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A comparison of finite elements for nonlinear beams: the absolute nodal coordinate and geometrically exact formulations

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Abstract

Two of the most popular finite element formulations for solving nonlinear beams are the absolute nodal coordinate and the geometrically exact approaches. Both can be applied to problems with very large deformations and strains, but they differ substantially at the continuous and the discrete levels. In addition, implementation and run-time computational costs also vary significantly. In the current work, we summarize the main features of the two formulations, highlighting their differences and similarities, and perform numerical benchmarks to assess their accuracy and robustness. The article concludes with recommendations for the choice of one formulation over the other.

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References

  1. Antman, S.S.: The Theory of Rods. Handbuch der Physik, vol. VIa/2. Springer, Berlin (1972) (Flügge, S., Truesdell, C. (eds.))

    Google Scholar 

  2. Argyris, J.H.: An excursion into large rotations. Comput. Methods Appl. Mech. Eng. 32, 85–155 (1982)

    Article  MATH  MathSciNet  Google Scholar 

  3. Avello, A., García de Jalón, J.: Dynamics of flexible multibody systems using Cartesian co-ordinates and large displacement theory. Int. J. Numer. Methods Eng. 32, 1543–1563 (1991)

    Article  MATH  Google Scholar 

  4. Bathe, K.J.: Finite Element Procedures. Prentice Hall, Englewood Cliffs (1996)

    Google Scholar 

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

    Article  MATH  Google Scholar 

  6. Bauchau, O.A., Theron, N.J.: Energy decaying scheme for non-linear beam models. Comput. Methods Appl. Mech. Eng. 134(1–2), 37–56 (1996)

    Article  MATH  MathSciNet  Google Scholar 

  7. Betsch, P., Menzel, A., Stein, E.: On the parametrization of finite rotations in computational mechanics. A clarification of concepts with application to smooth shells. Comput. Methods Appl. Mech. Eng. 155, 273–305 (1998)

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  9. Betsch, P., Steinmann, P.: Constrained dynamics of geometrically exact beams. Comput. Mech. 31, 49–59 (2003)

    Article  MATH  Google Scholar 

  10. Bonet, J., Wood, R.D.: Nonlinear Continuum Mechanics for Finite Element Analysis. Cambridge University Press, Cambridge (1997)

    MATH  Google Scholar 

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

    Article  Google Scholar 

  12. Cardona, A., Geradin, M.: A beam finite element nonlinear theory with finite rotations. Int. J. Numer. Methods Eng. 26, 2403–2434 (1988)

    Article  MATH  MathSciNet  Google Scholar 

  13. Crisfield, M.A., Galvanetto, G., Jelenić, U.: Dynamics of 3D co-rotational beams. Comput. Mech. 20(6), 507–519 (1997)

    Article  MATH  Google Scholar 

  14. Crisfield, M.A., Jelenić, U.: Objectivity of strain measures in geometrically exact 3D beam theory and its finite element implementation. Proc. R. Soc. Lond. Ser. A 455, 1125–1147 (1999)

    MATH  Google Scholar 

  15. Crisfield, M.A.: Non-linear Finite Element Analysis of Solids and Structures. Advanced Topics, vol. 2. Wiley, New York (1997)

    Google Scholar 

  16. Ericksen, J.L., Truesdell, C.: Exact theory of stress and strain in rods and shells. Arch. Ration. Mech. Anal. 1, 195–323 (1958)

    MathSciNet  Google Scholar 

  17. Gerstmayr, J., Matikainen, M.K.: Analysis of stress and strain in the absolute nodal coordinate formulation. Mech. Des. Struct. Mach. 34(4), 409–430 (2006)

    Article  Google Scholar 

  18. Gerstmayr, J., Shabana, A.A.: Efficient integration of the elastic forces and thin three-dimensional beam elements in the absolute nodal coordinate formulation. In: García Orden, J.C., Goicolea, J.M., Cuadrado, J. (eds.) Multibody Dynamics 2005. ECCOMAS Thematic Conference, Madrid, Spain, 21–24 June 2005

  19. Gerstmayr, J., Shabana, A.A.: Analysis of thin beams and cables using the absolute nodal coordinate formulation. Nonlinear Dyn. 45(1–2), 109–130 (2006)

    Article  MATH  Google Scholar 

  20. Green, A.E., Laws, N.: A general theory of rods. Proc. R. Soc. Lond. Ser. A 293, 145–155 (1966)

    Article  Google Scholar 

  21. Green, A.E., Naghdi, P.M., Wenner, M.L.: On the theory of rods I derivations from the three dimensional equations. Proc. R. Soc. Lond. Ser. A 337, 451–483 (1974)

    MATH  MathSciNet  Google Scholar 

  22. Green, A.E., Naghdi, P.M., Wenner, M.L.: On the theory of rods II developments by direct approach. Proc. R. Soc. Lond. Ser. A 337, 485–507 (1974)

