[en] This paper describes a dynamic formulation of a straight beam finite element in the setting
of the special Euclidean group SE(3). First, the static and dynamic equilibrium equations
are derived in this framework from variational principles. Then, a non-linear interpolation
formula using the exponential map is introduced. It is shown that this framework leads to a
natural coupling in the interpolation of the position and rotation variables. Next, the discretized
internal and inertia forces are developed. The semi-discrete equations of motion
take the form of a second-order ordinary differential equation on a Lie group, which is
solved using a Lie group time integration scheme. It is remarkable that no parameterization
of the nodal variables needs to be introduced and that the proposed Lie group framework
leads to a compact and easy-to-implement formulation. Some important numerical and
theoretical aspects leading to a computationally efficient strategy are highlighted and discussed.
For instance, the formulation leads to invariant tangent stiffness and mass matrices
under rigid body motions and a locking free element. The proposed formulation is successfully
tested in several numerical static and dynamic examples.
Disciplines :
Mechanical engineering
Author, co-author :
Sonneville, Valentin ; Université de Liège - ULiège > Département d'aérospatiale et mécanique > Laboratoire des Systèmes Multicorps et Mécatroniques
Cardona, Alberto; Universidad Nacional del Litoral > Centro Internacional de Métodos Computacionales en Ingeniería CIMEC-INTEC
Bruls, Olivier ; Université de Liège - ULiège > Département d'aérospatiale et mécanique > Laboratoire des Systèmes Multicorps et Mécatroniques
Language :
English
Title :
Geometrically exact beam finite element formulated on the special Euclidean group SE(3)
Publication date :
January 2014
Journal title :
Computer Methods in Applied Mechanics and Engineering
ISSN :
0045-7825
eISSN :
1879-2138
Publisher :
Elsevier Science, Lausanne, Switzerland
Volume :
268
Pages :
451-474
Peer reviewed :
Peer Reviewed verified by ORBi
Funders :
FRIA - Fonds pour la Formation à la Recherche dans l'Industrie et dans l'Agriculture [BE]
Géradin M., Cardona A. Flexible Multibody Dynamics: A Finite Element Approach 2001, John Wiley & Sons, Chichester.
Bauchau O.A. Flexible mutibody dynamics. Solid Mechanics and its Applications 2011, vol. 176. Springer.
Simo J. A finite strain beam formulation. The three-dimensional dynamic problem. Part I. Computer Methods in Applied Mechanics and Engineering 1985, 49:55-70.
Simo J., Fox D. On a stress resultant geometrically exact shell model. Part I: Formulation and optimal parametrization. Computer Methods in Applied Mechanics and Engineering 1989, 72:267-304.
Shabana A.A., Yakoub R.Y. Three dimensional absolute nodal coordinate formulation for beam elements: theory. Journal of Mechanical Design 2001, 123:606-613.
Romero I. A comparison of finite elements for nonlinear beams: the absolute nodal coordinate and geometrically exact formulations. Multibody System Dynamics 2008, 20:51-68.
Betsch P., Steinmann P. Frame-indifferent beam finite elements based upon the geometrically exact beam theory. International Journal for Numerical Methods in Engineering 2002, 54:1775-1788.
Brüls O., Cardona A., Arnold M. Lie group generalized-α time integration of constrained flexible multibody systems. Mechanism and Machine Theory 2012, 121-137.
O. Brüls, M. Arnold, A. Cardona, Two Lie group formulations for dynamic multibody systems with large rotations, in: Proceedings of the IDETC/MSNDC Conference, Washington D.C., U.S.
Brüls O., Cardona A. On the use of Lie group time integrators in multibody dynamics. ASME Journal of Computational and Nonlinear Dynamics 2010, 5:031002.
Cardona A., Géradin M. A beam finite element non-linear theory with finite rotations. International Journal for Numerical Methods in Engineering 1988, 26:2403-2438.
