a skew symmetric matrix, ! is the spatial angular velocity and Æ  is an arbitrary angular variation vector. Equation (13) introduces both the skew symmetric form of the rotation and the rotation vector which have the representations 2 b =4 !

0 !3 !2

!3 0 !1

!2 !1 0

3

2

5

and ! = 4

!1 !2 !3

3 5

(14)

Λ(t)

Current configuration

r(t) Λ0= I

R

Y,y

Reference configuration

X,x

Figure 2. Motion for rigid body in terms of translation and rotation

Assuming that body now becomes

R is placed at the center of mass, the momentum equation for a rigid

@ Æ = @t

[r x[r

Z h

Æ rT 0 r_ + Æ T (x



)T (

i

)! d

Æ ext = 0

(15)

Defining the total mass of the body as

m=

I=





0 d

(16)

[r x[r

and the spatial inertia tensor as Z

Z

0 (x

)T (

) d

(17)

Eq. (15) simplifies to

Æ  = Æ rT m r + Æ T

@ (I !) Æ ext = 0 @t

(18)

The expression for computing the spatial inertia tensor may be written as

I=

Z





0 (yT y) I

y yT



d where

y=x r

(19)

This also may be expressed in terms of the reference configuration inertia tensor J as

I =

T

Z



(Y

T

Y) I Y Y

T



d

 = T J 

(20)

where Y = X R. From the above we obtain the two Euler equations

dp d = fr and = mr (21) dt dt = m r_ is the translational momentum,  = I ! is the angular momentum, f r is

where p the resultant force on the body, and

mr is the resultant couple for an individual rigid body.

Integration of this form of the rigid body equations has been considered by Simo and Wong [5] and also used in the context of rigid-flexible systems by Chen [7]. We note that in the absence of external loadings that

p = Constant

and 

= Constant

(22)

and, thus, defines conservation of momentum properties for a rigid body. Conservation of kinetic energy requires

KE =

 1 T m r_ r_ + !T I ! = Constant 2

(23)

The above are generalized to multiple bodies in standard ways [7]. In developing discrete models for rigid-flexible bodies in terms of translations and rotations it is necessary to find consistent numerical time integration procedures. Failure to have consistency in discrete forms can lead to anomalous results as shown by Jelenic and Crisfield for rods [10]. One way to avoid such difficulties is to construct a formulation in terms of translation parameters only. For rigid body dynamics the enforcement of the constraint given by Eq. (10) can be achieved discretely [2], [4]. 2.3. Rigid bodies using translation parameters only If we initially represent the rigid body in terms of a simplex finite element (i.e., a 3node triangle in two dimensions or a 4-node tetrahedron in three dimensions) which is given by the isoparametric representation in terms of natural coordinates L  as [11], [12], [2]

X=

Nel X =1

  and L X

x=

Nel X =1

L x 

(24)

  and x  are positions of nodes in the reference and deformed coordinates, rein which X spectively, and Nel is the number of nodes required to describe the simplex (i.e., 3 or 4). The natural coordinates in this form are not all independent and are related through the constraint [11] Nel X =1

L = 1

(25)

A rigid body form is achieved by constraining the length of the simplex edges to retain their original length during all motions. Thus, the problem reduces to imposing a set of constraints i 1h T T     Ck = (x x  ) (x x  ) (X X ) (X X ) = 0 (26) 2 in which the points  and  represent the end points of an edge of the simplex element and k is the edge number. Application of the constraints ensures that E is zero throughout the

entire rigid body and, thus, Eq. (24) now describes a rigid body motion. For each rigid body we may now write the equations of motion in their weak form as

Æ =

Z

Æ UT 0 V_ d +

X k

Æk Ck +

X

where k is a Lagrange multiplier and Æk its variation.

k

ÆCk k

Æ ext = 0

(27)

The momentum for each master node  on the simplex of a rigid body a is given by

p (a) = M( a) V (a)

(28)

Conservation of the total linear momentum thus requires " X X a



p (a)

#

= Constant

(29)

similarly, the conservation of angular momentum requires X

x (a)  p (a)

a



= Constant

(30)

The kinetic energy for a system of rigid bodies is given by

KE =

1 2

X a

 (a) )T p (a) = Constant (V

(31)

We note that each of these forms is identical to results for a typical finite element system if we substitute nodal quantities of the finite element in place of the nodal quantities of the master simplex element.

3. Discrete equations for rigid bodies A time discretized system for the rigid body may now be constructed. In the present work we use an energy-momentum conserving scheme as introduced by Simo and Wong [5] and used successfully for rigid body analyses in terms of the constraint form Eq. (26) by Taylor and Chen [8] and Chen [7]. For each rigid body we may write the discrete weak form as

Æ =

Z



Æ UT 0 V_ (n+1=2) d +

X k

Æk Ck(n+1) +

X k

ÆCk(n+1=2) (kn+1)

n+1=2) Æ (ext =0

(32) where (
)(n) represents the discrete value of the quantity evaluated at time t n . The variation of the constraint Eq. (26) is given by

 ÆCk(n+1=2) = (Æ U

  )T (x(n+1=2) ÆU

x(n+1=2) )

(33)

In the above we note that the constraints are satisfied at the discrete times t n+1 whereas other quantities are evaluated at the mid-time interval t n+1=2 . The proof of satisfaction of momentum and energy conservation using this structure is given by Chen [7]. Introducing now the isoparametric expansions given by Eq. (24) and noting that nodal displacements on the simplex are defined by

x = X  + U  we can write the discrete form as

 T ÆU +

Z

X

k

(34)



(n+1=2) X _ L 0 L d V + Æk Ck(n+1)

 (Æ U

  )T (x(n+1=2) ÆU

k (  n+1=2) ) (kn+1)

x

 T F(n+1=2) = ÆU

(35)

in which F are the discrete forces from body and external loading effects. The discrete form of the momentum equations are now given by (n+1=2)

(n+1=2)

M V_

+ G(kn+1=2) (kn+1) = F(n+1=2)

(36)

Ck (U(n+1) ) = 0

(37)

and the constraint equations by

The mass matrix is computed from

M =

Z

L 0 L d I

(38)

with the constraint Ck given by Eq. (26) and the variation by

 T G(kn+1=2) = (Æ U  ÆCk = Æ U

  )T (x(n+1=2) ÆU

x (n+1=2) )

(39)

for each constraint. A Newton-Raphson scheme may be adopted to solve these nonlinear equations. Linearizing the momentum equation we obtain (n+1=2)

 ( n+1=2) + G(kn+1=2) d(kn+1) = R(n+1=2) M dV_ + K dU (40) _ and d denote incremental quantities, K is a geometric stiffness arising where dU, dV from the linearization of the variation of the constraints that may be expressed as

 T K dU  ( n+1=2) = k (Æ U  ÆU

  )T ( dU  (n+1=2) ÆU

 (n+1=2) ) dU

(41)

and (n+1=2)

R = F(n+1=2) M V_

G(kn+1=2) (kn+1)

(42)

is a residual for each iteration of the solution process. Similarly, the linearization of constraints gives the relations

 ( n+1) = rk(n+1) (G( kn+1) )T dU

(43)

where rk is a residual for the constraint equation (37). We next introduce a discrete time solution process and show that increments at time tn+1=2 and tn+1 are linearly related. Also, the increment in the velocity rate may be expressed in terms of an increment of the displacements. Accordingly, in this case we obtain a set of equations which may be expressed entirely in increments of displacements and Lagrange multipliers at tn+1 . 3.1. Time discrete solutions – energy-momentum conservation A time discrete form is introduced in which the conservation of linear and angular momentum is ensured. In addition, when combined with flexible bodies with fully elastic behavior, energy is conserved. The time integration procedure to be used for the rigid body is based on the energy-momentum conserving schemes used successfully to integrate rigid body motions [5], flexible systems composed of solids and rods [13], [14], [15], [16], [17], [6] and

combined rigid-flexible systems [8], [7], [4]. The discretization proceeds by approximating the translational acceleration by (n+1=2) 1 V_ =
t

where
t = tn+1



V (n+1)

V (n)



(44)

tn . The displacement is then advanced using the average velocity as   U (n+1) = U (n) + 12
t V n+1 + V (n) (45)

Thus, the deformed configuration is given in terms of nodal quantities on the simplex as

x (n+1) = X + U (n+1)

(46)

The mid-point position is defined by interpolation as

x (n+1=2) = 12 x (n) + x(n+1)




(47)

Using the above formulae, the residual for the discrete momentum equation may be written as

R(n+1=2) = F(n+1=2)

 1  (n+1) M  V
t

V (n)



G(kn+1=2) (kn+1) = 0

(48)

and the residual for the constraint equation as

rk = Ck (U(n+1) ) = 0

(49)

Using Eqs (44) to (47) the increments in the displacement may be related to the increment of the velocity by

 (n+1) = 1
t dV  (n+1) dx (n+1) = dU 2

(50)

and the mid-point solution values by

1  (n+1=2) = 1 dU  (n+1) dx (n+1=2) = dx (n+1) and dU 2 2

(51)

The linearization of Eqs (48) and (49) may now be written as 2 "

2
t2

M + 12 K G(kn+1=2) (G( ln+1) )T 0

#(

 ( n+1) dU d(kn+1)

)

(

=

R(n+1=2) rl(n+1)

)

(52)

The solution is then advanced using (

U ( n+1)

d(kn+1)

)

(

U ( n+1) (kn+1)

)

(

+

 ( n+1) dU d(kn+1)

)

(53)

and iteration continues until the residuals are zero to within a specified tolerance. As is common in energy-momentum algorithms, the above set of equations is unsymmetric. The asymmetry appears only through the manner in which the rigid constraint C k must be introduced in order to conserve energy. In the case of individual rigid bodies this fact 2

Note that for simplicity in notation we do not include an iteration index.

is not too severe since in two dimensions the M  and K arrays are given as 6  6 matrices and Gk is a 6  3 matrix (in three dimensions these matrices are 12  12 and 12  6, respectively). However, when combined with hyperelastic flexible bodies which are also treated by an energy-momentum conserving scheme further asymmetry will exist with respect to the tangent stiffness part of the solid bodies which now is much larger in general [13], [14], [18], [15]. A perturbed Lagrangian scheme may be introduced to eliminate the Lagrange multipliers locally for each rigid body. Accordingly, we then modify Eq. (52) to read "

2
t2

M + 12 K G(kn+1=2) 1 (G( ln+1) )T  Ælk

#(

 ( n+1) dU d(kn+1)

)

(

=

R(n+1=2) rl(n+1)

)

(54)

in which Ælk is a Kronecker delta and  is an augmented Lagrangian ‘penalty’ parameter. We then perform static condensation [19] using the second row to obtain the form h

2
t2

i

M + 21 K +  G(kn+1=2) (G( kn+1) )T

 ( n+1) = R(n+1=2) + (G( ln+1) )T rl(n+1) dU (55)

After convergence of the Newton-Raphson algorithm each Lagrange multiplier may be updated using the simple Uzawa formula [20], [12]

k

 ( n+1) k +  (G( kn+1) )T dU

(56)

and another pass through the iteration steps performed. This involves an extra iteration loop which may be used to satisfy the rigid constraints to any specified tolerance. An asymmetry of the tangent matrix is still encountered as indicated by the coefficient matrix in Eq. (55) but final matrices involved are of size M only. In results presented later the Lagrange multiplier form is used with a full Newton scheme. A solution tolerance on energy is set at half-machine precision (e.g., 10 16 ) [11], [12]. 4. Construction of the mass for rigid bodies In Section 3. we expressed the mass for the rigid body as

M =

Z

L 0 L d I

(57)

where L are the natural coordinates for the simplex master element. Here, we consider the case where a rigid body is described in terms of a finite element representation and is to be constrained to the simplex as shown in Figure 3. We assume that (e) the mass matrix is available for each finite element and is given by m  in which  and  denote the node numbers in the finite element mesh. Accordingly, we can relate the inertial forces for each form as

 T M x = ÆU

X e

~ T m(e) x~  ÆU

(58)

to complete the transformation it only remains to express the virtual displacements and accelerations of a node in terms of those on the simplex. Using Eqs (24) and (34) we obtain

~  = L Æ U   and x~  = L x  ÆU

(59)

Finite Element Mesh

Rigid Triangle

Figure 3. Finite element representation of a rigid body

in which L denotes the natural coordinates for simplex node  evaluated at the finite element node . These are determined by solving the linear equations relating the cartesian coordinates to their natural coordinates [e.g., Eq. (24)] and considering the constraint given by Eq. (25). For example, in two dimensions we solve the set of equations for node  which are given by 8 < :

1 ~ (X1 ) (X~ 2 )

9 = ;

2

=4

1 1 1    (X1 )1 (X1 )2 (X1 )3 (X 2 )1 (X 2 )2 (X 2 )3

38 < 5 :

L1 L2 L3

9 = ;

(60)

~ 1 ) and (X~ 2 ) denote the cartesian coordinates of node  and (X 1 )1 , (X 2 )1 , In the above (X etc. the cartesian coordinates of the simplex used to describe the motion of the rigid body. Introducing the solution for the natural coordinates and using Eqs (58) and (59) we can express the mass [using Eq. (57)] as

M =

X e

L m(e) L

(61)

in which summation over repeated indices is implied.

5. Flexible elements – no rotation parameters 5.1. Continuum elements – energy-momentum conserving Continuum elements which satisfy discrete energy-momentum conservation for hyperelastic formulations may be developed from Eq. (7). Here, we restrict attention to the case of a St. Venant-Kirchhoff material where, for simplicity, the stress-strain behavior is given in indicial form by

SIJ = CIJKL EKL

(62)

in which CIJKL are the elastic moduli. Generalizations to the development may be constructed by following the procedures summarized by Gonzalez [6]. The energy-momentum algorithm for a flexible body may be expressed as

1 Æ UT M
t

h

V

(n+1)

V

(n)

i

+

Z

Æ E(n+1=2) S(n+1=2) d = Æ UT F(n+1=2)

(63)

where, in index form (n+1=2) ÆEIJ = ÆFiI FiJ(n+1=2) + ÆFiJ FiI(n+1=2)

(64)

and (n+1=2) SIJ

= CIJKL



 1  (n) (n+1) E + EKL 2 KL



(65)

The development of a finite element from this form may now be constructed by introducing isoparametric interpolations for the position and displacements as

X=

Nel X 

~  and N () X

U

(n)

=

Nel X 

~ (n) N () U

(66)

~ , U ~  are element nodal in which  are natural coordinates, N are shape functions, and X quantities. A typical 4-node element of rectangular form is shown in Fig. 4. (n)

y (u1,u2) T

x

(u1,u2) H Figure 4. Solid element: + denotes Gauss point locations

5.2. Structural elements – energy-momentum conserving The above form is classical as far as the finite element treatment and energy-momentum conservation is concerned. However, the development of structural element types (e.g., rods or shells) in which no rotation parameters exist is not typical. Formulations for small deformation behavior of thin plates in which no rotation parameters exist was developed by Nay and Utku in 1972 [21]. More recently O˜nate and co-workers have developed thin plate and shell elements which have only translation degrees of freedom [22], [23], [24]. Such forms can be extended to large displacement-small strain applications using updated or co-rotational forms. Also recently, the development of shell elements which have no rotation parameters has been presented by Ramm and co-workers [25], [26] and Betsch and Stein [27]. These latter works are valid for both small and large displacement formulations. Here we adopt a simple form of that used for large deformation of shells to analyze twodimensional rod (beam) applications. In the form adopted, the nodal displacements for the isoparametric 4-node element described above are transformed to mid-surface displacements and top surface incremental displacements as shown in Fig. 5 In addition, it is well known that the ‘two-node’ beam formulation will experience shear locking unless reduced integration is used to compute the stiffness matrix. Accordingly, as also indicated in Fig. 5 we also use a reduced integration along the beam axis direction

y (∆u1,∆u2) x

T

(u1,u2) H

Figure 5. Beam element without rotation parameters: + denotes Gauss point location

to compute the stress residual and stiffness terms. This form permits the consideration of general constitutive behavior without further modification only if no coupling exists between the beam direction normal stresses and the through thickness normal stress. Accordingly, we consider only forms for which CIJKL is diagonal (e.g., Poisson ratio is zero). Enhanced strain forms can be introduced to remove this restriction [25], [27]. The above form is very easy to implement and leads to good performance for the problems considered in this study.

6. Rigid-flexible interface and joint descriptions The analysis of systems composed of rigid and flexible parts requires the treatment of two connection processes. In the most basic form a rigid body may be ‘bonded’ directly to a flexible body along an interface as shown in Fig. 6. Alternatively, we have situations in which two rigid bodies are constrained to move relative to one another in a specific manner. This latter behavior requires introduction of constraint conditions which are commonly referred to as joints. Here we describe briefly the treatment of both types of constrained conditions.

~ XJ

Xα

Figure 6. Rigid-flexible interface treatment

6.1. Rigid-flexible interface treatment The motion of nodes on the flexible body which lie on the interface of a rigid body, must satisfy the motion constraint given by Eq. (24). We assume that the nodes on the rigid body satisfy the set of constraints given by Eq. (26) and, thus, these nodes have the behavior

x~  =

X

U~  =

X



L x 

(67)

 L U

(68)

or 

with a similar computation procedure as defined in Eq. (60). Using these relations the residual and tangent matrices for a flexible element node on a rigid interface is transformed to be expressed in terms of the degrees of freedom on the master element. In this set of transformations one need not be concerned with the rigid body constraints as they only serve to   , etc. are indeed moving as a rigid body. The equations to be ensure that the quantities x transformed may be expressed as

~ T K? dU ~  = ÆU ~ T R ÆU

(69)

in which R is the element residual for node  and K? is the total tangent matrix from the flexible element (i.e., it includes inertia, internal force and geometric stiffness parts). For the energy-momentum method this matrix has the general form

K

? 

 2 (e) 1  (e) ( e ) = 2 m + k M + k G
t 2

(70)

with m(e) and k(e) denoting the element mass and tangent stiffness quantities, respectively. Accordingly, for nodes on the rigid interface we perform the transformation as 0

 T ÆU

@

X flex

~ + L K? dU

1

X rigid

 A = ÆU  T (L R ) L K? L dU

(71)

similarly for each flexible node on which a node is attached to a rigid body we transform as 0

~ T ÆU

@

X flex

K? dU~  +

X rigid

1

K? L dU A = ÆU~ T R

(72)

Thus, in matrix form, we have the transformed equations 

L K? L L K? K? L K?



 dU ~ dU



=



L R

R



(73)

in which the residual in the first row is added to the residual of the rigid body master node and the first term in the first row of the tangent matrix is added to the appropriate term in the rigid body tangent matrix. This step follow exactly the steps performed to combine flexible elements with rigid bodies described by the rigid motion Eq. (10) [see Taylor and Chen [8] or Chen [7]].

6.2. Multi-body coupling by joints Often it is desirable to have two (or more) rigid bodies connected in some specified manner. For example, in Fig. 7 we show a disk connected to an arm. Both are treated as rigid bodies but it is desired to have the disk connected to the arm in such a way that it can rotate freely about the axis normal to the page. This type of motion is characteristic of many rotating machine parts and it, as well as many other types of ‘connections’ are encountered in the study of rigid body motions [3], [28]. An interconnection of this type is commonly referred to as a joint. In quite general terms joints may be constructed by a combination of two types of simple constraints: translational constraints and rotational constraints. Combinations of these two constraint types may be used to describe a general library of constraints [29]. 6.2.1. Translation constraints The simplest type of joint is a spherical connection in which one body may freely rotate around the other but relative translation is prevented. Such a situation is shown in Fig. 7 where it is evident the spinning disk must stay attached to the rigid arm at its axle. Thus the disk can not translate relative to the arm in any direction (additional constraints are necessary to ensure it only rotates about the one axis – these are discussed in Section 6.2.2.). If full translation constraint is imposed a simple relation may be introduced as

Cj = x(ja) x(jb) = 0

(74)

where a and b denote two rigid bodies. Thus, addition of the Lagrange multiplier constraint

Æ j = Æ Tj

h

x

(a) j

x

(b) j

i

h

+ Æx

(a) j

i (b) T j

Æx

j

(75)

imposes a spherical joint condition. It is only necessary to define the location for the spherical joint in the reference configuration. Denoting this as X j (which is common to the two bodies) and introducing the rigid motion yields a constraint in terms of the rigid body positions as

Æ j = Æ Tj = Æ Tj

h

i

h

X(ja) + U(ja) X(jb) U(jb) + ÆU(ja) h i h iT U(ja) U(jb) + ÆU(ja) ÆU(jb) j

Æ U(jb)

iT

j (76)

The variation and subsequent linearization of this relation yields the contribution to the residual and tangent matrix for each body, respectively. This is easily performed using relations given above. ω

Figure 7. Spinning disk constrained by a joint to a rigid arm

If the translation constraint is restricted to be in one direction with respect to, say, body a it is necessary to track this direction and write the constraint accordingly. To accomplish this the specific direction of the body a in the reference configuration is required. This may be computed by defining two points in space X 1 and X2 from which a unit vector V is defined by 2 X1 V= jX X2 X1 j

(77)

v = j xx2 xx1 j 2 1

(78)

xi = Xi + Ui

(79)

The direction of this vector in the current configuration, v, may be obtained using the displacements of the points. Accordingly,

where

A constraint can now be introduced into the variational problem as

Æ j = Æj vT



x(ja) x(jb)



h

+ Æ vT



x(ja) x(jb)



+ vT



Æ x(ja)

Æ x(jb)

i

j

(80)

where, due to the fact there is only a single constraint direction the Lagrange multiplier is a scalar j and, again, Xj denotes the reference position where the constraint is imposed. The above constraints also may be imposed using a penalty function. The most direct form is to perturb each Lagrange multiplier form by a penalty term. Accordingly, for each constraint we write the variational problem as 

Æ j = Æ j Cj

1 2  2k j



(81)

where it is immediately obvious that the limit penalty parameter k ! 1 yields exact satisfaction of the constraint. Use of a large k and performing the variation with respect to  j gives 

Æj Cj



1  =0 k j

(82)

which may be easily solved for the Lagrange multiplier as

j = k Cj

(83)

When this ‘multiplier’ is substituted back into Eq. (81) we recover the classical penalty form

Æ j = ÆCj k Cj

(84)

An augmented Lagrangian form is also possible following the procedures summarized above for treating the constraint on rigid body motion (see also [11]). Implementation of the constraints is straight forward and follows the procedures introduced above to express points in terms of the degrees of freedom of the master simplex   ). If the constraint point is chosen at a node of the master element considelement (e.g., U   and U   may erable simplification occurs and direct substitution of the nodal quantities X be made. Thus, for example to introduce a spherical joint between two rigid bodies it is only necessary to place a node of the simplex for each body at the joint location and to assign to this node the same global node number in an analogous manner to connecting two normal finite elements at a node. This flexibility is not available when using traditional translation (r) and rotation parameters () for rigid bodies, thus providing an advantage for the rotation free formulation.

6.2.2. Rotation constraints A second kind of constraint that needs to be considered relates to rotations. We have already observed in Fig. 7 that the disk is free to rotate around only one axis. Accordingly, constraints must be imposed which limit this type of motion. This may be accomplished by constructing an orthogonal set of vectors V I in the reference configuration as described by Eqs (77) to (79) and tracking their orientation in the deformed body. A rotational constraint which imposes that axis i of body a remain perpendicular to axis j of body b may then be written as

(vi(a) )T vj(b) = ViT Vj = 0

(85)

More general constraints on rotation may be imposed by satisfying orthogonality conditions between the two rigid bodies. These constraints (assuming that V (a) = V(b) in the reference state) are given by conditions

Ck = (vi(a) )T vi(b) = 1 or Ck = (vi(a) )T vj(b) = 0

(86)

Indeed, it is not necessary to use unit vectors if Eq. (86) is modified to

Ck = (vi(a) )T vi(b) = VjT Vj

(87)

again assuming that the Vj reference vectors are equal in the two bodies. 6.2.3. Library of joints Translational and rotational constraints may be combined in many forms to develop different types of constraints between rigid bodies. For the development it is necessary to have only the three types of constraints described above. Namely, the spherical joint, a single translational constraint, and a single rotational constraint. Once these are available it is possible to combine them to form classical joint constraints and here the reader is referred to the literature for the many kinds commonly encountered [1], [3], [30], [7]. The only situation that requires special mention is the case when a series of rigid bodies is connected together to form a closed loop. In this case the method given above can lead to situations in which some of the joint equations are redundant. Using Lagrange multipliers this implies the resulting tangent matrix will be singular and, thus, one cannot obtain a unique solution. Here the penalty method provides a viable method to circumvent this problem. The penalty method introduces elastic deformation in the joints and in this way removes the singular problem. If necessary an augmented Lagrangian method can be used to keep the deformation in the joint loop to within required small tolerances. An alternative to this is to extract the closed loop rigid equations from the problem and use singular valued decomposition [31] to identify the redundant equations. These may then be removed by constructing a pseudo-inverse for the tangent matrix of the closed loop. This method has been used successfully by Chen to solve single loop problems [7].

7. Example solutions 7.1. Circular disk with flexible beam As an example for a coupled rigid-flexible system we consider the problem of a circular disk with an attached flexible beam. The configuration of the system in its initial state is shown in Fig. 8.

3 4 1

2 5 6 7 8 9 10 11 12 13 14

Figure 8. Circular disk with flexible beam attached. Initial state. 8 7

Angular momentum

6 5 4 3 2 1 0 0

20

40

60

80

100

Time − t

Figure 9. Circular disk with flexible beam attached. Angular momentum.

The radius of the disk is 3 units and the master triangle is embedded as shown. The beam is attached to the right in an initial horizontal position as shown, and has length 10 units and depth 0.1 units and, thus, the beam is quite thin. Each beam is modeled by an element in which the degrees of freedom are displacements at the centroid and relative displacements on a normal director as described in Section 5. Thus, this formulation also employs only translational displacements and their increments. The density of the rigid body is 0.2 units and that for the beam is 2.0 units. The beam is modeled using a St. Venant-Kirchhoff material with CIJKL expressed by a modulus, E , of 100,000 and a Poisson ratio,  , of 0. The body is completely unrestrained and is set in motion by a pair of equal and opposite forces applying a pulse over the first 4 units of time, after which the body is in free motion and thus should then conserve both momentum and energy.

0.8 0.7 0.6

Energy

0.5 0.4 Total Kinetic Stored

0.3 0.2 0.1 0 0

20

40

60

80

100

Time − t

Figure 10. Circular disk with flexible beam attached. Energy behavior.

4

Displacement − v

2

0

−2

−4 Center Right Top

−6

−8 0

20

40

60

80

100

Time − t

Figure 11. Circular disk with flexible beam attached. Vertical displacement at nodes of simplex.

A discrete solution is constructed using the energy-momentum conserving algorithm described above with a uniform time step size of 0.5 units. A plot of the angular momentum of the system is shown in Fig. 9 and is clearly conserved throughout the free motion. In Fig. 10 a plot of the kinetic energy, stored energy in the flexible beam, and total energy is presented. Again, it is evident that the total energy is conserved during free motion. In Fig. 11 we show the behavior of the vertical displacement of each of the nodes used to describe the master simplex element. In particular, it is noted that the vibration of the beam causes an oscillation of the center of the rigid disk. Finally, in Fig. 12 we show a cartoon of the deformed positions for several of the times during the first two revolutions in the solution. Here it is seen that the beam undergoes large oscillations while rotating about the rigid disk. 7.2. Two blocks with revolute joint The behavior of multiple rigid bodies requires the introduction of interconnections, generally called joints, to model the desired interactions. In the case of bodies modeled by the parent simplex use of such joints can be partially avoided by placing the nodes of the

(a) Initial state: t = 0

(d) t = 34

(b) t = 14

(e) t = 54

(c) t = 24

(f) t = 64

Figure 12. Circular disk with attached beam. Sequence of deformed states.

3

1

2

4

5

Figure 13. Two square blocks. Connection by revolute joint at node 2.

simplex at locations where a joint is needed. To demonstrate such situations we consider the simple problem of two square blocks which are interconnected by a revolute joint at the common corner as indicated in Fig. 13 In the case of a two-dimensional problem such revolute joint also defines a spherical joint since the blocks cannot displace out of plane. By placing a node at the location of the ‘joint’ the interconnection may be accomplished without addition of constraints – one merely assigns the same node number to the common node, as indicated in Fig. 13. This expedient is far simpler than that needed for a formulation in terms of the center of mass motion with an added orthogonal rotation [viz.[[q. (11] and indeed it is possible to have three such revolute joints on a single rigid body in two dimensions (provided they are not all on a straight line). Of course if more are added it is then necessary to add a joint constraint set to represent the interconnection as described in Section 6.2.. Further, if other types of joints are used it is necessary to specify the required constraints explicitly. For the two block form we excite the problem by forces applied to nodes 2,3, and 5 in the same manner as in the previous two problems. The forces at nodes 3 and 5 have equal components in the positive x and y directions and node 2 has twice these forces in the negative directions. Thus, no resultant force nor couple is applied to the system. The body 123 will rotate counterclockwise and the body 245 clockwise with equal angular velocity. A cartoon of the deformed shapes at various times during the first revolution is shown in Fig. 14.

(a) Initial state: t = 0

(d) t = 7.5

(b) t = 2.5

(c) t = 5.0

(e) t = 10.0

(f) t = 12.5

Figure 14. Two blocks with revolute joint. Sequence of positions.

15

10

Displacement −v

5

Node−1 Node−2 Node−3

0

Node−4 Node−5

−5

−10

−15 0

20

40

60 Time

80

100

120

Figure 15. Two square blocks. Vertical displacement of parent simplex nodes.

A plot of the vertical displacement for each node is shown in Fig. 15 where it is evident that several revolutions have occurred with pure cyclic motions. It will be noted that the node 2 also moves in a cyclic, though quite small manner in order to conserve the momentum and energy of the total system. This is apparent from the cartoon in Fig. 14 where we include a fixed node also – the node 2 is seen to oscillate around this point.

12

Kinetic Energy

10

8

6

4

2

0 0

20

40

60 Time

80

100

120

Figure 16. Two square blocks. Kinetic energy vs. time.

Finally, a plot of the energy with time is presented in Fig. 16 where is again observed that energy conservation for all times after the loads become zero (t = 4) is obtained. 8. Closing remarks This work has presented an introduction to using finite element models in which translational quantities are used to represent the motion of rigid and/or flexible bodies. It presents an alternative to using orthogonal matrices to describe the rotational behavior of rigid and structural elements. Here each rigid body is described in terms of a master simplex element using standard finite element isoparametric procedures. Each body is then required to be rigid by constraining the edges of the simplex to maintain their original length. While the method can successfully avoid the introduction of rotational parameters and, thus, the subsequent need to find compatible time discrete integration formulas for such parameters, it introduces

the need for additional constraints to impose the rigid body motion. Here we have used Lagrange multiplier methods to include such constraints, however, we also indicated that it is possible to use penalty or augmented Lagrangian methods for this purpose. The treatment of rigid multi-body response requires introduction of joints and we have shown how such joints also can be included without need for rotational constraints. Again we use Lagrange multiplier methods to embed the joint constraints for results presented. However, we note that the inclusion of a spherical joint can proceed without the need to introduce constraints provided we place a node of each simplex master element at the constraint location and give this node the same global number. This is a distinct advantage for the rotation free formulation since up to three such points can be defined for each two-dimensional rigid body (or four points in three dimensions). The numerical examples presented in this work are restricted to two-dimensional applications for simplicity. Extension to three dimensions presents no additional difficulties – contrary to the use of rotation parameters where the difficulties are significant to define the rotational update procedure. We believe the procedures offered in the present work provide a viable method to develop combined rigid-flexible body analysis systems. One final point is to reiterate the reason for eliminating rotational parameters. Here, we are motivated by the findings of Jelenic and Crisfield who observed lack of objectivity in the discrete translational and rotational response of a three dimensional beam which was rotated through several revolutions. It is our belief that the observed response is in part a discretization error between the rotational parameter descriptions and those of the translational parameters. Use of the formulation presented here avoids this discretization error by always using translational parameters.

References [1] A.A. Shabana. Dynamics of Multibody Systems. John Wiley & Sons, New York, 1989. [2] J. Garcia de J´alon and E. Bayo. Kinematic and Dynamic Simulation of Multibody Systems. SpringerVerlag, Heidelberg, 1994. [3] D.J. Benson and J.O. Hallquist. A simple rigid body algorithm for structural dynamics programs. International Journal for Numerical Methods in Engineering, 22 (1986) 723-749. [4] J.C. Garc´ia Orden and J.M. Goicolea. Dynamic analysis of rigid and deformable multibody systems with penalty methods and energy-momentum schemes. Computer Methods in Applied Mechanics and Engineering (2000). [5] J.C. Simo and K. Wong. Unconditionally stable algorithms for rigid body dynamics that exactly conserve energy and momentum. International Journal for Numerical Methods in Engineering, 31 (1991) 19-52. [Addendum: 33:1321-1323, (1992)]. [6] O. Gonzalez. Design and analysis of conserving integrators for nonlinear Hamiltonian systems with symmetry. Ph.D thesis, Department of Mechanical Engineering, Stanford University, Stanford, California, 1996. [7] A.J. Chen. Energy-momentum conserving methods for three dimensional dynamic nonlinear multibody systems. Ph.D thesis, Department of Mechanical Engineering, Stanford University, Stanford, California, 1998. (also SUDMC Report 98-01). [8] R.L. Taylor and A.J.H. Chen. Coupling of rigid and flexible structural components. In Proc. of the First South African Conference on Mechanics, Midrand, South Africa, 1996. [9] I.H. Shames and F.A. Cozzarelli. Elastic and Inelastic Stress Analysis. Taylor & Francis, Washington, D.C., 1997. (Revised printing). [10] G. Jelenic and M.A. Crisfield. Geometrically exact 3d beam theory: implementation of a strain-invariant finite element for statics and dynamics. Computer Methods in Applied Mechanics and Engineering, 171 (1999) 141-171. [11] O.C. Zienkiewicz and R.L. Taylor. The Finite Element Method: The Basis, volume 1. ButterworthHeinemann, Oxford, 5th edition, 2000. [12] O.C. Zienkiewicz and R.L. Taylor. The Finite Element Method: Solid Mechanics, volume 2. ButterworthHeinemann, Oxford, 5th edition, 2000.

[13] J.C. Simo and N. Tarnow. The discrete energy-momentum method. conserving algorithm for nonlinear elastodynamics. Zeitschrift fu¨ r Mathematik und Physik, 43 (1992) 757-793. [14] J.C. Simo and N. Tarnow. Exact energy-momentum conserving algorithms and symplectic schemes for nonlinear dynamics. Computer Methods in Applied Mechanics and Engineering, 100 (1992) 63-116. [15] J.C. Simo and O. Gonz´alez. Recent results on the numerical integration of infinite dimensional hamiltonian systems. In Recent Developments in Finite Element Analysis. CIMNE, Barcelona, Spain, 1994. [16] J.C. Simo and N. Tarnow. A new energy and momentum conserving algorithm for the non-linear dynamics of shells. International Journal for Numerical Methods in Engineering, 37 (1994) 2527-2549. [17] J.C. Simo, N. Tarnow, and M. Doblare. Non-linear dynamics of three-dimensional rods: Exact energy and momentum conserving algorithms. International Journal for Numerical Methods in Engineering, 38 (1995) 1431-1473. [18] N. Tarnow. Energy and momentum conserving algorithms for Hamiltonian systems in the nonlinear dynamics of solids. PhD thesis, Department of Mechanical Engineering, Stanford University, Stanford, California, 1993. [19] E.L. Wilson. The static condensation algorithm. International Journal for Numerical Methods in Engineering, 8 (1974) 199-203. [20] K.J. Arrow, L. Hurwicz, and H. Uzawa. Studies in Non-Linear Programming. Stanford University Press, Stanford, CA, 1958. [21] R. A. Nay and S. Utku. An alternative for the finite element method. Variational Methods in Engineering, 1 (1972). [22] E. O˜nate and M. Cervera. Derivation of thin plate bending elements with one degree of freedom per node: A simple three node triangle. Engineering Computations, 10 (1993) 543-561. [23] E. O˜nate, F. Zarate, and F. Flores. A simple triangular element for thick and thin plate and shell analysis. International Journal for Numerical Methods in Engineering, 37 (1994) 2569-2582. [24] E. O˜nate and F. Z´arate. rotation-free triangular plate and shell elements. International Journal for Numerical Methods in Engineering, 47 (2000) 557-603. [25] M. Braun, M. Bischoff, and E. Ramm. Nonlinear shell formulations for complete three-dimensional constitutive laws include composites and laminates. Computational Mechanics, 15 (1994) 1-18. [26] M. Bischoff and E. Ramm. Solid-like shell or shell-like solid formulation? A personal view. In W. Wunderlich, editor, Proc. Eur Conf on Comp. Mech (ECCM’99 on CD-ROM), Munich, September 1999. [27] P. Betsch, F. Gruttmann, and E. Stein. A 4-node finite shell element for the implementation of general hyperelastic 3d-elasticity at finite strains. Computer Methods in Applied Mechanics and Engineering, 130 (1996) 57-79. [28] H. Goldstein. Classical Mechanics. Addison-Wesley, Reading, 2nd edition, 1980. [29] A. Ibrahimbegovic, S. Mamouri, R.L. Taylor, and A.J. Chen. Finite element method in dynamics of flexible multibody systems: Modelling of holonomic constraints and energy conserving integration schemes. Multibody System Dynamics, 4 (2000) 195-223. [30] A. Cardona, M. Geradin, and D.B. Doan. Rigid and flexible joint modelling in multibody dynamics using finite elements. Computer Methods in Applied Mechanics and Engineering, 89 (91) 395-418. [31] G.H. Golub and C.F. Van Loan. Matrix Computations. The Johns Hopkins University Press, Baltimore MD, 3rd edition, 1996.

Dynamics of Complex Flexible Multibody Systems Undergoing Large Overal Motion Adnan IBRAHIMBEGOVIC and Said MAMOURIy Ecole Normale Sup´erieure de Cachan LMT, 61 av. Du pr´esident Wilson, 94235 Cachan France  UTC, D´ept. GSM. BP 529 60205 Compi`egne, Francey Abstract. In this work we discuss the application of the finite element method to modeling of flexible multibody systems, employing geometrically exact structural elements such as beams, shells or solids. Two different approaches to handle constraints, one based on the Lagrange multiplier procedure and another based on the use of release degrees of freedom, are examined in detail. The energy conserving time stepping scheme, which is proved to be well suited for integrating stiff differential equations, governing the motion of a single flexible link is appropriately modified and extended to nonlinear dynamics of multibody systems.

1. Introduction Currently accepted trend for treating the flexible members rather than the traditional rigid links has raised the level of modeling sophistication and has widely opened the door to employing the finite element method in multibody dynamics. In that respect we depart from the previous practice(so-called floating or shadow frame of [1] or [2] e.g. see [3] and extend the nonlinear formulation of structural dynamics problems in [4] to multibody dynamics. The latter is set in a fixed inertia frame, which drastically simplifies the structure of the finiteelement-based semi-discrete equations of motion in nonlinear structural dynamics resulting with a linear form of the inertia term and nonlinear form of the internal forces. Another key ingredient for development of such an approach to nonlinear structural dynamics problems are the so-called geometrically exact formulations of nonlinear structural theories (beams, plates and shells) capable of extracting the finite strain measures from the large overall motion, regardless of the size of displacements and rotations. The first formulations of this kind where provided for beams (e.g. see [5],[6], [7]), and subsequently extended to shells with drilling rotations (e.g. see [8]) and 3d solids with independent rotation field (e.g. see [9]). All these structural elements share the same configuration space, consisting of 3d displacement vectors and 3d finite rotation tensors, so that they can easily be combined within the same model of a complex mechanical system. The main difficulty remaining in such an approach to nonlinear structural dynamics, related to a non-vectorial character of 3d finite rotations (e.g. see [10] or [11]), is nowadays well under control, both in terms of a computationally convenient choice for rotation parameters (e.g. [12] or [13]) and in terms of a number of appropriate modifications (e.g. see [14] or [15]) of the standard time-stepping schemes for structural dynamics (e.g. Newmark or Wilson  , see [16]). In this work we extend this approach to dynamics of flexible multibody systems. To that end, several novel features which have attracted the attention of the current research are thoroughly discussed :

(i) The first objective of this work is to illustrate the suitability of the finite element method for constructing the computational model of a flexible multibody system of high level of complexity. The finite elements that can be used for such a propose should share the same nodal degrees of freedom, both for displacements and rotations. A more detailed description of the elements of this kind is given by Ibrahimbogevic and AlMikdad [15] for beams, Ibrahimbegovic, Brank and Courtois [17] and Brank, Mamouri and Ibrahimbegovic [18] for shells or Ibrahimbegovic [9] for solids. (ii) The second objective we have fixed in this work is to incorporate the holonomic constraints1 in a multibody dynamics formulation relying on the geometrically exact structural theories. The constraints to be discussed encompass all the standard types of joints (e.g. revolute, prismatic, spherical, cylindrical, plane, etc.), as well as the constraints representing a rigid component in a flexible multibody system. The first approach tested is the classical one employing the method of Lagrange multipliers, which was previously proposed by Cardona et al. [19] for joint constraint and extended to rigid component constraint by Taylor and Chen [20] and Chen [21]. Development of the pertinent equations of motion of a constrained multibody system using the Lagrange multiplier method to handle constraints is quite straight-forward, and can be carried out in a very systematic manner. However, this method does not necessarily lead to the most efficient implementation since the total number of unknowns is significantly increased by the presence of Lagrange multipliers. More importantly, as noted by Cardona and Geradin [22], the set of governing finite element semi-discrete equations of motion becomes of differential/algebraic type (with zero values of mass corresponding to constraint equations), which leads to unstable behavior of standard time-stepping schemes. A more efficient approach to handling constraints can be developed, as recently proposed by [23], where nonlinear kinematic relationship is employed to remove the constrained degrees of freedom from the global set of equations. The proposed procedure applies to all holonomic constraints (constraints on the configuration space) and includes all types of joints and rigid components. (iii) Presence of constraints is not the only reason for which the standard time-stepping schemes are not well suited for flexible multibody systems, and often experience unstable behavior. The latter can also be the consequence of a large difference in stiffness of particular members of a flexible multibody system or even a large difference in stiffness of the same member in particular modes of deformations (e.g. bending vs. axial deformation), which leads to a set of so-called stiff differential equations characterized by a very large difference in maximal and minimal eigenvalues of the tangent operator. In that respect, a set of differential/algebraic equations, arising from handling a constrained multibody system by the Lagrange multiplier method, can be considered as an ultimate case of stiff system with some frequencies (the frequencies associated with zero mass) taking an infinite value. The rich mathematical literature dealing with time-integrations schemes for stiff equations (e.g. see [24]) proposes universal remedies that are often only partially satisfying, since the physical nature of a particular problem is ignored. Therefore, we prefer to follow a number of very recent works carried out in the computational mechanics community, which discovered that the robust time-stepping schemes for stiff equations in nonlinear structural dynamics can be derived by enforcing the conservation of salient motion properties, such as the total energy or angular momentum (see [25], [11], [26] , [27] and [28], among others). The 1

holonomic constraint is an equality constraint which is enforced on the configuration space

latter complements the earlier findings (e.g. see [29]) that enforcing energy conservation can also improve the performance of the standard time-stepping procedures, such as the Newmark scheme . Our third objective in this work is thus directed towards extending the domain of application of the energy-conserving methods to multibody dynamics problems. The outline of the paper is as follows. In Section 2 we briefly recall the governing equations of nonlinear dynamics of a chosen model problem, a three-dimensional geometrically exact beam. Two different approaches to modeling the constraints in a flexible multibody system, based on the Lagrange multiplier technique and direct enforcement of nonlinear kinematic constraints are discussed in Section 3. In Section 4, we present the modification of the energy-conserving scheme of [25] pertaining to multibody systems. Several numerical simulations are presented in Section 5 in order to illustrate a very satisfying performance of the presented methodology. In section 6, we state some closing remarks.

