Skip to main content
SpringerLink
Log in

A frame-invariant scheme for the geometrically exact beam using rotation vector parametrization

  • Original Paper
  • Published:
Computational Mechanics Aims and scope Submit manuscript

Abstract

While frame-invariant solutions for arbitrarily large rotational deformations have been reported through the orthogonal matrix parametrization, derivation of such solutions purely through a rotation vector parametrization, which uses only three parameters and provides a parsimonious storage of rotations, is novel and constitutes the subject of this paper. In particular, we employ interpolations of relative rotations and a new rotation vector update for a strain-objective finite element formulation in the material framework. We show that the update provides either the desired rotation vector or its complement. This rules out an additive interpolation of total rotation vectors at the nodes. Hence, interpolations of relative rotation vectors are used. Through numerical examples, we show that combining the proposed update with interpolations of relative rotations yields frame-invariant and path-independent numerical solutions. Advantages of the present approach vis-a-vis the updated Lagrangian formulation are also analyzed.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Simo JC (1985) A finite strain beam formulation. The three-dimensional dynamic problem. Part I. Comput Methods Appl Mech Eng 49: 55–70

    Article  MATH  Google Scholar 

  2. Simo JC, Vu-Quoc L (1986) A three-dimensional finite rod model part II: computational aspects. Comput Methods Appl Mech Eng 58: 79–116

    Article  MATH  Google Scholar 

  3. Vu-Quoc L (1986) Dynamics of flexible structures performing large overall motions: a geometrically-nonlinear approach. Ph.D. thesis, UC Berkeley, Dissertation, ERL Memorandum UCB/ERL M86/36

  4. Cardona A, Géradin M (1988) A beam of the finite element nonlinear theory with finite rotations. Int J Numer Methods Eng 26: 2403–2438

    Article  MATH  Google Scholar 

  5. Simo J, Vu-Quoc L (1988) On the dynamics in space of rods undergoing large motions: a geometrically exact approach. Comput Methods Appl Mech Eng 66: 125–161

    Article  MATH  MathSciNet  Google Scholar 

  6. Simo JC, Vu-Quoc L (1991) A geometrically exact rod model incorporating shear and torsion warping deformation. Int J Numer Methods Eng 27: 371–393

    MATH  MathSciNet  Google Scholar 

  7. Ibrahimbegović A, Frey F, Kozar I (1995) Computational aspects of vector-like parametrization of three-dimensional finite rotations. Int J Numer Methods Eng 38: 3653–3673

    Article  MATH  Google Scholar 

  8. Ibrahimbegović A (1995) Finite element implementation of reissner’s geometrically nonlinear beam theory: three dimensional curved beam finite element. Comput Methods Appl Mech Eng 122: 10–26

    Google Scholar 

  9. Jelenić G, Saje M (1995) A kinematically exact space finite strain beam model finite element formulation by generalized virtual work principle. Comput Methods Appl Mech Eng 120: 131–161

    Article  MATH  Google Scholar 

  10. Smolenski WM (1999) Statically and kinematically exact nonlinear theory of rods and its numerical verification. Comput Methods Appl Mech Eng 178: 89–113

    Article  MATH  MathSciNet  Google Scholar 

  11. Mäkinen J (2007) Total lagrangian reissner’s geometrically exact beam element without singularities. Int J Numer Methods Eng 70: 1009–1048

    Article  Google Scholar 

  12. McRobie F, Lasenby J (1999) Simo-vu quoc rods using clifford algebra. Int J Numer Methods Eng 45(4): 377–398

    Article  MATH  MathSciNet  Google Scholar 

  13. Pimenta PM, Campello EMB, Wriggers P (2008) An exact conserving algorithm for nonlinear dynamics with rotational DOFs and general hyperelasticity. Part 1: Rods. Comput Mech 42(5): 715–732

    Article  MathSciNet  MATH  Google Scholar 

  14. Botasso CL, Borri M (1998) Integrating finite rotations. Comput Methods Appl Mech Eng 164: 307–331

    Article  Google Scholar 

  15. Betsch P, Steinmann P (2002) Frame-indifferent beam element based upon the geometrically exact beam theory. Int J Numer Methods Eng 54: 1775–1788

    Article  MATH  MathSciNet  Google Scholar 

  16. Atluri S, Cazzani A (1995) Rotations in computational solid mechanics. Arch Comput Methods Eng 1: 49–138

    Article  MathSciNet  Google Scholar 

  17. Pimenta PM, Campello EMB (2001) Geometrically nonlinear analysis of thin-walled space frames. In: Waszczyszyn Z, Stein E (eds) Second European conference on computational mechanics. Cracow, Poland (2001)