    MATH  MathSciNet  Google Scholar 

  23. Hughes, T.J.R.: The Finite Element Method. Prentice-Hall, Englewood Cliffs (1987)

    MATH  Google Scholar 

  24. Ibrahimbegovic, A.: On finite element implementations of geometrically nonlinear Reissner’s beam theory: three-dimensional curved beam elements. Comput. Methods Appl. Mech. Eng. 122, 11–26 (1995)

    Article  MATH  Google Scholar 

  25. Ibrahimbegovic, A., Al Mikdad, M.: Finite rotations in dynamics of beams and implicit time-stepping algorithms. Int. J. Numer. Methods Eng. 41(5), 781–814 (1998)

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  MATH  Google Scholar 

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

    Article  MATH  Google Scholar 

  28. Klinkel, S., Govindjee, S.: Using finite strain 3D-material models in beam and shell elements. Eng. Comput. 19(8), 902–921 (2002)

    Article  MATH  Google Scholar 

  29. Lens, E.V., Cardona, A.: A nonlinear beam element formulation in the framework of an energy preserving time integration scheme for constrained multibody systems dynamics. Comput. Struct. 86, 47–63 (2008)

    Article  Google Scholar 

  30. Maqueda, L.G., Bauchau, O.A., Shabana, A.A.: Effect of the centrifugal forces on the finite element eigenvalue solution of a rotating blade: a comparative study. Multibody Syst. Dyn. (2007). doi:10.1007/s11044-007-9070-6

    MATH  Google Scholar 

  31. Reissner, E.: A one-dimensional finite strain beam theory: the plane problem. J. Appl. Math. Phys. (ZAMP) 23, 795–804 (1972)

    Article  MATH  Google Scholar 

  32. Romero, I.: The interpolation of rotations and its application to finite element models of geometrically exact rods. Comput. Mech. 34(2), 121–133 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  33. Romero, I., Armero, F.: 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 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  34. Sansour, C., Wagner, W.: Multiplicative updating of the rotation tensor in the finite element analysis of rods and shells—a path independent approach. Comput. Mech. 31, 153–162 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  35. Schwab, A.L., Meijaard, J.P.: Comparison of three-dimensional flexible beam elements for dynamic analysis: finite element method and absolute nodal coordinate formulation. In: IDETC/CIE 2005 ASME International Design Engineering Technical Conferences, Long Beach, CA, Sept. 2005

  36. Shabana, A.A., Mikkola, A.M.: Use of the finite element absolute nodal coordinate formulation in modeling slope discontinuity. Trans. Am. Soc. Mech. Eng. J. Mech. Des. 125, 342–350 (2003)

    Google Scholar 

  37. Shabana, A.A., Yakoub, R.Y.: Three dimensional absolute nodal coordinate formulation for beam elements: theory. ASME J. Mech. Des. 123(4), 606–613 (2001)

    Article  Google Scholar 

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

    Article  MATH  Google Scholar 

  39. Simo, J.C., Tarnow, N., Doblaré, M.: Non-linear dynamics of three-dimensional rods: exact energy and momentum conserving algorithms. Int. J. Numer. Methods Eng. 38(9), 1431–1473 (1995)

    Article  MATH  Google Scholar 

  40. Simo, J.C., Vu-Quoc, L.: On the dynamics of flexible beams under large overall motions-the plane case II. Trans. Am. Soc. Mech. Eng. J. Appl. Mech. 53(4), 855–63 (1986)

    MATH  Google Scholar 

  41. Simo, J.C., Vu-Quoc, L.: A three-dimensional finite-strain rod model Part II: Computational aspects. Comput. Methods Appl. Mech. Eng. 58, 79–116 (1986)

    Article  MATH  Google Scholar 

  42. Simo, J.C., Vu-Quoc, L.: On the dynamics in space of rods undergoing large motions—a geometrically exact approach. Comput. Methods Appl. Mech. Eng. 66(2), 125–161 (1988)

    Article  MATH  MathSciNet  Google Scholar 

  43. Smoleński, W.M.: Statically and kinematically exact nonlinear theory of rods and its numerical verification. Comput. Methods Appl. Mech. Eng. 178, 89–113 (1999)

    Article  MATH  Google Scholar 

  44. Sopanen, J.T., Mikkola, A.M.: Description of elastic forces in absolute nodal coordinate formulation. Nonlinear Dyn. 34, 53–74 (2003)

    Article  MATH  Google Scholar 

  45. Yacoub, R.Y., Shabana, A.A.: Three dimensional absolute nodal coordinate formulation for beam elements: implementation and applications. Am. Soc. Mech. Eng. J. Mech. Eng. 123(4), 614–621 (2001)

    Google Scholar 

  46. Zienkiewicz, O.C., Taylor, R.L.: The Finite Element Method, 4th edn. McGraw-Hill, New York (1991)

    Google Scholar 

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  1. E.T.S. Ingenieros Industriales, José Guitiérrez Abascal 2, 28006, Madrid, Spain

    Ignacio Romero

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  1. Ignacio Romero
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Romero, I. A comparison of finite elements for nonlinear beams: the absolute nodal coordinate and geometrically exact formulations. Multibody Syst Dyn 20, 51–68 (2008). https://doi.org/10.1007/s11044-008-9105-7

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