2. Geometrically exact beam undergoing 3d finite rotations In this section we briefly recall the governing equations of the chosen model problem, the geometrically exact 3d beam. The model essentially represents a convenient reparametrization of the classical beam model of [30] and [31] proposed initially by [5] for straight beams and extended by [7] to space curved beams. The initial configuration of such a model is specified by the beam axis, a smooth curve which has a plane domain A referred to as the cross section, attached at each point. The orientation of the cross section is specified by its exterior unit normal vector 1;0 s . Without loss of generality, we can choose that the cross section is initially normal to the beam axis '00 s (1) 1;0 s

t ()

t ( )= ( )

()

()= ()

@  . A convenient way to construct the base where '0 s is the position vector and  0 @s vector of this local Cartesian frame is simply by rotating the Euclidean base vectors, i , by rotation matrix 0 ; i. e. ) i;0 0 i (2) 0 i;0 i



 =t

e

e

t = e

The key kinematic hypothesis of such a model is that each section is displaced with the beam axis remaining unchanged in shape. Therefore, the base vectors i;t s still form a local Cartesian basis, which can be defined by an orthogonal matrix

t ()

t(s) = ti;t ei =) ti;t = tei

(3)

Due to shear deformation the plane section in the deformed configuration no longer remains orthogonal to the beam axis. In order to preclude extreme values of shear deformation we require that '0t s
1;t s > (4)

() t () 0

In summary, the configuration space of the geometrically exact beam consists of one parameter family of 3d vectors and orthogonal matrices,

C := f('t ; t ) : [0; L]  [0; T ] =) R3  SO(3); '0t
t e1 > 0g (5) with time t 2 [0; T ] as the evolution parameter. The central problem of computational 453 dynamics is reduced to finding the time-history of the state variables by integrating their rate equations

= vt _ t = tW^ t ; W^ tb = Wt  b ; 8b 2 R3 '_ t

(6)

(_) = ( )

@ where  @t  is the partial derivative with respect time. The computed values of state variables should satisfy the weak form of momentum balance equations, which can be written (e.g. see [5] or [7]) as

ZL

ZL

('t ; t; v; w) := 0 (v
p_ t + w
r_ t )ds + 0 f(v0 w  '0t )
nt + w0
mtgds Gext = 0 (7) where w = t W and Wt and wt are material and spatial form of angular velocity. In (7) G

above,

pt and rt are linear and angular momenta, which can be written as pt = Avt ; rt = iwt

(8)

where A is distributed mass per unit length of the reference configuration A

=

Z

A

 dA

i is the moment of inertia Z X 3 3 T i = tIt ; I = A X ij [(ei
ej )I ei ej ]dA i=2 j =2 In (7) we denoted as nt and mt the stress resultant force and moment which derive from the whereas

linear elastic constitutive law

and

C

D

nt := tNt; Nt = CEt ; Et = Tt '0 T0 '00

(9)

mt := tMt; Mt = DKt ; Kt = T 0 T0 00

(10)

t

3 3

are constant  matrices. If the local ortho-normal frame, i , coincides where and with the principal axes of the cross-section, the constitutive matrices take a simple, diagonal forms, diag EA; GA2 ; GA3 diag GJ; EI3 ; EI3 (11)

C=

); D =

(

(

)

where E is the Young modulus, G is the shear modulus, whereas A i and Ii are section shear areas and moments of inertia. It can easily be shown by using the Hamilton principle of least action that in the absence of external forces the total energy of the chosen model is conserved. To that end, first recall that in accordance with the constitutive model in (9) and (10) the strain energy can be written as a quadratic form in the chosen strain measures.

(Et; Kt) = 21 (Et
CEt + Kt
DKt)

(12)

(vt ; Wt) = 12 (vt
Avt + Wt
I Wt)

(13)

W

Second, due to choice of inertia reference frame the kinetic energy can also be written as a quadratic form in linear and angular velocities K

Finally, by selecting the total energy as the corresponding Hamiltonian H

('t ; t ) =

ZL 0

fK (vt ; wt) + W (Et; Kt)gds

(14)

It can readily be shown that its time derivative amounts to the weak form of equations of motion in (7) written for Gext . It thus follows that the Hamiltonian is conserved by the solution of (6) in that

=0

@

@t

H

() = 0

As shown subsequently, enforcing the energy conservation in the discrete approximation leads to a very robust time-stepping scheme.

3. Flexible multibody system with holonomic constraints In this section we carry out the necessary development to make the presented model of geometrically exact beam applicable to modeling of multibody systems. More precisely, we discuss the joint constraints between flexible members modeled by the geometrically exact beams, as well as the connection of the flexible members to rigid components that might be integrated in the system. Two alternative methods to account for constraints are discussed, one using the Lagrange multipliers and the other employing nonlinear kinematic relations of master-slave type. 3.1. Joint constraints and Lagrange multiplier procedure We consider a flexible member attached to linkage assembly by a joint forcing it to follow the motion of the mechanism. To make this idea more precise, the linkage assembly node is refereed to as master and denoted with superscript ”m”, whereas the corresponding flexible member node is called slave, and denoted as ”s”. In view of the configuration space in (5), the motions of the master and slave nodes are specified by their position vectors and orthogonal matrices m m m m s 'm t ; t = [t1;t ; t2;t ; t3;t ] and 't ;

st = [ts1;t; ts2;t; ts3;t] (15) s m If the slave and master node are rigidly connected, then ' st = 'm t and t = t and the

motion of the slave node is identical to the motion of the master node. However, if the slave and master node are connected by a joint constraint (e.g. revolute, prismatic, or spherical), then some motion components of the slave node need not necessarily follow the motion of the master node. Hence, these motion components have to be obtained as a part of the global solution procedure. It is important to note that by using the method of Lagrange multipliers to handle constraints each pair of slave-master nodes doubles the number of the corresponding motion components to be computed, as well as adding a parameter for the multiplier itself. The role of the joint constraint is to enforce the equality of selected components. For example in the case of revolute joint constraint (see Figure 1) we can write that

'st

'm t

t
t1;t t
ts2;t m 3;t m 3;t

s

= 0 = 0 = 0

Figure 1. Revolute constraint.

(16)

Imposing these constraints by the Lagrange multiplier procedure leads to a modified weak form in (7) which can be written as Grev

:= G + v
@t@ ('st

'm t ) + w1

@ m m s s ( t ( t 3 ;t
t1;t ) + w 3 ;t
t2;t ) = 0 @t @t @

2

(17)

where v ; w1 and w2 are the corresponding Lagrange multipliers. The latter takes a clear physical interpretation as the reactive forces and moments enforcing the constraints. Carrying on with this approach for other types of joints one can see that only two different types of elementary constraints are needed to build any standard joint constraint. These elementary constraints impose the equality of the corresponding components of translational and rotational motion, i.e s m 'si;t 'm (18) i;t i;t
i;t

=0 ; t t =0

and each one introduces one Lagrange multiplier into the weak form. A complete lists of elementary constraints that need to be imposed for severals commonly used joints is given in [23]. 3.2. Joint constraints and relative motion In order to avoid an increase in number of global variables and resulting increase in computational effort which are inherent to the method of Lagrange multipliers, we propose an alternative approach to imposing the master-slave joint constraints, which introduces the relative motion. For example, going back to the revolute joint one can introduce the relative rotation angle  r (see Figure 1) orienting the slave triad with respect to the master one so that we can write

tmm1;t
ts s1;t = cos tr r tm2m;t
ts1s;t = sin trr t1;t
t2;t = sin t t2;t
t2;t = cos t

(19)

In view of constraints in (16), the last expression can thus be rewritten as

'st = 'm t

0 1 0C B r r s m ; t = t exp( | {zrt )} ; t = @ 0r A



(20)

t

t

By exploiting the last result the slave motion components can be completely eliminated from the global solution procedure, and recovered subsequently once this global computation, along with the local (element based) computation of the relative motion (see [23]), is completed. Such an approach does not increase the number of global equation and it leads to a very efficient computation. Carrying on with this type of consideration, we can find out (see [23]) that all standard types of joint constraints can be described by a generalized joint featuring the relative motion vectors r and  r such that

d

m r 'st = 'm t + t dt

d

r ; st = mt exp( | {zt )}



(21)

r t

The particular choices of r and  r for several most frequently used joints are given in [23]. Another type of kinematic constraint which we can also handle in a multibody system is the connection between a flexible and a rigid component (see [23]).

4. Energy conserving scheme Applying the finite element methods to dynamics of flexible multibody systems often leads to a set of stiff differential equations; or, when constraints are enforced by the Lagrange multiplier method, to a set of algebraic/differential equations. It has been noted in [32] that many standard implicit time-stepping schemes, might experience a loss of stability for the case when very high (or infinite) frequencies of a set of stiff equations are excited. This kind of instability might occur even for implicit time-stepping schemes, the ones which are unconditionally stable in linear analysis. The latter indicates that one ought to define the notion of stability appropriate for nonlinear analysis, which is typically related to energyconservation properties of the scheme. A number of schemes capable of enforcing the energy conservations have been proposed recently (e.g. see [25], [11], [26], [27] and [28], among others). In this work we choose the scheme proposed by [25], and devise the corresponding modification for the multibody dynamics problem on hand. 4.1. Energy conserving scheme for 3d geometrically exact beam The scheme is based on the mid-point approximation of the rate equation for state variables in (6) leading to the following relations

'n+1 where

'n = u ;

n+1 n = 12 (n+1 + n)^

(22)

u =
2t (vn+1 + vn) ;  =
2t (Wn+1 + Wn)

(23)

We note that (22b) leads to a corresponding update of large rotations featuring so-called 3 ! SO Cayley transformation (e.g. see [33], p 173), cay


[]:R (3) n+1 = ncay[] ; cay[] = [I + 21 ^ ][I 12 ^ ]

1

By exploiting the principal axis representation of the skew-symmetric matrix [12]), we can compute cay

where SO

[] = I + 1 +1  [^ + 12 ^ 2 ] 4 2

(24)

^ , (e. g., see (25)

 =k  k. It follows from the last expression that for any orthogonal matrix,  2

(3), we can state the following properties of the Cayley transform cay [] =  ; cay []T = cay []

(26)

With these results on hand we can develop an alternative form of the rotation update featuring the spatial representation of the rotation vector, # n , such that

=  n+1 = cay[#]n

and

n+1 n = 12 #^ (n+1 + n)

(27) (28)

It follows from (23b), (27) and (28) that

#

= n+1  n =
2t (wn+1 + cay[#]wn)

(29)

By making use of definition in (23a) and (29), we can write the mid-point approximation of the increment of kinetic energy in (13) as


K =
1t fu
(pn+1 pn ) + #
(n+1

 n )g = Kn+1

Kn

(30)

Similarly, the mid-point approximation for the increment of the strain energy can be written as


W = f(u0

#  '0n+ 1 )
n + #0
mg = (En+1 2

N= n M=

m



En)
N + (Kn+1 Kn)
M (31) = det(n+1 )n+T . The last result can be

1 and ? 1 ? 1 ; where 1 1 n+ 12 n+ 2 n+ 21 2 2 derived from (9) and (10) by exploiting the results in (23a) and (29) to get

En+1 En = Tn+1'0n+1 Tn '0n = Tn+ (u0

#  '0n+ 1 )

1 2

(32)

2

K^ n+1 K^ n = Tn+10n+1T Tn 0nT == Tn+ #^0n+ (33) ^ = FT a^F =) A = det(F)F T a, For any invertible matrix F and a; A 2 R3 , it holds that A | {z } and

1 2

1 2

F?

which implies that the last result can also be rewritten as

Kn+1 Kn = ?n+

1 2

#

(34)

If the constitutive equations in (9) and (10) are replaced with so-called algorithmic constitutive equations

N = C 21 (En+1 + En) ; M = D 12 (Kn+1 + Kn)

(35)

the increment of the strain energy in (31) can be rewritten as


W = 12 (En+1 En)
C(En+1 + En) + 21 (Kn+1 Kn)
D(Kn+1 + Kn) = Wn+1 Wn

(36)

From the results obtained in (30) and (36), we can see that the discrete approximation of the problem also ensured the energy conservation in the absence of external forces, in that Hn+1

= Hn

(37)

4.2. Energy conservation for multibody system with Lagrange multipliers If the joint constraints are imposed by the Lagrange multiplier procedure, the governing set of differential equations is accompanied by the corresponding set of algebraic equations. We will keep the same notation as in the previous section for displacement and rotations of s m two nodes related by constraints as ' st ; 'm t and t ; t , respectively, although in this case these nodes should not be referred to as the slave and the master, since they both feature in the list of unknowns at the global level. This list also contains the corresponding number of Lagrange multipliers, i.e the reactive forces enforcing the constraints. For example for a flexible multibody system with a revolute joint constraint described in Section 3.1, the weak form of the differential equation of motion can be written as

 

('t ; t; v; w)+ v
('_ st '_ mt )+ w (t_ m3;t
ts1;t + tm3;t
t_ s1;t)+ w (t_ m3;t
ts2;t + tm3;t
t_ s2;t) = 0

G

1

2

(38)

()

where G
is the weak form of the flexible components. The algebraic constraint equations are given as s m s v
'st 'm w1 m (39) t 3;t
1;t w2 3;t
2;t

(

= _

(t t ) + (t t ) = 0 = _ w . The mid-point approximation of the rate

)+

=_

where v v ; w1 w1 and w2 2 equations of the state variables, given in (23a) and (28), respectively, as well as the mid-point approximation of the weak form of the flexible part in (38), are constructed in the same way as already described for a single flexible component. For a multibody system it only remains to provide the corresponding mid-point approximation for the constraints. In particular, for the displacement constraint contribution to the weak form in (38) we can write

vnm+ ) =
1t v
(|'sn+1 {z 'mn+1} (|'sn {z'mn))}

v
(vns + 1

(40)

1 2

2

where we used the result in (23a). Furthermore using the results in (3) and (28) we can write the mid-point approximation for the rotational constraint contributions to the weak form in (38) as w1

(t_ m3;n+
ts1;n+ + tm3;n+
t_ s1;n+ ) = w (_ mn+ e3
sn+ e1 + mn+ e3
_ sn+ e1) w m m s s [(   = 2
n+1 n )e3
(n+1 + n )e1 + t (mn+1 + mn )e3
(sn+1 sn)e1) = 
wt [t| m3;n+1{z
ts1;n+1} t| m3;n{z
ts1;n}] (41) 1 2

1 2

1 2

1 2

1 2

1

1 2

1 2

1 2

1

1

We can also obtain by a similar consideration w2

(t_ m3;n+
ts2;n+ + tm3;n+
t_ s2;n+ ) = 
wt [t| m3;n+1{z
ts2;n+1} t| m3;n{z
ts2;n}] 1 2

1 2

1 2

2

1 2

(42)

We recall again that the results in (40), (41) and (42) provide the mid-point approximation to the constraint contribution to the weak form of the equations of motion for a multibody system. It is easy to see that by enforcing the constraint equations in (39) at t n+1 and tn all the contributions in (40) to (42) will vanish, thus it will not perturb the energy conservation which was already enforced for the flexible part of the system. 4.3. Energy conservation for multibody system with master-slave constraints For a multibody system with joint constraints modeled by nonlinear kinematic relationship accounting for relative motion between the master and slave nodes (see Section 3.3), the mid-point approximation given in (22a) and (22b) is used to compute the motion of each master node as well as the relative motion. In the presence of a revolute joint, the rate equations for the motion of the slave node are obtained by differentiating (21) with respect to time to get s m r m r 'st 'm (43) t t t t t t

_ = _ ; _ = _  +  _

The mid-point approximation of the rate equation in (43) then leads to

'sn+1

|

{zs

u

'sn = 'm n+1

} |

{zm

u

'm n

}

(44)

u

where m are obtained by the global mid-point method based solution procedure. It folm lows from (44) above that we can write s , if the displacement constraint in (20) are enforced both at tn+1 and tn

u =u

'sn+1 = 'm n+1

;

'sn = 'm n

(45)

By the same token, the mid-point approximation of the rate equation for the rotational motion of the slave node, leads to

sn+1 sn = (mn+1 mn)rn+ + mn+ (rn+1 rn) = mn+1rn+1 mnrn 1 2

1 2

(46)

Again, we note that the last result is true only if the rotational constraint in (20) is enforced both at tn and tn+1 , i.e.

sn+1 = mn+1rn+1 ; sn = mnrn

(47)

In view of the result in (27), it follows from (47) that cay

[#s]sn = cay[#m]mn cay[#r ] |mn{zT sn}



(48)

r n

Finally, by making use of the Cayley transform property indicated in (26), we can recast the last expression in the following form r #s = cay 1 [cay [#m ]cay [m n # ]]

(49)

Thus, the energy conservation for a multibody system with nonlinear master-slave joint constraints can be demonstrated by following the same line of arguments as for a single flexible components, leading to nnp X a=1

(

Ga 'n+ 1 ; 2

where, for any slave node, s simplify (50) accordingly.

=

n+ ; ua; #a) = Hn+1 1 2

Hn

(50)

a, relationships in (44) and (49) should be exploited to

5. Numerical Examples Two numerical simulations are presented in this section in order to illustrate a very satisfying performance of the proposed methodology for dynamics of flexible multibody systems with joint constraints and rigid links. All the computations are performed by an enhanced version of the computer program FEAP (e.g. see [35] and [16]) for the description of a basic version). 5.1. Spatial slider-crank mechanism The multibody system we study in this example (see Figure 2) is a three-dimensional version of the standard slider-crank mechanism. The system consists of two rigids links, initially placed in vertical position, joined by a flexible connecting rod, which is initially horizontal (see Figure 2).

E A=GA=1E+6 E I =E J =1E+5 Aρ =1 ; Jρ = diag(20,10,10) p=1000sin(π t) ; 0 t 2 3

3 B

1.5

A y

p

z x

Figure 2. Spatial slider-crank mechanism: mechanical and geometric characteristics.

Another flexible link is attached to the center of the connecting rod at one end and to a sliding support at another end. The mechanical and geometric properties of the flexible link 6, and connecting rod are chosen the same with axial and shear stiffness EA GA 5 and distributed mass A bending and torsional stiffness EI GJ and inertia diag ; ; . The connecting rod is attached to the rigid links by spherical joints, whereas the flexible link and the connecting rod are rigidly connected between them. In this manner the sliding motion of the flexible link can be transferred to the motion of the rigid links around their support point. In order not to prevent truely three-dimensional motion, both support points of the rigid links are provided by the spherical joints. We have first studied the motion induced by a forced sliding under a horizontally applied force F , with a given sinusoidal time variation for the first sec of motion, f t t; t 2 ; . After that time the force is removed and the system undergoes free vibrations. The finite element model used in the analysis consists of 12 two-node elements, 6 for the connecting rod and 6 for the flexible link. The constraints due to the spherical joints and rigid components are handled by master-slave logic explained in Section 3.2. The energy-conserving scheme is used to carry out the computation, with the time 2 sec: As shown in Figure 3, where we give the plot of the total energy versus step t time, the energy-conserving scheme fulfills its role in enforcing the salient feature of the corresponding continuum problem and preserving the total energy in the free-vibration phase, t > sec: In Figure 4, we present the displacement components at node A, the mid-point of

I=

(20 10 10)

= 1000 ( ) = sin [0 2]

=

= 10

= =1

2


= 10

2

Figure 3. Spatial slider-crank mechanism: Time history of the total energy.

= 10

Figure 4. Spatial slider-crank mechanism : Time history of components at A and B.

Figure 5. Spatial slider-crank mechanism: Deformed shapes at different stages of motion. the flexible link, and at node B, the point where the link is attached to the connecting rod. It can be seen that during the period of forced vibration (first 2 sec) and initial period of free vibration, the motion remains predominantly two-dimensional with essentially vanishing value of the out-of-vertical-plane displacement component. At approximately sec of motion, the out-of-plane motion components start participating in the motion, increasing their values towards the end of the time internal of interest. Namely, due to the presence of spherical joints the rigid links can change the axis of rotation, and that is precisely what happens at the later stage. The deformed shapes in Figure 5 confirm this finding. It is important to note, as indicated in Figure 3, the total energy remains conserved through all the different stages of the free-vibration phase. In the second part of the analysis we study the influence of the operating speed on the modeling assumption of the slider-crank mechanism as being either flexible or rigid. To that end we consider the second load case consisting of angular velocities applied at support point of one rigid link taking the value ! ; ; rad=sec. In order to be able to better represent different deformation modes, we have also refined the finite element model by increasing the total number of two-node beam elements to 15 for the flexible link and 15 for connecting rod. 2. The analysis is carried out by the energy-conserving scheme, using the time step t The results computed for the time history of the bending deformations of the flexible link at point A for different operating speeds are presented in Figure 6. They clearly illustrate that simplifying modeling assumption on neglecting the mechanism deformations becomes questionable at higher operating speeds.

5

= 5 10 20


= 10

Figure 6. Spatial slider-crank mechanism: Bending Deformation.

5.2. Closed loop multibody system In this example we study the forced and free vibrations of a closed loop mechanism. The mechanism takes a form of a closed frame of square shape, with each side equal to 10. A couple of revolute joints and a spherical joint are placed at the corners of the frame. See Figure 7. Three sides of frame are considered rigid, whereas the fourth is composed of a flexible component (whose lenght is equal to 6) attached to two rigid ends each of length equal to 2. Revolute (1)

Revolute (3)

3 1

Revolute (1)

2

Spherical

Figure 7. Closed loop mechanism : solid FE model. The mechanical chosen characteristics of the flexible component are : Young’s modulus

E

= 103 and Poisson’s ratio  = 0:25. The flexible and all rigid components are chosen of

unit cross-section and unit mass density. The mechanism is initially placed in a vertical plane and it is acted upon at its upper right corner by an out-of-plane force, which peaks to 10 at 0.5 sec, and goes to zero at 1 sec. Thereafter, the mechanism undergoes free vibrations. We have explored two modeling assumptions regarding the flexible component: The first is constructed with 8 two-node beam elements, whereas in the second model the flexible link is presented by 3d solid elements. The FE mesh of the second model is constructed by replacing each beam of the first model by 4 solid elements, i.e.  elements for the whole component. The analysis is performed by the presented energy-conserving scheme for the first 30 sec of motion, by using the time step t : . The deformed shapes computed with two FE models at every 10 sec of motion are presented in Figure 8. We can see from Figure 8 that the motion is truely three-dimensional, with the considerable values of the induced deformation. We can also see that the deformed shapes computed by two models are somewhat similar in nature. Far less agreement between the results obtained by two models is found when comparing the corresponding time histories of the total energy, which are given in Figure 9.

8 4 = 32


= 0 25

Figure 8. Closed-loop mechanism: Deformed shapes at each 10 sec. However, one can first note from Figure 9 that the presented energy conserving scheme is applicable to both models, in that it ensures the energy conservation in the free vibration phase for both models. The difference in the total energy computed values between the two models can be explained by the different manners to account for deformation and the resulting strain energy between beam and solid model (e.g. no warping of the cross-section and reduced integration for beam). Indeed, we have repeated the simulation by selecting the higher value of Young’s 6 , and thus significantly reducing the role played by the strain energy. modulus, E The total energy computed by the two models presented in Figure 10 shows much better correlation between the two models.

= 10

Figure 9. Closed-loop mechanism : Total energy time history for flexible model.

Figure 10. Closed-loop mechanism: Total energy time history for almost rigid model. The latter is due to the fact that in this case the deformation remains negligible and that both models, either beam or 3d solid, have the same behavior representing the assembly of rigid bodies.

6. Conclusions The major thrust in this work was directed towards enhancing the capabilities of the finite element models in dealing with the nonlinear dynamics of flexible multibody systems with rigid links and joint constraints. To that end, the most important modification of the standard finite element technology that we discussed pertain to handling the constraints between the flexible components. This can be accomplished either by introducing the method of Lagrange multipliers or by employing the nonlinear kinematic constraints and relative motions governed by the constraints. Although the Lagrange multipliers method is more general it is also less efficient, considering that the number of global equations should be increased to accommodate the multipliers. The method of relative motion is limited to holonomic constraints, but it can be rendered very efficient since it requires only the local (element-based) computations related to relative motion.

Nonlinear dynamics of multibody systems is often described by the stiff differential equations, arising either from the presence of the Lagrange multipliers or large difference in stiffness among different components of a multibody system, or even different deformation modes, which disqualifies the use of a great number of standard time-stepping schemes. It is shown that the proposed modification of the energy conserving scheme, which is constructed for both methods for handling the constraints, is quite capable of dealing successfully with the applications of this kind. We have also shown by means of numerical examples that at high operating speeds the mechanism deformations can play a significant role, and that, in general, it should not be ignored. As illustrated by the last example, the finite element methodology greatly simplify building of the models for a flexible multibody systems of any desired level of complexity.

Acknowledgments The financial supports from ABONDEMENT ANVAR and BFA program of FrenchAlgerian cooperation are gratefully acknowledged.

References [1] B.N. Fraeijs de Veubeke. The dynamics of flexible bodies. Int. J. Engng. Science, 14 (1976) 895-913. [2] T.R. Kane and D.A. Levinson. Simulation of large motions of nonunform beams in orbit, part i and ii. Int. J. Astro. Sceince., 29 (1981) 213-276. [3] T. R. Kane and D.A. Levinson. Dynamics Theory and Applications. Mc-Graw Hill, Newyork, 1985. [4] J.C. Simo and L. Vu-Quoc. A three dimensional finite strain rod model, part2: Computational aspect. Comput. Methods Appl. Mech. Eng., 58 (1986) 79-116. [5] J.C. Simo. A finite strain beam formulation. part 1. Comput. Methods Appl. Mech. Eng., 49 (1981) 53-70. [6] A. Cardona and M. Geradin. A beam finite element non-linear theory with finite rotations. Int. J. Numer. Methods Eng., 26 (1988) 2403-2438. [7] A. Ibrahimbegovi´c. On finite element implementation of geometrically nonlinear reissner’s beam theory: three dimensional curved beam element. Comput. Methods Appl. Mech. Eng., 112 (1995) 11-26. [8] A. Ibrahimbegovi´c. Stress resultant geometrically nonlinear shell theory with drilling rotations. part 1: A consistent formulation. Comput. Methods Appl. Mech. Eng., 118 (1994) 265-284. [9] A. Ibrahimbegovic. Finite elastic deformation and finite rotation of 3d continumum with independent rotation field. European J. Finite Element, 4 (1995) 555-576. [10] J. H. Argyris. Excursion into large rotations. Comput. Appl. Mech. Eng., 32 (1982) 85-155. [11] M. Geradin and D. Rixen. Parametrization of finite rotations in computational dynamics. European J. Finite Element, 5 (1995) 497-554. [12] A. Ibrahimbegovi´c, Frey F., and I. Kozar. Computational aspect of vector-like parameterization of three dimensional finite rotations. Int. J. Numer. Methods. Eng., 38 (1995) 3653-3673. [13] A. Ibrahimbegovi´c. On the choice of finite rotation parameters. Comput. Methods Appl. Mech. Eng., 149 (1997) 49-71. [14] J.C. Simo and L. Vu-Quoc. On the dynamics in space of rods undergoing large motions- a geometrically exact approach. Comput. Methods Appl. Mech. Eng., 66 (1988) 125-161. [15] A. Ibrahimbegovi´c and Almikdad M. Finite rotations in dynamics of beams and implicit time-stepping schemes. Int. J. Numer. Methods. Eng., 41 (1998) 781-814. [16] O.C. Zienkiewicz and R.L. Taylor. The finite element method. Volume 2:Solid and fluid mechanics, Dynamics and Non-linearity. Mc Graw-hill, 1991. [17] A. Ibrahimbegovic, B. Brank, and P Courtois. Stress resultant geometrically exact form of classical shell model and vector-like parametrization of constrained finite rotations. in press (1999). [18] B. Brank, S. Mamouri, and A. Ibrahimbegovic. Finite rotations in dynamics of shells and newmark implicit time-stepping schemes. in press (2000). [19] A. Cardona, M. Geradin, and D.B. Doan. Rigid and flexible joint modelling in multibody dynamics using finite elements. Comput. Methods Appl. Mech. Eng., 89 (1991) 395-418. [20] R. L. Taylor and A.J. Chen. Analysis of rigid flexible structural components. IACM expressions, 4/97 (1997) 8-11.

[21] A.J. Chen. Energy-momentum conserving methods for three dimensional dynamic nonlinear multibody systms. Ph.D Thesis, Stanford University (also SUDMC Report 98-01) (1998). [22] A. Cardona and M. Geradin. Time integration of the equations of motion in mechanism analysis. Comput. Struc, 33 (1989) 801-820. [23] A. Ibrahimbegovi´c and S. Mamouri. On rigid components and joint constraints in nonlinear dynamics of flexible multibody systems employing 3d geometrically exact beam model. Comput. Methods Appl. Mech. Eng., in press (2000). [24] E. Hairier and G Wanner. Solving ordinary differential equations: Stiff and differential-algebraic problems. Springer, Berlin, 1991. [25] J.C. Simo, N. Tarnow, and M. Doblare. Nonlinear dynamics of three-dimensional rods: Exact energy and momentum conserving algorithm. Int. J. Numer. Methods Eng., 38 (1995) 1431-1473. [26] D. Kuhl and E. Ramm. Constraint energy-momentum algorithm and its application to non-linear dynamics of shells. Comput. Methods Appl. Mech. Eng., 136 (1996) 293-315. [27] O.A. Bauchau, G. Damilano, and J. Theron. Numerical integration of non-linear elastic multibody systems. Int. J. Numer. Methods Eng., 38 (1995) 2727-2751. [28] C. L. Bottasso and M. Borri. Energy preserving/decaying schemes for non-linear beam dynamics using helicoidal approximation. Comput. Methods Appl. Mech. Eng., 143 (1997) 393-415. [29] T.J.R. Hughes, T.K. Caughey, and W.K. Liu. Finite element methods for nonlinear elastodynamics which conserve energy. ASME J. Appl. Mech., 45 (1978) 366-370. [30] E.H. Love. A treatise of the mathematical theory of elasticity. Dover, 1940. [31] E. Reissner. On one dimensional large-displacement finite strain beam theory. Struc. Appl. Math., 52 (1973) 87-95. [32] T.J.R. Hughes. Stablity, convergence and growth and decay of energy of the average acceleration method in nonlinear structural dynamics. Computers and Structures, 6 (1976) 313-324. [33] J. E. Marsden. Lectures on Mechanics. Combridge university Press, Combridge, 1991. [34] A. Ibrahimbegovi´c, Mamouri. S, R.L. Taylor, and J.A. Chen. Finite element method in dynamics of flexible multibody systems: Modeling of holonomic constraints and energy conserving integration schemes. J. Multibody System Dynamics, in press (spacial issue edited by A. Cardona, M. Geradin and A. Ibrahimbegovi´c) (2000) 4, 195-223. [35] O.C. Zienkiewicz and R.L. Taylor. The finite element method. Volume 1: basic formulation and linear prblems. Mc Graw-hill, 1989.

Implicit Kinematical Parameters and Sensitivity of Finite Rotation Shells Kris WISNIEWSKI, Michal KLEIBER and Ewa TURSKA IFTR, Polish Academy of Sciences, Swietokrzyska 21, 00-049 Warsaw, Poland Abstract. The paper is concerned with shell equations derived from a mixed 3D formulation with rotations incorporated via the rotation constraint equation. Two forms of rotations varying linearly over the thickness are verified: one with the in-plane twist parameter, and the other, with the in-plane twist and the warping parameters. These forms are combined with two generalized Reissner hypotheses, which include stretch parameters and, in the latter case, a term describing a bubble-like warping of a cross-section. Consequences of this new kinematics for shell equations, especially for rotation constrains and strain measures, are examined. The new kinematical parameters are treated as implicit ones, so that, externally, a 6-parameter formulation admitting C 0 approximations is retained. Next, the Design Sensitivity Analysis for finite rotation shells is presented. The design derivatives of displacements and rotations are calculated along the equilibrium paths in the whole pre- and post-critical range. Numerical solutions for structures of highly nonlinear equilibrium paths with singular points, and the corresponding sensitivity charts, are presented. In particular, the in-plane bending of a flat cantilever shell, which involves a finite drill rotation, is presented and compared with other results.

1. Introduction A new methodology of developing variants of shell equations with enriched kinematics from the 3D mixed formulation with explicitly incorporated rotations is presented. A crucial role in this plays the rotation constraint equation, obtained from a polar decomposition of a deformation gradient, as it enables the introduction of new kinematical parameters. Also, a proper balancing of assumptions for the deformation function and rotations is needed. The present paper discusses two variants of equations with additional rotational parameters. The drill rotation, which was a subject of interest in the recent years, see an overview e.g. in [1], is incorporated into both variants, but also new parameters associated with rotations varying over the thickness are examined. The first variant of equations is based on shell kinematics enhanced with the in-plane twist parameter, see [2]. For the classical Reissner kinematics, the rotation constraint yields two equations for the drill rotation. The use of both, the drill and in-plane twist, parameters has the following consequences: (a) the ambiguity of expressions for the drill rotation is eliminated, (b) the 1st order strain measure is enriched and includes a twist correction part, which can be expected to eliminate the in-plane twist rigid body mode of plane shell elements. The second variant of equations introduces a canonical rotation vector, linearly varying over the thickness, see [3]. To maintain a consistency between the deformation function and rotations, this new rotation vector necessitates a generalization of the Reissner kinematical hypothesis; a new term describes the bubble-like warping of a cross-section, and depends on

tangent components of the new rotation vector. Several consequences of this new kinematics can be observed; e.g. that it prevents the in-plane twist correction of the 1st order strain from vanishing for a negligible 0th order in-plane strain. In both variants, only the drill rotation is retained explicitly, while the other parameters are treated as implicit parameters. Therefore, externally, a 6-parameter formulation in terms of displacements and three-parameter rotations is retained, and the C 0 approximations can be used for the FE discretisation. In the second part of the paper the Design Sensitivity Analysis for finite rotation shells is presented. It is based on the Continuum Approach and the Adjoint System Method, see [4], [5], and follows the scheme of implementation described in [5]. The design sensitivities for rotational parameters of shells, including the drilling rotation, are necessary to enable a uniform treatment of all degrees of freedom, e.g. in optimization. Several numerical examples, displaying highly nonlinear equilibrium paths with singular points and the corresponding sensitivity charts, are presented. The design sensitivities of displacements and rotations are calculated along the equilibrium paths in the whole pre- and post-critical range. In particular, the in-plane bending of a flat cantilever shell, which involves a finite drill rotation, is presented and compared with other solutions.

=1 2

Notation. A convected orthonormal system of curvilinear coordinates S  ,  ; is used for the middle surface of a shell. The local orthonormal bases at the middle surface are: ft i g for the initial configuration, and fai g for the deformed configuration. Besides, small bold letters denote vectors, capital bold letters - 2nd rank tensors, dots ”
” - scalar products of vectors and tensors, ” ” - tensorial products. 2. 3D Formulation Including Rotations In this section a three-dimensional mixed formulation including rotations as an independent variable is presented. It is based on the rotation constraint equation, and is similar to that of [6] or [7], but with a constitutive equation for the Biot stress and the right stretch strain, see also [1]. To incorporate rotations into the formulation we make use of the rotation constraint equation. Let us first consider the polar decomposition of the deformation gradient, F RU, 1 where R FU 1 is a rotation tensor, and U  FT F 2 is the right stretching tensor (symmetric). If instead of R FU 1 we use an arbitrary Q 2 SO then the formula T Q F U can be re-written as follows

=

=

(

=

)

(

=

(3)

) + skew(QT F) = U: skew(QT F) = 0 implies that sym(QT F) = U. sym QT F

(1)

Note that the condition The equation T skew Q F is called the rotation constraint equation, and it provides a link between the deformation gradient and the rotations. Let us introduce the stress tensor T B  QT P, where P is the 1st Piola-Kirchhoff B stress tensor. The symmetric part of TB is called the Biot stress, i.e. TB s  symT . Also B the skew part, TB a  skewT , is used in the sequel. The set of local governing equations is as follows:

(

)=0

1. linear momentum balance equation:

Div[Q(TBs + TBa )] + Rb = 0

(2)

where R is the mass density for the reference (initial) configuration, and b is the body force. 2. rotational momentum balance equation:

[ ( + TBa )FT ] = 0 (3) det F > 0. Here  denotes the deformation function skew Q TB s

= Grad

 and where F which maps the reference configuration of the body configuration B .

B onto the current (deformed)

3. boundary conditions:

= u on @uB and QTB n = p^ on @ B (4) where @u B and @ B denote the disjoint parts of the boundary on which the deformation and traction boundary conditions are specified. The outward normal vector is ^ is the external load (surface traction). denoted by n, and p u

4. rotation constraint:

)=0

(

skew QT F

(5)

Besides, we postulate a constitutive equation for the symmetric Biot stress, assuming the existence of a strain energy function W U , which satisfies the frame indifference requirement, e.g. [8]. Then, the constitutive equation takes the form T B @ W U =@ U. On use of s the condition sym QT F U we can substitute for U, and then the constitutive equation takes the form TB (6) @ W sym QT F =@ sym QT F s

( )

(

=

)=

=

(

(

))

(

)

(

)

( )

The above equations, (2)-(6), furnish a mixed formulation in terms of fu; Q; T B a g. The weak form of the above equation can be obtained in a standard way, by first calculating the following scalar products: (a) of the linear momentum balance equation with a kinematically admissible variation Æ , (b) of the rotational balance equation with a skewsymmetric (right) tensor Æ , (c) of the constraint on rotations with a skew- symmetric tensor Æ TB a . Then, the virtual work equation is obtained as a sum of these scalars, Z h i TF B
skewÆ QT F B
skew QT F dV V TB
sym Æ Q T Æ T (7) ext Vboun s a a

~

(

B

)+

(

)+

=0

where Vext is a virtual work of the body force and external loads, and V boun of the displacement boundary conditions. The above equation is a variation of the potential Z h i B W sym QT F TBa
skew QT F dV ext boun (8) 3 u; Q; Ta

(

)=

B

(

(

)) +

(

)

+ +

where TB a plays the role of the Lagrange multiplier on use of which the rotation constraint is imposed. The strain measure can be deduced as the work conjugate to TB s . On the basis of T F in eq.(7), we can define the relaxed right stretch strain as follows TB
sym Æ Q s

(

)

= sym(QT F) I When the rotation constraint is satisfied then sym(Q T F) = U, H

obtained.

(9) and the standard definition is

Remark 1. The above formulation can be obtained from the four-field formulation of [9], based on the functional Z h i B (10) W U TB
QT F U dV ext boun 4 u; Q; U; T

(

)=

) + + ( )+ ( T T T B The Lagrange term can be split, TB s
[sym(Q F) U]+ Ta
skew(Q F). If U = sym(Q F) then the 1st term vanishes, and 4 reduces to the three-field 3 , eq.(8). B

Remark 2. In case of an isotropic material, TB a can be eliminated, and a two-field formulation obtained. Then, the potential energy is as follows, Z h i W sym QT F skew QT F
skew QT F dV ext  : (11) 2 u; Q

(

) = B ( ( )) + ( ) ( ) + ( ) where 2 (0; 1) is the penalty number, and the virtual work equation becomes Z h i T F) + skew(QT F)
Æ skew(QT F) dV TB sym ( Q Vext Vboun = 0
Æ s B

(12)

In the sequel the shell counterparts of the virtual work eq.(7) and (12) are obtained.

3. Rotations for Reissner Kinematics In this section we show that rotations R calculated by the polar decomposition of the deformation gradient for the classical Reissner kinematics, do depend on the thickness co-ordinate  . The position vector in the deformed configuration for this kinematics is as follows x x0  Q0 t3 (13)

( )= +

(3)

where Q0 2 SO is constant over the thickness, and, t 3 is a shell director in the undeformed configuration. Let us take, for simplicity, Q 0 as an elementary rotation around t1 . Then, the deformation gradient is

( ) = (x0;2

F

!;2 a2 ) t2 + a3 t3

(14)

where ! is a rotation angle. Here, we assumed that the position and rotation do not vary along the S 1 -co-ordinate. On use of the polar decomposition of the deformation gradient the rotation can be 1 calculated as R FU 1 , where U  FT F 2 is the right stretching tensor. First we calculate the right Cauchy-Green tensor,

=

(

)

= FT F = at2 t2 + b(t2 t3 + t3 t2) + t3 t3 (15) ca2 )
(x0;2 ca2 ), b = x0;2
a3 , and c = !;2. A general procedure

C

=(

where a x0;2 for finding U 1 is given e.g. in [10] and [11]. To avoid complicated expressions, we assume that x0;2
a3 , i.e. the transverse shear strain is equal to zero. Then, C has a diagonal representation and we can find

=0

= FU 1 = p1a (x0;2 ca2 ) t2 + a3 t3 1=pa and c=pa are (non-linear) functions of R

(16)

. where the coefficients Hence, we have proven that the rotation R is a function of  even for the kinematics based on the rotation Q0 , which is constant over the thickness. This result motivates the usage of  -dependent rotation tensors in the shell kinematics in the next sections.

4. Shell Kinematics with In-plane Twist Parameter 4.1. Kinematical assumptions In order to introduce the shell kinematics, in the present section we generalize the Reissner hypothesis as follows

( )  x0 +  d;

x

d

= d1 a1 + d2a2 + d3a3

(17)

where d is the deformed director, di are scalar parameters, and the thickness co-ordinate  2 h= ; h= . Besides, ai Q0 ti , and Q0 2 SO is a rotation constant over the shell thickness. In the current mixed formulation the rotations are introduced via the rotation constraint equation and are an independent field. Hence, their dependence on  must also be assumed. In this section we assume the following form of the rotations,

[

=

2 + 2]

(3)

( ) = Rt ( ) Q0

Q

(18)

( ) is the in-plane twist rotation defined as Rt ( ) = cos(!t )(a1 a1 + a2 a2 ) + sin(!t )(a2 a1

where Rt

a1 a2

) + a3 a3

and !t is the in-plane twist parameter, see Fig.1. Hence, a rotation around the shell director is a linear polynomial of  , i.e. !  !d !t , where !d is a drill rotation parameter. The gradient of the deformation function, eq.(17), is as follows

( )= +

F where F0

 ( )  @ (S @(yy);  (y)) = F0 +  (F; )0 + : : :

(19)

= x0; t + d t3;

(20)

(F; )0 = d; t

and y denotes the initial position vector. To obtain this expression, the basic definitions specified in Section 3.1 of [2] were used. In the current formulation fundamental is the product Q T  F  , as a basic component of the rotation constraint and the strain measure. For the defined kinematics, this product is a non-linear function of  , and it can be expanded in the Taylor series at  . The 0th order term of the expansion is as follows

() ()

=0

(QT ( ) F( ))0 = QT0 F0 = QT0 x0; t + QT0 d t3

Figure 1. In-plane twist by



3.

(21)

while the 1st order term is   QT  F  ; QT; 0 F0 0

= ( ) + QT0 (F; )0 = !tQT0 ST F0 + (d;
ai) ti t (22) a1 a2 is an elementary skew-symmetric tensor. The products of the

[ ( ) ( )] where S = a2 a1 2nd order effects,

!t d , are neglected in the first component.

4.2. Rotation constraints and strain measures

[ ( ) ( )] = 0

, and exNext we consider the rotation constraint equation, skew Q T  F  pand it into the Taylor series w.r.t.  at  . From the 0th order term of the expansion, skew QT0 F0 , we obtain the following scalar equations

[

=0

]=0

x0;1
a2

x0;2
a1

= 0;

x0;
a3

d = 0

(23)

where the first one is the constraint for the drill rotation, while from the second we can determine d . The 1st order term, skew QT; 0 F0 QT0 F; 0 , yields

[( ) !t (x0;1
a1 + x0;2
a2 ) + (d;1
a2

+ ( ) ]=0 d;2
a1 ) = 0; d;
a3 = 0

(24)

where the first equation is the constraint for the twist rotation, as we can calculate

!t =

d;1
a2 x0;1
a1

+

d;2
a1 x0;2
a2

(25)

+ (

)+

From the second one we can obtain a differential equation for d3 , i.e. d3; d3 a3;
a3 . For certain applications, such as multi-layer shells, this equation should be d a ;
a3 retained in the virtual work of rotation constraint. For many other, the plane stress condition can be used to calculate d3 , and d;
a3 is only used in 3 . Consider the right stretch strain, H  sym Q T  F  I  , and expand it into the Taylor series w.r.t.  at  . The 0th order term of the expansion, " sym Q T0 F0 I0 , is as follows h i " sym QT0 x0; t t QT0 d t3 t3 (26)

(

)=0

=0

=

(

=0 ( ) = [ ( ) ( )] ( ) = [ ] ) +( )

with the components in the basis ft i g,

"11 = x0;1
a1 "12 = "21 =

1;

"22 = x0;2
a2

1 (x
a + x
a ) ; 2 0;1 2 0;2 1

1;

"3 = "3 =

"33 = 

1

1 (x
a + d ) 2 0; 3 

(27)

They are not affected by the twist rotation, and are similar to the measures for a standard Reissner kinematics, excluding "3 in which d appears. The 1st order term of the expansion  sym Q T; 0 F0 QT0 F; 0 , can be written as the sum   , where

= +

=

 = sym[QT0 d; t ];

[( ) + ( ) ]

 = !t sym[QT0 S F0 ]

ftig are as follows 11 = d;1
a1 ; 22 = d;2
a2 ;

(28)

Components in the basis

12 = 21 =

1 (d
a + d
a ) ; 2 ;1 2 ;2 1

33 = 0

1 2

3 = 3 = d;
a3

(29)

 11 = !t (x0;1
a2 );

1 2

22 = !t (x0;2
a1 );

12 = !t (x0;1
a1

x0;2
a2

);

33 = 0

(30)

3 = 0

Note that  is the change of curvature strain of a formulation without the twist rotation. Besides, 33 , i.e. the linear part of the normal strain is equal to zero. Let us now consider the strain measures for a formulation with six degrees of freedom. Then, the drill rotation must be retained explicitly, while ! t and di should be implicit parameters. On use of the second of eq.(23) we obtain "3 "3 x0;
a3 , while the second of eq.(24) yields 3 3 . The other components are as given earlier. The objectivity of these strains is verified in [2].

