Abstract
In the present paper, we propose a representation of the discrete motion equations in structural nonlinear dynamics to obtain an improvement in the stability of time numerical integrations. A geometrically nonlinear total Lagrangian formulation for three-dimensional beam elements in the hypotheses of large rotations and small strains is presented. In this formulation, slopes are used instead of rotation parameters to compute the nonlinear representations of the strain measures in the inertial frame of reference. Such representations of the internal strains—rotations compatibility are then imposed in their time derivatives version. The results, related to Newmark approximations for the variations in the displacement and velocity vectors, show a significant increase in the range of stability of the time integration process and a reduction in the number of Newton iterations required in the time integration steps. The numerical tests, furthermore, show that the variation in the total energy in the time steps has bounded oscillations about the zero value.
Article PDF
Similar content being viewed by others
Explore related subjects
Discover the latest articles, news and stories from top researchers in related subjects.Avoid common mistakes on your manuscript.
References
Belytschko, T., Schoeberle, D.F.: On the unconditional stability of an implicit algorithm for nonlinear structural dynamics. J. Appl. Mech. 42, 865–869 (1975)
Hughes, T.J.R., Caughy, T.K., Liu, W.K.: Finite-element methods for nonlinear elastodynamics which conserve energy. J. Appl. Mech. 45, 366–370 (1978)
Simo, J.C., Tarnow, N.: The discrete energy-momentum method. Conserving algorithms for nonlinear elastodynamics. J. Appl. Math. Phys. 43, 757–792 (1992)
Simo, J.C., Tarnow, N.: A new energy and momentum conserving algorithm for the nonlinear dynamics of shells. Int. J. Numer. Methods Eng. 37, 2527–2549 (1994)
Kuhl, D., Ramm, E.: Generalized energy-momentum method for non-linear adaptive shell dynamics. Comput. Methods Appl. Mech. Eng. 178, 343–366 (1999)
Hoff, C., Pahl, P.J.: Development of an implicit method with numerical dissipation for generalized single step algorithm for structural dynamics. Comput. Methods Appl. Mech. Eng. 67, 367–385 (1988)
Chung, J., Hulbert, G.M.: A time integration algorithm for structural dynamics with improved numerical dissipation: the generalized-a method. J. Appl. Mech. 60, 371–375 (1993)
Zienkiewicz, O.C., Wood, W.L., Hine, N.W., Taylor, R.L.: A unified set of single step algorithms. Part I: general formulation and applications. Int. J. Numer. Methods Eng. 20, 1529–1552 (1984)
Lasaint, P., Raviart, P.A.: On a finite element method for solving the neutron transport equation. In: de Boor C. (ed.) Mathematical Aspects of Finite Elements in Partial Differential Equations, pp. 89–123. Academic Press, New York (1974)
Betsch, P., Steinmann, P.: Conservation properties of a time FE method. Part I: time-stepping schemes for N-body problems. Int. J. Numer. Methods Eng. 49, 599–638 (2000)
Lopez, S.: Changing the representation and improving stability in time-stepping analysis of structural non-linear dynamics. Nonlinear Dyn. 46, 337–348 (2006)
Gear, C.W., Petzold, L.R.: ODE methods for the solution of differential/algebraic systems. SIAM J. Numer. Anal. 21, 716–728 (1984)
Bachmann, R., Brüll, L., Mrziglod, T., Pallaske, U.: On methods for reducing the index of differential algebraic equations. Comput. Chem. Eng. 14, 1271–1273 (1990)
Mattsson, S.E., Söderlind, G.: Index reduction in differential-algebraic equations using dummy derivatives. SIAM J. Sci. Comput. 14, 677–692 (1993)
Belytschko, T., Hsieh, B.J.: Non-linear transient finite element analysis with convected co-ordinates. Int. J. Numer. Methods Eng. 7, 255–271 (1973)
Argyris, J.: An excursion into large rotations. Comput. Methods Appl. Mech. Eng. 32, 85–155 (1982)
Rankin, C.C., Nour-Omid, B.: The use of projectors to improve finite element performance. Comput. Struct. 30, 257–267 (1988)
Cardona, A., Geradin, M.: A beam finite element non-linear theory with finite rotations. Int. J. Numer. Methods Eng. 26, 2403–2438 (1988)
Crisfield, M.A.: A consistent co-rotational formulation for nonlinear three-dimensional beam elements. Comput. Methods Appl. Mech. Eng. 81, 131–150 (1990)
Ibrahimbegović, A., Shakourzadeh, H., Batoz, J.L., Al Mikdad, M., Guo, Y.Q.: On the role of geometrically exact and second-order theories in buckling and post-buckling analysis of three-dimensional beam structures. Comput. Struct. 61, 1101–1114 (1996)
Lopez, S., La Sala, G.: A vectorial approach for the formulation of finite element beams in finite rotations. In: Proceedings Tenth International Conference on Computational Structures Technology. Valencia (2010)
Lopez, S.: A three-dimensional beam element undergoing finite rotations based on slopes and distance measures. Internal Report, Dipartimento di Modellistica per l’Ingegneria, Università della Calabria, p. 53 (2010)
Betsch, P., Steinmann, P.: Constrained integration of rigid body dynamics. Comput. Methods Appl. Mech. Eng. 191, 467–488 (2001)
Betsch, P., Steinmann, P.: Constrained dynamics of geometrically exact beams. Comput. Mech. 31, 49–59 (2003)
Goldstein, H.: Classical Mechanics. Addison-Wesley, Reading (1980)
Arnold, V.I.: Mathematical Methods of Classical Mechanics. Prentice-Hall, Englewood Cliffs (1989)
Gonzales, O., Simo, J.C.: On the stability of symplectic and energy-momentum algorithms for non-linear Hamiltonian systems with symmetry. Comput. Methods Appl. Mech. Eng. 134, 197–222 (1996)
Crisfield, M.A., Shi, J.: An energy conserving co-rotational procedure for non-linear dynamics with finite elements. Nonlinear Dyn. 9, 37–52 (1996)
Kuhl, D., Ramm, E.: Constraint energy momentum algorithm and its application to non-linear dynamics of shells. Comput. Methods Appl. Mech. Eng. 136, 293–315 (1996)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Lopez, S. Relaxed representations and improving stability in time-stepping analysis of three-dimensional structural nonlinear dynamics. Nonlinear Dyn 69, 705–720 (2012). https://doi.org/10.1007/s11071-011-0298-6
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11071-011-0298-6