  18. Campello EMB, Pimenta PM, Wriggers P (2003) A triangular finite shell element based on a fully nonlinear shell formulation. Comput Mech 31(6): 505–518

    Article  MATH  Google Scholar 

  19. Nikravesh PE (1988) Computer aided analysis of mechanical systems. Prentice Hall, Englewood Cliffs

    Google Scholar 

  20. Géradin M, Cardona A (2001) Flexible multibody dynamics: a finite element approach. Wiley, New York, ISBN 0-471-48990-5

  21. Spring K (1986) Euler parameters and the use of quaternion algebra in the manipulation of finite rotations: a review. Mech Mach Theory 21: 365–373

    Article  Google Scholar 

  22. Géradin M, Rixen D (1995) Parametrization of finite rotations in computational dynamics: a review. Revue européenne des éléments finis 4: 497–553

    MATH  Google Scholar 

  23. Pimenta PM, Yojo T (1993) Geometrically exact analysis of spatial frames. Appl Mech Rev 46(11): 118–128

    Article  Google Scholar 

  24. Jagan Mohan S (2004) Group theoretic framework for FEM analysis of symmetric structures. Ph.D. thesis, Deptartment of Mechanical Engineerig, Indian Institute of Science, Bangalore, India

  25. Ritto-Corrêa M, Camotim D (2002) On the differentiation of the rodrigues formula and its significance for the vector-like parameterization of reissner-simo beam theory. Int J Numer Methods Eng 55(9): 1005–1032

    Article  MATH  Google Scholar 

  26. Ibrahimbegović A (1997) On the choice of finite rotation parameters. Comput Methods Appl Mech Eng 149: 49–71

    Article  MATH  Google Scholar 

  27. Ibrahimbegović A, Taylor RL (2002) On the role of frame invariance in structural mechanics models at finite rotations. Comput Methods Appl Mech Eng 191: 5159–5176

    Article  MATH  Google Scholar 

  28. Crisfield MA, Jelenić G (1999) Objectivity of strain measures in the geometrically exact three-dimensional beam theory and its finite element implementation. Proceedings of the Royal Society of London, Series A, Mathematical Physical and Engineering Sciences 455: 1125–1147

    Article  MATH  Google Scholar 

  29. Jelenić G, Crisfield M (1999) Geometrically exact 3D beam theory: implementation of a strain-invariant finite element for statics and dynamics. Comput Methods Appl Mech Eng 171: 141–171

    Article  MATH  Google Scholar 

  30. Romero I (2004) The interpolation of rotations and its application to finite element models of geometrically exact rods. Comput Mech 34: 121–133

    Article  MATH  MathSciNet  Google Scholar 

  31. Romero I, Armero F (2002) An objective finite element approximation of the kinematics of geometrically exact rods and its use in the formulation of an energy-momentum conserving scheme in dynamics. Int J Numer Methods Eng 54(12): 1683–1716

    Article  MATH  MathSciNet  Google Scholar 

  32. Ghosh S, Roy D (2008) Consistent quaternion interpolation for objective finite element approximation of geometrically exact beam. Comput Methods Appl Mech Eng 198(3–4): 555–571

    Article  MathSciNet  Google Scholar 

  33. Reissner E (1972) On one-dimensional finite-strain beam theory: the plane problem. J Appl Math Phys 23: 793–804

    Google Scholar 

  34. Antman SS (1976) Ordinary differential equations of one-dimensional elasticity: foundations of the theories of nonlinearly elastic rods and shells. Arch Rational Mech Anal 61: 307–351

    MATH  MathSciNet  Google Scholar 

  35. Varadarajan VS (1984) Lie Groups, lie algebras and their representation. Graduate Texts in Mathematics, vol 102. Springer, Berlin

  36. Jelenić G, Crisfield M (1998) Interpolation of rotational variables in nonlinear dynamics of 3D beams. Int J Numer Methods Eng 43: 1193–1222

    Article  MATH  Google Scholar 

  37. Engø K (2001) On the bch-formula in so(3). BIT Numer Math 41(3): 629–632

    Article  Google Scholar 

  38. Bathe KJ, Bolourchi S (1979) Large displacement analysis of three-dimensional beam structures. Int J Numer Methods Eng 14: 961–986

    Article  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Structures Lab, Department of Civil Engineering, Indian Institute of Science, Bangalore, 560 012, India

    S. Ghosh & D. Roy

Authors
  1. S. Ghosh
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. D. Roy
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to D. Roy.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Ghosh, S., Roy, D. A frame-invariant scheme for the geometrically exact beam using rotation vector parametrization. Comput Mech 44, 103–118 (2009). https://doi.org/10.1007/s00466-008-0358-z

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00466-008-0358-z

Keywords

Search

Navigation