=0

=

=

=0

=

5. Shell Kinematics with Warping and In-plane Twist Parameters 5.1. Kinematical assumptions In order to introduce the shell kinematics, in the present section we generalize the Reissner hypothesis as follows

( ) = x0 +  [d1a1 + d2 a2 + d3Q1( )a3] (31) where ai = Q0 ti , and Q0 2 SO (3) is a rotation constant over the shell thickness. The rotation depending on the thickness co-ordinate, i.e. Q 1 ( ) 2 SO (3), is defined below. x

Besides, di are scalar parameters. Because rotations are introduced via the rotation constraint equation, we also assume the  -dependence of the rotation tensor,

( ) = Q1 ( ) Q0 :

Q

(32)

While no restrictions are placed on Q0 , the  -dependent rotation is assumed in a linearised form, i.e. Q1   I  , which is valid only for small rotations. Here, I2 so , where is the canonical rotation pseudo-vector. Besides, we assume that varies linearly with  , i.e.    , where  is a vector of rotation coefficients. For so specified Q1  , the hypotheses (31) and (32) become

+ ~( ) ( )=

()

(3)

()

( ) = x0 +  d +  2d3( ~  a3);

x

~ =

~=

Q1

( ) = I +  ~ 

(33)

()

  I and d is defined in eq.(17). Note that the last term of x  is quadratic where  w.r.t.  and depends on the vectors normal to a 3 , i.e. it represents the bubble-like warping of a cross-section, see Fig.2. It can be shown that  are associated with the warping, while  3 with the in-plane twist rotation. The gradient of the deformation function, eq.(33), is as follows F where F0

 ( )  @ (S @(yy);  (y)) = F( ) = F0 +  (F; )0 + : : :

= x0; t + d t3;

(F; )0 = d; t + 2d3( ~  a3 ) t3

(34)

(35)

and y denotes the initial position vector. To obtain this expression, the basic definitions specified in Section 3.1 of [2] were used.

Figure 2. Bubble-like warping by



1

() ()

In the current formulation the product QT  F  , is fundamental, as a basic component of the rotation constraint and the strain measure. For the defined kinematics, eq.(33), this product is a non-linear function of  , and is expanded in the Taylor series at  . The 0th order term of the expansion is as follows

=0

(QT ( ) F( ))0 = QT0 F0 = QT0 x0; t + QT0 d t3

(36)

while the 1st order term is   QT  F  ; QT; 0 F0 0

[ ( ) ( )] = ( ) + QT0 (F; )0 = 1 A1 + 2 A2 3 A3 + D (37) where Ai  QT0 Si F0 , D = (d;
ai )ti t + 2d3 2 t1 t3 2d3 1 t2 t3 , the elementary Si = eijk ak aj aj ak 2 so(3), and eijk is a permutation symbol.

5.2. Rotation constraints

[ ( ) ( )] = 0

, and expand Next, we consider the rotation constraint equation, skew Q T  F  it into the Taylor series w.r.t.  at  . From the 0th order term of the expansion, skew QT0 F0 we obtain the scalar equations

[

]=0

=0

d = 0 (38) which are identical to eq.(23), and not affected by the form of Q1 ( ) and the warping term in x( ). The 1st order term, skew[(QT; )0 F0 + QT0 (F; )0 ] = 0, can be written in a matrix

form, as L

x0;1
a2

 = r, where 2 6 4

x0;2
a1

= 0;

x0;
a3

(x0;1
a1 + x0;2
a2) 37 ( ) x0;2
a3 L= (39) 5 ( ) x0;1
a3 r = f(d;1
a2 d;2
a1 ); (d;1
a3 ); (d;2
a3 )gT This form of L is obtained for d = x0;
a3 , implied by the 2nd of eq.(38). Then d = (x0;
a3 )a + d3 a3 , and the vector r also is affected. For  = f0; 0; 3 g the first equation is the constraint for the twist rotation, providing 3 = !t of eq.(25). Note that  x0;1
a3 x0;1
a2 x0;2
a2 d3

x0;2
a3 x0;1
a1 d3 x0;2
a1

treating the 1st order term as an equation for we must provide an additional equation for d3 , for instance such as the plane stress condition.

5.3. Strain measures

[ ( ) ( )] ( )

( )=

sym Q T  F  I  , and expand it into Consider the right stretch strain, H  the Taylor series w.r.t.  at  . The 0th order term of the expansion, " sym Q T0 F0 I0 , is as follows i h " sym QT0 x0; t t QT0 d t3 t3 (40)

=0 = (

)

+(

)

=

[

]

()

and it is identical to eq.(26), i.e. it is not affected by the form of Q 1  and the warping term in x  . The 1st order term,  sym QT0 F; 0 QT; 0 F0 , consists of two parts   , where the change of curvature strain is

()

=

 = sym[QT0 d; t ];

= +

[ ( ) +( ) ]  =

 1 symA1



 2 symA2

 3 symA3

(41)

The  is identical to the first of eq.(28), while  depends on the extended kinematics, what can be established by comparison with the second of eq.(28). Let us now find the components of  for a formulation with six degrees of freedom. Then, the drill rotation must be retained explicitly, while i and di should be implicit (internal) parameters. From the 2nd of the rotation constraints, eq.(38), d  x0;
a3 , and then the transverse shear strain is "3 "3 x0;
a3 , and the director d "3 a d3 a3 . Then,



=

11 =

  2 "13 + 3 (x0;1
a2 );

= =

=

22 = + 1 "23

 3 (x0;2
a1 );

33 =

+

  1 "23 + 2 "13

1 [+ "   (42) 2 1 13 2 "23 3 ("11 "22)] 1 13 = [ 1 (x0;1
a2 ) + 2 ("11 "33 ) + 3 "23 ] 2 1 23 = [ 1 ("22 "33 ) + 2 (x0;2
a1 ) 3 "13 ] 2  we additionally assume, just for a moment, that the 1st Remark. To grasp the meaning of  of rotation constraints, eq.(38), is also satisfied, what implies " 12 = "21 = x0;1
a2 = x0;2
a1 .  = sym(   "), i.e. is a product of the rotation  and the 0th order strain ". Then,   12 =

Example. Negligible in-plane 0th order strains. On use of the condition " we can obtain a simplified form of the matrix L, and next  and . Then, the L of eq.(39) becomes 2 3 "13 "23 6 "33 "23 75 (43) L 4



=0

= 0

2

0 "13 The rotation parameters are calculated as i = Di =D , where D = "33 ("213 + "223 + 2"33 ) D1 = [r1 "33 (d;1
a3 )"23 ] "13 (d;2
a3 )("223 + 2"33 ) D2 = [r1 "33 (d;2
a3 )"13 ] "23 (d;1
a3 )("213 + 2"33 ) D3 = [ r1 "33 (d;1
a3 )"23 + (d;2
a3 )"13 ] "33  , eq.(42), becomes and r1 = (d;1
a2 d;2
a1 ). The strain  11 = 2 "13 ; 22 = 1 "23 ; 33 = 1 "23 + 2 "13 "33

(44)

(45)

12 =

1[ 2

 1 "13

 2 "23 ] ;

13 =

1[ 2

  2 "33 + 3 "23 ] ;

23 =

1[ 2

 1 "33

 3 "13 ]

As we see, a different result was obtained than for the kinematics with the in-plane twist rotation parameter !t , only, see [2], Section 3.9, point b. Now, the strain  is nonzero, what is intuitively correct, and justifies the use of  instead of !t . If we additionally assume that the transverse shear strains " 3 , then the rotation parameters are as follows



=0

 (d;2
a3 ) ; 1= " 33

 (d;1
a3 ) ; 2= " 33

 3=



and the only non-zero components of the strain  are

 13 =

1 2

23 =

 2 "33 ;

1 2

1 (d
a 2 ;1 2

d;2
a1

 1 "33

)

(46)

(47)

The in-plane twist 3 is not present in the above formulas, but still they are not equal to zero. Besides, note that the role played by the normal strain stretch " 33 is important; if "33 then D , and the rotation parameters i are not uniquely determined.

=0

=0

6. Virtual Work Equation for Shell The virtual work equation for the shell is derived from the three-dimensional virtual work equation presented in Section 2 on use of the shell kinematics defined in Section 4 or 5. The corresponding formulas can be found in [2] and [3]. Here we only note that on use of the relations resulting from the rotation constraints, the implicit kinematical parameters, d i and  , can be expressed in terms of the explicit kinematical parameters, and eliminated from the formulas. Finally, for the shell, the virtual work equation accounting for all contributions, specified in the previous sub-sections, can be written as Z Z Æ 2d f Æ Æ RC Æ b ÆAg dS Æ A d@ S (48)

 =



S

+ 



@ S



where Æ is for stress and couple resultants, Æ b - for the body force, Æ A - for the external forces acting on the lower and upper bounding surfaces, Æ A - for the external forces acting upon the lateral boundaries. Besides, the rotation constraint contribution to the virtual work equation for the shell is as follows

Æ RC

= Æ(N12 C12 ); N12 = (NBa )12 , and C12 = 21 (x0;1
a2

Æ RC

= h C12 ÆC12 x0;2
a1 ), and 2 (0; 1).

(49)

where The first of the above expressions is for the three-field shell formulation, while the second for the two-field formulation valid for an isotropic material. Note that for the in-plane deformation,

2C12 = (u2;2 + u1;1 + 2) sin !d + (u2;1

u1;2)

cos !d :

and it links the drill rotation with the in-plane components of displacements.

(50)

7. Design Sensitivity for Rotational Parameters The subject of the Design Sensitivity Analysis (DSA) is the calculation of derivatives of state variables with respect to design variables. These derivatives, or the so called sensitivities, are subsequently used to modify (re-design) the structure in order to achieve a satisfactory performance, see e.g. [12]. The methodology of calculating the design derivatives has already been established for a wide class of materially and kinematically nonlinear problems, see e.g. [4] and [5]. When new values of the design parameters are sought as a solution of an optimization problem then the question of derivatives becomes further complicated. The optimization for nonlinear mechanical problems must be performed in the fh; q; g-space, where h is the design variable vector, q is the state variable vector,  is the load multiplier. For several reasons, such problems are solved by the staggered (bordering) solution schemes, and the design derivatives depend on the scheme which is applied, see [13] and [14]. Besides, equilibrium paths for nonlinear structures may possess extremum and bifurcation points, and additional nonlinear constraints are used to ensure stability of the solution. For six parameter shells, the state variable is z  fu; ! g, where u is the displacement vector, and ! is a vector of parameters for the rotation Q0 (constant over the thickness). To enable a uniform treatment of all degrees of freedom, for instance in optimization, we need to calculate also the design derivatives of rotational parameters, including the drill rotation. The design sensitivity analysis described in the present work is based on the Continuum Approach and Adjoint System Method (ASM). The shell thickness h is chosen as the design variable, and the design derivative is denoted as D h
 @
=@h. Within the Continuum Approach, first the design differential of the virtual work equation is calculated and next the obtained formula is discretized. For the virtual work given by eq.(48) it yields the sensitivity equation

()

@^h G +

()

@G
D q=0 @q h

=  ( )

(51)

^

where G Æ 2d h; q , and the explicit design derivative is denoted by @h . After the FEM discretisation, this equation can be re-written as K D h q @h r, where K is the tangent matrix and r is the residual of the equilibrium equation. Hence, we obtain D h q, which can be used to calculate the design derivative of a (scalar) performance h; q ,

= ^

( )

! @ ^ Dh = @h +
Dh q fem = @^h @q

" fem ="

@ @q

!T

^

K 1 @h r

(52)

denotes the Finite Element Method discretization. Within the Adjoint System where Method the design derivative of the performance, Dh , is evaluated in two steps: Step 1. 



= K 1 (@ =@ q)

Step 2.

Dh = @^h T @^h r,



()

where  is the adjoint solution. Note that we can take  q i , and use the ASM to calculate Dh q i without resorting to displacements, see e.g. [5]. As an example, consider a stretched tendon of the initial length L, the cross-section area A, and Young modulus E . One end tendon is fixed, and at the other a force P is applied. The Green strain  of the 2 1 L + u , where u is the axial displacement at the free end. Introducing is  2 L q u=L we obtain:  q 12 q 2 , and Æ Æu L1 q . The equilibrium equation is

()

=

= [

1]

= +

=

(1 + )

Figure 3. In-plane bending: initial and deformed mesh

r = (1 + q ) EA (q + 12 q 2 ) P = 0, and can be solved by the Newton-Raphson scheme, q i+1 = q i K 1 r, where K = EA (1 + 3q + 1:5q2 ). Assume that q is the performance, i.e. = q . Take A = bh, where b is the width and h is the height of the cross-section, and let h be the design variable. Within the ASM, first, the adjoint displacement is calculated for the adjoint unit load, resulting from @ =@q = 1, i.e.  = K 1 1. Then, the design sensitivity of performance is as follows

1 2

Dh = (1 + q ) Eb (q + q 2 )

(53)



=
(1+ ) + (
)


(1 + ) =


q , and is a As  denotes the displacement increment q due to a unit load, thus q 1 2 linear part of the strain increment,  q q 2 q . Hence, Dh L Eb , where L is due to the unit load. In this way, the sensitivities can be computed in terms of strains, without resorting to displacements.




7.1. Example 1. In-plane bending of a cantilever by a force This example establishes the performance of the element for large drill rotations, see the initial and deformed mesh in Fig.3. A cantilever is modeled by 40 quadrilateral shell elements, and the data is as follows: length L : in, width b : in, thickness h :

=10

=01

100

Load * 10

80

60

40

omegad, 4-n -u1, 4-n u2, 4-n beam

20

0 0

0.5

1 omegad, u1, u2

1.5

Figure 4. Drill rotation and displacements.

=01

100 4-node elmt beam elmt

Load * 10

80

60

40

20

0 0

0.2 0.4 d omegad / d h

0.6

Figure 5. Design derivative of drill rotation.

= 3 10

in, Young modulus E
7 lb/in2, Poisson ratio fixed while at the other an in-plane force is applied.

 = 0:3. One end of the cantilever is

Implementation of membrane part. The membrane part of the 4-node shell element is uniformly underintegrated (one-point rule), and the tangent matrix has 4 non-zero and 8 zero energy modes, of which 3 are the correct rigid body modes and 5 are spurious. Let us we denote the vectors of nodal values as s f ; ; ; g,  f ; ; ; g,  f ; ; ; g, and the hourglass mode h f ; ; ; g. Directly related to the rotation constraint is the energy mode s for the drill rotation; if this constraint is removed then one additional spurious mode appears. The spurious zero energy modes of the underintegrated element are as follows: ,  and h for the drill rotation, and h for each displacement component. To prevent hourglassing a stabilization was applied: the -method for the h modes, and a method based on the 1st order in-plane bending terms for the  and  mode.

= 1111 = 1 11 1

= 111 1

= 1 111

Remark. For the formulation based on the Green strain, the potential energy depends only on u, i.e. u , and the rotation constraint equation, c ! d ; u , plays a double role; it introduces !d and links both variables. If the rotation constraint is imposed via the penalty 1 2 method, u 2 c u; !d , then it can be shown that the solution u cannot be recovered. If the Allman shape functions are used then the potential depends on both variables, i.e. u; !d and the problem disappears.

( ) ( ) +

(

(

(

)=0

)

)

The solution curves and the sensitivity curves are calculated along the solution path, and compared with other solutions. The drill rotation and displacements are shown in Fig.4,

Figure 6. Thin-walled C-profile under a point load. Initial and deformed mesh.

Figure 7. Rotation X at the load point.

and the curve for the tip deflection coincides with the one presented in [15]. The sensitivity of the drill rotation with respect to the thickness of the shell is shown in Fig.5. The curve for the sensitivity of the drilling rotation has not yet been presented in the literature. Therefore, for comparison, the solutions obtained by the finite rotation/transverse shear beam element are also given. Both solutions are close, what confirms that the present implementation is correct.

Figure 8. Design derivative of rotation X at the load point.

7.2. Example 2. Thin-walled C-profile under a point load The geometry of the C-profile and the load are shown in Fig.6. The data is as follows; thickness h : in, length L in, height of flange b in, width of lower and upper 7 lb/in2, Poisson ratio  shelf a in, Young modulus E : . One end of the profile is clamped, at the other a vertical force is applied. 120 four-node shell elements are used. The load Pmax , and the critical load is about 509.6. The curves for the static analysis correspond to those presented in [16]. The shell thickness is the design parameter, see [5]. The nonlinear equilibrium path for the rotation X at the point where the force is applied, and the corresponding sensitivity chart are presented in Fig.7 and 8. Note that the rotation of the free end of the profile is finite. Besides, there are 3 asymptotic lines on the sensitivity chart, which correspond to the inflection point A, the maximum point B, and the minimum point C on the equilibrium paths. The points on the sensitivity chart appear in the following order: A - A’ - B - B’ - C - C’ - D.

=01 =2

= 36 = 10

=6

= 0 333

= 520

8. Final remarks In the paper the kinematics of finite rotation shells is extended by use of the mixed 3D formulation with rotations introduced via the rotation constraint equations. The newly introduced kinematical parameters are treated as implicit (intermediate) parameters, and do not enter the virtual work equation for the shell as independent variables. Two new variants of shell equations are derived, with the rotation tensor depending on the thickness co-ordinate assumed in the form of either the in-plane twist rotation or the canonical rotation vector. For the kinematics with the new rotation vector, it is necessary to add the bubble-like warping term to the Reissner hypothesis in order to keep consistency between the form of rotations and the deformation function. As a result of these assumptions, the classical bending strain  is corrected by a new term , the effects of which are as follows.





1. The normal strain 33 is non-zero, what eliminates the need for recovery of this strain, e.g. via the constant stress condition.



2. For the kinematics with in-plane twist rotation,  vanishes when the in-plane strain " , but for the kinematics based on the  -dependent rotation vector this strain is retained, in accord with expectations.

=0



3. For all the rotation constraints satisfied,  provides a coupling between the new rotation vector and the 0th order strain. Finally, we note that the applied methodology allows to retain a 6-parameter formulation, in terms of displacements and three-parameter rotations, admitting the C 0 approximations. Regarding the Design Sensitivity Analysis for the finite rotation shells, the continuum Adjoint System Method is used, which, as shown on a simple example, can be conveniently implemented without resorting to displacements. In the numerical examples two questions are exposed: the implementation of the membrane part of the element with the drilling rotation, and the design derivatives for rotational degrees of freedom. The formulation based on the right stretch strain introduces a strong coupling between rotations and displacements, differently than the formulation based on the Green strain, for

which the Allman shape functions are necessary. The locking is removed by a reduced integration, and more zero-energy modes have to be stabilized than in the case of the Green strain formulation. As indicate the tests of the element with the drilling rotation, correct sensitivities of the drilling rotation are obtained. Although the developed element performs well in several tests, it is likely that the Enhanced Assumed Strain method can bring further improvement.

References [1] K. Wisniewski. A shell theory with independent rotations for relaxed biot stress and right stretch strain. Computational Mechanics, 21(2) (1998) 101-122. [2] K. Wisniewski and E. Turska. Kinematics of finite rotation shells with in-plane twist parameter. in print, 1999. [3] K. Wisniewski and E. Turska. Warping and in-plane twist parameters in kinematics of finite rotation shells. submitted for publication, 2000. [4] M. Kleiber, H. Antunez, T.D. Hien, and P. Kowalczyk. Parameter Sensitivity in Nonlinear Mechanics. Wiley, 1997. [5] J.L.T. Santos, K. Wisniewski, and V. Apostol. Pre- and post-critical design sensitivity analysis for nonlinear structures undergoing finite rotations. In The First World Congress of Structural and Multidisciplinary Optimization, Book of Extended Abstracts-Lectures, number 83, pages 102–103, Goslar, Lower Saksony, Germany, May 28-June 2, 1995. [6] J.C. Simo, D.D. Fox, and T.J.R. Hughes. Formulations of finite elasticity with independent rotations. Comput. Methods Appl. Mech. Engng., 95 (1992) 227-288. [7] S.N. Atluri and A. Cazzani. Rotations in computational solid mechanics. Archives of Computational Methods in Engineering, 2(1) (1995) 49-138. [8] R. Ogden. Non-Linear Elastic Deformations. Ellis Horwood, Chichester, UK, 1984. [9] B. Fraeijis de Veubeke. A new variational principle for finite elastic displacements. Int. J. Engng Science, 10 (1972) 233-248. [10] T.C.T. Ting. Determination of c 1=2 , c 1=2 and more general isotropic tensor functions of c. J. Elasticity, 15 (1985) 319-323. [11] K.N. Morman. The generalized strain measure with application to nonhomogeneous deformations in rubber-like solids. Trans ASME, 53 (1986) 726-728. [12] E.J. Haug, K.K. Choi, and V. Komkov. Design Sensitivity Analysis of Structural Systems. Academic Press, New York, 1986. [13] K. Wisniewski and J.L.T. Santos. On design derivatives for optimization with a critical point constraint. Structural Optimization, 11 (1996) 120-127. [14] K. Wisniewski, E. Turska, and M. Kleiber. Comparison of two staggered schemes for optimization with a critical point constraint. Computer Assisted Mechanics and Engineering Sciences(4) (1997) 283-293. [15] F. Gruttmann, W. Wagner, and P. Wriggers. A nonlinear quadrilateral shell element with drilling degrees of freedom. Archive of Applied Mechanics, 62 (1992) 474-486. [16] J. Chroscielewski, J. Makowski, and H. Stumpf. Genuinely resultant shell finite elements accounting for geometric and material nonlinearity. Int. J. Num. Meth. Engng, 35 (1992) 63-94.

Part III Numerical Integration Methods for Rigid and Flexible Systems

Energy/Momentum Conserving Time Integration Procedures with Finite Elements and Large Rotations Michael A. CRISFIELD and Gordan JELENIĆ Department of Aeronautics, Imperial College of Science, Technology & Medicine London SW7 2BY, Great Britain. Abstract. The present paper is concerned with implicit techniques for dynamic non-linear finite element analysis. Probably the most popular technique is the Newmark trapezoidal rule [1], which may be augmented with the HHT-α procedure which provides numerical damping [2]. Work over the last few years has highlighted the fact that the Newmark trapezoidal rule is not unconditionally stable for non-linear systems and that the HHT-α method is not always guaranteed to provide dissipation [3]. As a consequence, much work has been devoted to the search for methods that are stable in the absence of dissipation; the aim being to provide the latter after stable algorithms have been developed. Much of the search for stable algorithms has been directed towards the preservation of key properties such as the energy, translational and angular momenta and symplectic nature of the system. The present paper will describe some of this work, with the emphasis being placed on the energy and the momenta. It is worth noting that we are here concerned with systems for which the strain energy plays a vital role. In these circumstances, it does not follow that the methods developed for rigid-body dynamics will provide the solution. The paper will start by considering solids and will then move on to beams and, finally, to flexible rotating systems including joints. Throughout the paper, emphasis will be placed on various forms of mid-point algorithm and the problems analysed will all involve large rotations. For solids, the variables will be the translations while, for the work on beams and flexible systems, the variables will include rotations. The latter will be treated via a multiplication of associated triads, although the stored quantities may be quaternions. For the joints, a non-linear master-slave approach [4] will be adopted. The procedure has been specially modified so that it preserves the energy and momenta of the system.

1.

Introduction

In this paper we cover three main topics and present some of our work over the past few years on several topics closely related to the design and implementation of numerical techniques for dynamic analysis of flexible dynamic systems featuring large spatial rotations. The first topic, which is addressed in Sections 1 and 2, relates to the development of numerically stable implicit dynamic integrators for 3D beams, typically through the algorithmic conservation of the constants of motion. Our work [5-8] follows on from the pioneering work of J.C. Simo's team, initially related to continua [9] and later to beams [10] and has links with other work in the field [11-15]. The second topic, presented in Section 3, deals with the dynamics of flexible systems with rotational degrees of freedom, mutually connected via different joints or hinges with specific kinematic properties. The practical analyses of structures made up of beams and joints differ depending on the area of applicability. Among other applications we can talk about robotics [16,17], design of mechanisms [18-23], aircrafts and satellites with

appendages [24,25], deployable structures [26-28] etc. The complexity in the analysis of these structures lies in the fact that the axes of kinematic releases translate and rotate along with the structure, which means that the joint kinematics has to be expressed in the bodyattached co-ordinate system. This proves to be a considerable problem in three-dimensional space, in particular if elasticity of the members is to be taken into account. A natural idea is to account for the joint kinematics through the use of Lagrange multipliers [20,21,23,2931]. However, instabilities can occur [29] and there can be difficulties introducing a timeintegration scheme, which takes account of the joint kinematics without leading to constraint violation [32,33]. Joints may be simulated using a penalty-type approach [18,34], but such a procedure is dependent on the choice of parameters, is prone to numerical illconditioning and lacks robustness [4,35]. A third idea is the introduction of the joint kinematics at a node prior to the assembly of the finite element mesh. In [4,35,36] this idea is called the ‘master-slave’ approach, while in [37] it was referred to as the ‘parent-child’ method. It has close links with the minimum set method, used with conventional multi-body dynamics techniques, although in the current context we emphasise its suitability within the finite element methodology, where the compatibility of displacements and rotations between elements is introduced depending on the type of joint being processed at a particular node. As a third topic, we will briefly describe the problem of strain invariance in 3D beams with large rotations [38,39] and comment on the complexities arising in the design of timestepping schemes which are both energy/momentum conserving and strain invariant.

2.

Local momentum conservation and continua

Our motivation in devising energy/momentum conserving algorithms is the desire to produce stable and effective time-marching procedures. However, this cannot be achieved by any algorithm that conserves these properties. This was demonstrated by Kuhl and Crisfield [40] in relation to the energy/momentum conserving procedures of Kuhl and Ramm [41]. The latter work ensures the conservation with the aid of Lagrange multipliers. Using this approach, the required properties are indeed conserved and the algorithm is stable in the sense that there is no spurious energy growth or momentum change. However, it is not effective because the stage can be reached where the time-marching process can proceed no further because the Newton-Raphson iterations fail to converge despite reductions in the time-step size. A key feature of the energy/momentum conserving algorithms that follow from the work of Simo and Tarnow [9] is the preservation of local element-level momenta. This feature will be explored here in relation to continua because the concepts are more easily visible in this context. We will initially consider a linear formulation and start with the dynamic equilibrium equations which are given by σ ∇ x + f − ρ !! x = 0,

(1)

where σ is a stress tensor, ∇xt = 〈(∂/∂x) (∂/∂y) (∂/∂z)〉, f is a vector of specific body forces, ρ is density of the material, a dot denotes a derivative with respect to time and x is position vector. As a start, we omit the inertia term and multiply equation (1) by the test functions, u=Iiui, where Ii are conventional isoparametric shape functions and a subscript-superscript pair of indices here and throughout the paper denotes a summation over all the components (here, over the nodes of the element). Following integration by parts and algebraic manipulation, we obtain

ui ⋅ (qki − qei ) = ui ⋅  ∫ σ J -1 (∇ξ I i )dV − ∫ I i f dV  V V 

(2)

= ui ⋅  ∫ σ (∇ξ I i )dV − ∫ I i f dV  = 0 V V 

where qki are the static internal forces, qei are the external forces at node i and V is the volume of the body. In addition, J is the Jacobian matrix which maps between the Cartesian co-ordinates, x, and the natural co-ordinates, ξ. The shape function property then ensures that ∑nodes∇ξ Ii = 0, so that ∑nodes qki = 0 and the static internal nodal forces for the element are in self-equilibrium. Again, for the static case, we can also show that the static internal forces satisfy rotational equilibrium, i.e. it can be shown that xi × qki = xi × ∫ σ (∇ξ I i )dV = 0

(3)

V

provided σ is symmetric. For the dynamic case (and without external forces), a similar procedure to that leading to equation (2) produces i qki + qm = ∫ σ (∇ξ I i )dV + ∫ ρ I i !! x dV = 0 V

(4)

V

The change of the translational momentum in time can now be expressed as L! = ∫ ρ ⋅1 ⋅ !! x dV = ∫ ρ ⋅ ( V

V

∑

I i ) !! x dV =

nodes

∑

i qm =−

nodes

∑

qki = 0

(5)

nodes

In equation (5), we have used the properties of the shape functions ∑nodes Ii =1, the dynamic equilibrium equation (4) and the earlier static self-equilibrium relationship ∑nodes qki =0. The change of the angular momentum, A = ∫ ρ x×x! dV , can be expressed as d A= dt

∫V ρ x! ×x! dV + ∫V ρ x×!!x dV = ∫V ρ ( I

i

i xi )×!! x dV = xi × qm = − xi × qki = 0

where we have finally used the static rotational self-equilibrium (3). If we now move to the non-linear case and attempt a similar argument with the implicit Newmark trapezoidal rule, we find that the difference of the angular momenta at times tn+1 and tn is not zero, i.e. An+1 − An ≠ 0. A solution can be found by using a form of mid-point procedure, which starts from the dynamic equilibrium equation

τ m ∇ xm −

ρ ∆t

( x! n+1 − x! n ) = (Fm S m FmT )∇ xm −

ρ ∆t

( x! n+1 − x! n ) = 0

(6)

which replaces equation (1). Here τ is the Kirchhoff stress, S is the second Piola-Kirchhoff stress, F is the deformation gradient and the subscript m denotes a “mid-point” state, so that Fm = ∂ xm/∂ X, where X is the vector of material co-ordinates. The precise form of Sm has yet to be defined. We can now apply a similar procedure to that used earlier to obtain qki ,m = ∫ τ m J m−1 (∇ξ I i ) dV = ∫ τ m (∇ξ I i ) dV . V

V

Very similar arguments apply to those given earlier and we can prove that the momenta are conserved provided Sm in equation (6) is symmetric. It can be shown that a particular choice of Sm that also ensures energy conservation [9] is Sm = 1/2C:(En+En+1), where C is the linear constitutive tensor and En and En+1 are Green's strain tensors at tn and tn+1. It is worth noting that the staggered or leapfrog explicit scheme (also called the Verlet method) also conserves the momenta, but not the energy.

3.

3D beams

3.1

Kinematics and strain-configuration relationships

In this section we briefly summarise the geometrically exact 3D beam theory. A more detailed account can be found in [10,11,38,39,42-48]. Let the line of centroids of cross sections of the undeformed beam element at time t0 be a straight line which coincides with the x axis of the element Cartesian frame (x,y,z) with G0,1, G0,2, G0,3 as the unit base vectors as drawn in Figure 1.

Figure 1. Initial and deformed configurations of the beam

At any time tk, the position vector of a material particle (x,0,0) on the line of centroids with respect to the inertial frame (X,Y,Z), with E1, E2, E3 as the orthonormal base vectors, is denoted by rk(x) ∀x ∈ [0,L] ⊂ ℜ, with L being the initial length of the beam. The cross sections of the undeformed beam in the co-ordinate plane x=const. at t0 are orthogonal to the line of centroids so that their normals coincide with the base vector G0,1(x) = r0′(x), where the dash (′) denotes a derivative with respect to the arc-length parameter x. The base vector Gn,1(x) in the deformed configuration is not necessarily tangential to the deformed centroid axis and the remaining two base vectors, Gk,2 and Gk,3, are directed along the principal axes of inertia of the cross section to make a right-handed orthonormal triad. The base vectors of the body-attached frame Gk,i; i=1,2,3 and the base vectors of the inertial frame Ei; i=1,2,3 are related via Gk,i(x) = Λ k(x) Ei ∀i=1,2,3, where Λ k(x) is an orthogonal transformation (direction cosine matrix), which will be referred to as the rotation matrix. Hence the configuration of the beam at x is completely defined by its position vector rk(x) and its rotation matrix Λ k(x). For any time tn > t0 the line of centroids of cross sections may deform into an arbitrary space curve as shown in Fig. 1. This curve is defined by the position vector rn(x)=r0(x)+un(x), where un(x) denotes a displacement of the material particle (x,0,0) at time tn. The nine components of a rotation matrix are mutually related by the conditions Λ −1 = Λ T and detΛ = 1, which leaves the rotation matrix dependent on only three parameters. In the

geometrically exact beam theory, it is convenient to parametrise the rotation matrix using the rotational vector ψ = ψi Ei ∈ ℜ3 as [44,49] 

sinψ



ψ

Λ = expψˆ Λ 0 ≡  Ι +

ψˆ +

1-cosψ

ψ2



ψˆ 2  Λ 0 ; ψ = ψ 

(7)

in which I is a 3×3 unit matrix. From here onwards, a superimposed hat (^) will always denote a skew-symmetric matrix of the hatted vector (note that for any v, w ∈ ℜ3 and Λ ∈ "v = Λ vˆΛT are valid). The rotation matrices at SO(3) the identities v×w = vˆ w = - wˆ v and Λ configurations n and n+1 are related via the incremental rotational vector θ which rotates the base vectors Gn,i onto the base vectors Gn+1,i around the axis θ/θ for the angle θ through Λ n+1 = exp θˆ Λ n. It is often useful to introduce a tangent-scaled incremental rotational vector ω, related to the unscaled incremental vector θ via ω=[tan(θ /2)/(θ /2)] θ. A direct calculation proves the relationship expθˆ = cay ω , where the Cayley transform cay ω is defined as [10] cay ω = I +

1 1+

1ω2 4

(ωˆ + 12 ωˆ 2 ) ; ω = ω

In the Reissner-Simo beam theory, the translational and rotational (material) strain measures γ and κ, defined with respect to the body-attached frame at x are related to r and Λ through the equations given in Tables 1 and 2 of [44] as 1    γ = Λ r ′ − 0  , κˆ = ΛT Λ ′ 0  T

(8)

A convenient expression for the vector of rotational strains at configuration n+1, which is related to the configuration n and the rotational vector θ between the two configurations is [10,45,46,48] κn+1 = κn + Λ nT T−T (θ)θ ′ = κn + Λ nT S−T (ω)ω ′, where T−1(θ) and S−1(ω) are defined as T −1 (θ ) =

1  sinθ 1 − θ θ2  S −1 (ω ) =

sinθ 1 − cos θ ˆ  I+ θ θ ⊗ θ + θ θ2  1

1 + 14 ω 2

( I + 12 ωˆ )

(9)

(10)

The linear elastic material is taken to be defined by a relationship between the stress and stress-couple resultants N, M and the adopted strain measures so that N=CN γ and M=CM κ , where CN = diag 〈 EA GA2 GA3 〉 and CM = diag 〈 GJt EI2 EI3 〉. Here, E and G denote the elastic and shear moduli of the material, A is the cross-sectional area, A2 and A3 are the shear areas in the directions of principal axes of inertia of the cross section, Jt is the torsional inertial moment of the cross section and I2 and I3 are the cross-sectional inertial moments about the principal axes of inertia of the cross section. It will be useful to introduce the specific angular momentum with respect to the centroid of the cross section π, which is defined as π = Λ Jρ W. In these equations, we have denoted the tensor of the mass moments of inertia as Jρ = ρ diag 〈 I2+I3 I2 I3 〉, while the angular velocity with components in the body attached frame (material angular velocity) is denoted as W.

3.2

Newmark's end-point scheme [47,48]

The end-point equilibrium at node i is given as gin+1 ≡ qik,n+1 + qim,n+1 − qie,n+1 = 0, where qik,n+1, qim,n+1 and qie,n+1 (which will not be considered in this paper) are the nodal vectors of internal, inertial and external forces respectively. Here, the nodal internal and inertial forces qik,n+1 and qim,n+1 follow from Appendix C of [48] as qki ,n+1

i qm ,n +1

=∫

L

0

L 

 i′  0  ΛN   I I   dx  i # i′  Λ M n+1 , ′ − I I r I n +1  

 dx , =∫  i ˆ J W + J W! )  0 I Λ (W ρ ρ  n +1 !! I i Aρ u

where the shape functions I i(x) are polynomials of degree N−1, which satisfy the standard conditions I i(xj) = δ ij , ∑Ni=1 I i(x) = 1 ∀x ∈ [0,L] ; i,j=1, …,N, where δ ij = 1 for i=j and δ ij = 0 otherwise. The end-point equilibrium qik,n+1 + qim,n+1 = 0 can be generalised along the lines of the α-method of Hilber, Hughes and Taylor [2] in several ways [10,29,47]. Here we follow our earlier approach [47] and set up the dynamic equilibrium as (1+α)qik,n+1−α qik,n+qim,n+1 = 0. Velocities and accelerations at time tn+1 are in Newmark's scheme defined using the displacements, velocities and accelerations at time tn, the displacements at time tn+1 and the parameters β and γ as [1] u! n+1 =

γ γ γ (un+1 − un ) + (1 − ) u! n + ∆t (1 − ) u!!n 2β β ∆t β

!!n+1 = u

γ

 un+1 − un − ∆t u! n − ∆t 2 ( 1 − β ) u !!n  , 2   β ∆t 2

!!n , u!n +1 and u !!n+1 are the velocities and accelerations at times tn and tn+1. where u! n , u As pointed out in [12], the application of such an integration to angular velocities and accelerations only makes sense if performed in a body-attached frame. In this way we obtain Wn+1 = W! n+1 =

γ γ γ ! ΛTn θn + (1 − ) Wn + ∆t (1 − ) Wn 2β β ∆t β γ

 ΛTnθ − ∆t Wn − ∆t 2 ( 1 − β )W! n  2 , β ∆t  2

where Wn , W! n , Wn+1 and W! n+1 are angular velocities and accelerations at times tn and tn+1, the components of which are given in body-attached frame Λ n (or Λ n+1). 3.3

Momentum conserving mid-point scheme based on difference of momenta [7]

As discussed earlier, the end-point dynamic equilibrium approach given in the previous section provides the conservation of the total translational momentum but does not provide the conservation of the total angular momentum. Following the directions indicated in Section 1, the latter important property can be conserved by redefining the nodal inertial force vector, so that it be based on the difference of the specific translational and angular

momenta at the end and the beginning of a time step divided by the time step length, rather than on the time derivative of the specific momenta at the end of the time step, i.e. i =∫ qm

L

0

1  I i Aρ (u! n+1 − u! n )    dx ∆t  I i (π n+1 − π n ) 

Also, it is necessary to provide the average position vector derivative ′ 1 ) 2 in the definition of the nodal internal force vector, i.e. rm′ = (rn′ + rn+  i′  L I I 0 n qki = ∫    dx 0  i$ i′   m   − I rm′ I I 

where n and m are arbitrarily chosen algorithmic spatial stress and stress-couple resultants, which may (but do not have to) be defined as n = Λ n+1 Nn+1 and m = Λ n+1 Mn+1 (as in the previous section). For purposes which will become apparent later, it is convenient to define them as n = Λ m Nm and m = T-1(θ) Λ n Mm, where (•)m=((•)n+(•)n+1)/2 and T-1(θ) is defined by equation (9). Finally, in order to guarantee the momentum conservation, the velocity update has to be performed as in Newmark's scheme with β = 1/4 and γ = 1/2. This algorithm does not conserve the total energy of the system, but the algorithmic spatial stress and stress-couple resultants given here, are specifically chosen to make the energy error as small as possible [50]. Two ways of adding the full energy conservation to this algorithm are given in the following two sections.

3.4

Energy and momentum conserving mid-point algorithm based on averaged stress resultants and tangent-scaled rotations [10]

In contrast to the algorithms given earlier, in this energy and momentum conserving algorithm it is essential to interpolate the tangent-scaled incremental rotations. Under this condition, the definition of the nodal inertial force vector remains the same as in the earlier momentum conserving algorithm, while the nodal internal force vector must be defined as  i′  L  0   Λm N m I I qki = ∫  dx   -1 0  i$ i′   S (ω ) Λn M m   − I rm′ I I 

where S−1(ω) is given by equation (10) and the angular velocity update has to be performed via Wn+1 = 2 ∆t ΛTnω − Wn , while the translational velocity update remains the same as in the momentum conserving algorithm given earlier. Some of the problems related to the angular velocity update, required in this algorithm, are analysed in [51].

3.5

Energy and momentum conserving mid-point algorithm based on adjusted averaged stress resultants

An alternative energy and momentum conserving algorithm to that given in the previous section can be provided on the basis of the momentum conserving algorithm given in Section 2.3. In this algorithm, there are no restrictions on the interpolation of rotations, which are typical of the energy and momentum conserving algorithm given in Section 2.4, and, in particular, there is no need to introduce tangent-scaled rotations. Here, the definition of the nodal inertial force vector and the velocity update is kept the same as in the momentum conserving algorithm, while the nodal internal force vector is redefined as

qki

=∫

L

0

 i′  * 0   Λm N m   I I  -1  dx  i $ i′  T (θ ) Λn M m   − I rm′ I I 

with N∗m = Nm+β Nd and Nd = Nn+1−Nn. The nodal internal force vector which depends on N∗m and Mm can thus be conveniently expressed as * qki ( N m , M m ) = qki ( N m , M m ) + β qki ( N d , 0 )

where the newly introduced coefficient β, which varies from element to element, is used to ensure the full energy conservation over a time step via * i  , M m ) + qm pi ⋅ qki ( N m = H n+1 − H n  

with Hn and Hn+1 being the total energies computed at the beginning and the end of the time step and pi being the complete set of incremental displacements and rotations at node i. The last two equations give i  H n+1 − H n − pi ⋅ qki ( N m , M m ) + qm   β= i pi ⋅ qk ( N d , 0 )

whenever pi qik (Nd,0)≠0. By denoting the kinetic energy at the beginning and the end of the time step as Tn and Tn+1, respectively, and by noting that the algorithm provides pi . qim=Tn+1−Tn by definition, an alternative way of computing the energy conserving coefficient β may be stated as

β=

φn+1 − φn − pi ⋅ qki ( N m , M m ) pi ⋅ qki ( N d , 0 )

where φn and φn+1 are the strain energies at the beginning and the end of the time step, respectively. 3.6

Invariant interpolation of large 3D rotations

The strain measures (8) satisfy the required invariance properties, but this is no longer true once the conventional finite element interpolations are added [44-47]. In order to preserve the invariance property, we decompose the rotation matrix via

Λ ( x ) = Λr exp Ψˆ l ( x )

(11)

where Λ r is a reference triad which can be taken as one of the nodal triads, although for two-noded elements, which will be chosen to run the numerical examples, it is more appropriate to define it as a triad defining the 'midpoint' rotation between the triads at two chosen nodes I and J via Λr = ΛI exp ( 1 2 φˆIJ ) , where the relative rotation between the nodes I and J is extracted from exp φˆ = ΛΤ Λ . In equation (11), Ψ l(x) are the local IJ

Ι

J

rotations with respect to the rotating reference frame, Λ r, which will be interpolated using standard Lagrangian polynomials and nodal local rotations Ψ li, extracted from exp Ψˆi l = ΛΤr Λi . An analytical proof that this interpolation provides invariant strain measures is given in [38] while numerical evidence is given in [39]. An alternative invariant, but path-dependent, formulation may be be defined in an incremental sense via

Λn+1 ( x ) = Λr,n+1 exp Θˆ l ( x) ΛΤr,n ( x) Λn ( x)

(12)

where the incremental local rotations Θ l(x) with respect to the rotating reference frame Λ r,n are interpolated using standard Lagrangian polynomials and incremental nodal local rotations Θ li, extracted from exp Θˆιl = ΛΤr,n+1 Λi,n+1 ΛΤi,n Λr,n . Both of these interpolations are applicable to the algorithms given in Sections 2.2 [39], 2.3 and 2.5. Due to the additional condition on the interpolation of rotations, in the case of the algorithm given in Section 2.4, only the incremental form of the invariant interpolation is applicable, and it must be defined using the incremental local tangent-scaled rotations l Ω l ( x) = 2 l tan(θ 2 )Θ l ( x) , so that exp Θˆ l ( x) in equation (12) is substituted with θ

cay Ω l (x). Furthermore, due to the additional problems related to the specific angular velocity update in this algorithm [51], this interpolation does not fully restore the invariance of strain measures. 3.7

Numerical example

We analyse the free flight of a flexible beam depicted in Fig. 2 and verify the conserving properties of the algorithms presented in Sections 2.3-2.5. The initial velocities, finite element mesh and the geometric and material properties of the beam are given in Fig. 2. This problem was solved in [47] using Newmark's trapezoidal rule and Hilber, Hughes and Taylor's method with α = −0.05 and α = −0.33. Here, we solve it by applying the momentum-conserving algorithm given in Section 2.3 and the two energy and momentum conserving algorithms, given in Sections 2.4 and 2.5. In all three cases we use the invariant interpolation of incremental local rotations described in Section 2.6 (note that, as mentioned at the end of Section 2.6, the invariant interpolation alone does not fully restore the strain invariance in the algorithm from Section 2.4 [51]). The dynamic equilibrium at a time step will be taken to be satisfied when the mean residual norm, defined as the square root of the mean value of the squares of the force components of the dynamic residual becomes smaller than 10−5, and when the displacement norm, defined as the square root of the sum of the squares of the iterative displacements as percentage of the square root of the sum of the squares of the total displacements becomes smaller than 10−9. Material Properties E = 1010 G = 5 ·109 ρ=1

Geometric Properties A = A2 = A3 = 1 I2 = I3 = 1/12 It = 1 / 6

Finite element mesh: four linear isoparametric elements Initial time step: ∆t = 0.008 Figure 2. Free flight of a flexible beam - geometry, initial conditions, properties and discretisation

All the three algorithms successfully solve the problem and all of them occasionally hit convergence problems in trying to satisfy dynamic equilibrium. When no converged solution is achieved, the time step is halved and then gradually increased in the consecutive steps until the original time-step length has been restored. If there were no time-step halving, the problem would be solved in exactly 25000 steps. Here, the momentum-conserving algorithm from Section 2.3 solves it in 25029 steps, the energy and momentum conserving algorithm

from Section 2.4 solves it in 25023 steps and the energy and momentum conserving algorithm from Section 2.5 solves it in 25090 steps.