Crisfield M., Jelenic G. Objectivity of strain measures in the geometrically exact three-dimensional beam theory and its finite-element implementation. Proceedings of the Royal Society of London A 1999, 455:1125-1147.
Borri M., Bottasso C. An intrinsic beam model based on a helicoidal approximation - Part I: formulation. International Journal for Numerical Methods in Engineering 1994, 37:2267-2289.
Cesarek P., Saje M., Zupan D. Dynamics of flexible beams: finite-element formulation based on interpolation of strain measures. Finite Elements in Analysis and Design 2013, 72:47-63.
McCarthy J. An Introduction to Theoretical Kinematics, Cambridge 1990, MIT Press, Mass.
Murray R.M., Li Z., Sastry S.S. A Mathematical Introduction to Robotic Manipulation 1994, CRC Press.
Angeles J. Fundamentals of Robotic Mechanical Systems: Theory, Methods, and Algorithms 2003, Springer-Verlag, Berlin.
Selig J.M. Geometric Fundamentals of Robotics. Monographs in computer science 2005, Springer, New York.
Crouch P., Grossman R. Numerical integration of ordinary differential equations on manifolds. Journal of Nonlinear Science 1993, 3:1-33.
Munthe-Kaas H. Runge-Kutta methods on Lie groups. BIT 1998, 38:92-111.
Bottasso C., Borri M. Integrating finite rotations. Computer Methods in Applied Mechanics and Engineering 1998, 164:307-331.
Park F. Distance metrics on the rigid-body motions with applications to mechanism design. Transactions of the ASME 1995, 117:48-54.
Zefran M., Kumar V. Interpolation schemes for rigid body motions. Computer-Aided Design 1998, 30:179-189.
Sander O. Geodesic finite elements for Cosserat rods. International Journal for Numerical Methods in Engineering 2010, 82:1645-1670.
Selig J.M., Ding X. A screw theory of Timoshenko beams. Journal of Applied Mechanics 2009, 76:031003.
Iserles A., Munthe-Kaas H., Nrsett S., Zanna A. Lie-group methods. Acta Numerica 2000, 9:215-365.
Boothby W. An Introduction to Differentiable Manifolds and Riemannian Geometry 2003, Academic Press. 2nd ed.
V. Sonneville, O. Brüls, Formulation of kinematic joints and rigidity constraints in multibody dynamics using a Lie group approach, in: Proceedings of the 2nd Joint International Conference on Multibody, System Dynamics (IMSD).
Bathe K.-J., Bolourchi S. Large displacement analysis of three-dimensional beam structures. International Journal for Numerical Methods in Engineering 1979, 14:961-986.
Simo J., Vu-Quoc L. A three-dimensional finite-strain rod model. Part II: computational aspects. Computer Methods in Applied Mechanics and Engineering 1986, 58:79-116.
Jung P., Leyendecker S., Linn J., Ortiz M. A discrete mechanics approach to the Cosserat rod theory - Part 1: static equilibria. International Journal for Numerical Methods in Engineering 2011, 85:31-60.
A. Dhooge, W. Govaerts, Y.A. Kuznetsov, W. Mestrom, A.M. Riet, B. Sautois, MATCONT and CL MATCONT: continuation toolboxes in matlab, , 2006. http://www.matcont.ugent.be.
Ibrahimbegović A., Mikad M.A. Finite rotations in dynamics of beams and implicit time-stepping schemes. International Journal for Numerical Methods in Engineering 1998, 41:781-814.
Lens E., Cardona A. A nonlinear beam element formulation in the framework of an energy preserving time integration scheme for constrained multibody systems dynamics. Computers and Structures 2008, 86:47-63.
V. Sonneville, O. Brüls, Sensitivity analysis for multibody systems formulated on a Lie group, Multibody System Dynamics, 2013, in press.
Park J., Chung W. Geometric integration on Euclidean group with application to articulated multibody systems. IEEE Transactions on Robotics 2005, 21:850-863.