Figure 3. a) Total energy against the response time in the three algorithms; b) Energy conservation numerical error in the two energy conserving algorithms

The last two algorithms conserve the total energy of the system by design and the momentum conserving algorithm manages to keep the total energy within a narrow band, as can be seen in Fig. 3.a. A closer inspection into the numerical conservation of the total energy in the two energy conserving algorithms shows that in fact there is some error, apparently due to the chosen convergence norms (see Fig. 3.b). It cannot be predicted what the effect of the accumulation of the numerical error in longer-term solutions would be. Despite a very good showing of the momentum conserving algorithm in this example, it must be kept in mind that, in this algorithm, there is no guarantee that the energy will remain bounded. Indeed, by just modifying a sequence of arithmetic operations in the computation of the position vector derivatives, a numerical instability is triggered which makes the total energy blow up before the quarter of the total response time has elapsed, which causes severe problems in achieving converged solution with multiple time-step reductions. The energy history in this case is plotted in Fig. 4.a. It should be noted that throughout the process the components of the total angular momentum remain conserved. See Fig. 4.b.

Figure 4. a) Total energy against the response time in the three algorithms; the momentum conserving algorithm uses an alternative arithmetic (*) in the computation of the position vector derivative; b) Components of the total angular momenta in the momentum conserving algorithm which uses an alternative arithmetic (*) in the computation of the position vector derivative

4.

3D beams with end releases

4.1

Joint kinematics

The principle of the master-slave approach is illustrated in Fig. 5.a. A beam element is assumed to be unconnected to the adjoining elements prior to the assembly of the structural equilibrium. In contrast to the standard assembly procedure, which at this stage introduces the conditions of compatibility of deformations at structural nodes, in the master-slave approach we introduce the existing joint kinematics. The node which is initially shared by a number of elements, one of which is not fully connected to the others, is in the deformed configuration no longer completely shared and from Fig. 5.a we establish the following relations: d = dm + ρ

Λ = Λ∗Λm

(13)

where dm and Λ m define the displacement vector and rotation matrix of a node taken to be the master node, and d and Λ define the displacement vector and rotation matrix of the disconnected (at least partially) slave node. The rotation matrices Λ m=[q1m q2m q3m] and Λ=[q1 q2 q3] are shown in Fig. 5.a in terms of their three constituent orthonormal vectors, while the translational displacements at the slave node, d, differ from those at the master node, dm, by the relative displacement vector, ρ and the rotation matrix (triad) Λ ∗ in equation (13) defines an equivalent multiplicative relative rotation.

(a)

(b)

(c)

Figure 5. a) Master, slave and released nodal variables; b) Released rotation in a revolute joint; c) Released translation in a prismatic joint

When modelling different types of joints, the master variables, dm and Λ m, are generally not entirely independent of the slave variables, d and Λ . Depending on the type of joint, some of the components of the displacement vectors, dm and d, and/or parameters of the rotation matrices, Λ m and Λ , can be the same. Different types of joints are defined by releasing displacements and/or rotations along/around chosen axes. However, in a geometrically nonlinear environment, these axes rotate together with the structure. In the present master-slave approach we define the master triad as the one along/around which base vectors q1m, q2m, q3m the actual releases take place. For translational joints, this means that the difference vector, ρ, between the master and slave variables is, when transformed into co-ordinates defined by the master triad, equal to the vector of released displacements s = Λ Tm ρ, where the vector of released displacements, s, has zero components in non-released directions. In a similar fashion, if from the rotation difference matrix, Λ∗ = exp ϕˆ * , we extract the rotational vector ϕ ∗, and transform it into co-ordinates defined by the master triad, the result must be equal to the rotational vector of released rotations ϕ , i.e. ϕ = Λ Tm ϕ ∗, where ϕ has

zero components in non-released directions. For a revolute joint [52] which allows rotation about the rotating vector q1m, we would only consider its first component to be non-zero. This is illustrated in Fig. 5.b. An example of a prismatic joint allowing for a translation along the rotating vector q2m is given in Fig. 5.c. More details on the master-slave kinematics can be found in [4]. Following on from this discussion, equation (13) may be rewritten as d = d m + Λm s

Λ = Λm exp ϕˆ ≡ Λm cayψ

In order to reduce problems associated with excessive magnitudes of ψ and, in addition, to pave the way for the later work on dynamics, it is useful to work with incremental (over a time step), rather than the total released rotations (from now on, we will use only the form with tangent-scaled rotations). This gives Λ n+1 = Λ m,n+1 cayαr Λ Tm,n Λ n, where αr is the incremental released tangent-scaled rotation. 4.2

End-point dynamic equilibrium based on virtual work

The relation between the master and the released variations on the right-hand side and the slave variations on the left-hand side for each node i on the beam follows as

δ pi = H i δ pmr ,i

(14)

T T where δ piT = δ d iT δβ iT , δ pTmr ,i = δ siT δα rT,i δ d m ,i δβ m,i , indices m and r stand for β is the “spin” variable coming from δΛ = δβˆ Λ and “master” and “released”, respectively, δβ

 Hi =  

Λm,i

0

Ι

# −Λ m,i si

0

Λm,i S −1 (α r ,i )

0

Ι

  

Application of the principle of virtual work to a general N-noded beam element gives δ p gi = 0, where gi is the nodal dynamic residual or “out-of-balance” load vector. We recall the important principle of the master-slave approach: An existing element is taken as it is and subjected to additional processing at the point of assembling the structural equilibrium. This processing depends on the choice of unknowns and their variations and in the present work the formulation has been applied only to particular type of rotational variations - the spin variables δβ. Therefore, for present purposes it is not necessary to look closely into the definition of gi - any nodal dynamic residual can be processed within the present master-slave methodology provided it is work conjugate to δ pTi = 〈δ dTi δ βiT〉. The introduction of equation (14) into δ pTi gi = 0 now leads to the virtual work equation T i

( H δ pmr )i ⋅ gi = 0 This equation must hold for any virtual variables, δ pmr,i (i=1, …,N) and hence we can obtain the equilibrium equation at every node i as

(

)

i

i gmr ≡ HT g = 0

4.3

Mid-point dynamic equilibrium based on incremental energy

In contrast to the earlier standard approach, the energy and momentum conserving algorithms do not originate from the principle of virtual work. Instead, the energy conservation implies that the nodal dynamic residual gi should be extracted from the condition that the change in total energy H over a time step remains equal to zero [3,915,50]. The relation between the incremental master and released displacements and tangent-scaled rotations on the right-hand side and the incremental slave displacements and tangent-scaled rotations on the left-hand side for each node i on the beam can therefore be expressed as pi = Ei pmr ,i T T where pi = uiT α iT , pmr ,i = urT,i α rT,i um ,i α m,i ,

C 0 I Di  Ei =  i   0 Ai 0 Bi 

the matrix coefficients Ai, Bi, Ci and Di must satisfy the kinematic conditions

αi =

1−

1α 4 m,i

(

1 α m,i + Λm,i,n α r ,i + 21 α m,i × Λm,i,n α r ,i ⋅ Λm,i ,n α r ,i

)

# ui = um,i + Λm,i ,n+ 1 ur ,i − Λ s α m,i ,n + 1 i , n + 1 m,i 2

2

2

i

For any nodal dynamic residual g that is energy conserving, a generalised energy conserving dynamic residual can be expressed as

(

)

i

i gmr ≡ ET g = 0

The conservation of the total momenta and the total energy is provided for any momentum and energy conserving gi by the following choice of the matrix coefficients Ai =

Ι + αˆ m,i + 14 α m,i ⊗ α m,i Λm,i,n 1 − 14 α m,i ⋅ Λm,i ,n α r ,i

Bi = I C i = I − 14 αˆ m2 ,i Λm,i ,n+ 1 2 1 # # Di = − Λm,i , n si ,n + Λm,i ,n+1 si ,n+1 2

(

(

)

)

More detail on the master-slave dynamics can be found in [35].

4.4

Numerical example

We analyse the motion of a double flexible pendulum under the impact of the linearly varying initial velocities given in Fig. 6. This example was solved in [30,31] by using a staggered explicit time integration scheme with good energy-conserving properties. In [31] it was suggested that for complex flexible multi-body systems with different types of joints ‘the most one can realistically hope for is the preservation of the system energy’. However, we will demonstrate that the conservation of both the energy and the momenta for such

systems is possible. A formal proof, based on the theory outlined in Section 3.3, can be found in [35].

Figure 6. Flexible double pendulum: properties and initial conditions

Each beam is modelled using eight two-noded beam elements and we have adopted a time-step length ∆t = 0.008, which is approximately five times smaller than the period of the second bending mode of natural vibration. No detail was given in [30,31] as to the time-step size applied therein, but due to the explicit nature of the integration applied to kinematic quantities (implicit integration was used for the Lagrange multipliers) it had to be much smaller. The analysis is run for a total response time t=8. For the given geometric and material parameters, the initial energy H0 (which is equal to the initial kinetic energy T0) and the components of the initial angular momentum J around the co-ordinate axes Y and Z can be computed as H0 ≡ T0 = 891 000, Jy0 = 641 520 and Jz0 = 106 920.

Figure 7. Total energy for the flexible double pendulum

The following results have been obtained by utilising isoparametric strain-invariant elements [39] in conjunction with the Newmark's trapezoidal rule [1] and Hilber, Hughes and Taylor's α-method [2] with α = −0.05 (described in Sections 3.2, 3.6 and 4.2) and also with Simo, Tarnow and Doblare's energy-momentum method [10] applied to the conserving master-slave technique (described in Sections 3.4 and 4.3).

Figure 8. Non-zero components of the total angular momentum for the flexible double pendulum

We use the following two convergence criteria: (i) the displacement norm, defined in Section 2.7, must be less than 10−9, and (ii) the residual norm, defined as the square root of the sum of the squares of the nodal residual forces (the force part of the nodal residual vector gi) over all the nodes in the structure as percentage of the square root of the sum of the squares of the total reactions, must be less than 10−6. Newmark's trapezoidal rule experiences a well-known numerical instability which results in the sudden growth of the total energy near the completion of the first revolution of the pendulum. Fig. 7 shows that here the α-method successfully solves the problem, but in so doing dissipates almost a third of the total energy of the system, despite the fact that only 5% of numerical damping has been applied. In contrast, the proposed conserving methodology for joints successfully solves the problem and by design conserves the total energy of the system. As shown in Fig. 8, the two non-zero components of the total angular momentum are also conserved by the proposed formulation. A similar graph could also be plotted for the remaining component of the total angular momentum. Neither the α-method nor the proposed conserving method have required any time-step halving in order to complete the analysis. Fig. 9 shows the history of motion of the pendulum during the first and the second revolution, obtained with the proposed conserving formulation. Due to its simple kinematics, the spherical joint can be easily simulated through the use of linear constraint equations, whereby the two elements joining at the spherical joint would inititaly be left unconnected and then the three components of displacement at the joint would be constrained to be the same at both elements. This fact was also recognised in [31]. However, this technique cannot be applied to the more sophisticated revolute or universal joints or those that release translational degrees of freedom, because in these joints the axes of release rotate. The present master-slave technique, however, is equally applicable to all types of joints and in [35] the problems where the spherical joint between the two beams is substituted with (i) a revolute joint with the released component of the rotation around the axis that initially coincided with the global Y-axis and , (ii) a prismatic joint with the released component of the translation along the axis that initially coincided with the global X-axis were successfully solved.

Figure 9. Deformed configurations of the flexible double pendulum during the first and the second revolution (conserving formulation)

5.

Conclusions

In this paper we have presented some of our recent work on the development of conserving dynamic intergrators for flexible systems with large rotations. We have emphasised the problems in which large spatial rotations exist as independent degrees of freedom. Further to the already accepted conclusion that the energy conservation is on its own too weak a condition to ensure a stable and robust solution procedure, our results demonstrate that the momentum conservation alone, is also not sufficient to provide a stable solution. Design of algorithms which conserve both the momenta and the energy is, in the case of systems with large rotations, far from being straightforward. Additional complexities arise when a strain invariant solution is required. In this work, we have presented two of the possible approaches at designing energy and momentum conserving algorithms. One of them was already given in [10], and is based on the use of tangent-scaled rotations. This choice brings additional difficulties in any attempts at providing strain invariant solutions, but enables an establishment of straightforward kinematic relations, which extends the conservation properties to systems with end releases. The other approach is based on the unscaled rotations and adjusted algorithmic stress resultants.

Acknowledgements

The second author is supported by Engineering and Physical Sciences Research Council under Advanced Research Fellowship AF/100089. The support is gratefully acknowledged.

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EfÞcient and Robust Computational Algorithms for the Solution of Nonlinear Flexible Multibody Systems Gregory M. HULBERT Mechanical Engineering Department, University of Michigan 2250 G.G. Brown, Ann Arbor, Michigan 48109 USA Abstract. Many numerical methods have been proposed to solve problems in multibody dynamics and in nonlinear continuum mechanics. The numerical challenge remains one of efÞcient and robust solution of the underlying nonlinear systems. This paper presents an examination of two classes of numerical methods for solving nonlinear ßexible multibody systems; namely, energy and momentum conserving algorithms and numerically dissipative schemes from the so-called alpha family of methods. Performance of the algorithms is studied in the context of two simple benchmark problems. The aim of this paper is to provide some guidance to the analyst for choosing appropriate numerical methods for solving nonlinear ßexible multibody systems.

1. Introduction Nonlinear structural systems with large rigid body motions embody mechanics from the multibody dynamics community and from the continuum mechanics community. With the notable exception of Benson and Hallquist [1], until the early 1990's, each community independently developed formulations for dynamical systems and the numerical solution of the time-dependent equations that emanated from spatial and temporal discretization of such dynamical systems. From the multibody dynamics community, formulations and solution methodologies were proposed and implemented to treat geometric nonlinearity, arising from the large rotations inherent in three-dimensional rigid body dynamics, and to to treat nonlinearities due to joints connecting multibodies. From the continuum mechanics community, formulations and solution techniques were developed to handle geometric and material nonlinearities. During the past 15 years, numerous researchers have traveled between both communities to develop formulations and solution methodologies that can treat the full spectrum of nonlinear behavior associated with nonlinear structural systems undergoing large geometric motions. Despite the proliÞc body of reported research related to ßexible multibody systems, there remains the need to address efÞcient and robust solution of the underlying nonlinear systems. For the solution of fast transient problems, such as crashworthiness, the explicit central difference scheme remains the canonical algorithm, at least in the context of nonlinear continuum

mechanics. In the multibody dynamics community, explicit algorithms of choice typically emanate from the Runge-Kutta family. For problems in which the time scales are on the order of global motion, i.e., structural dynamics (vibration), implicit algorithms have been shown to be more efÞcient, in general. This notion of time scale leads to an intuitive and fundamentally correct classiÞcation of stiff/non-stiff systems. A stiff system is one in which there exists fast-scale dynamics that are not of fundamental interest or importance. An example of a stiff system relevant to nonlinear ßexible multibody dynamics is that engendered by discretizing the spatial domain using standard Þnite element techniques. The high-frequency modes of the Þnite element model are poorly resolved; thus they are not of fundamental interest. However, the same Þnite element model, when employed in crashworthiness studies, does not produce a stiff system of equations because of the fast-scale dynamics associated with crush and element/node contact. Although crashworthiness problems typically are solved using explicit time integration schemes and structural dynamics problem typically are solved using implicit time integration schemes, the deÞnition of system stiffness should not refer to the type of numerical integration scheme employed. There are many examples in the literature (and in daily practice) that employ explicit time integration methods for non-stiff systems, e.g., metal forming. There have been two different trends towards developing time integration algorithms for ßexible multibody systems. One approach has been to develop enhanced time integration algorithms with numerical dissipation that damps out spurious high-frequency oscillations. Such oscillations may emanate from the Þnite element discretization of the structure, or may result from temporal discretization of constraint equations associated with modeling joints. Inclusion of joints through the use of constraint equations and Lagrange multipliers leads to well known differential-algebraic equation (DAE) systems. Numerical dissipation has been shown to provide a stabilizing mechanism for solving DAE's, see, e.g., [2-5], and has been widely used in the structural dynamics community [6-7] and in commercial Þnite element codes, such as ABAQUS, [8]. The second modern trend in time integration algorithms for ßexible multibody systems is the development of methods that inherit the conservation properties of Hamiltonian dynamical systems, namely, linear momentum, angular momentum, and energy. Conservation of such properties can be of importance when studying the long-term dynamics of systems. Conservation of these quantities also implies algorithmic stability, at least in the context of the classical energy notion of stability. References 9-12 provide some examples of such conservation algorithms. Results presented in these references clearly demonstrate the conservation performance of this class of algorithms. There appears to be a dichotomy between these two approaches to time integration of ßexible multibody systems. On the one hand, numerical dissipation has been clearly shown to be of beneÞt for the stiff systems that arise from ßexible multibody system formulations. On the other hand, with the exception of the trapeeze problem presented in [4], the conserving schemes do not appear to suffer from the absence of numerical dissipation. Clearly, part of the robustness of the conserving algorithms is due to the careful construction of the algorithms to maintain the conservation properties, rather than adding the conservation properties as an afterthought. In this regard, the modern conserving algorithms should not be grouped with more general time integration methods that add conservation rules as, e.g., constraint equations.

In light of this wide array of numerical methods and published research, the following questions naturally arise: 1. Should conserving algorithms or dissipative algorithms be employed? 2. If conserving algorithms are chosen, what quantity or quantities should be conserved? 3. What is the role of numerical dissipation? 4. Do kinematic constraints, which arise from joint connections, reduce solution robustness? 5. Does the range of eigenvalues inßuence solution robustness? 6. Are there simple benchmark problems with which to assess the performance of different algorithms? This present study is directed towards addressing these equations by evaluating the robustness of a small subset of dissipative algorithms and conserving algorithms. The evaluation is performed using several problems that involve the dynamics of a three-dimensional nonlinear rod formulation. The nonlinear rod was chosen due to its inherent nonlinear geometric behavior, speciÞcally the large rotation aspects common to all multibody systems. In addition, the nonlinear rod has a simple mathematical representation and implementation into Þnite element codes so that the study of robustness is not obscured by other complex mechanics and the questions of implementation of such mechanics. In section 2, a brief overview of the nonlinear rod formulation from [9] is provided. Section 3 presents the time integration algorithms to be studied; their performance on 2 model problems is detailed in section 4. The Þnal section provides a summary and some conclusions of this study, with the goal to provide guidance to the analyst in choosing appropriate numerical methods for solving nonlinear ßexible multibody systems. 2. Governing equations for three-dimensional rod dynamics The rod conÞguration is deÞned in terms of the position of the line of centroids, φ(S, t), where t denotes time and S denotes the coordinate along the line of centroids of the undeformed rod. The reference conÞguration is denoted by φ0 (S). Each cross-section of the rod is identiÞed by a set of body-Þxed axes, t1 , t2 , with t3 = t1 × t2 tangent to the line of centroids. The orientation of this body-Þxed frame relative to an inertial Þxed basis, e1 , e2 , e3 , is speciÞed via the orthogonal transformation, tI

=

ΛeI ,

for I = 1, 2, 3

(1)

The velocity Þeld of the points on the line of centroids and the spatial angular velocity are given by: v = φ˙ (2) ˆ ˆ = ΛΩ Λ˙ = ωΛ where the dot superscript denotes differentiation with time and the superposed hat denotes the skew-symmetric matrix associated with the corresponding vector. Thus, ω and Ω are the spatial angular velocity and body-Þxed angular velocity, respectively. Let Aρ denote the mass per unit length associated with a cross-section of the rod while J denotes the inertia dyadic in which J33 denotes the polar moment of inertia of the cross

section, while Jαβ , α, β = 1,2, denotes the inertia moments of inertia of the cross-section relative to the initial orientations t1 (0), t2 (0). The spatial inertia dyadic is given by j

=

ΛJ ΛT

(3)

With these deÞnitions, the linear and angular momenta are given as p = Aρ v ,

π = jω

(4)

Denoting the resultant force and unit couple per unit of reference arc-length as n and m, respectively, the linear and angular momentum balance equations are 0

p˙

=

¯ n +n

π˙

=

¯ m +φ ×n+m

0

0

(5)

¯ and m ¯ are the respective externally applied force and couple (per unit of reference in which n arc-length), and where the prime superscript denotes differentiation with respect to S. Boundary conditions and initial conditions (φ0 , Λ0 , v0 , ω0 ) complete the deÞnition of the initial boundary-value problem. For the problem at hand, the strain measures and the force and couple resultants are deÞned as: 0 0 Γ = ΛT φ − ΛT 0 φ0 (6) c = ΛT Λ0 − ΛT Λ0 W 0 0 n m

= =

ΛCN Γ ΛCM W

(7)

where CN = DIAG[GA1 , GA2 , EA] ,

CM = DIAG[EI1 , EI2 , GJ]

(8)

in which E is the Young's modulus, G is the shear modulus, A is the cross-sectional area of the rod, E1 and E2 are the principal area moments of inertia of the cross-section relative to t1 (0), t2 (0), and J is the polar moment of inertia (for torsion). 3. Time Integration Algorithms For this study, attention is focused on two types of single-step, multi-value time integration algorithms, an energy-momentum conserving algorithm, and variants of the so-called alpha family of algorithms. In the following, ∆t denotes the time step, i.e., ∆t = tn+1 − tn , n denotes the time step number, and subscript n denotes the approximation to the subscripted Þeld quantity at time tn . 3.1. Energy-Momentum Conserving (EMC) Algorithm The energy-momentum conserving algorithm employed in this study is presented in [9]. Rather than reiterate the details, the salient features of the algorithm are summarized below.

Update equations for the position and rotation matrix are given as φn+1

=

Λn+1

=

ϑ

=

∆t (vn + vn+1 ) 2 cay[ϑ]Λn ∆t (ωn+1 + cay[ϑ]ωn ) 2 φn +

(9)

in which ϑ is the spatial incremental rotation vector, and cay[ϑ]

=

2 I+ 1 1 + 4 ||ϑ||2

µ

1 ˆ 1 ˆ2 ϑ+ ϑ 2 4

¶ (10)

The discretized momentum balance equations are 1 (pn+1 − pn ) ∆t 1 (πn+1 − πn ) ∆t

= =

0

¯ n +n 0

(11)

0

¯ m +φ ×n+m

in which n

=

Λn+1/2 N

m

=

Λ∗n+1/2 M

N

=

M

=

Λn+1/2

=

Λ∗n+1/2

=

φ

=

1 CN (Γn+1 + Γn ) 2 1 CM (Wn+1 + Wn ) 2 1 (Λn+1 + Λn ) 2 det[Λn+1/2 ]Λ−T n+1/2

(12)

1 (φn+1 + φn ) 2

3.2. Hilber-Hughes-Taylor-α (HHT-α) Algorithm This implementation of the HHT-α algorithm follows that suggested in [9]. The algorithmic form of the momentum balance equations is given as 1 (pn+1 − p˜n ) γ∆t 1 ˜ n) (πn+1 − π γ∆t

= =

0

¯ n+αH nn+αH + n 0

mn+αH

0

(13)

¯ n+αH − φn+αH × nn+αH +m

where p˜n

=

pn + (1 − γ)∆tAρ an

˜n π

=

πn + (1 − γ)∆t (jn αn + ωn × jn ωn )

(14)

The spatial force and couple resultants are evaluated using nn+αH

=

Λn+αH CN Γn+αH

mn+αH

=

Λn+αH CM Wn+αH

where

0

(15)

0

Γn+αH

=

T ΛT n+αH φn+αH − Λ0 φ0

cn+α W H

=

T ΛT n+αH Λn+αH − Λ0 Λ0

φn+αH

=

αH φn+1 + (1 − αH )φn

Λn+αH

=

exp(αH ϑ)Λn

0

and

0

(16)

(17)

The external force and couple are evaluated via ¯ n+αH n

=

¯ n+αH ) = n(α ¯ H tn+1 + (1 − αH )tn ) n(t

¯ n+αH m

=

¯ n+αH ) m(t

(18)

Note that in the implementation studied herein, the resultant forces and resultant couples are evaluated using a Òmidpoint"-type approximation rather than a trapezoidal-type approximation. To complete the algorithmic description, four additional update equations are required; these are the standard Newmark formula: µ ¶ 1 2 φn+1 = φn + ∆tvn + ∆t ( − β)an + βan+1 2 vn+1

=

ωn+1

=

αn+1

=

vn + ∆t ((1 − γ)an + γan+1 ) µ ¶ β − 12 γ γ T β−γ ωn + ∆t αn + ϑn Λn+1 Λn β β β∆t µ ¶ 1 1 1 T 2 −β ωn − αn Λn+1 Λn ϑn − ∆t2 ∆t β

(19)

Following [7], the HHT-α parameters may be deÞned in terms of ρ∞ , the algorithmic spectral radius at ω∆t → ∞. The algorithmic parameters are given as µ ¶2 2ρ∞ 3 1 1 +γ , γ = − αH , β= (20) αH = 1 + ρ∞ 2 4 2 For linear problems 1 ≥ ρ∞ ≥ algorithm.

1 2

deÞnes the range of unconditional stability for the HHT-α

3.3. Simo-Tarnow-Doblare-α (STD-α) Algorithm In [9], a variant of the α algorithm was proposed. Although not explicitly stated, it can be inferred that the algorithmic parameters γ and β are given in terms of α as γ β

= =

3 −α 2 µ ¶2 1 1 +γ 4 2

(21)

with α > 1/2. The upper limit on α is not speciÞed; for this study, 1/2 ≤ α ≤ 2/3. 3.4. Generalized-α (g-α) Algorithm The generalized-α algorithm was originally developed for linear structural dynamics in [7]. It was shown to have optimal dissipation properties in the sense that for a given value of high-frequency dissipation, ρ∞ , the low-frequency dissipation is minimized. For the present study, the g-α implementation follows the same form as (13)-(19), with αH replaced by αf and the left-hand sides of (13) replaced by: 1 (αm pn+1 − (1 − αm )p˜n ) γ∆t 1 ˜ n) (αm πn+1 − (1 − αm )π γ∆t

(22)

The algorithmic parameters are deÞned as αm

2 − ρ∞ = , 1 + ρ∞

2ρ∞ αf = , 1 + ρ∞

1 γ = − αf + αm , 2

1 β= 4

µ

1 +γ 2

¶2 (23)

4. Numerical Results In all of the numerical experiments, Þxed time step sizes were employed. NewtonRaphson iteration was used so that a fair comparison of algorithmic performance could be achieved without the need to consider additional iterations due to non-use of consistent tangent matrices throughout. Convergence of the iteration loop was controlled by the L2 norm of the residual of the momentum balance and constraint equations. An error tolerance of 10−8 times the initial residual was used for convergence. Calculations terminated if the number of residual evaluations exceeded 20 in any given time step. Robustness is assessed by comparing the number of steps required to converge the Newton iteration loop. 4.1. ÒFlying spaghetti" with uniform mesh The so-called Òßying spaghetti" problem of [9] was Þrst examined. The initially straight beam, of length 10, is aligned with the e3 axis at t0 . Forces and couples are applied to one end of the beam; their values linearly increase until t = 2.5, and then linearly decrease to zero at t = 5, after which time there are no applied forces or couples. The values of the applied forces and couples are given by M1

=

−p(t)

M2

=

0.3p(t)

M3

=

0.4p(t)

F2

=

0.08p(t)

F3

=

−0.06p(t)   80t

for 0 < t ≤ 2.5

 200 − 80(t − 2.5) 0

for 2.5 < t ≤ 5 5 0.6. This may be observed in Figure 1 as the opposite curvature of the HHT-α method energy time history, when compared to the other dissipative methods. Table 1 provides the total number of function evaluations and the maximum number of function evaluations in a time step for the different algorithms and algorithmic parameters, for the ∆t = 0.2 case and with the Þnal time, Tf inal = 100. Note that the EMC algorithm has the smallest maximum number of function evaluations; this suggests that a different predictor scheme may be more appropriate for the dissipative methods, at least for the HHT-α and g-α methods. The most highly dissipative method, the STD-α scheme, requires the fewest total number of iterations; this results from the dissipation smoothing out the dynamics after the external loading is removed (t > 5). The less dissipative α methods require almost 2 extra iterations per time step than the other methods. Thus, for this smaller time step size, the price of the robustness gained for larger time steps is paid in loss of efÞciency at smaller time step sizes. Table 1. Summary of Number of Function Evaluations for ∆t = 0.2 Algorithm Algorithmic Number of Maximum Number of Parameters Function Evals Function Evals EMC 2508 6 STD-α α = 0.51 2367 7 ρ∞ = 0.6 3444 8 HHT-α HHT-α ρ∞ = 0.5 3346 8 g-α ρ∞ = 0.9 3612 8 ρ∞ = 0.8 3618 8 g-α g-α ρ∞ = 0 2561 7

EMC

700

g-α ρ∞ = 0.80

Total Beam Energy

600

HHT-α ρ∞ = 0.5

500 400

HHT-α ρ∞ = 0.6

g-α ρ∞ = 0

300 200 STD-α α = 0.51

100 0 0

20

40

60

80

100

Time Figure 2. Total beam energy computed from different algorithms, ∆t =0.2

Similar behavior was observed when employing a smaller time step size of ∆t = 0.1. For this case, the EMC algorithm requires the fewest iterations since the dissipative schemes have less numerical dissipation with smaller time step sizes.

4.2. ÒFlying spaghetti" with non-uniform mesh and impulsive loading To assess the inßuence of high-frequency oscillations on the performance of the various time integration algorithms, the ßying spaghetti problem was simulated with the addition of an axial initial speed of -100 units/sec on the end node of the beam opposite to that of the applied loads; this speed is the axial wave speed in the beam (bar wave speed). Thus, the addition of this initial velocity condition injects a initially compressive impulsive load on the end of the beam. Such loading is well-known to excite the high-frequency modes of a structure. To accurately capture the dynamics of such impulsive loadings, the explicit central difference method is the canonical algorithm of choice. For the present study, however, the primary interest is on the performance of the time integration algorithms, with secondary interest on the momentum and energy measures of algorithmic response. To create a wider spread in the discretized beam's natural frequencies, the Þrst element of the beam (that which has the applied loads) has a length of 0.05, the last beam element has length 0.1; the remainder of the 10 unit length is uniformly discretized with 8 elements. The natural frequencies of this model range from .22 Hz to 487 Hz. With ∆t = 0.2, the EMC algorithm cannot converge beyond t = 3.6. Convergence is attained when ∆t = 0.1. Table 2 summarizes the total number of function evaluations and the maximum number of function evaluations in a time step for the different algorithms and algorithmic parameters, for this second problem to advance from t = 0 to t = 50. For this problem, the dissipative schemes not only are more robust, with respect to time step size convergence, but also, for the step sizes reported, show increased efÞciency as well, compared to the EMC algorithm. The dissipative algorithms annihilate the high-frequency energy induced by the impulsive loading on the beam, and, depending on the value of algorithmic parameters, dissipate at various rates the energy associated with the external loading. It is again noted the the STD-α method dissipates more energy than the g-α method, which is to be expected as the g-α method is designed to minimize low-frequency dissipation. Table 2. Summary of Number of Function Evaluations for Problem 2 Algorithm Algorithmic ∆t Number of Maximum Number of Parameters Function Evals Function Evals EMC 0.2 no convergence EMC 0.1 3421 9 STD-α α = 0.51 0.2 1362 8 α = 0.51 0.1 2407 6 STD-α HHT-α ρ∞ = 0.6 0.2 1707 8 HHT-α ρ∞ = 0.6 0.1 2833 6 g-α ρ∞ = 0.5 0.2 1708 8 g-α ρ∞ = 0.5 0.1 2843 6 g-α ρ∞ = 0 0.2 1346 7 ρ∞ = 0 0.1 2557 6 g-α

4.3. ÒFlying spaghetti and meat ball" loading To assess the inßuence of kinematic constraints on the robustness of the above time integration schemes, a rigid body was attached, using a spherical joint, to the end of the beam where the load (24), was applied. The joint formulation employed Lagrange multipliers on the position constraints. No condensation of Lagrange multipliers was employed; thus the

system can be classiÞed as an Index 3 problem (see [5] for an excellent discussion of system Index). The joint was parameterized using global position degrees-of-freedom rather than relative coordinates. Thus, the problem retains the physical simplicity of the beam model while providing a direct assessment of the inßuence of kinematic constraints on solution robustness. The mass of the rigid body is 10., the inertia dyadic is diagonal with value 20. for each diagonal. The initial position of the mass center of the rigid body was located one unit from the end of the beam along the beam axis. With this formulation, the robustness of the algorithms was identical to that obtained from problem 4.1. That is, the introduction of the joint constraints had no inßuence on the performance of the algorithms studied. 5. Summary and Conclusions The performance of an energy and momentum conserving (EMC) algorithm, and several different dissipative algorithms, were evaluated in the context of physically simple, geometrically nonlinear example problems. It is apparent that for the problems studied, the EMC algorithm could attain convergence when the step size was reduced. As the frequency content of the excitation increased, computational efÞciency of the EMC method decreased, when measured with respect to the number of iterations required for convergence. The dissipative methods all proved to be more robust than the EMC method for large time step sizes, although it is important to note that these methods require some time step size restriction to control solution accuracy. The HHT-α algorithm, as implemented in this paper, exhibits abnormal behavior when ρ∞ exceeds 0.6. Similar abnormality is observed with the generalized-α method when ρ∞ is close to 1. The STD-α method showed no such abnormal response; however, it is by far the most dissipative scheme in the low-frequency domain of all the dissipative methods studied. Conserving algorithms do provide some inherent robustness compared to the more traditional time integration schemes that embodied conservation properties for linear systems, e.g., trapezoidal rule. However, the implementational overhead of conserving algorithms is fairly high as they require careful formulation of the element technology, which requires a re-coding in most Þnite element programs. Numerical dissipation was seen to play no role when kinematic joint constraints were introduced. This suggests that numerical dissipation is not a requirement for systems with kinematic constraints, provided that the underlying time integration scheme is robust. It has been demonstrated clearly that numerical dissipation is necessary when the underlying algorithm does not possess the robustness of conserving schemes. Based upon this study, it does not appear that numerical dissipation enhances robustness for stiff systems. However, numerical dissipation is important for damping spatially unresolved deformations. Such spurious oscillations may lead to solution termination if they are not suppressed. The results presented in this paper suggest that an Òoptimal" time integration method for nonlinear ßexible multibody systems would embody the best of both the dissipative schemes and the conserving schemes. The dissipative methods could be employed when the conserving schemes cannot reach convergence with a given step size. Then, using some appropriate algorithmic measure, the conserving schemes could be used again when the solution

was deemed to lie in the smoother solution domain. Alternatively, methods could be developed that selectively dissipate unresolved physics in the model, rather than simply all high-frequencies. The problems proposed here can provide a starting testbed for researchers to study other time integration algorithms, joint formulations and Þnite element formulations. One simple enhancement to the proposed set would be to employ a revolute joint instead of a spherical joint. The present study has left unanswered two important questions: 1. What quantities are important to enforce, preserve, conserve, or maintain as accurately as possible? 2. What are the costs and beneÞts of particular computational algorithms for ßexible, nonlinear multibody dynamics? The results presented in this paper can provide some initial thrusts towards answering these critical questions. Acknowledgment The author gratefully acknowledges partial support of this research by the U.S. Army Tank-Automotive Research, Development and Engineering Center, through the Automotive Research Center, a U.S. Army Center of Excellence, under Department of Defense contract number DAAE07-98-3-0022. References [1] D. J. Benson and J. O. Hallquist, A Simple Rigid Body Algorithm for Structural Dynamics Programs, International Journal for Numerical Methods in Engineering 22 (1986) 723-749. [2] A. Cardona and M. Geradin, Time Integration of the Equations of Motion in Mechanism Analysis, Computers and Structures 33 (1989) 801-820. [3] B. Simeon, Numerical Analysis of Flexible Multibody Systems, In: J. A. C. Ambr—sio and W. O. Schiehlen (eds.), Advances in Computational Multibody Dynamics, Euromech Colloquium 404, 1999 pp. 397-416. [4] O. A. Bauchau and N. J. Theron, Energy Decaying Scheme for Nonlinear Elastic Multi-Body Systems, Computers and Structures 59 (1996) 317-331. [5] K. E. Brenan, S. L. Campbell and L. R. Petzold, Numerical Solution of Initial-Value Problems in DifferentialAlgebraic Equations, ISBN:0444015116, Elsevier, 1989. [6] H. M. Hilber, T. J. R. Hughes and R. L. Taylor, Improved Numerical Dissipation for Time Integration Algorithms in Structural Dynamics, Earthquake Engineering and Structural Dynamics 5 (1977) 283-292. [7] J. Chung and G. M. Hulbert, A Time Integration Algorithm for Structural Dynamics With Improved Numerical Dissipation: The Generalized-α Method, ASME Journal of Applied Mechanics 60 (1993) 371-375. [8] HKS, Inc., ABAQUS User's Manual, HKS, Inc., Pawtucket, R. I., 1993. [9] J. C. Simo, N. Tarnow and M. Doblare, Non-Linear Dynamics of Three-Dimensional Rods: Exact Energy and Momentum Conserving Algorithms, International Journal for Numerical Methods in Engineering 38 (1995) 1431-1473. [10] A. J. Chen, Energy-Momentum Conserving Methods for Three-Dimensional Dynamic Nonlinear Multibody Systems, Ph. D. thesis, Stanford University, 1998. [11] A. Ibrahimbegovic and S. Mamouri, Modeling of Holonomic Constraints and Energy Conserving Integration for Flexible Multibody Systems, In: J. A. C. Ambr—sio and W. O. Schiehlen (eds.), Advances in Computational Multibody Dynamics, Euromech Colloquium 404, 1999 pp. 397-416. [12] O. A. Bauchau, Computational Schemes for Flexible Nonlinear Multi-Body Systems, Multibody System Dynamics 2 (1998) 169-225.

Sensitivity Analysis and Design Optimization of Differential-Algebraic Equation Systems Linda PETZOLD, Radu SERBAN, Shengtai LI, Soumyendu RAHA and Yang CAO Department of Mechanical and Environmental Engineering University of California, Santa Barbara, CA 93106, USA

Abstract. We report on our progress in developing algorithms and software for sensitivity analysis and design optimization of differential-algebraic equation systems.

1. Introduction Differential-algebraic equations (DAEs) arise in a wide variety of engineering and scientific problems, including the modeling of multibody and flexible systems. Much work has been devoted to understanding these systems and developing numerical methods and software for the simulation problem[4], although some substantial technical challenges remain. In this paper we report on our progress in developing algorithms and software for sensitivity analysis and optimization of DAE systems. These computations are an order of magnitude more complex than simulation, require highly efficient and robust methods for simulation, and are of critical importance throughout engineering design. In section 2, we outline algorithms and software for sensitivity analysis of large-scale DAE systems, via both the forward and reverse (adjoint) modes. The new software can handle DAE systems of index up to two (in Hessenberg form), and solves for consistent initial conditions for both the state and sensitivity systems. In section 3, we formulate the general problem of design optimization and optimal control for DAE systems, and describe our algorithm based on a modified multiple shooting method for solving these problems. An important issue for this type of method is the handling of constraints. In particular, given an equality constraint, it can be included in the dynamic optimization problem either as part of the DAE, or be handled directly by the optimizer. This choice can affect both the index and stability of the resulting DAE system. In section 4, we describe our tool for analyzing the DAE structure and finding a stable partitioning of the constraints.

2. Sensitivity Analysis Sensitivity analysis for DAE systems is important in many engineering and scientific applications. The information contained in the sensitivity trajectories is useful for parameter estimation, design optimization, optimal control, model reduction and experimental design.  This

research was supported by NSF Multidisciplinary Challenge Grant CCR-98-9896198, NSF/DARPA Virtual Integrated Processing program, DOE DE-FG03-98ER25354 and by LLNL ISCR 00-15.

Here we present algorithms and software for sensitivity analysis of large-scale DAE systems of index up to two. There are two basic types of sensitivity approaches. The first, which we call the forward mode (sometimes referred to as the Tangent Linear Model) is very efficient for computing sensitivities of a potentially large number of output variables with respect to relatively few input variables. The second, which we call the reverse mode or adjoint sensitivity analysis, is advantageous when it is required to find the sensitivity of a single or low-dimensional output variable with respect to a large number of input variables. Thus the approaches are complementary. We have recently developed algorithms and software for both, and will outline them here. 2.1. Forward Mode To illustrate the basic approach for sensitivity analysis, consider the general DAE system with parameters, F t; x; x0 ; p ; x x0 p (1) where x 2 Rnx ; p 2 Rnp . Here nx is the number of time-dependent variables x as well as the dimension of the DAE system, and n p is the number of parameters in the original DAE system. Sensitivity analysis entails finding the derivative of the solution x with respect to each parameter. This produces an additional ns np
ny sensitivity equations which, together with the original system, yield

(

)=0

(0) = ( ) =

F (t; x; x0 ; p) @F @F @F si + 0 s0i + @x @x @p

where si

= 0; = 0;

= dx=dpi. Defining 2

X=

6 6 6 6 4

x s1 .. .

snp

3

2

7 7 7 7 5

6 6 6 6 6 4

; F=

(2a)

i = 1; :::; np;

F (t; x; x0 ; p) @F s + @F s0 + @F @x 1 @x 1 @p1 0

@F s @x np

.. .

+ @x@F s0np + @p@Fnp

(2b)

3 7 7 7 7 7 5

0

the combined system can be rewritten as 2

F(t; X; X 0 ; p) = 0;

X (0) =

6 6 6 6 6 4

x0

dx0 dp1

.. .

dx0 dpnp

3 7 7 7 7 7 5

:

This system can be solved by the k -th order BDF formula with step size h n+1 to yield a nonlinear system

G(Xn+1 ) = F tn+1 ; Xn+1

; X 0 (0)

n+1

s (X hn+1 n+1

!

Xn(0)+1

); p = 0;

(3)

where Xn+1 and X 0 n+1 are predicted values for Xn+1 and Xn0 +1 , which are obtained via polynomial extrapolation of past values [4]. Also,  s is the fixed leading coefficient which is defined in [4]. Newton’s method for the nonlinear system produces the iteration (0)

(0)

Xn(k+1+1) = Xn(k+1)

J

1

G(Xn(k+1) );

where

2

J

and and 

  

=

=

6 6 6 6 6 6 6 4

J J1 J J2 0 J .. .

Jnp

.. .

.. .

3

..

.

0


0

J

7 7 7 7 7 7 7 5

(4)

@J @J @F @F J =  0 + ; Ji = si + @x @x @x @pi

s =hn+1 . There are three well-established methods to solve the nonlinear system (3): Staggered direct method, described in [8]. Simultaneous corrector method, described in [24]. Staggered corrector method, described in [10].

Analysis and comparison of the performance of these three methods have been given in [10, 20], where it was found that the relative efficiencies of the methods depend on the problem and on the number of parameters. Here we describe briefly the three methods. The staggered direct method first solves equation (3) for the state variables. After the Newton iteration for the state variables has converged, the sensitivity equations in (3) are updated with the most recent values of the state variables. Because equation (2b) is linear with a matrix J for the sensitivity equations, it is solved directly without Newton iteration. However, to solve the linear system in this way requires computation and factorization of the Jacobian matrix at each step and also extra storage for the matrix @F=@x 0 . Since the Jacobian is updated and factorized only when necessary in DAE solvers such as DASSL and DASPK [4], the additional matrix updates and factorizations may make the staggered direct method unattractive compared to the other methods. However, if the cost of a function evaluation is more than the cost of factorization of the Jacobian matrix and the number of sensitivity parameters is very large (see [20]), the staggered direct method is more efficient. We have modified the implementation of [8] to make the staggered direct method more reliable for ill-conditioned problems. The simultaneous corrector method solves (3) as one whole nonlinear system, where Newton iteration is used. The Jacobian matrix J in (4) is approximated by its block diagonal in the Newton iteration. Thus, this method allows the factored corrector matrix to be reused for multiple steps. It has been shown in [24] that the resulting iteration is two-step quadratically convergent for full Newton, and convergent for modified Newton iteration. The staggered corrector method lies in between the staggered direct method and the simultaneous corrector method. Instead of solving the linear sensitivity system directly as in the staggered direct method, a Newton iteration is used

s(ik+1) = s(ik)

J 1 Gsi (s(ik) );

(5)

where Gsi is the residual for the i-th sensitivity and J is the factored Jacobian matrix which is used in the Newton iteration for the state variables. Like the simultaneous corrector method, this method does not require the factorization of the Jacobian matrix at each step. One of the advantages of the staggered corrector method is that we do not need to evaluate the sensitivity equations during the iteration of solving for the state variables. This can reduce the

computation time if the state variables require more iterations than the sensitivity variables. After solving for the state variables in the corrector iteration, only the diagonal part of J in (4) is left. We can expect that the convergence of the Newton iteration will be improved over that of using an approximate iteration matrix in the simultaneous corrector method. This has been observed in our numerical experiments. Several approaches have been developed to calculate the sensitivity residuals that may be used with either the staggered corrector or the simultaneous corrector methods. Maly and Petzold [24] used a directional derivative finite difference approximation. For example, the ith sensitivity equation may be approximated as

F (t; x + Æi si ; x0 + Æi s0i ; p + Æi ei ) F (t; x; x0 ; p) = 0; Æi

(6)

where Æi is a small scalar quantity, and ei is the ith unit vector. Proper selection of the scalar Æi is crucial to maintaining acceptable round-off and truncation error levels [24]. If F t; x; x0 ; p is already available from the state equations, which is the case in the Newton iteration of DASPK, (6) needs only one function evaluation for each sensitivity. The main drawback of this approach is that it may be inaccurate for badly scaled problems. The selection of the increment Æi for equation (6) in our current software is an improvement over the algorithms of [24] which was suggested by Hindmarsh [15]. The increment is given by Æi jpij; =jjuijj2 (7)

(

where

)

=
max(


is a scale factor,

1

)

ui = W T inx+j =W T j : j = 1; :::; nx ; 



and W T is a vector of weights determined by the relative and absolute user error tolerances and the solution x, W T j RTOLj
jxj j ATOLj :

=

+

Alternatively, the sensitivity residuals can be evaluated analytically by an automatic differentiation tool such as ADIFOR [3] or other automatic differentiation (AD) methods. We recommend using AD to evaluate the sensitivity equations. Even for some well-scaled problems, the ADIFOR-generated routine has better performance in terms of efficiency and accuracy than the finite difference approximation. We have recently developed new software DASPK 3.0 [17] for solution and forwardmode sensitivity analysis of DAE systems based on the above methods. The software makes use of the basic methods of DASSL [4] for time integration, and includes as an option the preconditioned iterative methods of DASPK [5] which are needed for solving large-scale DAE systems. DASPK 3.0 provides for consistent initialization of the solutions and the sensitivities, interfaces seamlessly with automatic differentiation for the accurate evaluation of the sensitivity equations, and is capable via MPI[9] of exploiting the natural parallelism of sensitivity analysis as well as providing an efficient solution in sequential computations. The DASPK 3.0 software can be found at http://www.engineering.ucsb.edu/˜cse. 2.2. Reverse Mode Some problems require the sensitivities of a single or small-dimensional output with respect to a large number of parameters. For these problems, particularly if the number of

state variables is also large, the forward sensitivity approach is intractable. For a general DAE (1) and a function given by Z T

G(p) =

0

(

g (p; x; t)dt

(8)

)

or alternatively by the scalar function g p; x; T at time T , these problems take the form: find dG or dg , where p is a potentially large number of parameters. dp dp The adjoint problem for (8) is derived as follows. Linearize the DAE (1) with respect to p to obtain Fp Fx xp Fx_ xp ; xp x0p (9)

+

+ _ =0

(0) =

where subscripts on functions such as F or g are used to denote partial derivatives. Introducing the multiplier , we get Z

T 0

 (Fp + Fx xp + Fx_ x_ p )dt = 0

(10)

where  denotes the conjugate transpose. Integrating by parts, the last term in the above integral becomes Z

T 0

where

dF  x_ dt

 Fx_ x_ p dt = ( Fx_ xp )jT0 "

=

Z

_  F

T 0

!

 dFx_ x dt; x_ +  p

(11)

dt

# d          h    i h    i F  = Fx_  t + Fx_  x x_ + Fx_  x_ x : dt x_

(12)

A bar over a variable indicates that the variable is held fixed for the purpose of the current differentiation. Without loss of generality, we can assume that F depends linearly on x and therefore the last term in (12) is zero. Indeed, any other case can be reduced to this one by introducing the additional variables y x. Note that, in the worst case, the problem size is increased by x_ rank(Fx_ ). So from now on, we calculate  dF dt by

_

=_

dF  x_ dt

=

"

# d          h    i F  = Fx_  t + Fx_  x x_ : dt x_

Thus we have Z

T 0

"

 (F

p

+ Fxxp)

dF _  Fx_ +  x_

!

dt

#

xp dt + ( Fx_ xp )jT0

=0

(13)

= 0:

(14)

which can be written as Z

T 0

Now letting

"

 F

p

dF _  Fx_ +  x_

x

dt

(

_  Fx_ + 

h

!

 F dFx_ dt

Fx

i

(Fx_ )jt=T

#

xp dt + ( Fx_ xp )jT0

= gx = 0;

(15)

we obtain the equation for dG dp

dG Z T = 0 (gp dp

 Fp) dt + ( Fx_ xp )jt=0

(16)

which can be written alternatively as

dG Z T = 0 (gp dp

 Fp ) dt + ( Fx_ )jt=0 x0p :

(17)

Equations (15) are the so called adjoint equations for dG dp . Note that the boundary condition

(Fx_ )jt=T = 0 is applicable only for DAEs with index up to one. In [7], we develop boundary conditions that are valid also in the index-two (Hessenberg) case. For dg dp , we have

dg d dG = dp dT dp

so

dg = (gp dp

 F

p

Z

)(T )

T 0

T Fp dt + (T Fx_ )jt=0 x0p ;

(18)

@ . The corresponding adjoint equations are where T denotes @T "

_  F

T x_

#

+  dFx_ T

Fx

dt

= 0:

(19)

For DAEs of index up to one, to find the boundary condition for this equation, we write  as  t; T because it depends on both t and T . Then

( )

 (T; T )Fx_ jt=T

= 0:

Taking the total derivative, we obtain

_

(t + T )(T; T )Fx_ jt=T + (T; T ) dFdtx_ = 0:

Since t is just , and using the previously derived expression for dF x_ =dt, we have the boundary condition " #

(T Fx_ )jt=T =

 (T; T )

dFx_ _  +  Fx_ dt



t=T

:

(20)

According to equation (15), the upper condition is

(T Fx_ )jt=T = (gx  Fx)jt=T : In the case that Fx_ is invertible, we have (T; T ) = 0, which leads to T = _ .

(21)

Boundary conditions for the index-two (Hessenberg) case are derived in [7]. We have been developing software called ADJOINTDASPK for the reverse mode sensitivity problem. Our goal for the adjoint sensitivity calculation has been algorithms and software that are as reliable, efficient, and easy to use as the current algorithms and software for forward sensitivity analysis. An efficient formulation of the adjoint equations can be generated by the current generation of automatic differentiation software. For example, TAMC [11] and ADIFOR 3.0 [16] implement the forward and reverse modes that are needed for this task. ADJOINTDASPK makes use of the reliable and efficient methods in DASPK 3.0 for solving the adjoint equations, and for determining a consistent set of initial conditions. Several research issues are under investigation. For large-scale problems whose solution and forward sensitivities are best computed in DASPK via preconditioned iterative

methods, we need to ensure that these same ‘matrix-free’ methods and corresponding preconditioners are accessible for the solution of the adjoint equations. Predictable and compact means of storing the forward solution information required by the adjoint computation are needed. We have been investigating the use of reduced order models both for storing the forward solution information and for highly efficient sensitivity solution in applications where repeated forward and reverse sensitivity solves are required. Determination of consistent initial conditions for the adjoint system for general DAE systems is also an issue that is in progress.

3. Design Optimization We consider the differential-algebraic equation (DAE) system

F(t; x; x0 ; p; u(t)) x(t1 ; r)

= 0 = x1 (r)

(22)

where the DAE is index one (see [4] or [1]) or Hessenberg index-two Hessenberg, and the initial conditions have been chosen so that they are consistent (so that the constraints of the DAE are satisfied). The control parameters p and the vector-valued control function u t must be determined such that the objective functional

()

Z

tmax

(t; x(t); p; u(t)) dt

t1

is minimized

and some additional equality and/or inequality constraints

G(t; x(t); p; u(t))  0

()

are satisfied. The optimal control function u  t is assumed to be continuous. In several of our applications, the DAE system is large-scale. Thus, the dimension N x of x is large. However, the dimension of the control parameters and of the representation of the control function u t is much smaller. To represent u t in a low-dimensional vector space, we use piecewise polynomials on t1 ; tmax , their coefficients being determined by the optimization. For ease of presentation we can therefore assume that the vector p contains both the parameters and these coefficients (we let M denote the combined number of these values) and discard the control function u t in the remainder of this section. Also, we consider that the initial states are fixed and therefore discard the dependency of x1 on r. Hence we consider

[

()

()

]

()

F(t; x; x0 ; p) = 0; Z

tmax

(t; x(t); p) dt g(t; x(t); p)  0: t1

x(t1 ) = x1 ; is minimized,

(23) (24) (25)

There are a number of well-known methods for direct discretization of this optimal control problem, for the case that the DAEs can be reduced to ordinary differential equations (ODEs) in standard form. The single shooting method solves the ODEs (23) over the interval t1 ; tmax , with the set of controls generated at each iteration by the optimization algorithm. However, it is well-known that single shooting can suffer from a lack of stability and robustness [2]. Moreover, for this method it is more difficult to maintain additional constraints and

[

]

to ensure that the iterates are physical or computable. The finite-difference method or collocation method discretizes the ODEs over the interval t 1 ; tmax with the ODE solutions at each discrete time and the set of controls generated at each iteration by the optimization algorithm. Although this method is more robust and stable than the single shooting method, it requires the solution of an optimization problem which for a large-scale ODE system is enormous, and it does not allow for the use of adaptive ODE or (in the case that the ODE system is the result of semi-discretization of PDEs) PDE software. We thus consider the multiple-shooting method for the discretization of the optimal control problem. In this method, the time interval t 1 ; tmax is divided into subintervals titx ; titx+1 (itx ; : : : ; Ntx ), and the differential equations (23) are solved over each subinterval, where additional intermediate variables Xitx are introduced. On each subinterval we denote the solution at time t of (23) with initial value X itx at titx by x t; titx ; Xitx ; p . Continuity between subintervals is achieved via the continuity constraints

[

[

]

]

[

=1

]

(

Citx 1 (Xitx+1 ; Xitx ; p)  Xitx+1

)

x(titx+1 ; titx ; Xitx ; p) = 0:

For the DAE solution to be defined on each multiple shooting subinterval, it must be provided with a set of initial values which are consistent (that is, the initial values must satisfy any algebraic constraints in the DAE). This is the case even for feasible methods, since the optimizer does not see the DAE constraints. To begin each interval with a consistent set of initial values, we first project the intermediate solution generated by SNOPT onto the DAE constraints, and then solve the DAE system over the subinterval. In the case of index-1 problems with well-defined algebraic variables and constraints such as the problem considered in this paper, this means that we perturb the intermediate initial values of the algebraic variables so that they satisfy the DAE constraints at the beginning of each multiple shooting subinterval. The additional constraints (25) are required to be satisfied at the boundaries of the shooting intervals

Citx 2 (Xitx ; p)  g(titx ; Xitx ; p)  0:

Following common practice, we write

(t) =

Z

t t1

(; x( ); p) d;

(26)

 ( ) = ( ( ) ) ( ) = 0

t; x t ; p , t1 . This introduces another equation and which satisfies 0 t variable into the differential system (23). The discretized optimal control problem becomes X2 ;:::;min XNtx ;p (tmax )

(27)

subject to the constraints

Citx 1 (Xitx+1 ; Xitx ; p) Citx 2 (Xitx ; p)

= 

0; 0:

(28) (29)

This problem can be solved by an optimization code. We use the solver SNOPT [13], which incorporates a sequential quadratic programming (SQP) method (see [14]). The SQP methods require a gradient and Jacobian matrix that are the derivatives of the objective function and constraints with respect to the optimization variables. We compute these derivatives via our differential-algebraic equation (DAE) sensitivity software DASPK 3.0 described earlier. Our algorithms and software for the optimal control of dynamical systems are described in detail in [29].

This basic multiple-shooting type of strategy can work very well for small-to-moderate size ODE systems, and has an additional advantage that it is inherently parallel. However, for large-scale ODE and DAE systems there is a problem because the computational complexity grows rapidly with the dimension of the ODE system. The difficulty lies in the computation of the derivatives of the continuity constraints with respect to the variables X itx . The work to compute the derivative matrix @ x t =@ X itx is of order O Nx2 , and for the problems under consideration Nx can be very large (for example, for an ODE system obtained from the semidiscretization of a PDE system, Nx is the product of the number of PDEs and the number of spatial grid points). In contrast, the computational work for the single shooting method is of order O Nx Np although the method is not as stable, robust or parallelizable. We reduce the computational complexity of the multiple shooting method for this type of problem by modifying the method to make use of the structure of the continuity constraints to reduce the number of sensitivity solutions which are needed to compute the derivatives[12]. To do this, we recast the continuity constraints in a form where only the matrix-vector products @ x t =@ Xitx wj are needed, rather than the entire matrix @ x t =@ Xitx . The matrixvector products are directional derivatives; each can be computed via a single sensitivity analysis. The number of vectors wj such that the directional sensitivities are needed is small, of order O Np . Thus the complexity of the modified multiple shooting computation is reduced to O Nx Np , roughly the same as that of single shooting. Unfortunately, the reduction in computational complexity comes at a price: the stability of the modified multiple shooting algorithm suffers from the same limitations as single shooting. However, for many DAE and partial differential algebraic equation (PDAE) systems, where simulation of the forward problem is stable, this is not an issue, and the modified method is more robust for nonlinear problems. The inherent parallelism of the multiple shooting algorithm is also lost by the modified method. In the context of the SQP method, the use of modified multiple shooting involves a transformation of the constraint Jacobian. The affected rows are those associated with the continuity constraints and any path constraints applied within the shooting intervals. Path constraints enforced at the shooting points (and other constraints involving only discretized states) are not transformed. The transformation is cast almost entirely at the user level and requires minimal changes to the optimization software, which is important because software in this area is constantly being modified and improved. Gill et.al. ([12]) have shown that the modified quadratic subproblem yields a descent direction for the ` 1 penalty function. DAOPT is a modification to the SNOPT optimization code that uses a merit function based on an ` 1 penalty function. A limitation of the basic COOPT software arises for PDAE systems because the method of lines is used to solve the PDAE. This does not allow for an adaptive grid in space, which would be needed if there are steep spatial gradients moving in time. We have developed methods and software based on adaptive mesh refinement (AMR) for systems of partial differential equations[18]. The software has been designed to be flexible, to allow the user to ‘plug and play’ existing non-adaptive simulation software, and to exert direct control over the adaptivity when needed. Recently we have shown how to compute sensitivity derivatives via AMR [19], and developed a capability for doing dynamic optimization including AMR for time-dependent PDE systems[21].

()

(

( )

)

( ()

)

( ) ( )

()

u2 z-axis theta

tau

z

d

r

x x-d

p

u1

Figure 1. Planar Crane Model

4. Partitioning for Structure and Stability An important issue for shooting or multiple shooting-type methods in design optimization is the handling of constraints[27]. Given an equality constraint, it can be included in the dynamic optimization problem either as part of the DAE, or to be handled directly by the optimizer. If it is included as part of the DAE, then there is a possibility that it could alter the index (mathematical structure) of the DAE, making it potentially more difficult to solve; on the other hand, if it is an important physical constraint, then its inclusion into the DAE where it will always be enforced should help the optimizer to avoid non-physical solutions. In our work, this question has been studied from the point of view of mathematical structure (index), and stability/conditioning of the DAE. It turns out that including some constraints with the DAE can alter the DAE stability, in a way which may be either favorable or unfavorable. Algorithms have been developed for partitioning the constraints to lead to a stable DAE system of index two or lower (which can be solved by the existing software). The logarithmic norm, which is closely related to the pseudoeigenvalues, is used to measure the stability for potential partitionings. We have developed a partitioning tool for analyzing the DAE structure and determining a stable partitioning of the constraints. The dilemma of how best to treat the constraints arises often in the dynamic optimization of path-constrained dynamics systems. For example, a planar rigid body model of a crane manipulating a payload which must be lifted along a specified trajectory, is described below [6]. 4.1. Example: Planar Model of a Crane The crane is a mechanical system in planar Cartesian co-ordinates, x and z , denoting the horizontal and vertical positions of the payload. The other state variables are d, the horizontal distance traveled by the crane trolley from the origin and r , the paid out cable length. The equations are

M2 x

=

 sin()

(30a)

18

2.5

x 10

error in scleronomic constraints

2

1.5

1

0.5

0

−0.5

0

0.2

0.4

0.6

0.8

1

1.2

1.4

time

Figure 2. Partial plot of errors neglecting scleronomic constraint

M2 z M1 d J r

= = = 0 = 0 = x = z =

 cos() + mg C1 d_ + u1 +  sin() C2 r_ C3 u2 + C32   tan 1 ((x d)=z ) r2 (x d)2 z 2 1 (t) 2 (t)

(30b) (30c) (30d) (30e) (30f) (30g) (30h)

where M1 , M2 , m are the masses of the trolley, cable and payload system and the payload alone, respectively. C1 , C2 , C3 are constants, and J is the mass moment of inertia for the pulley paying out the cable. The algebraic variables are  , the tension in the cable and  , the cable angle with vertical. The horizontal force driving the trolley (u 1 ) and torque driving the pulley in the winch (u2 ) are the two control variables. Equations (30g) and (30h) describe the specified path of the payload. The initial values were taken to be x : ;x : ;z

=30 _ =27 = 21:0; z_ = 2:0; d = 1:0; d_ = 2:7; r = ((x d)2 + z2 )0:5 ; r_ = 2:0;  = 980:0; u1 = 0; u2 = 0;  = tan 1 (x z d) , on the interval of integration from t0 = 0 to t nal = 8. 



The path-constrained crane model 30 is an index-5 system. Thus it is not directly solvable by any of the methods described earlier. Rt f inal Figure 2 illustrates the error ( t0 jjr 2 x d 2 z 2 jj2dt) in simulating the planar crane problem with an index 1 partitioned DAE (pDAE) consisting of (30a - 30e) only and excluding equation (30f). While the physical constraints inherently should be included with the pDAE, they can raise the index of the pDAE to higher than 2. As an example, equation (30f) in the crane example when included in the pDAE (30a - 30e) raises the index of the system to 3. When these constraints raise the index of the pDAE to higher than 2, a suitable index reduction method can be adopted. For the crane model, constraints (30e) and (30f) are both scleronomic constraints, i.e.,

(

)

)

crane example: state variables 25 desired z desired x z x error

20

positions and errors

15

10

5

0

−5

0

1

2

3

4 time

5

6

7

8

Figure 3. Crane problem using equations (30a)-(30e) and (31) in the pDAE.

they describe some geometric structure of the physical problem. Constraint (30e) introduces the measurement of  , the angle by which the payload deviates from the vertical. Constraint (30f) connects the payload to the pulley, by relating its horizontal and vertical positions to the length of the cable paid out. Violation of constraint (30f) in the pDAE would produce an unphysical simulation since the payload is now modeled as detached from the cable. This is evident from the plot of x, z , d and r in Figure 5. Applying our partitioning algorithm to this model, constraint (30e) is included in the pDAE since it satisfies the index one structure and stability criterion. The angle of deflection from vertical  is chosen as the index 1 variable. The resulting underlying ODE in x does not have large eigenvalues in C + at t t0 . For a practical physical application, the eigenvalues of the pDAE system can be expected to be of moderate size and the system can be expected to be stable. The constraint equation (30f) is an index 3 scleronomic constraint (known from the physics of the problem) and is reduced to its index 2 form via one differentiation of constraint (30f) with respect to t

=

rr_

(x

d)(_x d_) z z_ = 0

(31)

or to index 1 via two differentiations of constraint (30f) with respect to t. (if the available DAE integration software can integrate index 0 or 1 systems only). The other constraints (pre-programmed trajectory of the payload) (30g and 30h) are handled by the SQP method. The search for index 2 constraints returns the empty set. To search for index 3 scleronomic constraints, the remaining algebraic equations are differentiated using automatic differentiation once with respect to time. The index-2 search algorithm is re-applied and the algorithm returns the differentiated form of constraint (30f), i. e., (31) as an index-2 constraint suitable for inclusion in the pDAE system. The tension in the cable  is returned as the corresponding index 2 variable. Even though the scleronomic constraint (31) slightly raises the logarithmic norm estimate for the index 2 pDAE system thus constructed, this partition leads to a more physical and well conditioned problem. The plots in Figures 3 and 4 are the results obtained

crane example: controls 400

Force on the Cart Pulley Torque

300

200

100

controls

0

−100

−200

−300

−400

−500

−600

0

1

2

3

4 time

5

6

7

8

Figure 4. Crane controls using equations (30a)-(30e) and (31) in the pDAE.

5

2

x 10

z d x r

1

position variables

0

−1

−2

−3

−4

0

1

2

3

4 time

5

6

7

Figure 5. Crane problem using equations (30a)-(30e) in the pDAE.

8

from a multiple shooting type scheme using DASPK 3.0 [17] as the DAE integrator and SNOPT as the NLP method [26]. In this case the path constraints are chosen as p 1  x : t2 :t : and p2  z : : t2 :t . The optimizer SNOPT stops at an acceptable point which cannot be improved further after 4 major iterations (with 188, 182, 112 and 1 minor iterations). The objective function is given R by the error integral tt0f inal p21 p22 0:5 dt. The time interval is  t  . The controls (u 1 and u2 ) have been modeled as quadratic polynomials over each of the 8 shooting intervals 7 and atol 7 used for the problem. The tolerances for DASPK3.0 were rtol 5 . The control and state continuity constraints across and all tolerances for SNOPT were the shooting intervals have been satisfied to less than or equal to : when the optimizer has stopped. With the same parameter settings for the numerical methods, a pDAE formulation leaving (30f) with the optimizer as a path constraint and treating  as an additional control variable has been attempted. The plot in Figure 5 is obtained from the results. The solution is incorrect. After 7 major iterations in SNOPT (162, 50, 72, 8, 1, 9 and 22 minor iterations), the optimizer fails to find a descent direction. Clearly the stopping point is an unacceptable solution to the physical problem. Many of the continuity constraints in the multiple shooting method are unacceptably violated.

( 0 0675 + 2 7 + 3 0) = 0 ( + )

10

( 21 0 0 05 + 2 0 ) = 0 0

8

= 10 01

= 10

References [1] U. M. Ascher and L. R. Petzold, Computer Methods for Ordinary Differential Equations and DifferentialAlgebraic Equations, SIAM, Philadelphia, Pennsylvania 1998. [2] U. M. Ascher, R. M. M. Mattheij and R. D. Russell, Numerical Solution of Boundary Value Problems for Ordinary Differential Equations, SIAM, Philadelphia, Pennsylvania 1995. [3] C. Bischof, A. Carle, G. Corliss, A. Griewank and P. Hovland, ADIFOR—Generating Derivative Codes from Fortran Programs, Scientific Programming 1 (1992) 11-29. [4] K. E. Brenan, S. L. Campbell and L. R. Petzold, Numerical Solution of Initial-Value Problems in Differential-Algebraic Equations, SIAM, Philadelphia, Pennsylvania 1995. [5] P. N. Brown, A. C. Hindmarsh and L. R. Petzold, Using Krylov Methods in the Solution of Large-Scale Differential-Algebraic Systems, SIAM J. Sci. Comput. (1994) 1467-1488. [6] S. L. Cambell, High Index Differential Algebraic Equations, Mech. Struct. and Mach. 23(2) (1995) 199222. [7] Y. Cao, S. Li, L. Petzold and R. Serban, Adjoint Sensitivity Analysis for Differential-Algebraic Equations: Part I, The Adjoint System, in preparation. [8] M. Caracotsios and W. E. Stewart, Sensitivity analysis of initial value problems with mixed ODEs and algebraic equations, Computers and Chemical Engineering 9(4) (1985) 359-365. [9] N. Doss, W. Gropp, E. Luck and A. Skjellum, A model implementation of MPI, Technical report Argonne National Laboratory, 1993. [10] W. F. Feehery, J. E. Tolsma and P. I. Barton, Efficient sensitivity analysis of large-scale differentialalgebraic systems, Applied Numerical Mathematics 25 (1997) 41-54. [11] R. Giering and T. Kaminski, Recipes for adjoint code construction, ACM Trans. Math. Software 24 (1998) 437-474. [12] P. E. Gill, L. O. Jay, M. W. Leonard, L. R. Petzold and V. Sharma, An SQP Method for the Optimal Control of Large-Scale Dynamical Systems, J. Comp. Appl. Math. 20 (2000) 197-213. [13] P. E. Gill, W. Murray, and M. A. Saunders, SNOPT: An SQP Algorithm for Large-Scale Constrained Optimization, Numerical Analysis Report 97-2, Department of Mathematics, University of California, San Diego, La Jolla, CA, 1997. [14] P. E. Gill, W. Murray and M. H. Wright, Practical Optimization, Academic Press, London and New York, 1981. [15] A. C. Hindmarsh, personal communication. [16] P. Hovland, Argonne National Laboratory, personal communication, 1999. [17] S. Li and L.R. Petzold, Software and Algorithms for Sensitivity Analysis of Large-Scale Differential Algebraic Systems, to appear, J. Comp. Appl. Math.. [18] S. Li, Adaptive Mesh Methods and Software for Time-Dependent PDEs, Ph.D. thesis, Department of

Computer Science, University of Minnesota, 1998. [19] S. Li, L. R. Petzold and J. M. Hyman, Solution Adapted Nested Grid Refinement and Sensitivity Analysis for Parabolic PDE Systems, in preparation. [20] S. Li, L. R. Petzold and W. Zhu, Sensitivity analysis of differential-algebraic equations: A comparison of methods on a special problem, Applied Numerical Mathematics 32 (2000) 161-174. [21] S. Li, R. Serban and L. Petzold, Optimal control for time-dependent partial differential equations with implicit adaptive mesh refinement, in preparation. [22] M. Lo and S.Ross, Low Energy Interplanetary Transfers Using Invariant Manifolds of L1, L2, and Halo Orbits, AAS/AIAA Space Flight Mechanics Meeting, Monterey, Ca, 9-11 February 1998. [23] R. Serban, W. S. Koon, M. Lo, J. E. Marsden, L. R. Petzold, S. D. Ross and R. S. Wilson, Optimal Control for Halo Orbit Missions, Proc. IFAC Workshop on Lagrangian and Hamiltonian Methods for Nonlinear Control, Princeton University, March 16-18, 2000. [24] T. Maly and L. R. Petzold, Numerical methods and software for sensitivity analysis of differentialalgebraic systems, Applied Numerical Mathematics 20 (1996) 57-79. [25] S. E. Mattson and G. and S¨oderlind, Index reduction in differential-algebraic equations using dummy derivatives, SIAM J. Sci. Comput. 14(3) (1993) 677-692. [26] L.R. Petzold, J. B. Rosen, P. E. Gill, L. O. Jay and K. Park, Numerical Optimal Control of Parabolic PDEs using DASOPT, Large Scale Optimization with Applications, Part II: Optimal Design and Control, Eds. L. Biegler, T. Coleman, A. Conn and F. Santosa, IMA Volumes in Mathematics and its Applications, 93 (1997) 271-300. [27] S. Raha, Constraint Partitioning for Solution of Path-Constrained Dynamic Optimization Problems, Ph.D. Thesis, University of Minnesota, Scientific Computation, 2000. [28] L. Raja, R. Kee, R. Serban and L.R. Petzold, Dynamic Optimization of Chemically Reacting Stagnation Flows, 1998 Electrochemical Society Conference, Boston, Ma., 1998. [29] R. Serban, COOPT - Control and Optimization of Dynamic Systems - Users’ Guide, UCSB, Department of Mechanical and Environmental Engineering, Report UCSB-ME-99-1,1999.

Multi-Timestep Integration in Computational Dynamics William J.T. DANIEL Department of Mechanical Engineering, University of Queensland, St. Lucia Campus 4072 Queensland ,Australia. Abstract. Subcycling is the use of different timestep sizes to integrate different degrees of freedom in a model - this can result in a spatial adaption of timestep size, as opposed to the more common temporal adaptivity. Its use arises naturally in finite element modelling of interface problems where there are strikingly different time constants or natural frequencies. For instance, multi-timestep integration has been applied successfully to acoustic/structural interaction in aeroelasticity by Piperno et al in [1], using a different timestep in the fluid to that used in the structure. Coupled thermomechanical problems have been solved effectively using different timesteps for temperature and displacement by Armero and Simo [2]. Where there is no interface, the purpose of subcycling in a single field simulation has been to circumvent the need for a large number of cycles of integration, due to the small critical timestep values that can occur in codes that use explicit integration. Explicit integration is often favoured for short duration simulations, due to its robustness in handling nonlinearities. For subcycling to be effective, there must be a spatial localisation of rapid response in a first-order system, or of high frequency response in a second order system. This is often associated with a fine finite element mesh in one small region of a model, the local fine mesh limiting the timestep that can be used. If subcycling is adopted, larger timesteps can be used elsewhere, either in larger elements, or where properties change, causing the local single-element estimate of critical timestep to increase. For first order diffusion problems, the use of multiple timesteps in explicit analyses is well established, from work by Belytschko, Liu, and Smolinski [3,4,5,6] and Daniel[7] and others. A number of different approaches to multi-timestep explicit integration of diffusion problems can be proved to be stable. Daniel has shown that subcycling of implicit integration is also possible in diffusion problems [7]. The use of subcycling is also attractive in solving some dynamics problems, and has been applied to finding the response to an impact, and to simulation of metal forming, both of which are analysed using explicit finite element codes derivative from DYNA3D. Such impact problems possess a spatial localisation of high frequency response, which can be exploited by subcycling. The second order equations of dynamics pose more problems for successful design of a subcycling algorithm. This lecture will describe some of the algorithms that have been proposed, and the problems encountered in obtaining a truly robust subcycling algorithm for dynamics problems.

1.

Introduction

Time integration of dynamic finite element models has been done principally using algorithms of second-order accuracy, as higher order accuracy can only be obtained by solving larger sets of equations each timestep, or by computing quantities at sub-steps. Algorithms of higher accuracy are of debatable merit, especially in explicit analysis. It is

pointless to track every last wiggle in a solution, when the highest frequencies present are artifacts of the discretization. Hence the popular algorithms for small deformation have been the Newmark family of algorithms, including the central difference method, and the generalized α method, generalized by Chung and Hulbert [8] from the Newmark algorithm to dissipate high frequencies effectively. Recent work has focussed on finite deformation problems, for which the Newmark approach has been found inadequate, and has considered how to obtain appropriate conserving properties in such algorithms. Typically midpoint evaluation of internal forces (or moments) has been used to find velocities (or angular velocities) and midpoint evaluation of velocities has been used to find displacements, in order to conserve energy. Explicit subcycling of second order algorithms for structural dynamics is considered here. It should be noted though, that the period of time that can be simulated accurately is limited by this choice. While subcycling of implicit algorithms for structural dynamics is possible, assumptions made at a timestep interface tend to limit the accuracy of implicit formulations, if the timesteps used exceed explicit stability limits. To introduce concepts, the discussion focusses mainly on small deflection elastic analysis, and simple model problems.

2.

The Belytschko, Yen, Mullen Algorithm

Subcycling algorithms make use of either a nodal interface, in which a particular timestep is associated with a set of nodes, or an element interface, in which a particular timestep is associated with a group of elements. A simple, but usually accurate, example of a nodal interface was proposed by Belytschko, Yen and Mullen [9] and can be described as follows. A dual timestep situation is described, but this can be generalized to more timestep sizes. S is used to indicate small timestep nodes and L to indicate large timestep nodes. The start of the current major timestep will be arbitrarily made state 0, subcycle states being counted as 1, 2, 3 up to n. The central difference method can be generalized as follows. a, v, and u are used for acceleration, velocity and displacement. FINT is an internal force sum, FEXT are external forces found for the current cycle. M is a lumped mass matrix. Viscous damping is omitted for simplicity of presentation. A major cycle becomes the following. (a) A large timestep update of L nodes (timestep n ∆t) - a central difference update. aL0 = ML –1 ( - FLINT(uS0, uL0) + FLEXT 0) vLn/2 = vL-n/2 + aL0 n ∆t L n

L 0

L n/2

u =u +v

n ∆t

(1) (2) (3)

(b) n subcycle updates of S nodes. In order to estimate internal forces at S nodes on the timestep interface, linear interpolation on old and new uL values is used to give an intermediate displacement at the start of subcycle i of uLi = uL0 + vLn/2 i ∆t. In practice, vLn/2, which is held constant, may be used to find changes in internal forces. For i = 1 to n-1 aSi = MS-1 ( - FSINT(uSi , uLi) + FSEXT i) v

S

= v i-1/2 + a i ∆t = uSi + vSi+1/2 ∆t

i+1/2 S i+1

u

S

S

(4) (5) (6)

Figure 1 One-dimensional model problems used to study comparisons between algorithms

This algorithm has been applied successfully to large finite element simulations of impact problems. It does however have two limitations, which limit its application to a more general class of problems. One of these concerns a lack of energy conservation at the timestep interface and the other a lack of momentum conservation. Both become apparent in the simplest of dynamic systems, but are much less obvious in large models.

3.

Statistical Stability

The above algorithm, and other similar algorithms which use a nodal interface and update the large timestep side first, such as the constant acceleration timestep interface presented in [10], are not stable in a classical sense. This is easily demonstrated by considering a two degree of freedom problem with a small mass (the S node) connected by a spring to a large mass (the L node) as in figure 1(a). The natural frequency of the S partition is that with the L node fixed ie ωS = √(2k/mS) and that of the L partition is ωL = √(2k/mL). The plot in figure 2 applies to the case of n = 10 and uses Courant number axes where Ω S = ωS ∆t and Ω L = ωL n ∆t. Below the expected stability limits where Ω S or Ω L equals 2, narrow timestep ranges are unstable, instability being plotted as 1 and stability as 0. Figure 2 is obtained by rewriting equations (1) to (6) to form an amplification matrix representing a major cycle. If any eigenvalue of this matrix exceeds 1 in magnitude, instability is predicted. Analysis of this situation in [11] shows that these unstable timestep ranges are associated with the nonlinearity of switching between using a discrete model of the whole system and one of the S partition only. The boundaries of these unstable regions are functions of the highest natural frequencies of the S partition and those of the whole model with the mass of the L region scaled down by the factor n. This factor arises as the large timestep interface node effectively accelerates to its new velocity in the first subcycle, not over a major timestep. As more degrees of freedom are introduced, these frequencies become more and more similar, leading to the unstable timestep ranges becoming more and more narrow, so that in a large finite element model, a timestep causing instability is very unlikely to be selected. Energy dissipation in a model, such as the presence of plastic deformation, also removes the instabilities. The unstable timestep ranges also become narrower as the number of subcycles is increased. The worst case, showing the broadest range of instability, is the two-mass problem with 2 subcycles. It gives the “ridges” of instability visible in figure 2(a).

Figure 2

Unstable timestep ranges for the Belytschko, Yen, Mullen algorithm applied to the model problem of figure 1(a). 0 = stable. 1 = unstable. (a) 2 subcycles (b) 10 subcycles.

A cautious approach that minimizes the odds of instability while maintaining accuracy can be adopted by grading timestep changes, so that any two nodes within the same element only differ in timestep by a factor of 2. As well, the timestep interface can be placed cautiously. For instance, an increase in timestep may be made to lag spatially an increase in element size. In a case with only two timesteps in use, the first occurrence of instability for a linear elastic system is when the smallest timestep ∆t is √2/ωn. ωn here is the highest natural frequency of K u = ωn2 [MS + ML/2] u

(7)

This eigenvalue problem describes the effective dynamic system being modelled during the first update of a major cycle, when both partitions of the model are updated together. If more than two timestep sizes are used, narrow bands of instability do in exist at lower timestep sizes (with no energy dissipation present), but they can be very narrow, even in quite a small system, such as that of figure 1(b), for which the unstable timestep ranges correspond to ∆Ω of the order 10-3, if the element length ratio f is set to two. To avoid such a problem, either an energy-conserving algorithm is needed or an algorithm which dissipates high frequencies.

4.

Momentum Conservation

The timestep interface in the above algorithm does not necessarily conserve momentum. This can be demonstrated by a thought experiment on the free-free two mass problem of figure 1(c). Suppose an initial velocity acting to the right is applied to the small timestep mass mS. In the first major cycle, the large timestep mass mL will not move, as initial displacements are zero. However, in the meantime, the small timestep mass can bounce off the large timestep mass, and when its displacement is sampled to perform the second large timestep update, it may be now moving to the left. This leads to the wrong momentum being transferred to the large mass, and to it moving in the wrong direction. This situation is artificial, and can be avoided in a finite element code by setting the timestep of a node to be the minimum of the timesteps of elements connected to that node. Hence considering the present case as modelling a rod element with a lumped mass added to one end, use of the large timestep would not be permitted. Nevertheless, it is desirable to avoid this problem, which is caused by sampling the small timestep state only once per major cycle. A related case, which shows any problems with lack of momentum conservation at a timestep interface, is to support a one-dimensional model with a weak spring, as in figure 1(d). The accuracy of the low frequency mode of vibration that results can then be investigated. Hence to achieve a robust subcycling algorithm for explicit structural dynamics, it is desirable to have a major timestep update at a timestep interface depend on all subcycle states. It is also desirable to either avoid the problem of statistical stability or to introduce high frequency energy dissipation to prevent potential instabilies and to damp out spurious oscillations.

Figure 3

Information flows across a nodal partition over a major cycle, for the energy-conserving subcycling algorithm.

5.

An Energy Conserving Nodal Interface

A nodal interface can be set up in a way that preserves some of the energy conserving properties of the central difference method, and hence avoids statistical stability problems. This approach still is limited by the problem of momentum conservation, as in order to conserve energy, it is necessary to sample the small timestep state only once per major cycle [7]. Hence cautious placement of the timestep interface is needed. For instance, small timestep nodes on the timestep interface, should be capable of being integrated using the large timestep without accuracy being lost. The subcycling algorithm in reference [7] makes symmetrical use of midpoint values in a leapfrog fashion when updating the S or L partitions, the information flow being illustrated in figure 3. An accurate explicit algorithm for elastic problems can be written assuming constant displacement at neighbouring nodes across the timestep interface when updating either partition. The algorithm can be described as follows for a typical major cycle. Note the algorithm being generalised here is not the central difference method, as it uses midpoint estimation of internal forces. For the case of a linear elastic system, stiffness can be partitioned into submatrices KSS for interactions between S nodes, KLL for L node interactions and KSL or KLS for coupling terms. (a)

(b)

n subcycles starting at state -n/2, ending with state n/2, but using state 0 (the midpoint) to estimate the L partition velocity. For i = -n/2 to n/2 -1, splitting the displacement update into two stages: uSi+1/2 = uSi + vSi ∆t/2

(8)

vSi+1 = vSi + MS–1{-KSS uSi+1/2 - KSL uL0 + FSEXT i+1/2} ∆t

(9)

uSi+1 = uSi+1/2 + vSi+1 ∆t/2

(10)

an update of the L partition, using the (current) midpoint displacement uSn/2 uLn/2 = uL0 + vL0 n∆t/2

(11)

vLn = vL0 + ML–1{-KLL uLn/2 - KLS uSn/2 + FLEXT n/2} n∆t

(12)

uLn = uLn/2 + vLn n∆t/2

(13)

This procedure maintains the stability of the central difference method in each partition and conserves the following pseudo-energy over a major cycle, for a linear elastic problem: E1 = uTKu + vS T(MSS - KSS∆t2/4) vS + vL T(MLL – KLL(n∆t)2/4) vL

(14)

This algorithm differs from that in reference 7 in using midpoint evaluation of internal loads everywhere (for consistency with the handling of internal forces on the timestep interface). It can be remarkably accurate on simple elastic problems. An example is a twomass system (one S mass, one L mass) like that of figure 1(a). If two subcycles are used, with an initial velocity on the left mass, then the algorithm of (8) to (13) is more accurate than use of the small timestep only, showing the same second-order convergence. However, an initial velocity on the right, large timestep mass, does make a small timestep solution more accurate. The algorithm does better than the Belytschko, Yen, Mullen algorithm at capturing rigid body motion due to an initial velocity on the small timestep mass of the model problem of figure 1(c), with two subcycles, but still shows some inaccuracy on this problem. With plastic deformation however, the assumption of constant displacement can lead to permanent strain due to one side of the timestep interface moving and then the other,

when both should move together. Hence a constant velocity approach is clearly more appropriate in an elasto-plastic impact problem. The constant displacement algorithm can be programmed fairly simply. Update a clock at half the minimum nodal timestep in use. For all nodes to be updated, which is those half a nodal timestep behind a clock, predict the midstep displacements as u+v∆tn/2. Using these and other current displacements of neighbouring nodes, find impulses of internal forces and external forces using the nodal timesteps ∆tn. These add to predict a velocity change at each node updated. The new velocity is then used to complete the displacement update (u+v∆tn/2 again). To obtain a constant velocity approach, one option is to compute changes in acceleration found from velocity values – essentially replacing displacement by velocity and velocity by acceleration, in the equations above. This is described in [7]. It maintains central difference stability in each partition but can show overshoot of displacements at large timesteps. It also can tend to the wrong static solution when the dynamics are numerically damped. This can occur as errors in displacement are not corrected when computing acceleration changes from velocities. An alternative is to use the known velocities to interpolate the large timestep state. This will approximate energy conservation, but leads to much more limited instability on a plot like that in figure 2(a) – see figure 4(a). Note that the instabilities are now confined to where Ω S+Ω L>2. The velocity update equations for the S partition then be modified using the following, for i = -n/2 to n/2-1. uLi+1/2 = uL0+ vL0 (i + ½)∆t

(15)

vSi+1 = vSi + MS –1{-KSS uSi+1/2 - KSL uLi+1/2 + FSEXT i+1/2} ∆t The L update equations are unmodified, as the midpoint value of displacement u available.

Figure 4

(16) S n/2

is

Spectral radius of modified energy-conserving algorithm on the problem of figure 1(a). (a) no dissipation (truncated at ρ = 1.2). (b) β1 = 1.5, γ1 = 2, α = -1 – see section 10.

To generalise this algorithm to non-linear behaviour, the nodal interface is better thought of as an overlapping element interface. The internal forces in interface elements joining S and L nodes are evaluated from stress changes in both small and large timestep updates. These elements effectively belong to both partitions. In a set of S updates, the state of interface L nodes is held at constant velocity during these updates. The stress in these interface elements is being updated with both minor and major timesteps – this inefficiency is inherent in the nodal interface. That is, while the stress in interface elements is updated each minor cycle, at state i+1/2; in order to perform the large timestep update, the internal forces are needed at n/2, the midpoint of a major cycle, which is not the midpoint of a minor cycle. This leads to an additional evaluation of internal forces. It is possible to add

corrector cycles to the algorithm above, to correct the estimates of the midpoint displacements in (9) and (12). Convergence to conservation of physical energy is then possible, but the timestep interface still approximates the exchange of momentum.

6.

A Momentum Conserving Element Interface

A subcycling algorithm taking all minor timestep states into account in making the major timestep update is desirable but will retain the problem of statistical stability. The problem of statistical stability derives from the interaction of the highest modes of the S partition with those of an equivalent problem involving all degrees of freedom, and the super-harmonics generated. Hence it is of interest to examine the effect of introducing high frequency energy dissipation, as is done in the generalized α algorithm. Consider nodes on a timestep interface held at constant velocity while the small timestep partition is updated. Each minor cycle, there will be reaction forces on interface L elements, due to the constraint of constant velocity. These can be computed from the internal forces on S partition interface elements and added to give a total impulse over a major cycle. This impulse P is then applied to the L partition, when it is updated, giving loads of P/(n ∆t). This procedure can be implemented with the explicit version of the generalized α method [13] and is programmed by deciding first which elements to update each cycle, and then whether their nodes are due for an update. The internal forces acting on each element are found, and used to accumulate impulses at nodes not due to be updated.

Figure 5

Spectral radius ρ of the explicit generalised alpha method subcycled by impulse summation on the model problem of figure 1(a). 2 subcycles. ρb = 0 in both partitions. Unstable states at A plotted as ρ = 2.

The resulting algorithm is not as accurate as the Belytschko, Yen, Mullen algorithm. However, it performs well in capturing the rigid body motion of the free-free system of figure 1(c), and when the spectral radius at bifurcation parameter ρb that controls high frequency energy dissipation in the α algorithm is reduced to zero (maximum dissipation), statistical stability problems are well controlled. The worst case of a two mass system as in figure 1(a), with two subcycles is illustrated in figure 5, which is plotted on Courant number axes. The figure shows if Ω Lte j then new internal forces are found to contribute to the velocity updates such as equation (22). If t=tn i+∆tn i then the new displacement is found at a node.

8.

Adding Energy Dissipation

Energy dissipation, algorithmic or otherwise, will improve the probability of stability, and remove spurious responses at high frequency caused by discretization. One way of introducing high frequency dissipation is to use the explicit generalized α method, which can also be subcycled this way. The appropriate partial velocities to add (to achieve stability but avoid adding excessive energy dissipation) are those associated with the parameter β of the Newmark algorithm eg for the S element partition vSβ = vSi + (β aSi+1 + (1/2 - β) aSi) ∆t

(23)

where aSi+1 is found from the S partition internal forces at step i and from aSi, e.g., see equation (40). Thus the displacement update becomes ui+1 = ui + (vSβ + vLβ) ∆t

(24)

where the velocity held constant is vLβ = vL0 + (β aLn + (1/2 - β) aL0) n ∆t

(25)

To program this algorithm previous partial velocities and accelerations need to be calculated and stored, as a change in the partial velocity at a node, associated with one element, depends on the present partial velocity and acceleration values there, as well as on the impulses of internal and external forces. This is inconvenient, and does not permit an elegant update of velocities as in equation (22), e.g., for the S partition, substituting for aSi+1 in equation (23): vSβ = vSi + (-β/(1 − α) M-1 FS + (1/2 - β - α/(1 − α)) aSi) ∆t

(26)

An example of the dissipative properties of the resulting algorithm is shown in figure 10. It applies to the single interface freedom model of figure 1(g), the horizontal axis being (natural frequency of the full model) x (minor timestep). Spectral radius is plotted on the vertical axis for a setting of ρb = 0. Two subcycles are used. The behaviour of the parent algorithm with small and large timesteps is shown. As the natural frequency of the S partition is less than that of the full model, using the full mass of the interface node, the stability of the subcycled algorithm is slightly better than that of its parent algorithm. The energy dissipation is similar, but not precisely controlled to give a smooth reduction of spectral radius.

Figure 10

Example of decay of spectral radius with increasing timestep size. Explicit generalised alpha method applied to the model of figure 1(g). A – small timestep. B – large timestep. C – subcycled, 2 subcycles, partial velocity method.

On the problem of figure 1(h), a row of elastic/perfectly plastic rod elements, with short elements in the centre, algorithmic energy dissipation can cause error in estimating the correct energy dissipation due to plasticity. The displacement of the left-hand end due to an initial velocity is plotted on figure 11. If ρb = 0 is used in both small and large timestep partitions, with the parameters given in Figure 11, elastically the solution is just more filtered due to subcycling (figure 11(a)), but with plastic behaviour, an early reflection is predicted on figure 11(b). This problem disappears if the algorithmic energy dissipation is removed from the large timestep partition, by increasing ρb. In general, the larger the local timestep size, the higher the local value of ρb that can be used, to make the filtering of high frequencies more consistent across the mesh. To maintain stability, it is also appropriate to give the smallest timestep the lowest ρb. A typical structural problem, involving plastic bending of plates or beams, shows a response much more dominated by low frequencies - in cases such as this, use of ρb = 0 is not a problem.

Figure 11

Explicit generalised alpha method subcycled using the partial velocity method on the mesh of rod elements of figure 1(i). A – small timestep only. B – 4 subcycles.

9.

A Simpler Method of Adding Energy Dissipation

It is convenient for simplicity of programming, to retain the elegance of the velocity update of equation (22), and then compute the displacements in a way that introduces high frequency energy dissipation. A way to do this, that is simple, but cannot achieve a spectral radius of zero is the following. For node k with timestep ∆t=∆tk, ukt+∆t = ukt + vkt+∆t/2 ∆tk + α1(vkt+∆t/2 – 2vkt-∆t/2 + vkt-3∆t/2) ∆tk

(27)

Note if vkt+∆t/2=vkt-∆t/2=vkt-3∆t/2 corresponding to a low frequency, then the last term does not affect the update. If the velocity oscillates in each cycle so that vkt+∆t/2=–vkt-∆t/2=vkt-3∆t/2 then the term has a maximum effect. The correction term is a measure of jerk. To study the effect of this correction, amplification matrices can be written for the single degree of freedom system of figure 1(a), for a single timestep, and for a 2 subcycle partial velocity solution. In the single timestep case, the velocity updates are vkt+∆t/2∆tk = vkt-∆t/2∆tk - Ω 2 ukt vkt-∆t/2∆tk = vkt-3∆t/2∆tk - Ω 2 ukt-∆t

(28) (29)

If equation (29) is used to eliminate vkt-3∆t/2∆tk in equation (27) then equations (27) to (29) can be written as  u t +∆t  1 − (1 + α )Ω 2 α Ω 2 1 1    t = u 1 0    2  vt +∆t / 2   −Ω 0   

t 1  u    0  u t - ∆t  1   vt - ∆t / 2   

(30)

Plotting the spectral radius ρ (the maximum eigenvalue of this amplification matrix) in figure 12, it can be seen that high frequency dissipation is achieved. α1 = 0.04 gives the lowest spectral radius value of 0.5 at Ω = ωn ∆t = 1.8, where bifurcation occurs. As shown on the figure, higher values of α1 reduce stability further and have a higher minimum value of ρ. It is interesting to compare the explicit generalized alpha method with spectral radius at bifurcation ρb set to 0.48 (figure 13). The variation of spectral radius with Ω is almost identical, as is the response viewed in the time domain. However, with a higher ρb of 0.68, the explicit generalized alpha method does give more high frequency dissipation than the present method with α1 = 0.01. This approach requires α1 to be larger than 0.04 to stabilize models with a small number of degrees of freedom. However, with many degrees of freedom leading to narrow unstable timestep ranges in the undamped case, α1 = 0.04 is sufficient to ensure stability and limit spurious oscillation.

10.

Adding Energy Dissipation to the Energy Conserving Algorithm

The algorithms of equations (8) to (13) or (8), (15), (16), (10), (11), (12), (13) can have high frequency energy dissipation added by adapting the explicit generalised alpha algorithm to evaluate internal forces mid-step. The displacements in the original algorithm are reinterpreted as ui+vi ∆t/2 and velocities are reinterpreted as vi+ai ∆t/2. For a single timestep, the modified algorithm is as follows. FEXT are also evaluated mid-step. aα = M-1{-K (ui + vi∆t/2) + FEXT i+1/2} vi+1 = vi + (γ1 ai+1 + (1 - γ1) ai) ∆t

(31) (32)

where aα = (1 − α) ai+1 + α ai

(33)

ui+1 = ui + vi ∆t + (β1 ai+1 + (1/2 - β1) ai) ∆t2

(34)

and

To achieve an algorithm with identical properties (the same eigenvalues of an amplification matrix) we set γ1 = γ - ½ and β1 = β - γ/2 + ¼. α, γ and β are the parameters of the original algorithm estimated from ρb , the spectral radius at bifurcation. Note for no dissipation, γ1 = ½ and β1 = ¼, although the above algorithm is still explicit. When applied to the algorithm of equations (8), (15), (16), (10), (11), (12), (13), on the model problem of figure 1(a) with 2 subcycles, the variation of spectral radius of figure 4(b) results for ρb = 0. This shows dissipation when both timesteps approach their stability limits, which effectively removes the instabilities visible in figure 4(a).

Figure 12

11.

Spectral radius obtained using the α1 correction compared to that using the explicit generalised alpha method. A-α1 = 0.01, B-α1 = 0.04, C-α1 = 0.1, D-EGα, ρb = 0.68, E-EGα, ρb=0.48.

Predictor-corrector implementations of the partial velocity approach

A predictor-corrector subcycling algorithm is not quite a contradiction in terms. Without repeating an entire major cycle, it is not possible to correct the assumptions made on a timestep interface, however extra corrector cycles can help stabilize an algorithm to cope with viscous damping better, and to deal with the statistical stability problem. Since the aim of a subcycling algorithm is to minimise the number of evaluations of internal forces in explicit integration, it is not desirable to introduce more correction than necessary. 11.1 Dealing with viscous damping A weakness of explicit algorithms is the limited ability to cope with viscous effects, for instance in viscoelastic materials, without a reduction in the stable timestep size. A solution adapting the central difference method, is to adopt a predictor-corrector formulation and use two corrector cycles. In the single timestep case, this restores the stability of the undamped central difference method [16]. The partial velocity algorithm can also be put into a form

where multiple corrector cycles can be associated with any update to correct the partial velocity in the partition being updated. Table 1: Predictor-corrector partial velocity algorithm for significant damping

vL = vL0 + n/2 aL0∆t For n subcycles find predictors vSi+1 = vSi + aSi ∆t/2 vi+1 = vSi + vL = vPi+1 ui+1 = ui + vPi+1 ∆t aPi = 0 For j = 0, m-1 corrector cycles within each subcycle ∆aS = FSEXT - FSINT(ui+1) – FSVISC(vPi+j) – M aPi+j aPi+1 = aPi+j + ∆aS vPi+1 = vi+1 + aPi+1∆t/2 To complete each subcycle, aSi+1 = aPi+1 vSi+1 = vSi+1+ aSi+1 ∆t/2 After n subcycles, and aPn= 0 vn = vSn + vL = vPn For j = 0, m-1 corrector cycles ∆aL = FLEXT – FLINT(un) – FLVISC(vPn) – M aPn aPn = aPn + ∆aL vPn = vn + aPnn∆t/2 so that new partial accelerations and velocities are found for the large timestep partition. aLn = aPn and vLn = vL + aLnn∆t/2 The dual timestep case is as given in table 1 above. This algorithm can be tested on the single degree of freedom model problem of figure 1(e). If damping is added to the small timestep partition to give a damping ratio of 0.5, then with m=1, both a single timestep solution and one with two subcycles show the lowered stability limit on figure 13(a). This Figure plots stable states as zero and unstable states as one on Ω S, Ω L axes. With m=2, both cases regain the stability limit at A of figure 13(b), expected with no damping, but the subcycled case shows a lowered stability limit at B as well. The stability of the solution is still much better than that with m=1. If the damping is shared equally either side of the timestep partition, then with m=2 the single timestep case has the stability limit of figure 13(c). This is the same limit as that at A on Figure 13(b), but the subcycled case exceeds Ω = 2 on both axes, being more stable than its parent algorithm using the minor timestep only (figure 13(d)). On the other hand, if damping giving a damping ratio of 0.5 is added to the large timestep partition only, then stability is poor with n=2 and m=2. Experiments with other numbers of subcycles indicate that damping distributed to each partition in the same ratio as the distribution of stiffness tends to maximize stability. If damping is due to viscoelastic material properties, this situation will tend to be the case.

Figure 13

Stability of subcycling the system of Figure 1(e) with viscous damping giving ζ = 0.5 added. 0 = stable, 1 = unstable. (a) m=1, n=2. (b) m=2, n=2, S partition damped. (c) m=2, n=1, S and L partitions equally damped (not subcycled). (d) m=2,n=2, S and L partitions equally damped (subcycled).

11.2 An Alternative Predictor-Corrector With Algorithmic Dissipation Added An alternative iterative explicit algorithm for structural dynamics involves evaluating internal forces using displacements averaged over a step. A tangent matrix is found [17], which can be used in corrector cycles to correct predictions of both displacement and velocity values. Where the tangent matrix is estimated ignoring the elastic effects, and depends only on the mass matrix, an explicit analysis is obtained, and effectively, it is the residual impulse R found each cycle that is used to correct velocity and displacement values. Use of multiple corrector cycles with a single timestep size, and mid-step evaluation of internal forces converges to exact conservation of energy and momentum [17], although slowly, if the timestep is close to the stability limit. This approach has been used in conjunction with finite rotation updates each cycle to predict large rigid body motions. Such an approach can be subcycled using a partial velocity element interface in a way that converges to momentum conservation, and energy conservation in each timestep partition, but is still limited by approximations made to achieve a timestep interface. The statistical stability problem can be avoided in two ways: by dissipating energy when using a single corrector cycle, or by use of at least two corrector cycles. The algorithm proposed makes updates in a leapfrog fashion in a manner similar to the nodal interface in equations (8) to (13) above, but with the added benefit of taking account of all subcycle states in updating displacements on the timestep interface. Use of two corrector cycles in each partition removes the problem of statistical stability, even when there is no energy dissipation present. The dual timestep case is as follows. Consider state the S partition that has been updated to state zero using partial velocities on the large timestep side of a timestep

partition, vL–n/2. vL–n/2 can now be updated using several different predictors of un/2. Firstly, old displacements u-n/2, half a major cycle ago, but current partial velocities vS0 could be used. Alternately, the displacements u0 are currently known, and correspond to the midpoint of the major cycle. These could be used directly to evaluate internal forces, or to estimate a prediction of un/2. un/2 P = u-n/2 + (vS0 + vL –n/2) n∆t

(35)

½(un/2 P + u-n/2) = u0

(36)

un/2 P = u0+ (vS0 + vL –n/2) n∆t/2

(37)

or or

v

L n/2 P

L –n/2

=v

(38)

For 1 to m corrector cycles, using αL to get a generalised midpoint value : uLP = un/2 P(1 - αL) + u-n/2 αL

(39)

RL = {FLEXT - FLINT(uLP)} n∆t + M (vL–n/2 - vLn/2 P)

(40)

vLn/2 P = vLn/2 P + M-1 RL

(41)

un/2 P = un/2 P + M-1 RL n∆t/2

(42)

vLn/2 = vLn/2 P

(43)

ui+1 P = ui + (vSi + vLn/2) ∆t

(44)

vSi+1 P = vSi

(45)

end Subcycle updates use vLn/2 . For i = 0, n-1 (subcycles)

For 1 to m corrector cycles: uSP = ui+1 P(1 - αS) + ui αS S

S

R = {F

(u P)} ∆t + M (v i – v

S

EXT

-F

S

S

INT

(46) S

)

i+1 P

(47)

vSi+1 P = vSi+1 P + M-1 RS

(48)

ui+1 P = ui + (vLn/2 + (vSi + vSi+1 P)/2) ∆t

(49)

ui+1 = ui+1 P

(50)

end S

v

i+1

=v

S i+1 P

(51)

end Equation (49) can also be written in the manner of equation (42). However, equation (49) makes it clearer that while the S partition is being updated, the partial velocity from the L partition is used but not corrected. Note that for timestep interface nodes, the new displacement uLP found with the major timestep, is only used to correct the update of vL. The highest stable values of αS and αL are ½. Using the older state to predict uLP equation (35), with αS = ½ and αL = ½, n = 2 and m = 1, the algorithm is dissipative enough to suppress the instability at Ω S = √2, in the single degree of freedom, two subcycle case. This is shown in the plot of spectral radius of a major cycle: Figure 14(a). Note, as with

other leapfrog interfaces, the dissipation occurs only when both partitions approach their stability limits. The algorithm is stable until Ω S = 2 or Ω L = 2 with these α’s – that is, no loss of stability occurs due to coupling the timestep partitions. If more corrector cycles are used, there is a tendency to reduce this dissipation of high frequencies, as the corrector cycle is not dissipative, and stability is maintained. With m = 1, there is a loss of accuracy, compared to use of the small timestep only, in the region where the algorithm becomes dissipative. Examination of convergence to the exact solution shows little loss of accuracy due to subcycling with when Ω S+Ω L2. With extra corrector cycles, this “ridge” of statistical stability disappears.

With α’s equal to zero, maximum dissipation of high frequencies results with m = 1 – with a loss of stability. If the algorithm is to be programmed by accumulating velocity changes in a clock-driven implementation, the use of an uncorrected uLP = u0 is clearly most convenient, as previous displacements do not then need to be stored. An attractive possibility, if two corrector cycles are to be used at timestep interface nodes, is to use the non-dissipative setting αS = αL = ½ and add dissipation of high frequencies to the corrector cycles by other means, rather than by using the α parameters, as extra corrector cycles tend to undo the dissipation introduced with the α parameters. One way is to use modify equation (49) to resemble equation (27), so the small timestep update of displacement in the above algorithm becomes ui+1 P = ui + (vLn/2 + (1+ α1) vSav i – 2α1 vSav i-1 + α1 vSav i-2) ∆t

(52)

where vSav i=(vSi+vSi+1 P)/2 and the other averages are from the final corrector cycles of previous S updates. Another way to add dissipation is to use the modified explicit generalised alpha approach modified to enable midstep evaluation of internal forces, as in equations (31) to (34). The implementation of this using u0 to estimate the large timestep update and just one corrector cycle is given in table 2, where γS or γL and βS correspond to γ1 and β1 in the single timestep case of section 10. Table 2

Partial velocity algorithm with leapfrog treatment of the timestep interface and generalised α dissipation.

% L update from state –n/2 to n/2 aLn/2 = 1/(1 - αL) M-1{FLINT(u0) + FLEXT 0} - αL/(1 - αL) aL-n/2 vLn/2 = vL-n/2 +{γL aLn/2 + (1 - γL) aL-n/2}n ∆t % n S updates aSi+1 = 1/(1 - αS) M-1{FSINT(ui + (vSi + vLn/2) ∆t/2) + FSEXT i+1/2} - αS/(1 - αS) aSi vSi+1 = vSi + (γS aSi+1 + (1 - γS) aSi) ∆t ui+1 = ui + (vSi + vLn/2) ∆t + {βS aSi+1 + (1/2 - βS) aSi } ∆t2

14.

Conclusions

Subcycling of non-linear explicit dynamics problems can greatly reduce the computation needed to solve an impact problem. A truly robust algorithm needs some high frequency energy dissipation available, to ensure stability where the material behaviour does not dissipate energy sufficiently. As well, it is desirable to have a major timestep update dependent on all minor timestep states at a timestep interface. This will ensure momentum is transferred across the timestep interface correctly. Approaches to subcycling the explicit generalized alpha method that satisfy these criteria are to sum impulses at the timestep interface, on nodes yet to be updated, or to add partial velocities or accelerations. The partial velocity or acceleration approaches tend to be more accurate, as more “up-to-date” midpoint information from the large timestep side of an element interface is being used. However, extra variables (partial velocities and accelerations) must be stored. The algorithms that hold velocity constant discussed here tend to be more robust than ones that hold an acceleration constant. If the central difference method is extended by partial velocity subcycling, the need to store such quantities can be avoided by accumulating impulses associated with different element timesteps. High frequency dissipation can be added to

such an algorithm by altering the way that displacements are found from velocities. If a timestep change is placed cautiously, it can be satisfactory to sample the small timestep state once per major cycle, in which case the energy conserving algorithm presented could be used. It is accurate and has no risk of instability for a linear elastic case. With use of a constant velocity assumption on a timestep interface, high frequency damping is needed, but can be added effectively to this algorithm. Viscous damping effects need not lead to a deterioration of stability if an extra corrector cycle is added to each update. The situation modelled may require an explicit algorithm with multiple corrector cycles, as when large viscous effects are present. If this is so then a subcycling algorithm can still be used (provided that the use of the larger timestep would have been valid at the timestep interface). Use of two corrector cycles in a partial velocity algorithm can remove the problem of statistical stability, associated with parts of the model being effectively switched on and off during subcycling. One potential application of these algorithms is to particle mechanics models.

References [1] S. Piperno, C. Farhat, and B. Larrouturou, Partitioned procedures for the transient solution of coupled aeroelastic problems. Part I: Model problem, theory and two-dimensional application, Comput. Methods Appl. Mech. Engrg. 124(1995) 79-112. [2] F. Armero, and J.C. Simo, A new unconditionally stable fractional step method for non-linear coupled thermomechanical problems, Int. Jnl. for Num. Meth. in Engrg. 35 (1992) 737-766. [3] W.K. Liu, Development of mixed time partition procedures for thermal analysis of structures, Int. Jnl. for Num. Meth. in Engrg. 19 (1983) 125-140. [4] T. Belytschko and P. Smolinski and W.K. Liu, Stability of multi-time step partitioned integrators for first-order finite element systems, Comput. Methods Appl. Mech. Engrg. 49 (1985) 281-297. [5] P. Smolinski, T. Belytschko, and M. Neal, Multi-time-step integration using nodal partitioning, Int. Jnl. for Num. Meth. in Engrg. 26 (1988) 349-359. [6] T. Belytschko and Y.Y. Lu, Convergence and stability analysis of explicit multi-timestep algorithm for parabolic systems”, Comput. Methods Appl. Mech. Engrg. 102 (1993) 179-198. [7] W.J.T. Daniel, Subcycling first- and second order generalizations of the trapezoidal rule, Int. Jnl. for Num. Meth. in Engrg. 42 (1998) 1091-1119. [8] J. Chung and G.M. Hulbert, A time integration algorithm for structural dynamics with improved numerical dissipation: the generalized-α method, ASME J. Appl. Mech. 60 (1993) 371-375. [9] T. Belytschko, H.-J. Yen and R. Mullen, Mixed methods for time integration, Comput. Methods Appl. Mech. Engrg. 17/18 (1979) 259-275. [10] W.J.T. Daniel, Analysis and implementation of a new constant acceleration subcycling algorithm, Int. Jnl. for Mum. Meth. in Engrg. 40 (1997) 2841-2855. [11] W.J.T. Daniel, A study of the stability of subcycling algorithms in structural dynamics, Comput. Methods Appl. Mech. Engrg. 156 (1998) 1-13. [12] P. Smolinski, S. Sleith and T. Belytschko, Stability of an explicit multi-time step integration algorithm for linear structural dynamics equations, Comput. Mech. 18 (1996) 236-244. [13] G.M. Hulbert and J. Chung, Explicit time integration algorithms for structural dynamics with optimal numerical dissipation, Comput. Meth. Appl. Mech. Engrg. 137 (1996) 175-188. [14] W.J.T. Daniel, Conserving momentum while subcycling structural dynamics, to appear. [15] P. Smolinski and Y.S. Wu, Stability of explicit subcycling time integration with linear interpolation for first-order finite element semidiscretizations, Comput. Methods Appl. Mech. Engrg. 151 (1998) 311324. [16] T.J.R. Hughes, The Finite Element Method: Linear Static And Dynamic Finite Element Analysis, Prentice-Hall, Englewood Cliffs, New Jersey, 1987. [17] L. Briseghella, C.E. Majorana and C. Pellegrino, Conservation of angular momentum and energy in the integration of non-linear dynamic equations, Comput. Methods Appl. Mech. Engrg. 151 (1999) 247-263.

Part IV Advanced Methods in Systems with Large Rigid Body Motion

Nonlinear Structural Multibody System Simulation Using Structural and Rigid Body Dynamic Analysis Software Weidong PAN and Edward J. HAUG National Advanced Driving Simulator and Simulation Center, The University of Iowa, 2401 Oakdale Blvd, Coralville, IA 52241, U.S.A. Abstract. Three approaches are presented to integrate a multibody dynamics code and a nonlinear finite element code for structural multibody system dynamic simulation. The first approach is the most general and requires implementation of a new joint library to model constraints between bodies that are modeled using nonlinear finite element methods (called general bodies) and rigid or linear elastic bodies that are modeled using multibody dynamics. The second approach avoids such a library, with a modest assumption. It also reduces the degree of coupling between the multibody dynamics and nonlinear finite element formulations, which leads to efficient implementation. The third approach assumes that inertia effects of general bodies in a system are negligible and/or not of interest. This approach leads to a formulation with minimal coupling between multibody dynamics software and nonlinear finite element analysis software. With the third approach, multibody dynamics software such as DADS and nonlinear finite element analysis software such as ABAQUS can be integrated without accessing source code, other than standard user-defined subroutines. Numerical examples are presented for each of the three approaches.

1. Introduction Virtual prototyping is becoming increasingly important in mechanical system design and evaluation. With virtual prototyping, information that was traditionally gathered in a test lab, in test flights, or on a test track with physical prototypes can now be obtained more quickly and efficiently on the computer. Virtual prototyping can thus reduce the requirement for building and testing numerous physical prototypes. It can also lead to significant cost savings and reductions in time-to-market. Depending on simulation objectives, the multibody model; i.e., the virtual prototype of a mechanical system, can have different levels of resolution. The lowest resolution model is a multibody dynamics model in which all bodies are modeled as rigid. The highest resolution model is a model in which all bodies are modeled as capable of performing arbitrary large translations and rotations, as well as geometric and material nonlinear deformation. A mid-resolution model is one that consists of bodies capable of large translations and rotations, but small elastic deformation. It is appropriate to model different bodies in a multibody system with different resolutions. This is because most physical multibody systems are not homogeneous in terms of rigidity; i.e., some bodies are nearly rigid and/or are subjected to low load, so that modeling them as rigid bodies is adequate. Other bodies, however, may be very flexible and/or subjected to severe loading, so that large deformation occurs. In this respect, an

accurate computer model of a multibody system is composed of three types of bodies; rigid bodies, flexible bodies that undergo large gross motion accompanied by small elastic vibration, and flexible bodies that undergo large gross motion and geometric and material nonlinear deformation. In this paper, the second and third types of bodies are called linear elastic bodies and general bodies, respectively. A multibody system consisting of rigid bodies, linear elastic flexible bodies, and general bodies is called a general flexible multibody system. For rigid bodies, computational efficient formulations and codes have existed for two decades [1,2]. The key component that distinguishes these formulations and codes is the choice of generalized coordinates. There are two basic classes of generalized coordinates; topologically-based coordinates and topologically-independent coordinates. Joint coordinates are the most commonly used topologically-based coordinates [3]. Cartesian generalized coordinates, on the other hand, are the most commonly used topologically-independent coordinates [1]. Cartesian generalized coordinates are adopted in this paper for deriving equations of motion of rigid bodies. For linear elastic bodies, the deformation mode superposition approach, or modal flexibility approach [4-7], has proved to be accurate and efficient and is widely used. The modal approach is employed in this paper, so a linear elastic body is called a modal body. Among the most important issues for this approach are generation of component deformation modes [6] and formation of lumped inertia matrices [7, 8]. The modal flexibility approach, though computationally efficient, is not valid for modeling geometric and material nonlinear deformations. Due to two decades of development of formulations and associated solution methods, highly sophisticated commercial finite element analysis software such as ABAQUS is used routinely for nonlinear deformation problems. The maturity of nonlinear finite element formulations and solution methods; coupled with improvements in computer speed and memory/disk sizes, makes it possible for multibody dynamic analysis to rigorously include nonlinear deformation. Multibody dynamics of mechanisms and nonlinear finite element structural dynamics are commonly regarded as two distinct fields, both theoretically and in computational practice. In addition, there exists a large body of nonlinear finite element theory and methods that has not been brought to bear on multibody dynamics. In two recent papers [9, 10], formulations of multibody dynamics and nonlinear finite element methods were unified, resulting in formulations and associated solution methods for dynamic simulation of general flexible multibody systems. The objective of this paper to present computer implementations of the formulations and solution methods presented in Refs. 9 and 10. As noted in Ref. 9, the process of developing these formulations included consideration that their implementation should be able to leverage existing implementations of multibody dynamic analysis and nonlinear finite element analysis software. Implementations of the formulations presented in this paper are achieved by integrating existing, general-purpose, large-scale finite element analysis codes and multibody dynamics codes. The finite element analysis codes chosen are an in-house pilot nonlinear finite element code called GFLEX and the commercial ABAQUS/standard code. The multibody dynamics code used in this paper is the commercial DADS code. Since rigid and modal bodies are already available in DADS, they are collectively called conventional bodies. This paper is organized as follows. In Section 2, formulations presented in Refs. 9 and 10 are reviewed, leading to three sets of differential-algebraic equations (DAE) that are respectively called DAE-1, DAE-2, and DAE-3. In Section 3, the organizations of DADS and GFLEX are described. Section 4 presents an implementation of DAE-1 by integrating GFLEX into DADS as a general body module. To enable modeling of constraints between general bodies and conventional bodies new joint modules are implemented into DADS.

Section 5 implements DAE-2 using an approach similar to the one used for DAE-1. Section 6 implements DAE-3 by linking the DADS and ABAQUS/standard. Numerical examples that demonstrate these three implementations are presented in Section 7. Conclusions are presented in Section 8.

2.

Formulation

2.1

DAE-1

The variational equation of motion of all rigid and modal bodies in a multibody system is obtained by summing variational equations of motion of rigid bodies [1] and modal bodies [7,11]. It can be written in compact form as !! − G ) = 0 δq T (Mq

(1)

where q is the vector of Cartesian generalized coordinates and modal coordinates; M is the inertia matrix; G is the generalized force vector that excludes workless constraint reaction forces; and δq is a vector of kinematically admissible variations of q. The variational equation of motion of all general bodies in a multibody system is [9]

!! g − G g ) = 0 δq g (M g q T

(2)

where q g is the vector of nodal coordinates of general bodies; M g is the finite element mass matrix; G g is the vector of nodal forces; and δq g the vector of kinematically admissible variations of q g . The superscript g stands for general body. The generalized force vector G g can be expressed as G g = Qg − P g

where Q g is the vector of external nodal forces, and P g is the vector of internal nodal forces. Equation (2) forms the basis for the displacement-based finite element analysis procedure [12–14]. Hybrid and mixed methods, with discontinuous fields across interelement boundaries, lead to elemental equations in the form of equation (2), so they can be incorporated into existing single-field programs with minimal modification [15, 16]. The corotational approach yields exactly the same form of equations as equation (2) [17]. In conclusion, equation (2) encompasses a large part of the existing finite element technology. Constraints between bodies in a general multibody system are represented by a set of non-linear algebraic equations as  Φ(q,t ) Φ q,q ,t = Φ cg q,q g ,t  Φ g q g ,t a

(

g

)

( (

)

 =0  

)

(3)

where Φ is the collection of constraints among the rigid and modal bodies, Φ cg is the collection of constraints between any pairs of rigid or modal bodies and general bodies, and Φ g is the collection of nonlinear constraint equations among general bodies. The variable t stands for time. It should be emphasized that Φ g contains nonlinear constraints only. This is because linear constraints can be imposed during the finite element assembly process, by distributing terms associated with slave degrees of freedom to appropriate locations of global matrices that correspond to master degrees of freedom [9].

Taking the variation of equation (3) yields Φqaδq + Φqa g δq = 0

(4)

where Φ (a) = ∂Φ a / ∂( ) . By introducing a vector λ a of Lagrange multipliers, equations (1), (2), and (4) can be combined to obtain

(

)

(

)

!! − G + Φ qa λ a + δq g M g q !! g − G g + Φ qa λ a = 0 δq T M q T

T

T

g

(5)

where δq and δq g are arbitrary. Thus, !! + Φqa λ a = G Mq

(6)

!! g + Φ qa g λ a = G g M gq

(7)

T

T

Taking the second time derivative of equation (3) yields !! + Φ qa g q !! g = γ a Φ qaq

where Φ (a)( ) = ∂ 2 Φ a / ∂ ( )∂ ( ) and

(

(

)

γ a = − Φ qa q! q q! − 2Φ qat q! − Φ qa g q! g

)

qg

(8)

q! g − 2Φ qa g t q! g − Φ tta

Combining equations (6), (7), and (8) yields M   0  a Φ q

(Φ )

a T q a T

0 Mg

(Φ )

Φ qa g

0

qg

 q !! G   g   g  !! = G  q   λ a   γ a      

(9)

Equation (9), along with equation (3), comprise a set of DAE of the general flexible multibody system. This set of DAE is called DAE-1 in this paper. 2.2

DAE-2

DAE-1 is general. However, it has two potential disadvantages. The first disadvantage is that for many applications in which conventional bodies are connected with general bodies through a large number of nodes, the dimension of the DAE to be solve is huge. The second disadvantages is that in this formulation the equations of motion of conventional bodies and the equations of motion of general bodies are kinematically coupled, thus requiring a very involved implementation. A revised general flexible multibody dynamics formulation that remedies these two problems is presented as follows. By exploiting modeling options, nonlinear constraint equations between nodes of general bodies can often be avoided [10]. In such cases, Φ g is empty, and equation (3) reduces to  Φ(q,t )  (10) Φ a q,q g ,t =  cg =0 g Φ q,q ,t  ~ that does not appear in any constraint The vector q g can be partitioned into a vector q equation and a vector q that appears in constraint equations; i.e., ~ q (11) qg →   q 

(

)

(

)

The second constraint equation in equation (10), Φ cg = 0 , can be written in the form Φ cg (q,q ,t ) = 0

(12)

Using the partitioning defined in equation (11), M g and G g in equation (2) can be partitioned into ~  M MU  g (13) M → L  M M ~ G  g G =  (14) G  respectively. Equation (2) can then be rewritten as T ~  ~ ~ !!  G q   M M U  q δ~    !!  −    = 0 δ q   L M  q  G     M

(15)

Using equations obtained by taking the first variation and second time derivative of !! from equation (15), and then following the same equation (12) to eliminate δ q and q approach that leads to equation (9) yields [10] ~ ~ ~ !!     M M U H1 0  q G − MU H 2    T L T T   T T !!  = G + H 1 G − H 1 M H  (16) M + H 1 MH 1 Φ q  q H1 M   0 0   λ   Φq γ   where

( ) Φ = −(Φ ) γ

H1 = Φ cg q H2

−1

cg −1 q

cg q cg

(17) (18)

It is shown in Ref.10 that H 1 and H 2 are well-defined and can be analytically expressed, thus requiring no numerical inversion. Equation (16), along with the constraint equations Φ=0

(19)

comprise a new set of DAE of the general flexible multibody system. Compared to DAE-1, this set of DAE is of smaller dimension. The reduction in dimension is significant when conventional bodies and general bodies are connected through a large number of nodes. Moreover, since Φ cg does not appear in this DAE, there is no need to implement a new joint library, hence requiring less implementation effort. An important feature of equation (16) is that it allows further simplification by choosing appropriate inertia formulation for general bodies. The lumped inertia formulation is widely used and has proven to be effective in structural dynamics [14] and flexible multibody dynamics [4, 7]. If the lumped inertia formulation is used for finite elements that are connected to conventional bodies; i.e., at least one nodal coordinate of each finite element is included in q , then M U = 0, M L = 0

(20)

Finite elements that are not connected to conventional bodies can use either the lumped or consis-tent inertia formulation. There are inertia lumping methods that generate lumped

inertia matrices using consistent mass matrices. Examples are the row-sum method [14] and Hinton’s method [18]. By applying these methods to a row of the global mass matrix of a general body, lumped inertia can be obtained for a specific nodal coordinates; i.e., use the lumped inertia formulation for nodal coordinates in q and the consistent inertia ~ . Since the nodal coordinates q of general bodies formulation for nodal coordinates in q are defined by generalized coordinates of conventional bodies through Φ cg = 0 , the lumped inertia associated with them can be considered as part of the inertia of the conventional bodies to which they are connected. Thus, by proper preprocessing, the lumped inertias can be accounted for in M, in such away that M=0

Substituting equations (20) and (21) into equation (16) yields ~ !~! ~ Mq =G M  Φ q

Φ Tq  !q! G + H 1T G  =   0  λ   γ 

(21)

(22) (23)

Equation (22) is a set of ordinary differential equations (ODE) that governs the dynamics of general bodies in the multibody system. Equation (23), along with Φ = 0 , is a set of DAE that governs the dynamics of rigid and modal bodies in the multibody system. In this paper, equations (22), (23), and (19) are called DAE-2. ~ The right side of equation (22), G , depends on q, and G on the right side of ~ ; i.e., equation (23) depends on q ~ ~ ~ G = G (q , q (q ), t ) (24)

~, q (q ), t ) G = G (q

(25)

Thus, equations (22) and (23) are force-coupled. In other words, general bodies can be considered as a force element in conventional multibody dynamics. However, this force element is different from conventional translational-spring-damper-actuators (TSDA), rotational-spring-damper-actuators (RSDA), and simple tire models because it contains the ~ of internal states. Unlike kinematic and inertia coupled formulations, the force vector q ~! associated with general bodies and coupled formulation allows accelerations !q !! and λ associated with conventional bodies be accelerations and Lagrange multipliers q ~ solved for independently. In finite element methods, the sparsity pattern of mass matrix M is well-defined and an efficient sparse linear solver [19] can be used to solve equation (22). ~ When the lumped inertia formulation is used for general bodies; i.e., M is diagonal, ~ is trivial, solving for q ~ G q~α = ~ α M αα where no summation is performed over repeated Greek indices. The weak coupling ~ and q also allows effective use of dual-rate integration methods [20, 21], in between q ~ and slow integration is performed for q. which fast integration is performed for q Furthermore, compared to formulations with tighter coupling, the integration Jacobian matrix for implicit numerical integration is easier to derive and cheaper to evaluate [22].

2.3

DAE-3

There are applications in which inertia effects in general bodies are not important. For example, in the aerospace industry, most mechanism analyses involve “slow speed” movement of flaps, landing gears, doors, etc. Another example is vehicle dynamic simulation of ride and handling in which the small amplitude, high frequency internal vibration of tires is not of interest and has little effect on overall simulation fidelity. In these and many other engineering applications, the associated inertia effects can be ignored ~ !~! by setting Mq = 0 . Thus, equation (22) reduces to a set of nonlinear algebraic equations ~ ~ G (q ,q) = 0 (26) General bodies governed by equation (26) are called quasi-static general bodies in this paper. When q is known, q can be calculated from Φ cg = 0 . Equation (26) can then ~ , using a nonlinear solver. With both q and q ~ calculated, G can be be solved for q evaluated. Thus, general bodies act exactly as a nonlinear force element of a multibody system, without internal state variables. In this paper, equations (23), (19), and (26) are called DAE-3. In finite element analysis (FEA) software, q is called a vector nodal coordinates with speci-fied displacements. A quasi-static analysis, carried out using FEA software, ~ and forces and moments required to produce the specified values of q , calculates q denoted as F . For general bodies to be in equilibrium, F must be in balance with the internal force; i.e., P = − F . Also, note that for a degree of freedom with specified displacement, the external force associated with it should be zero; i.e., Q = 0 . In other words, external force should not be specified on a degree of freedom that has specified displacement. Thus, G =Q−P =F

(27)

Quasi-static general bodies are computationally efficient, since they do not introduce high frequency states into the equations of motion of a multibody system. Thus, compared to general bodies, quasi-static general bodies allow use of substantially larger integration stepsizes. Another advantage of using quasi-static general bodies is that they allow integration of multibody dynamics software and nonlinear finite element analysis software, without accessing source code other than a small set of user-defined subroutines. To be more specific, if finite element software provides a means for user-defined subroutines to specify displacements, as well as a way of accessing reaction forces due to specified displacements, then it can be brought into a multibody dynamics code for dynamic analysis of systems containing conventional bodies and quasi-static general bodies.

3.

Organization of DADS and GFLEX

The organization of commercial DADS software and an in-house pilot nonlinear finite element analysis software, called GFLEX, are described in this section, laying the base for implementations presented in the following sections. 3.1

DADS

A modular approach is employed in DADS. Schematically, the structure may be represented as shown in figure 1. To the left of the junction subroutine is the library of

modules that provide modeling capabilities. Modeling modules are mutually transparent; i.e., there is no need to know the details of existing modules in order to implement a new module. To the right of the junction subroutine are analysis modules that comprise the central processing algorithm for the solution of constrained dynamic system assembles and integration of the equations of motion of the system. Modeling modules and analysis modules are separated by and communicate through the junction subroutine. This separation is beneficial, in that (1) the solver is transparent to researchers who implement and/or enhance modeling capabilities and (2) modeling activities are transparent to researchers who implement and/or enhance solvers. This provided greater flexibility and maintainability of a code that continues to evolve. Each modeling module represents either an inertial, constraint, force, or control element in the system. A module for a particular type of elements in the system must contain subroutines that perform the proper tasks of input, output, pointer definition, and evaluation of force, mass, and constraint terms that are required for that element to be completely represented in the system. There are at most eight basic tasks that a DADS module may need to perform. These tasks are listed in table 1. For each task that must be performed, a junction subroutine that loops through the modules is called. In most tasks, the modules accumulate the number of generalized coordinates, the number of constraint equations, and the number and addresses (row and column numbers) of nonzero entries in Jacobian matrices and mass matrices.

Start

Read .fm3 Create .ZYX

ISTAT

Conventional joints

Read data for each module

ISTAT F, F q

Model Assembly .

(q, q) n ISTAT

Other

DAE solver M G Fq n, g

.

..

(q, q, q, l) n)1 ttT no End

Figure 1: Computational flow chart of commercial DADS code.

yes

n+1 → n

Modal body module

Junction

Rigid body module

Table 1 Basic tasks that a DADS module performs.

ISTAT

Description

1

Data input.

2

Perform misc. functions, such as check on input data consistency.

3

Transfer values of generalized coordinates and velocities of the assembled model to arrays Q and QD, respectively. This is in effect a procedure of specifying initial conditions.

4

Generate row and column pointers of nonzero terms of Jacobian matrix or mass matrix.

5

Depending on the value of a flag, evaluate constraint function, right side of velocity equations, or right side of acceleration equations.

6

Evaluate generalized forces.

7

Evaluate mass matrix.

8

Output.

The central processing algorithm is briefly described as follows. The first step of the algorithm is to read a formatted model definition file (with extension .fm3) and store all non-comment, non-blank lines in an external file with extension .ZYX. In this file, lines are linked together so that all elements of a particular module are linked together for processing by the appropriate input routine of that module. Pointers are also defined for each module present in the model, identifying the start of that module's data in the memory allocated. The algorithm then passes ISTAT=1 to the junction subroutine, to direct all modules to read data from the .ZYX file and store them in the memory. The model assembly phase is then carried out using assembly minimization methods [1]. At each iteration of the minimization process, requests are made to the junction subroutine to generate the constraint equations and Jacobian information that are required for the minimization computation. Following successful assembly, an analysis subroutine carries out a computational check on the rank of the constraint Jacobian, to identify any redundant constraints that may exist. Redundant constraints are automatically removed. If a feasible model has been specified; i.e, Φ = 0 is satisfied to within a specified tolerance, analysis proceeds. If not, the user is informed that refinement to the model or estimated data is required. If a feasible model has been specified, the DAE solver is started to perform dynamic analysis. The DAE solver employs the generalized coordinate partitioning algorithm. A detailed description of the DAE solver can be found in Ref. 1. 3.2

GFLEX Code

GFLEX has the structure of a general-purpose, large-scale, nonlinear finite element analysis program, although its element and material libraries are not as extensive as commercial code such as ABAQUS. It adopts a modular design, employs a dynamic memory allocation scheme, and is able to generate the sparsity patterns of mass and stiffness matrices. The input file is in a keyword-based free format. Models with simple geometry can be generated using its automatic node and element generation capability. Complex models can be created using the commercial PATRAN software and imported into GFLEX.

The structure of GFLEX is shown in figure 2. It has both modeling and analysis capabilities. Thus, it is self-sufficient and can be used as stand-alone software for nonlinear finite element analysis. As in the structure of DADS, modeling and analysis are separated and communicate through a junction subroutine. This feature, which is very standard among commercial finite element software, allows trivial extraction of its modeling capability; i.e., the modules to the left the junction subroutine, for integration into DADS. Start

Other

Junction

Element library

Read Gflex input file

ǒq g, q. gǓ

n

IND ODE Solver Mg Gg g g K 5 ēGg ēq g C g 5 ēG. g ēq

ǒq g, q. g, q.. gǓ ttT

n)1

n+1 → n

IND

Material library

yes

no End

Figure 2: Computational flow chart of the Glfex code.

4.

An Implementation For DAE-1

The modular structure of DADS lends itself well to the task of adding new functionality into it by adding new modules. Implementation of DAE-1 is achieved by adding a general body module and modules of new joints to the existing library of DADS modules. The resulting implementation is shown in Figure 3, where the new modules are grayed. The central processing algorithm of DADS is used without any change. Implementation of the new modules are described in the subsections that follow. 4.1

General Body Module

The general body module consists of the GFLEX modules and a module master subroutine. This master subroutine is called by the DADS junction subroutine and performs the following tasks according to ISTAT: • ISTAT=1: Call GFLEX modules to read GFLEX input file. • ISTAT=2: Call GFLEX modules to perform initialization, such as node resequencing for minimization of bandwidth of global matrices, formation of destination array, etc. Input data consistency checking is also performed. • ISTAT=3: Transfer initial values of q g and q! g to the appropriate positions of arrays Q and QD, respectively. • ISTAT=4: Generate row and column indices of nonzero terms of mass matrix M g . These indices are used to assemble nonzero terms of M g at proper locations of the coefficient matrix in Equation 9.

• ISTAT=6: Evaluate generalized force vector G g . • ISTAT=7: Evaluate mass matrix M g . !! g to output file. Stress and strain information can • ISTAT=8: Report q g , q! g , and q also be generated and reported if requested. Start

Read .fm3 Create .ZYX

Rigid body module

ISTAT

Conventional joints

Read data for each module

ISTAT Fa ƪFaq, Faqgƫ

Model Assembly

ǒq, q. , q g, q. gǓ

n

ISTAT New Joints

ƪ

M 0 0 Mg

Other

ƪGG ƫ

ƫ

g

ƪFaq, Faq ƫ

DAE solver

ǒq, q. , q.. , q g, q. g, q.. g, l aǓ n)1 ttT

yes

no

g

n a, g a

n+1 → n

General body module

Junction

Modal body module

End

Figure 3: Computational flow chart for generating and solving DAE-1.

Another alternative is to implement general body capability into the existing modal body module. To do so, a flag is introduced for each body to indicate if it is a general body. When this flag is set true, then the above functionalities are performed. If the flag is set to false, then the modal body functionalities are performed. A modal body, like a rigid body, has a body reference frame with which seven generalized coordinates is associated. In contrast, a general body does not have an associated body reference frame, so these seven generalized coordinates are redundant for the general body and must be eliminated. This is easily done by using the ``fixed to ground'' option provided by DADS. This option automatically introduces seven constraints into the system to eliminate artificial degrees of freedom; i.e,

Φ fix

e02 + e12 + e 22 + e32 − 1   x cg − c1     y cg − c 2   = z cg − c 3 =0   e1 − c 4   e 2 − c5     e3 − c 6  

where ci , i=1, 2, 3, 4, 5; 6, are constants that depend on the initial position and orientation of the body reference frame.

Although it is intuitive and convenient for a user to define as many general bodies as appropriate, there is no need to define more than one general body in a multibody system from computation point of view. This is because general bodies do not have body reference frames and nodal coordinates of all general bodies are defined with respect to the global reference frame. The advantage of defining only one general body is that it saves bookkeeping. For the sake of user friendliness, it is a good idea to provide an interface for a user to define a body reference frame for each general body, to create the model and interpret results using that reference frame. To do so, a pre-processor must be provided to collapse all general bodies into an internal one-general-body representation, and a post-processor must be provided to convert analysis results back to original settings. 4.2

New Joint Modules

To model kinematic constraints between general bodies and conventional bodies, new joint modules need to be implemented. Joint formulations are derived in this section. Implementation of these joints is standard. The interested reader is referred to Ref. 23. For planar multibody dynamic analysis, only bracket joints between a general body and a rigid body are treated here. This, however, does not compromise the modeling capability, compared to implementing all types of joints in DADS. Its justification is as follows. By attaching (using the present bracket joint), an auxiliary rigid body to a node of a general body, any constraint applied at that node of the general body is equivalent to a constraint applied at the auxiliary rigid body. Thus, all types of joints currently supported by DADS can be applied to a general body through an auxiliary rigid body. To ensure that the auxiliary rigid body does not change the dynamic behavior of a system, the center of mass of the auxiliary rigid body is chosen to be at the node to which it is attached. Its mass and rotary inertia are chosen to be a fraction of the nodal mass and rotary inertia of the node to which it is attached. For spatial dynamic analysis, only spherical joints between a general body and a rigid or modal body are treated here. As in planar dynamic analysis, this in no way compromises the modeling capability. This is because whenever a joint that is of a type other than a spherical joint is needed, an auxiliary rigid body can be introduced. The auxiliary rigid body is fixed to the general body by connecting three or more non-collinear points of it to the corresponding nodes of the general body using spherical joints. This way, any constraint applied at the small region bounded by these three nodes of the general body can be approximated by a constraint applied on the auxiliary rigid body. Thus, any type of joint currently supported by DADS can be applied to a general body, through the use of auxiliary rigid bodies. To ensure that the auxiliary rigid body does not change the dynamic behavior of a system, the center of mass of this body is chosen to be at the centroid of the region, and its mass and rotary inertia are chosen to be a fraction of the mass and rotary inertia of the region. First, consider a bracket joint for planar multibody dynamic analysis. Let a rigid body i and a general body j be connected at a common point P, through a bracket joint. Let the node number of point P in the general body be I. Assume that there are three nodal coordinates associated with node I. The bracket joint can be mathematically represented by three constraint equations,

(

r + A i s′i P − X Ij + u Ij Φb =  i φ i − Θ Ij + θ jI 

(

)

) = 0  

where ri and A i are the current location and orientation of the body reference frame of body i with respect to the global reference frame; s ′i P is the location of point P relative to

the body reference frame of body i; X Ij and Θ Ij are the initial location and orientation angle of node I with respect to the global reference frame, respectively; and u Ij and θ jI are nodal displacements and nodal rotation of node I. The Jacobian matrix, right side of the velocity equation, and right side of the acceleration equation are Φ

 I B i s ′i P  =  1  0

b qi

Φ bq I = −I j

ν ( ij ) = 0 2 γ (ij ) = A i s ′i P (φ i ) T

respectively, where qi = [riT φ i ]T , q Ij = [uiI θ Ij ]T , and B = dA i / dφi . Next, consider a spherical joint for spatial multibody dynamic analysis. Let a conventional body i and a general body j be connected at a common point P, through a spherical joint. Let the node number of point P in the general body be I. The spherical joint can be mathematically represented by the constraint equation Φ s = ri + A i s ′i P − (X Ij + u Ij ) = 0

When the conventional body is rigid, the Jacobian matrix, right side of the velocity equation, and right side of the acceleration equation are Φ s = I − 2A ~s ′ P G p

[

qi

i i

i

]

Φ qs I = −I j

γ (ij )

qi = [riT

ν ( ij ) = 0 ~ ′ω ~ ′s ′ P = −A ω i

i i

respectively, where pi ] , = [−ei −e" i + e0i I ] . When the conventional body is a modal body, point P should correspond to a node in the finite element model of the body, and the position vector s ′i P of node P is T

G ip

s ′i P = s ′i 0P + Ψ P a

where s ′i0P is the nodal position measured in the body reference frame, in the undeformed configuration; Ψ P is the modal matrix corresponding to node P; and a is the vector of modal coordinates. The Jacobian matrix, right side of the velocity equation, and right side of the acceleration equation can be similarly derived.

5. An Implementation For DAE-2 In DAE-2, there are no kinematic constraints between general and conventional bodies. New joint modules are thus not needed. An implementation for DAE-2 is shown in Figure 4. The general body module is implemented as in the implementation of DAE-1 shown ~ in figure 3, except that M is used in the place of M g and additional work of evaluating and assembling H1T G is implemented. The latter can be efficiently accomplished [10].

Start

Read .fm3 Create .ZYX Rigid body module ISTAT

Read data for each module

ISTAT Model Assembly

F, F q

ǒq, q. , q g, q. gǓ

Conventional joints

ISTAT

Other

ƪM0 M0 ƫ

n

DAE solver

~

ƪ

~

G G ) H T1G Fq n, g

ƫ

ǒq, q. , q.. , q g, q. g, q.. g, lǓ n)1 ttT

n+1 → n

General body module

Junction

Modal body module

yes

no End

Figure 4: Computational flow chart for generating and solving DAE-2.

6.

An Implementation For DAE-3

Implementation of DAE-3 is the simplest, in terms of coding effort. In DAE-3, general bodies act as forces elements, from the point of view of multibody dynamics. Since user-defined force elements is a standard feature in multibody dynamic analysis software such as DADS, finite element software can be brought into a multibody dynamics code for dynamic analysis of systems containing conventional bodies and quasi-static general bodies, provided the software provides a means for user-defined subroutines to specify displacements, as well as a way of accessing reaction forces due to specified displacements. In ABAQUS/standard [24], there are two user-defined subroutines that provide exactly these desired functionalities. The disp.f subroutine can be used specified values of a set of selected nodal coordinates, q . The urdfil.f subroutine can be used to retrieve reaction forces due to the specified values of q . Thus, implementation of DAE-3 can be accomplished by linking DADS and ABAQUS/standard. The computational flow of the resulting software is shown in figure 5. The upper part represents the ABAQUS/standard process, and the lower part represents the DADS process. These two processes exchange data through a data pool, which can be either a disk file or shared memory. Process management and inter-process communication functions provided by computer operating systems are used to achieve the integration. Since DADS uses Euler parameters for parameterization of finite rotation [1] and ABAQUS uses a rotational pseudo vector [25, 26], conversion between these two representations needs to be defined. A detailed description of the integration can be found in Ref. 27.

7.

Numerical Examples

7.1

DAE-1

Two examples are used to validate the DAE-1 implementation. More examples can be found in Ref. 9. The first example is a planar multibody system. In this example, shown in figure 6, the left ends of bodies 1 and 2 are fixed to ground. Bodies 1, 2, 3, and 4 are modeled as general bodies using beam elements. Bodies 5, 6, and 7 are modeled as rigid bodies. All joints shown in figure 6 are revolute joints. A torque of magnitude 30 Nrad is applied at the crank (body 7). Results of dynamic simulation are shown in figure 7, where twelve snap-shots of the system at times t = 0:8, 1.6, 2.4,…, and 9.6 are shown. ABAQUS

Start

Element library

Incremental Static Analysis

ǒq~ , FǓ

N)1

Other

yes

N t N max User–defined subr. q

N=N+1

Material library

Junction

Read ABAQUS input file

no End

F

Data pool Start

F

Read .fm3 Create .ZYX

User–defined force Rigid body module

Conventional joints Other

Junction

Modal body module

ISTAT

Read data for each module

ISTAT F, F q

Model Assembly .

(q, q) n ISTAT DAE solver M G Fq n, g

.

..

(q, q, q, l) n)1 ttT

yes

no

DADS

End

Figure 5: Computational flow chart for geneating and solving DAE-3.

n+1 → n

q

5 1

3

6 7 4

T

2

Figure 6: A flexible seven-body system.

The second example simulates dynamic settling of a passenger car. The car body structural model (including chassis) and front suspension systems are shown in figure 8. The rear suspension systems are similar to the front suspension systems, except that each bushing component is approximated by a revolute joint [7]. The car body is modeled as a modal body, whose flexibility data are obtained in Ref. 8, in which inertia is lumped at 250 nodes. Piston rods, lower control arms, and strut-spindle assemblies of front suspension systems are modeled as rigid bodies. The lower control arm of each front suspension system is connected to the chassis through a bolt and a bushing component. Bolts are modeled as rigid bodies and bushing components are modeled as general bodies. A 3 ×3 ×12 mesh of 8-node solid elements is used to discretize each bushing component. Nodes on the outer surface of a bushing component are connected to a lower control arm through spherical joints. Nodes on the inner surface of a bushing component are connected to a bolt through spherical joints. Bolts are fixed to the chassis using bracket joints. Springs and shock absorbers in the struts are modeled using translational-spring-damper-actuator (TSDA) elements. In total, there is one modal body, two general bodies, fourteen rigid bodies, four TSDAs, four translational joints, four distance constraints, two revolute joints, and two hundred spherical joints in this system. Time histories of the three components of the resultant external force acting on the inner surface of the bushing component of the right front suspension system are shown in figure 9. The deformed configuration at time t=0.1, which is when the bushing components are subjected to the most severe loading, is shown in figure 10. 7.2 DAE-2 The implementation of DAE-2 presented in Section 5 is validated using the planar High Mobility Multipurpose Wheeled Vehicle (HMMWV) model shown in figure 11. There are three rigid bodies in the model, a chassis and two wheels. Wheels are directly connected to the chassis using revolute joints. The suspension system is not modeled. The front and rear tires are modeled as general bodies, using nonlinear beam elements. The simulated event is to accelerate the vehicle by applying torque on rigid wheels and let the vehicle run over a sinusoidal pothole. The trajectory of the tires is shown in figure 12. Zoom-in snapshots of the deformed rear tire are shown in figure 13. For the purpose of comparison, simulations are performed using both the DAE-1 implementation and the DAE-2 implementation. Simulation results obtained using these two implementations are almost identical. However, computational costs are different. When the DAE-1 implementation is used, tire rims are connected to rigid wheels, using bracket joints. Execution time on an HP9000/770 workstation is about 2 hours per second of real time. When the DAE-2 implementation is used, execution time on the same workstation is about 2 minutes per second of real time. Thus, the DAE-2 implementation is sixty times faster than the DAE-1 implementation, for this model.

t=0.8s

t=1.6s

t=2.4s

t=3.2s

t=4.0s

t=4.8s

t=5.6s

t=6.4s

t=7.2s

t=8.0s

t=8.8s

t=9.6s

Figure 7: Snap-shots of dynamic simulation of the flexible seven-body system.

3 Piston Rod

Strut–Spindle Assembly Lower Control Arm

Figure 8: A car model

50

Fz Fy

0

Fx, Fy, Fz (lbf)

-50 -100

Fx

-150 -200 -250 -300

0

0.25

0.5

0.75

1

Time (s)

Figure 9:

Components of the resultant external force acting on the inner surface of the bushing component of the front right suspension system.

Z

X

Y

Figure 10: Deformed configurations of bushing components.

Figure 11: A planar HMMWV model

Figure 12: Trajectory of front and rear tires

Figure 13: Two configurations of the deformed rear tire.

7.3

DAE-3

The spatial HMMWV model, shown in figure 14, is used to demonstrate the implementation of DAE-3. Nineteen rigid bodies are used to model the chassis, steering, and suspension systems. Four general bodies are used to model the tires. TSDAs are used to model the compliance of suspension systems. The nineteen rigid bodies in the model are as follows. B1 chassis B11 steer link B2-B5 lower control arms B12-B15 upper control arms B6-B9 axle assemblies B16-B19 wheel assemblies B10 pitman arm Kinematic joints and constraints that are used to connect rigid bodies are specified in table 2. In Cartesian coordinates, with Euler parameters for orientation, this model is described by 133 rigid body generalized coordinates, 99 joint constraints, and 19 Euler parameter normalization constraints, resulting in 15 articulated degrees of freedom. In the finite element model of terrain and tires, each tire general body contains about 800 nodes and 500 solid elements. Terrain is modeled as a rigid surface. Tire inflation pressure is modeled using follower surface load, and interaction between tires and terrain is modeled using contact elements. In this example, it is assumed that internal vibrations of tires are not of interest, so the general body tires are considered as quasi-static. The equations of motion of the system is thus of type DAE-3, and the implementation presented in Section. 6 can be used. It should be noted that analytical and semi-empirical tire models commonly used in multibody dynamics codes such as DADS are also quasi-static, since they take position, orientation, and velocity of the center of a rigid wheel as input and calculate forces and moments acting on the wheel center as output. There are no generalized coordinates associated with tire inertia, and coupling between tire force and wheel acceleration is ignored. Dynamic settling simulation is performed. The vertical response of the chassis is shown in figure 15. Execution time on an SGI workstation is 1.5 hours per second of real time. Table 2: Joints in the spatial HMMWV model

Type

Body i

Body j

Type

Body i

Body j

1

Revolute

B1

B2

15

Spherical

B6

B2

2

Revolute

B1

B3

16

Spherical

B6

B12

3

Revolute

B1

B4

17

Spherical

B7

B3

4

Revolute

B1

B5

18

Spherical

B7

B13

5

Revolute

B1

B12

19

Spherical

B8

B4

6

Revolute

B1

B13

20

Spherical

B8

B14

7

Revolute

B1

B14

21

Spherical

B9

B5

8

Revolute

B1

B15

22

Spherical

B9

B15

9

Revolute

B1

B10

23

Distance

B1

B18

10

Revolute

B6

B16

24

Distance

B1

B19

11

Revolute

B7

B17

25

Distance

B11

B6

12

Revolute

B8

B18

26

Distance

B11

B7

13

Revolute

B9

B19

27

Universal

B10

B11

14

Rev.-Sph

B1

B11

1

18

S

2

3

4

6

7

8

12

13

14

16

17

18

R 14

8

10

R

11

S 4 R

R 15

chassis frame LC arms 9 axles pitman arm steering link 15 UC arms 19 wheels 5

D

S R 19

5

9

S R

S

16 R

1

12

D

6

R S 2 R

R

13 D

R

S

10 R 17 R

7

R

3

U 11 S

S

idler arm

D

Figure 14: Multibody model of HMMWV chassis and suspension systems.

34.2

HMMWV chassis vertical position (in)

34 33.8 33.6 33.4 33.2 33 32.8 32.6 32.4 32.2 32 0

0.5

1

1.5

2 2.5 time (seconds)

3

3.5

4

4.5

5

Figure 15: Time hisory of HMMWV chassis vertical position.

8.

Conclusions

By taking advantage of the modular design of modern software, formulations of general flexible multibody dynamics can be implemented by combining existing multibody dynamics and nonlinear finite element software. Thus, the existing body of work is leveraged to minimize implementational effort. Among the three approaches presented in this paper, the first approach is the most general, but requires most effort, both computationally and in implementation. The second approach is significantly more

efficient and involves significantly less implementational effort, without compromising accuracy. The third approach is valid only for applications involving general bodies that can be modeled as quasi-static. However, it is the most computationally efficient and requires the least implementational effort in combining commercial multibody and finite element analysis codes. The three implementations are validated by numerical examples. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15]

[16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27]

E. J. Haug, Computer Aided Kinematics and Dynamics of Mechanical Systems. Allyn and Bacon, Boston, 1989. W. Schiehlen (ed), Multibody Handbook, Springer, Heidelberg, 1990. J. Wittenburg. Dynamics of Systems of Rigid Bodies. B. G. Teubner, Stuttgart, 1977. W. S. Yoo and E. J. Haug, Dynamics of articulated structures. part I. theory. Journal of Structural Mechanics, 14(1), (1986) 105-126. S. C. Wu, E. J. Haug, and S. S. Kim, A variational approach to dynamics of flexible multibody systems. Mechanics of Structures and Machines, 17(1), (1989), 3-32. W. Pan and E. J. Haug, A system-level component approach for flexible multibody dynamic simulation, Mechanics of Structures and Machines, 25(3), (1997), 335-356. W. Pan and E. J. Haug. Flexible multibody dynamic simulation using optimal lumped inertia matrices. Computer Methods in Applied Mechanics and Engineering, 173(1999) 189-200. E. J. Haug and W. Pan, Optimal inertia lumping from modal mass matrices for structural dynamics, Computer Methods in Applied Mechanics and Engineering, 163(1-4), (1998), 171-191. W. Pan and E. J. Haug, Dynamic simulation of general flexible multibody systems, Mechanics of Structures and Machines, 27(2), (1999), 217-251. W. Pan and E. J. Haug, A minimum-coupled approach for general flexible multibody dynamics. In preparation, 2000. W. Pan, S. Mao, and D. Solis, Efficient modal approach for flexible multibody dynamic simulation. NSF I/UCRC technical paper, National Advanced Driving Simulator and Simulation Center, 1999. T. Belytschko and T. J. R. Hughes, Computational Methods for Transient Analysis, Elsevier Science Publishers., 1983. K.J. Bathe, Finite Element Procedures in Engineering Analysis, Prentice-Hall, Englewood Cliffs, NJ, 1982. T. J. R. Hughes, The Finite Element Method: Linear Static and Dynamic Finite Element Analysis, Prentice-Hall, Englewood Cliffs, NJ, 1987. A. K. Noor. Multifield (mixed and hybrid) finite element models. In: A.K. Noor and W.D. Pilkey (eds.) State-of-the-art surveys on finite element technology, American Soc of Mechanical Engineers (ASME), 1983, pp. 127-156. T. H. H. Pian and P. Tong, Mixed and hybrid finite element methods. In: H. Kardestuncer (ed.) Finite Element Handbook , McGraw-Hill, Inc., 1987. M. A. Crisfield and G. F. Moita, A unified co-rotational framework for solids, shells and beams, Int. J. Solids Structures, 33(20), (1996), 2969-2992. E. Hinton, T. Rock, and O. C. Zienkiewicz, A note on mass lumping and related processes in the finite element method, Earthquake Engineering and Structural Dynamics, 4(3), (1976), 245--249. N. E. Gibbs, W. G. Poole, and P. K. Stockmeyer, An algorithm for reducing the bandwidth and profile of sparse matrix. SIAM Journal of Numerical Analysis, 13, (1976), 236-250. C. W. Gear and D. R. Wells, Multirate linear multistep methods, BIT, 24 (1984) 484-502. A. Rukgauer and W. Schiehlen, Simulation of modular mechatronic systems with application to vehicle dynamics, Acta Mechanica, 125 (1997) 183-196. W. Pan and E.J. Haug, Integrated nonlinear finite element tire modeling in vehicle dynamic simulation, NSF I/UCRC technical paper, National Advanced Driving Simulator and Simulation Center, 1999. Computer Aided Design Software, Inc., 2651 Crosspark Road, Coralville, IA. DADS/User-Defined Subroutines Manual, Revision 8.5, 1997. Hibbitt, Karlsson,& Sorensen, Inc., Pawtucket, RI, ABAQUS/Standard User's Manual, Volume III, Version 5.8, 1998. J. H. Argyris, An excursion into large rotation, Computer Methods in Applied Mechanics and Engineering, 32 (1982) 85-155. Hibbitt, Karlsson,& Sorensen, Inc., Pawtucket, RI. ABAQUS Theory Manual, Version 5.8, 1998. W. Pan, M. Qidwai, and D. Solis Linking DADS and ABAQUS/std, NSF I/UCRC technical paper, National Advanced Driving Simulator and Simulation Center, 2000.

Nonlinear Structural Behavior, Analysis and Design of Deployable Structures Charis J. GANTES Department of Civil Engineering, National Technical University of Athens 42 Patision Street, GR-106 82 Athens, Greece Abstract. Deployable structures are prefabricated structures that can be transformed from a folded, compact configuration to a predetermined, expanded form, in which they are stable and can carry loads. The deployable structures investigated in the present paper are stress-free in these two shapes, but exhibit strong geometric nonlinearity when they are forced to switch from one state to the other. Issues pertaining to this nonlinear behavior are discussed from an analysis and design point of view.

1.

Introduction to the Concept of Deployable Structures

Deployable structures is a generic name for a broad category of prefabricated structures that can be transformed from a closed, compact configuration, which is used for storage and transportation, to a predetermined, expanded form, in which they are stable and can carry loads. Due to this inherent transformability, deployable structures can be considered a special case of the broader class of adaptive structures, which are characterized by their ability to adapt their shape, mechanical and physical properties and their overall behavior to the external excitations and the requirements emanating from their use at any given time. The interest in deployable structures is due to their promising applications and the advantages they offer compared to conventional, non-deployable structures for certain types of applications, particularly the temporary construction industry and the aerospace industry. For such applications, their potential for compact storage, transportability, and easy erection and dismantling is of primary importance, and outweighs the restrictions imposed by the need for complex design and detailing, necessary to achieve deployability. Deployable structures can be classified in several categories based on different criteria [1]. One such criterion is the environment of application, according to which we differentiate between structures that are used on the earth [2] as opposed to others, which are deployed in space [3]. This distinction is associated with many differences in assumptions that are adopted for design, including types of loading, values of factors of safety, requirement for reliability, degree of automation of the deployment and dismantling processes etc. Another criterion for categorization is the type of structural members employed in the structure. There is the option of using surface structures, which consist of 2-D or 3-D building modules [4], and strut structures where the basic modules are 1-D bars. The members may be stiff compression struts and plates or flexible cables and membranes that resist only tension [5]. Such tensile structures must be prestressed in order to acquire stiffness and be able to carry loads. This prestressing is achieved either through the application of forces or via internal pressure, in which case we refer to pneumatic

structures. Combinations of the above are also possible, resulting for example in the socalled tensegrity structures [6] or other novel structural systems. The strut structures can be further classified into groups, on the basis of their structural behavior. Some of them, called manually locking deployable structures, behave as structural mechanisms during deployment, and need additional members in order to be stable in the deployed configuration [2,7,8]. This increases the effort required for erection and dismantling, and is therefore undesirable. Other designs, called self-locking deployable structures, avoid this problem but have curved members and residual stresses in the deployed configuration due to geometric incompatibilities between the bar lengths [9]. This decreases their load bearing capacity and makes them more susceptible to catastrophic failures due to member buckling.

Figure 1: The umbrella, an example of a deployable structure in every day life

By the satisfaction of certain geometric constraints, other, more recent and improved self-locking structures can be stable in the deployed configuration having straight and stress-free members, except for dead weight and live load effects [1,10,11,12]. During deployment however, geometric incompatibilities result in the development of second order strains and stresses and a snap-through type of behavior that ‘locks’ the structure and assures its stability in the deployed configuration. On the other hand, linear elastic material behavior has to be achieved during deployment in order to avoid any reduction of the load bearing capacity in the deployed configuration. Two examples of deployable structures of this type are shown in figures 1 and 2.

Figure 2: Folded and deployed configuration of a deployable dome

The present paper aims at presenting the qualitative features of the nonlinear structural response of these structures during the deployment process and at discussing how these features are treated for analysis and how they affect the design. 2.

Snap-through Type Deployable Structures

As mentioned above, the prefabricated, deployable space frames investigated here, utilize geometric nonlinearity and snap-through as a means of self-stabilization. They consist of straight bars linked together in the factory as a compact bundle, which can then

be unfolded into large-span, load bearing structural forms by simple articulation. A fundamental design requirement is that these structures are self-standing and stress-free when fully closed or fully deployed. However, at intermediate geometric configurations during the deployment process, incompatibilities between the member lengths lead to the occurrence of strains and stresses resulting in a snap-through phenomenon that "locks" the structures in their deployed configuration. Snap-through buckling is usually very destructive; therefore it should, in general, be avoided for engineering structures. There are cases, however, where this phenomenon may in fact be utilized in a creative manner, and this type of deployable structures is such an example. The prerequisite for that is that the structure must exhibit some usefulness in two distinct geometric configurations, the unloaded one, and the stable, post-snap-through equilibrium state, in which it must, in addition, be able to have a desired load bearing capacity. Thus, loading of the structure to the level of snap-through can be regarded as a form of prestressing it, so that it can carry loads in its new geometric configuration, usually for different boundary conditions. This process may be reversible and repetitive, provided that the material remains in the elastic range. In order for that to be achieved, it is usually necessary to have at least some connections between the structure’s members which allow for large relative rotations, as otherwise inelastic strains and stresses are bound to occur in the region of the joints.

Figure 3: A typical scissor-like-element (SLE)

This type of deployable structures is based on the so-called scissor-like elements (SLEs), shown in figure 3, pairs of bars connected to each other at an intermediate point through a pivotal connection which allows them to rotate freely about an axis perpendicular to their common plane but restrains all other degrees of freedom, while, at the same time, their end points are hinged to the end points of other SLEs. Several SLEs are connected to each other in order to form units with plan views of regular polygons (figure 4). These polygons, in turn, are linked in appropriate arrangements constituting deployable structures, which are either flat or curved in their final deployed configuration (figure 5). A strict geometric design is needed in order to achieve the desired behavior [11].

Figure 4: Plan views and perspective views of polygonal deployable units

Figure 5: Flat and curved deployable structures

The snap-through character of the response of this type of deployable structures has first been exhibited experimentally. The curved pentagonal unit shown in figure 6, made of high-density polyethylene (HDPE), was subjected to testing involving its dismantling process in order to obtain its load-displacement curve. The lower center hub of the unit was fixed against vertical and horizontal translations, while the upper center hub was attached to the loading piston. A displacement-controlled test was carried out. The load-displacement curve obtained is shown in figure 7. The beginning of the curve corresponds to the stress-free deployed configuration. The snap-through character of the structural behavior is clearly illustrated. In addition, one can observe the strong influence of friction in the last part of the curve, near the collapsed configuration. This friction and the finite dimensions of member cross-sections prevent a complete collapse of the structure. The curve of figure 7 provides a basis for comparison of numerical results obtained with finite element analysis, as will be described in the next section. The design of deployable structures is a very complicated process. The very different nature of the structural behavior in the two phases of the life of such a structure, during deployment and in the deployed configuration under service loads, prevent the formulation of any simple and straightforward design methodology. An acceptable design has to provide satisfactory trade-off between desired stiffness in the deployed configuration and desired flexibility during deployment [12]. Since these two objectives contradict each other, iterations are necessary until a balanced behavior is achieved. The present paper will mostly focus on one part of the design process, namely the simulation of the deployment process.

Figure 6: Experimental set-up for testing during the dismantling process

Figure 7: Experimental load-displacement curve

3. Simulation and Analysis during Deployment Analyzing the structure during deployment is a rather complicated problem. The response of the deployable structures during deployment is geometrically nonlinear, and therefore a large displacement - small strain finite element formulation [13] implemented in ADINA [14] and MSC-NASTRAN [15] is used for the numerical modeling of the problem [16-18]). Experimental observations showed that the stresses occurring during deployment are very sensitive to changes in geometry or member properties, and can become comparable to, or even much larger than stresses due to service loads. Assuming that the structure is designed to resist service loads, low deployment stresses are desired, so that the members behave elastically. Plastic material behavior during deployment should be avoided, since it would result in reduced load-bearing capacity in the deployed configuration. Similar observations are true for the load required for deployment and dismantling. In less carefully designed structures, which exhibit a very stiff response during deployment, this load can be quite high. This makes the deployment process cumbersome, and creates a need for more elaborate mechanisms and equipment. In conclusion, failure to take the deployment response into account during design can result in inefficient and expensive solutions, and in making the structure’s feasibility questionable. Hence, both a qualitative understanding of the behavior and a quantitative evaluation of stresses occurring during deployment and required loads constitute an integral part of the design of deployable structures. In this section, the numerical modeling for the simulation of the deployment analysis is explained in detail, starting from the initial simple model and going through all stages of the model refinement. The effects of mesh refinement, discrete joint dimensions, geometric imperfections, thin member cross-sections and friction are included. The numerical results produced by several recent studies have been compared with the corresponding test results, and are found to be in excellent agreement. 3.1 Basic considerations for single units Initially, a relatively simple unit with a square plan-view, as shown in figure 8, consisting of eight SLEs has been analyzed. All members are made of HDPE (Young's modulus of 0.8GN/m2), and have square cross-sections of 9mm by 9mm. The simplest possible method of deployment is applied to this single polygonal unit. The lower center node of the unit is considered fully supported, while the upper center node is free to move

vertically only and is subjected to a vertical concentrated load. All other nodes are free. This deployment procedure offers the important advantage of symmetry, and therefore simplifies the analysis considerably, since only part of the structure has to be analyzed. Nevertheless, there are many different potential deployment procedures that can be used. A good deployment method must have practical advantages by providing for an easy way to attach cables at selected nodes and apply the necessary deployment forces. In addition, the stresses that develop in the members of the structure during deployment must be within some acceptable limits. The selection of the most convenient method, both from a practical and from a structural point of view, is still an open question. The nature of the strains and stresses that develop in the members of the structure during deployment defines the type of kinematic assumptions that have to be made for this problem. The strains and stresses result from compatibility requirements between the members of inner and outer SLEs. The members have to deform in such a way that the plan view of the structure remains square throughout the deployment procedure. At the same time, the vertical distance between upper and lower circumferential nodes has to be the same for both inner and outer SLEs. Hence, the resulting strains and stresses are due to second order effects, and a “large displacement - small strain” finite element formulation is appropriate [13]. In the collapsed configuration, all nodes of the structure lie theoretically on a straight line. Furthermore, a small deformation has to take place before the structure can carry any loads. Therefore, the deployed configuration is almost always used as the initial state for the analysis, i.e. dismantling is simulated instead of deployment.

Figure 8: A flat square unit and the method of deployment used

Nonlinear Hermitian beam elements [13] have been initially used to model the members. One element was used between a pivotal and a hinged connection, so that four beam elements were needed to model an SLE. After introducing auxiliary coordinate systems, the master node/slave node technique was used to model the pivotal connections. The assumption of equal radial displacements of upper and lower circumferential nodes is generally adopted. This assumption, while not necessarily correct for a single unit, helps in simulating the behavior of a unit within a larger assemblage, where the upper and lower corner nodes are forced to have equal horizontal displacements due to symmetry considerations. Due to the symmetry of both the structure and the loading about the vertical axis connecting the two center nodes, all nodes are only free to translate radially and vertically, and to rotate about the tangential axis. This reduces the number of degrees of freedom for the problem. Furthermore, only one fourth of the structure need be analyzed, as illustrated in figure 9. This reduces the number of degrees of freedom of the numerical model and, hence, the cost of the analysis. Therefore, only one fourth of the structure was modeled for all subsequent analyses. A comparison of the load-displacement curve for the full and the quarter-unit is shown in figure 10.

Figure 9: Symmetry considerations and the finite element mesh

The next step was to switch from Hermitian to 2-node isoparametric beam elements [13], in order to model the inner SLEs. When used for nonlinear analyses, the Hermitian beam element requires a high order of numerical integration, and is therefore very expensive. Moreover, it neglects the degradation of bending stiffness due to axial forces. By using 2-node isobeam elements, the cost of the analysis has been reduced significantly. Numerical integration of first order, which is equivalent to mixed interpolation for the transverse displacements and the shear deformation, has been used along the longitudinal axis of the 2-node isoparametric beam elements, in order to avoid shear locking. According to the initial analysis, outer SLEs have very small shear stresses, and are only subjected to almost pure axial stresses; therefore, truss elements have been used to represent them.

Figure 10: Comparison of P-u graphs for full and reduced model

Special attention has been paid to the boundary conditions at the intersection of the structure with the symmetry axes, so that the symmetry conditions would not be violated. No tangential translation was allowed, and the vertical translation was appropriately constrained to the vertical translation of upper and lower corner nodes of the structure, so that overall stability of the structure was achieved. The automatic step incrementation method has been used in most analyses [13,19]). After a better understanding of the behavior has been acquired, the more economical BFGS method can be employed ([13,20]). In some cases line search is required in order to achieve convergence.

Curves describing the variation of the required external load and corresponding internal member forces, as the structure deforms from the deployed configuration to the collapsed one, have been presented by Gantes in [10] and subsequent work. The loaddisplacement curve indicates the snap-through type of behavior for the structure. The above-simplified analysis is useful in order to gain initial insight into the structural behavior, and the results agree qualitatively with the observed behavior of physical models. However, the actual values of the required deployment load are smaller for the numerical model than for a corresponding experimental one. This indicates that more sophisticated modeling is required for design purposes. Therefore, an extensive effort has been undertaken to refine the model, including all necessary factors, in order to obtain a reliable numerical tool for preliminary and final design. The influence of the mesh refinement of the finite element model is illustrated in figure 11. The model developed for the analysis of single units for flat structures has been extended to describe curved spherical structures as well. The only change in the model are the nodal coordinates, which should be such that lines connecting corresponding lower and upper nodes meet at the same point, the center of the sphere. In figure 12, the loaddisplacement curve for this unit is illustrated. The type of response is qualitatively the same. A snap-through type of behavior can be observed. The almost vertical slope of the curve in the collapsed configuration corresponds to the sum of the axial stiffnesses of the members, which by then lie practically on a straight line.

Figure 11: Influence of mesh refinement

The next development in the gradual refinement of the model has been the modeling of discrete hub size. This concept was motivated by experimental observations, which have shown the influence on the structural response of ignoring the joint size during geometric design. This led to the conclusion that the behavior of deployable structures is very sensitive to member lengths and, hence, inclusion of the hubs in the numerical model is necessary for an accurate simulation. Figure 13 shows the type of hinge that has been used for the physical models. It also illustrates the idea of modeling this hinge as a stiff grid composed of short 2-node isoparametric beam elements. Figure 14 shows the actual finite element mesh used. In figure 15, the corresponding series of deformed configurations is presented.

Figure 12: The load-displacement curve for the finite element model of a single curved unit

Figure 13: A real hinge and its finite element model

Figure 14: The finite element mesh with discrete joint dimensions

Figure 15: Deformed configurations of model with discrete joint dimensions

The influence of the stiffness of the isobeam elements that constitute the grid has been investigated next. This influence is insignificant for the major part of the loaddisplacement curve, but the last part of the curve is affected, and local disturbances occur (figure 16) which have been attributed to geometric incompatibilities and to the fact that the members that are connected to the hinge are not concurrent during the deployment process. A plot of the stress variation in the truss elements has revealed that these disturbances are associated to a sudden increase in tension. This led to a slight increase in the length of these members, which resulted indeed in obtaining a smooth load-displacement graph (figure 16). However, this change in length creates compression of the truss members and a change in sign of the load-displacement curve before the structure is completely dismantled. This detail is not of great importance for the final model, which also includes the effect of friction between members and joints. In conclusion, including the discrete hubs in the numerical model has an overall stiffening effect, which is larger when the members are relatively short, while the hubs remain approximately the same. By observing the deformation of the members of various experimental models used in previous studies [10], one can identify the behavior as a combination of in-plane bending and out-of-plane buckling. This behavior is very much dependent upon the ratio of in-plane to out-of-plane stiffness of the members of radial scissors, a response that could not be captured with the “perfect” geometry used for the initial finite element models. During the analysis of these “perfect” models, the members remain in their plane and are subjected only to in-plane bending and axial compression. As a result, the response is stiffer than in reality.

Figure 16: Influence of discrete joint size on load-displacement curve

In order to model the real mode of behavior of the structure during deployment, an initial imperfection has been imposed on the members of radial scissors, in the form of distorted out-of-plane initial node coordinates corresponding to the out-of-plane buckling mode obtained from a linearized buckling analysis of the structure in its deployed configuration. This causes the radial scissors to deform both in-plane and out-of-plane during the deployment process, and to respond in a more flexible manner. The distorted initial mesh as well as the influence of this type of imperfection on the load-displacement path is illustrated in figure 17. The quantitative influence of the imperfections is small, as is usually the case for systems that lose their stability via a limit point as opposed to a bifurcation point.

Figure 17: Influence of imperfections on the structural response

These studies have been performed using isoparametric beam elements with square cross-section. To verify the expectation that member imperfections would have a more significant effect on structures with weak out-of-plane members, an analysis has been attempted where the ratio of width to height of the cross-section was one to two. More specifically, rectangular cross-sections have been used instead of square ones, with such dimensions that the cross-sectional areas were almost the same. When there were no geometric imperfections, the structure with rectangular cross-sections exhibited, as expected, a much stiffer nonlinear response. The reason was that the behavior had been characterized by in-plane bending, which made the moment of inertia of the bars a very important factor. It had already been observed that out-of-plane imperfections have a small effect on the structure with square cross-sections. For rectangular cross-sections, a much more significant influence has been recorded at the beginning of the curve, with an approximately 15% reduction in the maximum required dismantling load. Convergence difficulties related to stiffening of the response that made no physical sense were observed. These problems have finally been attributed to an insufficient (for this problem) formulation of warping that is used for the isoparametric beam elements.

Figure 18: Finite element mesh with shell elements

Following this conclusion, 9-node shell elements [13] have been successfully used to avoid this problem. Figures 18, 19 and 20 show, respectively, the finite element mesh used, a series of plots of successive deformed configurations from deployment to collapse, and the load-displacement curve that has been obtained for a structure with rectangular crosssections and without geometric imperfections. Figure 21 shows that this mesh can indeed handle geometric imperfections for members with “thin” cross-sections, and that the influence of such member imperfections is indeed more important here than for members with square cross-sections. Despite successive refinements, the finite element model did still not show good agreement with experimental results because it did not account for friction between the members of the structure during deployment and dismantling. Therefore, it was deemed necessary to incorporate this effect. The macroscopic effect of friction can be taken into account by using appropriate nonlinear rotational springs or pairs of translational springs at all pivotal connections [21]. Deployable structures consist of scissor-like elements which allow relative rotations of their two bars about the pivotal connection. The two bars of an SLE are theoretically straight lines lying in a plane. In reality, however, the bars have cross-sections with discrete widths, while their end nodes are forced to lie in a common plane due to boundary conditions. This causes deformation of the bars, as illustrated in figure 22, which shows a perspective and a top view of a scissor-like element. Because of this deformation, transverse forces between the two bars are generated. When there is a relative rotation between the two bars about the pivotal connection, these forces produce friction forces over the contact surface of the bars. These friction forces add up to a total friction moment that resists rotation. This happens with all SLEs of the structure during deployment, and is the main mechanism of friction.

Figure 19: Successive deformed configurations of the shell mesh

The modeling approach used to account for frictional effects on the macroscopic response is illustrated in figure 23. It consists of adding at all pivotal connections nonlinear rotational springs, which simulate the effect of friction. Such an approach can be sufficient to describe the overall influence of friction on the structural response, although it cannot capture local effects. A more refined model based on improved contact algorithms could be used in the future to identify areas of local wear that require strengthening.

Figure 20: The load-displacement curve for the mesh with shell elements

Figure 21: Influence of imperfections for square and rectangular cross-sections

The overall approach to the problem consists of two steps [21]. The first is the derivation of the frictional moments that resist rotation at the pivotal and hinged connections as functions of the constant geometry and material characteristics, and of the varying angle between the two bars of an SLE during deployment. The friction at the hinges between bars and hubs produces a moment that is usually smaller than 5% compared to the moment at the pivot. Therefore, the friction at the hubs was neglected for the rest of the calculations. The variation of the moment at the pivot during dismantling is shown in figure 24 for two different approximations [21]. A reasonably good agreement between “exact” and “approximate” models can be observed. As expected, the moment is minimum for φ =90°, when the contact area between the two bars is minimum. The moment increases rapidly near the folded configuration, when the discrete width of the bars causes large transverse forces between them, and prevents them from folding completely.

Figure 22: Perspective and top view of SLE with discrete member width

Figure 23: The proposed friction model for SLE

Figure 24: Variation of friction moment during dismantling

The second step is the derivation of appropriate stiffness functions for a nonlinear rotational spring or a pair of nonlinear translational springs which can be included in the numerical model and produce resisting moments equal to those due to friction (figure 25).

Figure 25: Springs that simulate friction at pivotal connection

Implementing this friction model into the finite element model gave a loaddisplacement curve in excellent agreement with the experimental results as shown in figure 26. Dismantling load is recorded on the vertical axis, while change of distance between center lower and upper node is recorded on the horizontal one.

Figure 26: Comparison of numerical and experimental load-displacement curves

In conclusion, the effects of mesh refinement, discrete joint dimensions, geometric imperfections, thin member cross-sections, and friction need to be included in the model simulating the deployment process. Figure 27 shows the influence of gradual refinement of the numerical model used for a single unit, taking into account all crucial factors for the design and analysis of deployable structures.

Figure 27: Influence of gradual refinement of numerical model

3.2 Deployment results for multi-unit structures Once the accuracy of the numerical model was verified for single-unit structures, it was extended and used for the deployment analysis of structures consisting of many units. An initial realization was that a different deployment mechanism had to be used for this new structure. It would be practically impossible to fix the lower center nodes and pull upwards the upper center nodes of all units. Application of radial horizontal loads at all corner nodes while the lower corner nodes were supported against vertical displacements was selected instead. In order to reduce the cost of the analysis, discrete joint sizes, imperfections, and friction were not included in the model. Figure 28 shows the deployed configuration of a flat slab consisting of nine square units in a three by three layout. The dismantling mechanism used is also illustrated in that figure and consists of eight concentrated horizontal loads applied at the eight lower and upper corner nodes in a radial direction. None of the lower nodes are allowed to move vertically. Figure 29 shows successive deformed configurations of the slab during the dismantling process as they were obtained from the numerical analysis.

Figure 28: The dismantling mechanism for a multi-unit slab

Figure 29: Successive deformed configurations during deployment of a flat slab

Figure 30 illustrates the variation of the required dismantling load as a function of the in-plane displacement of a corner node towards the center of the slab during folding. The load of figure 30 is one of the eight loads applied in order to achieve dismantling. Finally, figure 31 shows the change of axial stress in one of the members during the deployment process. It is interesting to note that the snap-through character of the response is maintained for structures consisting of many units.

Figure 30: Load-displacement curve for the multi-unit slab

It has been found that the results of the analysis of multi-unit structures, both for member stresses and for required load, can be predicted quite well by using the corresponding results from the deployment analysis of one unit. Figure 32 compares axial stresses at a given cross-section in the single-unit structure, and the corresponding crosssection in a three-by-three unit slab. The agreement is quite good, at least for preliminary design purposes. In addition, the required dismantling load for the three-by-three unit structure turned out to be three times the one for the single unit, as shown in figure 33.

Figure 31: Stress variation for a member of the multi-unit slab

Figure 32: Comparison of stress variation curves for 1 and 3x3 units

Figure 33: Comparison of load-displacement curves for 1 and 3x3 units

4.

Example of Application

Figure 34 illustrates an example of a deployable scaffolding structure that has been designed following the approach described above, while figure 35 shows successive deformed configurations of one of the units.

Figure 34: A photograph of a model of a deployable scaffolding system

Figure 35: Successive deformed configurations during deployment of one unit of scaffolding system

5.

Summary and Conclusions

Deployable structures, which make use of snap-through buckling occurring during their deployment process as a means of self-stabilization in the deployed configuration, have been investigated. Emphasis has been placed on the simulation and analysis of such structures during deployment. In that phase the response is highly geometrically nonlinear, while linear elastic material behavior is a fundamental design requirement. The effects of mesh refinement, discrete joint dimensions, geometric imperfections, thin member crosssections, and friction have been included in the numerical model. Single units as well as multi-unit deployable structures have been analyzed. The modeling details have been presented and a comparison with experimental measurements has been used to verify numerical analysis with very good results. A short example of a deployable scaffolding system has been briefly described.

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An Embedded Projection Method for Index-3 Constrained Mechanical Systems Marco BORRI, Carlo L. BOTTASSO and Lorenzo TRAINELLI Politecnico di Milano, Dipartimento di Ingegneria Aerospaziale, Via La Masa 34, 20158, Milano, Italy. Abstract. General constrained systems, such as those modeling the dynamics of multibody systems, control systems and chemical processes, typically involve the solution of differential-algebraic equations (DAEs). The reliable solution of high-order DAEs can still be considered an unsolved problem under many respects. In this paper we present the ‘Embedded Projection Method’ (EPM), a novel approach to the solution of index-3 constrained mechanical systems designed to eliminate most of the numerical difficulties related to the solution of DAEs, while preserving generality. The constraints are enforced at both the position and velocity levels employing differentiated multipliers, resulting in a complete uncoupling of the algebraic and differential parts of the problem. This way, the two subproblems can be solved separately. Furthermore, one obtains an ordinary differential equation (ODE) involving a new, modified state vector representing an unconstrained quantity. Any suitable ODE integration algorithm can then be used, by-passing the need for specialized DAE solvers.

1. Introduction The development of efficient and reliable numerical methods for the simulation of constrained dynamical systems impacts many diverse applications in several scientific disciplines. These systems are generally modeled by sets of differential-algebraic equations (DAEs), i.e. ordinary differential equations (ODEs) coupled with algebraic equations which account for the constraints. Solving general DAE systems still represents an open field of research, since their intrinsic numerical difficulty has prevented to date from reaching the same degree of maturity achieved in the numerical treatment of ODE systems. This difficulty is usually measured by the differential index, a concept discussed in refs. [17, 12, 9]. While for DAEs of index 1 a vast class of suitable numerical methods is now available [15], for DAEs of differential index greater than 1, it is generally recognized that obtaining a good numerical solution can still represent a serious task. The source of most difficulties encountered in the numerical treatment of DAE problems can be traced to the inability of standard ODE integrators to exactly solve algebraic equations. Furthermore, a degraded accuracy with respect to the unconstrained case can often be observed [17]. Much work has been done in order to reformulate the problem. Among the different methodologies [11, 2], we cite here the lowering of the differential index [12, 9], the differentiation of the constraints in order to eliminate the Lagrangean multipliers and solve the resulting underlying ODE, the Coordinate Projection Method [10], and the Projected Invariants Method [1]. However, some drawbacks are connected with these methods, which in

some cases give rise to a typical drift, i.e. the inability of the solution to stay on the configuration manifold of the system [2, 15]. Different stabilization techniques have been proposed in order to control the drift [4, 9]. The Embedded Projection Method, EPM in short, presented here represents a departure from the common approaches to the problem. We completely uncouple the DAE system into separate algebraic and differential parts, solving them independently from one another. This is accomplished by the definition of a new variable, the ‘modified state’, which is an unconstrained quantity in all respects and may be integrated using any suitable ODE solver. The original state is then recovered by means of an ‘embedded projection’ of the modified state onto the constraint manifold. To this end, the constraints at both position and velocity levels are appended to the governing equations, reducing the index of the DAE set from 3 to 2. The numerical preservation of both the position and velocity constraints is desirable in holonomically constrained mechanical systems for precision requirements. However there may be more to it. In fact, it is usually argued that the presence of constraint conditions amounts to the introduction of infinite frequencies in the system. This in turns requires the use of high frequency damping integrators that can remove these frequencies from the computed response, stabilizing the numerical procedure. It will be shown in the examples, that the EPM can be successfully used without the need for high frequency damping. In the proposed framework, we consider both constraints in view of the introduction of multipliers in differentiated form. These were first introduced in refs. [6, 5] to deal with nonholonomic constraints and, independently, in ref. [12] as a method for index reduction. Here, they allow for the definition of the modified state and, consequently, they lead to the uncoupling process. The computed solution attains the same order of accuracy that is obtained when the solver is applied to purely differential systems, and the need for designing specialized DAE solvers is altogether by-passed. In the following, we formulate the methodology for general constrained mechanical systems within the Hamiltonian framework. The Lagrangean version is recovered in the end with the specific aim of clarifying the meaning of the multipliers with respect to reaction forces. The application of the EPM to the simple model pendulum problem is presented to ease the comprehension of the main features of the procedure. Numerical results are shown for the case of a planar multibody system commonly used as a benchmark problem in the DAE numerics literature. These results clearly show that the EPM has a valuable impact on high-index DAE numerical stabilization, and that it provides the solution with the correct order of accuracy of all field variables, namely states and reactions.

2. Governing Equations 2.1. Unconstrained Dynamical Systems Let us consider a generic holonomic dynamical system with n degrees of freedom, where q 2 R n represents the vector of Lagrangean coordinates describing the system configuration. We assume a very general form for the Lagrangean function L q; q; t as

( )

L = 12 q_
M q_ + m
q_ + L0;

(_

)

(1)

where M q; t is the symmetric and positive-definite matrix coefficient of the quadratic part of L with respect to q, termed the inertia matrix, m q; t is the vector coefficient of the

_

( )

( )

linear part, while the homogeneous part is denoted by L 0 q; t . The Lagrange equations for the system read

d L _ L = Q; dt q q

(2)

where Q 2 R n denotes the Lagrangean external force conjugated to the Lagrangean coordinates q. Next, we move to the Hamiltonian framework defining the momentum p L q_ and obtaining the Hamiltonian function H q; p; t as

H = 21 (p

(

:=

)

m)
M

1

(p

m) + H0 ;

(3)

( )=

( )

by means of a standard Legendre transformation. Note that H 0 q; t L0 q; t and
Hq q; p; t Lq q; Hp q; p; t ; t . The state of the system in phase space is represented by vector x q p 2 R 2n . Note that, throughout this work, we make use of the above notation with the semi-colon to indicate ‘composed’ column vectors instead of the more cumbersome xT qT ; pT . The unconstrained system dynamics is described by the canonical equations:

(

)= ( := ( ; ) := ( )

)

p_ q_

Hq + Q = 0 ; Hp = 0 : n

(4)

n

We rewrite this ODE system in the following, more compact form,

S x_ + X = 02 ;

(5)

n

where X represents the total Hamiltonian force defined as X

S :=



O +I

n n

I O

n



:= (Q Hq ; Hp) 2 R 2

n

, and (6)

n

is the unit symplectic matrix. We use the symbol 0 k for the null vector in the space R k , Ik and Ok for the identity and null matrices in R kk , respectively, and finally Onm for the null matrix in R nm . 2.2. Holonomically Constrained Dynamical Systems We now enforce m < n independent holonomic constraints on the dynamical system considered thus far. We cast the governing equations for general holonomically constrained dynamical systems in view of a numerical solution that preserves the constraints at both position and velocity levels. This implies that the velocity-level constraint is not ‘inherited’ from the position-level constraint, but enforced a priori as an independent constraint, resting on the fact that the position q and momentum p constitute a set of independent coordinates. We consider a constraint manifold defined by the algebraic equation

(q; t) = 0 ; m

where  matrix A

(7)

: R  R ! R , represents the position-level constraint function. The gradient := q 2 R  has full-row rank by the assumed independence of the constraints, n

m

T

n

m

:=

and define a t constraint equation

2R

m

. The total time derivative of equation (7) yields the velocity-level

A(q; t) q_ + a(q; t) = 0 : T

(8)

m

This differential equation in the position q becomes an algebraic equation when cast in both q and p as independent variables:

(q; p; t) = A(q; t) Hp(q; p; t) + a(q; t) = 0 ; (9) where Hp = M 1 (p m) as derived from equation (3). We denote by : R  R  R ! R the momentum-level constraint function. The equations  = 0 and = 0 may well be looked at as independent constraints for the state vector x, contemporarily acting on the system. Stacking them into a state-level constraint function  : R 2  R ! R 2 defined as  := ( ; ), we get the algebraic T

m

n

n

m

m

m

n

m

constraint equation for the state x as

(x; t) = 02 :

(10)

m

:=

For further convenience, we define the gradient C Tx 2 R 2n2m , clearly with full rowrank, and c t 2 R 2m . In this case, the following differential constraint equation

:=

C(x; t) x_ + c(x; t) = 02 : T

(11)

m

is obtained considering the total time derivative of equation (10). Now we are ready to formulate the governing equations for holonomically constrained dynamical systems in the traditional framework. In fact, denoting by  2 R 2m the Lagrangean multiplier vector associated to the constraint equation (10), we get the following DAE set in the unknowns x;  :

(

)

S x_

C  + X = 02 ;  = 02 : n

(12)

m

This system cannot, in general, be successfully solved by applying a standard ODE integrator.

3. The Embedded Projection Method 3.1. Splitting into Subproblems Seeking a new, more convenient form of the governing equations, we consider the constraint equation in differential form (11) and introduce a new multiplier vector  2 R 2m , defined as   . This way we obtain the ODE set in the unknowns x;  :

_=

S x_

C _ + X = 02 ; C x_ + c = 02 : n

T

(

)

(13)

m

In this form the problem is no longer differential-algebraic. Nevertheless, we do not directly integrate this system, since in general this procedure would result in the drifting of the solution from the system configuration manifold. Next, we define the following algebraic problem, linear in the multiplier  :

S (x

x~ )

= 02 ;  = 02 :

C

n

m

(14)

~ = (~ ; ~ )

q p plays the role of a new vector of state variables. It Here the modified state x represents an unconstrained quantity, designed in order to implicitly satisfy the constraints. Clearly, if the constraint equation is satisfied by the state x, the multiplier  vanishes and the modified state x coincides with the ‘original’ state x. Equation (14) represents the ‘embedded projection’. We may rewrite it as

~

y = f (~x; t);

(15)

:= ( ; )

with y x  , at least whenever the Jacobian of equation (14) is non–singular. Similarly, the differential equation (13) may be rewritten as

y_ = g(~x; t):

(16)

_

(~ )

Solving equation (15) for y as a function of x; t allows the solution of equation (16) for y. We remark that g 6 f , in other words, y is computed independently from y as a function of x; t . With these functional equations understood, the constrained differential problem associated to equation (13) reduces to the following unconstrained equation

=_

(~ )

_

S x~_ + C_  + X = 02 :

(17)

n

_

(~ )

X and x must be considered as functions of x; t through the functions f and g introduced above. A clarification of the functional dependencies of the second term at the left-hand side is obtained by writing C_ 

= D x_ + d; (18) := @ (C  )=@ x and d = @ (C  )=@t are also understood as functions of

=

where D DT x; t . This way, equation (17) may be rewritten, with a certain abuse of notation, as

(~ )

S x~_ + D(~x; t) x_ (~x; t) + d(~x; t) + X(~x; t) = 02 : n

(_ _)

(19)

Note that, had we not expressed the derivatives x;  through function g, equation (17) would have resulted in an implicit differential equation for x, giving rise, consequently, to considerable computational drawbacks. Equation (19) represents a vectorial ODE that, in principle, can be integrated using a vast class of numerical algorithms, resting on the fact that x represents an unconstrained variable under all respects. Clearly, when equation (19) has been solved for x, the reaction forces XC C  can be recovered by the solution of equation (13). We remark that the numerical integration of equation (19) not only yields a computed state x that stays exactly on the constraint manifold defined by  0 2m , but also a computed state derivative x that stays exactly on the tangent bundle defined by  02m . In fact, the method contemporarily takes into account the constraints at the position (), velocity ( ), momentum ( ), and acceleration ( ) levels, as shown by equations (13) and (14). Note also that the multiplier  may be rescaled to zero at the end of each time step, and the next time step started with x x.

~

~

~_

:= _

=

_

_

_=

_

~=

3.2. Exploiting Block Sparsity The matrices involved in the present formulation exhibit a typical structural sparsity, which leads to different sparsity characteristics of the algebraic and differential subproblems.

In fact, since the position-level constraint function  does not depend on the momentum p, the Jacobian C has a block upper triangular matrix form:

C=





A O n

B ; M 1A

m

(20)

:= q , b := , and accounted for p = qM 1 , getting _ = A M p_ + B q_ + b. We set  = ( ; ), where  and  denote the multipliers associated to the constraint equations  = 0 and = 0 , respectively. Now, subproblem (14) may be split into the

where we defined B 1

T

T

t

T

m

m

following consecutive sets of algebraic equations:

q~ = 0 ; =0 ;

M 1A 

q

n

(21)

m

B  + p~ = 0 ; =0 :

A

p

n

(22)

m

We remark that only the first set (21) is non-linear. This may be solved with respect to the unknowns q;  as functions of q; t . After this has been done, the second set (22) is simply a linear algebraic problem in the unknowns p;  as functions of q; p; t . Analogously, the dynamic equation (13) may be split into the following consecutive sets of equations:

(

)

(~ ) q_

(

(~ ~ )

)

M 1 A _ Hp = 0 ; A q_ + a = 0 ; n

(23)

T

m

p_

A_ B_ Hq + Q = 0 ; A M 1 p_ + B q_ + b = 0 : n

T

(24)

T

m

(_ _) (_ _)

Both of them are simply linear algebraic problems in the unknowns q;  , p;  , respectively, as functions of q; p; t .

(~ ~ )

3.3. Sketching an Efficient Solution It is important to understand that the multiplier  must be considered as a small correction term due to the numerical inconsistencies resulting from the time discretization process. In fact, it would vanish exactly for the exact solution. It is intuitive that the magnitude of this multiplier is related to the accuracy of the numerical integration scheme employed and that it can be controlled by a suitable choice of the scheme characteristics and of the time step size. To this regard, note that the solution of equation (23) yields a vanishing time derivative , since the state x exactly satisfies the constraint at the momentum level. In fact by equation (23) we get

_

_ = M (A Hp + a) = M ; T

where the ‘reduced’ inertia matrix M sis laid on A and M.

:= (A

T

M 1 A)

1

(25)

is positive definite by the hypothe-

These considerations lead to an efficient solution of the problem. In fact, solving the algebraic subproblem (21) with respect to the unknown z (1) q  by the application of a Newton-type iteration with an initial prediction q q;  0m , requires the solution of the following linear system

:= ( ; ) ( =~ = )

T
z + r = 0 ; (26) is the increment of z , while r and T are the residual vector and the tan(1)




(1)

(1)

p

where z(1) (1) (1) (1) gent matrix of problem (21), respectively. We denote as subproblems. Matrix T(1) is given by

T :=



I

(1)

:= (

E

n

A

:= n + m the dimension of the

p



M 1A ; O

T

(27)

m

)

@ M 1 A  =@ q. Now, given the solution z(1) of the previous non-linear problem, with E the solution of the algebraic subproblem (22) requires to solve the following system of linear algebraic equations in the unknown z(2) p,

:= ( ; ) T z +r =0 ; (2)

(2)

(2)

(28)

p

where the tangent matrix T(2) is given by

T :=



I +E A M 1

(2)

(



A O

n

T

m

;

(29)

)

:= (~ ;

+

p AT M 1 m since E  @ B  =@ p and the right-hand side vector is defined as r(2) a. We observe that matrix E linearly depends on , a small corrective term. Therefore, one could neglect E in eqs. (27) and (29) so that T (1) becomes equal to the transpose of T(2) and coincides with one of the coefficient matrices of the derivative of the unknowns in both differential subproblems (23) and (24). In fact, these may be conveniently rewritten as

)

A z_ + Z = 0 ; B z_ + Z = 0 ; (1)

(1)

p

(30)

(1)

(2)

p

(31)

A z_ where the coefficient matrices A and B are defined as T

A :=

T

(2)



I A

T

B :=





M 1A ; O

n

(32)

m

O B

B O

n T

m

:= (



;

; )

(33)

:= (

; )

Hp a and Z(2) QT Hq b . and the forcing terms are given by Z(1) Matrix A is non-singular due to the full row-rank requirement of the constraint function . Its inverse can therefore be written as A := 1

where A

:= M

1



A M M

P M A

T

A, A := A M , and P := I

n



;

M 1 A A . T

(34)

= = =

Note that matrix A belongs to the same space spanned by A and satisfies A T A Im . Matrix P is nilpotent, or P2 P . Furthermore, P M 1 is symmetric, or P M 1 On and PT A Om . M 1 PT , and also orthogonal to both A  T and A , or P A Therefore, to solve problems (30) and (31) we only need the computation of the ‘reduced’ inertia matrix M . Since this matrix is positive definite, it can be obtained in a factorized form, for example by using Choleski factorization. A further enhancement in numerical efficiency can be gained from a modification of the solution strategy described up to here. In fact, since the multiplier  is viewed as a small correction term, it is envisaged that we may conveniently solve the first algebraic subproblem (21) with the approximate tangent matrix T (1)  A, i.e., solve the problem

=

=

A
z + r = 0 : (1)

(1)

(35)

p

For the same reason, the solution of the second algebraic subproblem (22) can be iteratively obtained solving the linear problem

A

T

^ := (~

;

z(2) + ^r(2) = 0 ;

(36)

p

+ )

p E p AT M 1 m a . where r(2) This way we have to compute only the matrix A and solve different linear problems with the same coefficient matrix or its transpose. The numerical implementation of the procedure can also take advantage of the particular block structure of the matrix A and of the sparsity of the gradient A. To this regard, we remark that the sparsity of the algebraic and differential subproblems are quite different, so that exploiting these different structures in the numerical solution appears highly desirable. 3.4. Lagrangean Approach The aim of this paragraph is to present the Lagrangean version of the governing equations within the framework just established, in order to clarify the meaning of the multipliers  and  in connection with the reaction forces. Indeed, the procedure followed in this work, starting with the Hamiltonian approach and only now reverting to the Lagrangean approach, is somewhat unusual. Nevertheless, in this case it seems to be the most convenient, given the independent conservation requirements for both the position and velocity (and, consequently, momentum) constraints. The first equation of the set (23) may be recast as

p = M q_ + m + A _ ;

=

(

)

(37)

since Hp M 1 p m . In order to obtain the Euler-Lagrange equations for the constrained system we must perform the change of state variables from q; p to q; q . This may be accomplished by deriving equation (37) and substituting the right-hand side for p in the first equation of the set (24). The result is

(

) ( _)

_

d (Mq_ + m) + A  + A_ _ H = Q + A _ + B _ : (38) q dt Now, replacing the partial derivative Hq by Lq , (M q_ + m) by Lq_ , and noting that B coin_ , when expressed in terms of (q; q_ ), we get the Lagrangean dynamic equation cides with A d L _ L = Q + A (_  ): (39) dt q q

Although we do not use this equation as a basis for a numerical integration algorithm, we point out that it clarifies the meaning of the multipliers introduced previously with respect to the constraint reaction forces. The latter are given by QC Tq   . Therefore, it appears that only the difference   has a direct physical meaning. For this reason, the Lagrangean setup just recovered can be understood as a sort of ‘Lagrangean multiplier splitting’. Separately, each of the multipliers  and  control the constraint equations at the velocity and acceleration levels. We have also seen that the multiplier  has the meaning of a small correction term while its time derivative  vanishes. It seems natural to expect that  converges to zero whenever the numerical solution converges to the exact solution. Thus, the constraint reaction term still holds its usual meaning, i.e. the meaning assumed when enforcing only the position-level constraint, as it is done in conventional formulations. Multiplying this equation by q leads to the energy conservation law naturally inherited by equation (39)

(_

:=

)

_

(_

)



_

_

d (_q
L L) = q_
(Q + Q ) L : (40) q_ dt We end this section remarking that the function ( q_
Lq_ L) corresponds to the Hamiltonian C

t

of the constrained system only if the constraint equations are satisfied at both position and velocity levels.

4. The Model Pendulum It is interesting at this point to consider a very simple example as an exercise to get familiar with the proposed formulation. We refer to the classical case of the model pendulum. Let l and m denote the length and mass of the pendulum, q q 1 q2 the Cartesian coordinates of the moving extremity, p p 1 p2 the related momenta. We write the constraint function   in the following form

=( ; )

=( ; )

=

1 (l2 2

q
q);

= q and B = = (q ; p) as 1 q = q~ ;

In this case we have A lem we get the state x

and the multiplier 

=( ; )

= ( ; ) as



=

q~
p~

 l2

;

=

m 1 q
p:

(41)

m 1 p. From the solution of the algebraic subprobp=

2

1

~; P p  

 = m (

1);

(42)

(43)

where for convenience we have set

(~ q
q~ )1 2 = ; =

l

P = I2

q~ q~ : q~
q~

(44)

The dynamic equations are expressed as

p_ q_

_ q m 1 _ p + Q = 02 ; m 1 _ q m 1 p = 02 ;

(45)

=(

; 0)

where Q mg denotes the external force, being g the acceleration of gravity. The components of the time derivative  of the constraint function read

_ = q
q_ ;

_

_=

m

1

(_q
p + q
p_ ):

(46)

_

These equations lead to the following solutions for the time derivative of the state x as 

q_ = m p;

p_ = I2

1

_

q q q
q



Q + c q;

(47)

and of the multiplier  as

_ =

=(

)

q
Q

c;

l2

_ = 0;

(48)

m 1 =l2 p
p is the centrifugal stiffness and the state x is understood as a function where c of the modified state x as given by equation (42). Eventually, for the present case, we get the following ODEs

~

p~_ = Q +  q_ + m 1  p_ ; q~_ = m 1 p m 1  q_ ;

(49)

_

where the time derivative of the state, x, is understood as given by equation (47) and the multiplier  as given by equation (42). 5. Numerical Studies To illustrate the features of the proposed scheme, we apply the EPM to Andrews’ squeezer mechanism. The problem involves a planar multibody system, depicted in figure 1, consisting of 7 rigid bodies fK i gi=1;::: ;7 connected by joints without friction, loaded by a linear spring acting on body K3 , and actuated through a constant torque driver acting on crank K1 . This model is a classical test case for numerical codes for constrained system dynamics. Details are given in ref. [15] (chapter VII.7 Computation of Multibody Mechanisms). We implemented Andrews’ squeezer as a general 2-dimensional multibody system using the set x; y;  composed of the standard Cartesian coordinates plus the in-plane rotation. The constraints for each joint were treated as previously discussed. All the data are taken from the cited reference. We implemented the formulation using two families of finite-element-in-time schemes, namely the bi-discontinuous (BD) and singly-discontinuous (SD) schemes of arbitrary order using Gauss-Legendre quadrature. Both schemes are unconditionally stable in the linear regime. However, the former has a unitary spectral radius, while the latter is characterized by asymptotic annihilation ( 1 : ). Since the BD scheme provides no high frequency damping, it would not be an ideal candidate for solving multibody systems with a classical approach. The equivalence of the schemes to collocated Runge-Kutta methods was discussed in refs. [7, 8]. The results obtained by the second order schemes of the conservative and dissipative families were compared with those obtained from the general purpose research codes DYMORE [3] and MBDyn [14]. The first implements a family of non-linearly unconditionally stable, second/third order, single-step dissipative schemes, while the second features a family of linearly unconditionally stable, second order, multi-step dissipative schemes. Both

(

)

= 00

C

B K3

D

K5 K4

K2

A

O

K1

K6 K7 Figure 1. Andrews’ squeezer mechanism: assembled system configuration.

are based on a ‘traditional’ formulation with standard Lagrange multipliers in the equilibrium equations plus appended position-level constraint equations, and rely on high frequency damping for numerical stabilization. Figures 2 and 3 show the time history plots of variables and Æ as defined in ref. [15], 0.5 0.45 0.4 0.35

γ

0.3 0.25 0.2 0.15 Embedded Projection with BD Embedded Projection with SD MBDyn ρ∞=0.6

0.1 0.05 0 0

DYMORE ρ∞=0.0 0.01

0.02

0.03

0.04

0.05

t

Figure 2. Andrews’ squeezer mechanism: time history of the orientation of body K 3 .

0.53 0.525 0.52 0.515

δ

0.51 0.505 0.5 0.495 Embedded Projection with BD Embedded Projection with SD MBDyn ρ∞=0.6

0.49 0.485 0.48 0

DYMORE ρ∞=0.0 0.01

0.02

0.03

0.04

0.05

t

Figure 3. Andrews’ squeezer mechanism: time history of the orientation Æ of body K 5 .

i.e. the orientations of bodies K3 and K5 as computed by the second order EPM BD and SD schemes, by the schemes of DYMORE with asymptotic spectral radius  1 : , and MBDyn with asymptotic spectral radius  1 : . Figures 4 and 5 show the Cartesian components of the constraint reaction between bodies K4 and K6 , as computed by EPM and the scheme featured in MBDyn. The complete agreement between all the methods analyzed is clearly apparent. However, it must be stressed that the methods based on conventional formulations do need a considerable amount of dissipation to perform their task, while in the EPM framework we were able to successfully use the BD scheme, which has a constant unitary spectral ra-

=00

= 06

40

20

λ•

x

0

−20

−40

−60

−80 0

Embedded Projection with BD Embedded Projection with SD MBDyn ρ∞=0.6 0.01

0.02

0.03

0.04

0.05

t

Figure 4. Andrews’ squeezer mechanism: time history of the horizontal component of the reaction force _ x exerted between bodies K 4 and K6 .

100 80 60 40

λ•

y

20 0 −20 −40 −60 Embedded Projection with BD Embedded Projection with SD MBDyn ρ =0.6

−80

∞

−100 0

0.01

0.02

0.03

0.04

0.05

t

Figure 5. Andrews’ squeezer mechanism: time history of the vertical component of the reaction force _ y exerted between bodies K 4 and K6 .

dius and therefore provides no high frequency numerical damping. In fact, when an attempt was made to solve the problem using the two reference multibody codes with  1 :, large oscillations in the reaction forces were observed. These oscillations eventually lead to the blow-up of the simulation around time t : s. In particular, for MBDyn the choice 1 : amounts for using the classical Crank-Nicholson scheme, i.e the trapezoidal rule, which also coincides with the second order BD scheme. This example shows that even the trapezoidal rule, widely recognized as a poor choice for solving constrained mechanical systems, can be successfully used in the proposed EPM framework. Therefore, it may be argued

= 10

= 05

=10

0

10

Embedded Projection with BD Embedded Projection with SD −1

10

−2

λ• magnitude error

10

−3

10

−4

10

−5

10

−6

10

3

2

−7

10

3

10

4

10 1/h

5

10

Figure 6. Andrews’ squeezer mechanism: convergence of the relative magnitude error of the reaction force _ at point O.

0

10

Embedded Projection with BD Embedded Projection with SD −1

10

−2

λ• phase error

10

−3

10

−4

10

−5

10

−6

10

3

2

−7

10

3

10

4

10 1/h

5

10

Figure 7. Andrews’ squeezer mechanism: convergence of the relative phase error of the reaction force _ at point O.

that the EPM provides an intrinsic means of numerical stabilization. All the simulations were performed selecting a fixed time step h : 4 s. This value roughly coincides with the limit value that ensures convergence for the two ‘conventional’ methods. It is worth mentioning that the two schemes implemented within the EPM framework proved capable of bearing a much larger time step, up to more than one order of magnitude higher. Finally, consider figures 6 and 7, depicting the accuracy realized by the EPM BD and SD schemes. The first shows the logarithmic plot of the magnitude error of the constraint reaction at point O , while in the second the phase error is considered. The reference converged values of the solution for computing the errors were taken from the IVP Testset [16]. In both cases the convergence lies within second and third order, as we expect from the second order BD and SD schemes.

= 1 0E

6. Conclusions The Embedded Projection Method for index-3 constrained dynamical systems has been presented in this work. We showed how the governing differential-algebraic equations may be split into uncoupled algebraic and differential parts. This process, which involves the definition of a modified, unconstrained state, leads to the formulation of an ODE which can be solved by any suitable standard numerical integrator, by-passing the need for specialized ODE solvers. An ‘embedded projection’ onto the constraint manifold allows to recover the original state, which satisfies the constraints both at the position and velocity levels. The application to a common benchmark test case, the so-called ‘Andrews’ squeezer mechanism’, was presented as an aid for discussing the characteristics of the new formulation. While on the side of implementation and application further work is needed in order to save computational costs, as well as to test the methodology on large scale, complex applications of deformable multibody systems, on the theoretical side we envisage an extension of the proposed framework encompassing arbitrarily high index DAE sets. In this sense, the

EPM may be seen as a general procedure for reducing the index from any value to 1. These ideas will be explored in a forthcoming paper.

References [1] U. Ascher, Stabilization of invariants of discretized differential systems, Numerical Algorithms 14 (1997) 1–23. [2] U. Ascher, H. Chin, L.R. Petzold and S. Reich, Stabilisation of constrained mechanical systems with DAEs and invariant manifolds, J. Mech. Struct. Machines 23 (1995) 135–158. [3] O.A. Bauchau, C.L. Bottasso and Y.G. Nikishkov, Modeling rotorcraft dynamics with finite element multibody procedures, Math. Comput. Modeling, accepted, in press. [4] J. Baumgarte, Stabilization of constraints and integrals of motion in dynamical systems, Comp. Math. Appl. Mech. Eng. 1 (1972), 1–16. [5] M. Borri, C.L. Bottasso and P. Mantegazza, A modified phase space formulation for constrained mechanical systems – Differential Approach, Eur. J. Mech., A/Solids, 11 (1992) 701–727. [6] M. Borri and P. Mantegazza, Finite time element approximation of dynamics of nonholonomic systems, A.M.S.E. Congress, Williamsburg, VA 1986. [7] C.L. Bottasso, A new look at finite elements in time – A variational interpretation of Runge–Kutta methods, Appl. Num. Math., 25 (1997) 355–368. [8] C.L. Bottasso and M. Borri, Some recent developments in the theory of finite elements in time, Comp. Mod. Sim. Engrg., 4 (1999) 201–205. [9] K. E. Brenan, S.L. Campbell and L.R. Petzold, Numerical solution of initial-value problems in differential-algebraic equations, Elsevier Science Publishing Co., 1989. [10] E. Eich, Convergence results for a coordinate projections methods applied to constrained mechanical systems with algebraic constraints, SIAM J. Numer. Anal. 30 (1993) 1467–1482. [11] C. F¨urer and B. Leimkuhler, Numerical solution of differential-algebraic equations for constrained mechanical motion, Numerische Mathematik, 59 (1991) 55–69. [12] C.W. Gear, Differential-algebraic equation index transformations, SIAM J. Sci. Stat. Comput. 9(1) (1988) 40–47. [13] C.W. Gear, G.K. Gupta and B.J. Leimkuhler, Automatic integration of the Euler-Lagrange equation with constraints. J. Comp. Appl. Math., 12 (1985) 77–90. [14] G.L. Ghiringhelli, P. Masarati and P. Mantegazza, Multi-body analysis of a tiltrotor configuration, Nonlinear Dynamics, 19(4) (1999) 333–357. [15] E. Hairer and G. Wanner, Solving ordinary differential equations – II: Stiff problems Springer-Verlag, Berlin, Germany 1991. [16] W.M. Lioen, J.J.B. De Swart and W.A. Van Der Veen, Testset for IVP solvers, Technical Report NMR9615, CWI, Amsterdam, The Netherlands 1996. [17] L.R. Petzold, Order results for implicit Runge-Kutta methods applied to differential/algebraic systems, SIAM J. Numer. Anal.. 23(4) (1986) 837–852. [18] L.R. Petzold, L.O. Jay and J. Yen, Numerical solution of highly oscillatory ordinary differential equations, Acta Numerica (1997) 437–484. [19] J. Yen and L.R. Petzold, Convergence of the iterative methods for coordinate splitting formulation in multibody dynamics, TR 95-052, Tech. Rept., Dept.of Comput. Sci., University of Minnesota, 1995.

Localized Formulation Of Multibody Systems K. C. PARK*, Carlos A. FELIPPA* and Roger OHAYON+ Department of Aerospace Engineering Sciences and Center for Aerospace Structures, University of Colorado, Campus Box 429, Boulder, CO 80309, U.S.A + Structural Mechanics and Coupled Systems Laboratory, Conservatoire National des Arts et Métiers (CNAM), 2, rue Conte 75003, Paris, France *

Abstract. When one applies the classical method of Lagrange multipliers to model multibody systems whose linkages or multi-particle bonding and contact involve more than two bodies or particles at a number of constraint points, the resulting equilibrium equations can lead to singular systems with a large number of redundancies. In addition, the resulting interaction constraints and the associated classical Lagrange multipliers become tightly coupled, thus making physical interpretation of the model solution difficult. This paper presents a localized version of Lagrange multipliers that leads to a unique and nonsingular localized set of equilibrium equations for systems with multiple constraints, bonding and/or contacts. The resulting model is easy to formulate, provides direct physical insight, and is conducive to regularization when the bonding or contact forces are widely different among lattices and/or granular particles. Model problems are used to illustrate the present localized formulation in detail.

1.

Introduction

The classical method of Lagrange multipliers is widely adopted for the enforcement of kinematical and boundary constraints in mechanics. When one models a multibody system that can be viewed as a succession of two-body or two-particle problems in bonding or constraint, an elementary application of Newton's laws yields the nonsingular system equilibrium equations with linearly independent constraint conditions. However, when one applies Newton's laws to model multi-body linkages or multi-particle bonding and contact problems, the resulting equilibrium equations can lead to singular systems with a large number of redundancies. In addition, the resulting interaction constraints and the associated classical Lagrange multipliers become tightly coupled, thus making physical interpretation of the model solution difficult. In other words, when more than two bodies are constrained together at a point, ambiguities occur in selecting an independent set of possible constraint conditions. Otherwise a numerical procedure is needed to eliminate all of the redundant constraints such as a master-slave concept, a Kirchhoff analogy and rank elimination. Second, the constraint conditions thus constructed inherently relate from a point in one body to a point in another body, whose path must include the constraint points. Therefore, even though the inertia and internal forces can be formulated locally for each of the multi-bodies, the formulation of constraint conditions must consider their interdependencies. In other words, while the inertia and internal forces can be formulated in an element-by-element manner, the constraint formulation must be carried out in a global node-by-node manner.

This paper describes a localized version of the method of Lagrange multipliers that alleviates the above mentioned constraint formulation ambiguities and that leads to an element-by-element constraint formulation. Benefits that accrue by employing the proposed localized Lagrange multipliers method include: uniqueness in the resulting constraint conditions, element-by-element construction of constraint conditions without having to consider inter-body connectivities, and generality of the resulting governing equations of motion that can accommodate existing as well as new solution algorithms. In particular, the resulting formulation is shown to offer an effective regularization for systems that consist of widely varying compliances, or very stiff components. The proposed localized Lagrange multipliers has been applied to the modeling of multi-point constraints that emanate from contact-impact problems and partitioned structural analysis using parallel computers. Example problems are given to illustrate the present localized formulation and its regularization feature for highly stiff systems.

2.

Classical Formulation of Lattice Problems

This section reviews the use of Newton's 3rd law and the principle of virtual work to obtain the conditions of constraints, which leads to the classical method of Lagrange multipliers. It is shown that, when more than two lattice elements are bonded together, the resulting constraint conditions are redundant. To this end, consider a two-lattice problem bonded together as shown in Fig. 1. Let us assume that the equilibrium equations of individual lattices when they are completely free are given by For lattice 1: S1 ( x1 ) = 0 For lattice 2: S2 ( x2 ) = 0

(1)

When the two lattices are partitioned subject to unknown bonding forces, the governing equations may be expressed as For lattice 1: S1 ( x1 ) = λ12 For lattice 2: S2 ( x2 ) = λ21

(2)

where index {ij} refer to the bonding force acting on lattice {i} due to its interaction with lattice {j}.

Figure 1. Classical modeling of two lattices bonded together

The virtual work of the above system, equation (2) may be stated as

δ W ( x1, x2 , λ12 , λ21 ) = δ x1 ⋅ [ S1 ( x1 ) − λ12 ] + δ x2 ⋅ [ S2 ( x2 ) − λ21 ]

(3)

In order to reveal the system kinematical constraint from the virtual work statement, equation (3), we invoke Newton's 3rd law

λ12 + λ21 = 0

(4)

Figure 2. Classical modeling of three lattices bonded together

and eliminate λ21 in the above virtual work to obtain:

δ W ( x1, x2 , λ12 ) = δ x1 ⋅ [ S1 ( x1 ) − λ12 ] + δ x2 ⋅ [ S2 ( x2 ) + λ12 ]

(5)

The preceding equation can be rearranged to consist of the virtual work for two completely free lattices, plus the condition of constraint that is multiplied by an unknown coefficient, viz., a Lagrange multiplier λ12:

δ W ( x1, x2 , λ12 ) = {δ x1 ⋅ S1 ( x1 ) + δ x2 ⋅ S2 ( x2 )} − λ12 ⋅ δ ( x1 − x2 )

(6)

We observe that Lagrange's condition of constraint is given by

x1 + x2 = c12

(7)

where c12 can be viewed as a gap constant between the two lattices at the bonding point. Equations (6) and (7) state that, when the governing equations obtained by Newton's 2nd law and the law of moment are used to construct the corresponding virtual work expression by invoking Newton's 3rd law, one can obtain the system kinematical constraints given by equation (7). For the two-lattice problem, the interaction force λ12 acts as the Lagrange multiplier. Now consider a three-lattice problem as shown in Fig. 2. The governing equations for this case are given by For lattice 1: S1 ( x1 ) = λ12 + λ13 For lattice 2: S2 ( x2 ) = λ21 + λ23 For lattice 3: S3 ( x3 ) = λ31 + λ32

(8)

along with three equations among the interaction forces due to Newton's 3rd law:

λ12 + λ21 = 0 λ13 + λ31 = 0 λ23 + λ32 = 0

(9)

The virtual work that combines equations (8) and (9) can be expressed as

δ W ( x1, x2 , x3 , λ12 , λ23 , λ31 ) = {δ x1 ⋅ S1 ( x1 ) + δ x2 ⋅ S2 ( x2 ) + δ x3 ⋅ S3 ( x3 )} − −λ12 ⋅ δ ( x1 − x2 ) − λ23 ⋅ δ ( x2 − x3 ) − λ31 ⋅ δ ( x3 − x1 )

(10)

where the first three terms account for the virtual work of three completely free lattices, and the last three terms yield the kinematical constraints that we are seeking. Hence, we obtain the following three conditions of constraints:

x1 + x2 = c12 x2 + x3 = c23 x3 + x1 = c31

(11)

where {cij, (i,j) = 1,2,3} are gap constants. We note that the third condition, x3 − x1 = c31, in the preceding set of the conditions of constraints is obtained by a linear combination of the first two since the three gap magnitudes form a closed triangle to have

c31 + c23 + c12 = 0

(12)

Therefore, we conclude that one of the three constraint conditions is redundant and hence must be deleted in order to yield a linearly independent set of system equations. For example, if x3 − x1 = c31 is deleted, then the resulting system equations become:

S1 ( x1 ) = λ12 S2 ( x2 ) = −λ12 + λ23 S3 ( x3 ) = −λ23 x1 − x2 = c12 x2 − x3 = c23

(13)

Observe that, with the deliberate deletion of x3 − x1 = c31, the corresponding Lagrange multiplier λ31 is also deleted. While this deletion is mathematically necessary to achieve linear independence, the physical link between node 3 and 1 is now lost. Two other alternative system equations, although they are equivalent to equation (13), can be generated by deleting either the first or the second constraint in equation (12). Extensions to multi-lattice problems lead to similar linear dependency issues. For an n-lattice problem there are (n−1)(n−2)/2 redundancies which must be deleted or eliminated in order to recover full rank in the resulting governing equations. We now offer the following observations. 1. A straightforward application of the principle of virtual work to the equilibrium equations leads to a linearly dependent set of constraint equations for the number of lattices bonded at a point exceeding 2. 2. Choosing a linearly independent set of constraints can vary from one modeler to another. 3. For the example 3-lattice problem, the physical bonding forces f are given by  f1   λ12       f 2  = −λ12 + λ23   f3   −λ23 

(14)

which indicates that a different choice of linearly independent constraint set will lead to different expressions. 4. The constraint conditions cij are characterized by the double indices {i} and {j}, which implies that the construction of constraints must take the inter-body connectivities, viz., how lattice {i} is related to lattice {j}. In other words, the constraints are necessarily non-local or are expressed in a global node-to-node manner. This has a far reaching implication on the formulation and regularization

which plays a key role when lattices with severely different bonding forces are bonded together.

3.

Procedures for Generating Rank-Sufficient Classical Constraint Conditions

The redundancies and non-uniqueness issues associated the classical method of Lagrange multipliers have been well-known especially to the multibody mechanicians. Of various remedial approaches proposed so far, we present three below. 3.1

Master-Slave Procedure

In this procedure one of the constraining nodes is chosen as a master node, say, node 1. All of the remaining nodes are constrained to this node. For a n-lattice problem, if one selects node n to be the master node, the constraint conditions are constructed as

c1n = x1 − xn = 0 c2 n = x2 − xn = 0 c3n = x3 − xn = 0 ! c(n−1) n = xn−1 − xn = 0

(15)

While this procedure yields a rank-sufficient constraint conditions, it suffers from the lack of explicitness that node 1 is also directly constrained to nodes 2, 3, ..., (n-1). In general, for node k (k