An Asymptotic Numerical Method for non-linear transient dynamics

Authors

  • Bruno Cochelin Laboratoire de Mecanique et d'Acoustique, UPR CNRS 7051 Ecole Superieure de Mecanique de Marseille 1MT-Technopole de Chateau Gombert F- 13451 Marseille cedex 20
  • Christophe Compain Laboratoire de Mecanique et d'Acoustique, UPR CNRS 7051 Ecole Superieure de Mecanique de Marseille 1MT-Technopole de Chateau Gombert F- 13451 Marseille cedex 20

Keywords:

Perturbation methods, Non-linear Algorithm, Structural Dynamics, F.E.M, Series

Abstract

The main objective of this presentation is to show that a perburbation method can be very effective for solving a large class of transient non-linear dynamic problems. We describe an algorithm which has three part : apply a perturbation technique to transform a non-linear problem into a series of linear ones, use an FEM and a time stepping scheme to solve the linear problems, perform the summation of the series to get the solution. Usually, the perturbation series has a finite radius of convergence, and the algorithm has to be restarted several times to get the solution on the whole time interval. However, as compared to a classical conbination of time stepping and Newton-Raphson method, the present algorithm requires much less stiffness matrix evaluations and triangulations. The performances of the proposed algorithm will be demonstrated with an example.

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References

[AMM 96] AMMAR S., Methode asymptotique perturbee appliquee a Ia resolution de problemes

non lineaires en grandes rotations et deplacements, PhD Thesis, Laval University,

QUEBEC, 1996.

[AZR 93] AZRAR L., COCHELIN B., DAMIL N., POTIER-FERRY M., "An asymptotic numerical

method to compute the post-buckling behavior of elastic plates and shells", Int. J.

Numer. Meth. Eng., vol. 36, 1993, p. 1251-1277.

[BRA 97] BRAIKAT B., DAMIL N., POTIER-FERRY M., "Methodes asymptotiques numeriques

pour Ia plasticite", Revue Europeenne des Elements Finis, vol. 6, 1997, p. 337-

[COC 94a] COCHELIN B., "A path following technique via an asymptotic numerical method",

Computers & Structures, vol. 29, 1994, p. 1181-1192.

[COC 94b] COCHELIN B., DAMIL N., POTIER-FERRY M., "The Asymptotic-NumericalMethod:

an efficient perturbation technique for nonlinear structural mechanics", Revue

Europeenne des Elements Finis, vol. 3, 1994, p. 281-297.

[COC 94c] COCHELIN B., DAMIL N., POTIER-FERRY M., "Asymptotic-Numerical Methods

and Pade approximants for nonlinear elastic structures", Int. J. Numer. Meth. Eng., vol. 37,

, p. 1181-1192.

[CRI 97] CRISFIELD M.A., Nonlinear Finite Element Analysis of Solids ans Structures, Vo/.2,

John Wiley and Sons, I st edition, 1997.

[DAM 90] DAMIL N., POTIER-FERRY M., "A new method to compute perturbed bifurcations:

application to the buckling of imperfect elastic structures", International Journal of

Engineering Sciences - N9, vol. 28, 1990, p. 943-957.

[ELH 98] ELHAGE-HUSSEIN A., DAMIL N., POTIER-FERRY M., "An asymptotic numerical

algorithm for frictionless contact problems", Revue Europeenne des Elements Finis, vol. 7,

, p. 119-130.

[FAF 97] FAFARD M., AMMAR S., HENCH! K., GENDRON G., "Application of an asymptotic

method to transient dynamic problems", Journal of Sound and Vibration, vol. 208, 1997,

p. 73-99.

[GAL 75] GALLAGHER R., "Perturbation procedures in nonlinear finite element structural

analysis", Computational Mechanics - Lecture Notes in Mathematics, vol. 461, 1975,

p. 75-89.

[KHU 96] KHUL D., RAMM E., "Constraint Energy Momentum Algorithm and its application

to nonlinear dynamics of shells", Comput. Methods Appl. Mech. Engrg., vol. 136, 1996,

p. 293-315.

(NAJ 98) NAJAH A., COCHELIN B.AND DAMIL N., POTIER-FERRY M., "A critical review

of Asymptotic Numerical Methods", Archives of Computational Methods in Engineering,

vol. 5, 1998, p. 31-50.

[NAY 73) NAYFEH A., Perturbation methods, John Wiley and Sons- New York, 1973.

[NAY 89) NAYFEH A., MOOK T., Nonlinear Oscillations, Whiley Classic Library, 2nd edition,

[NOO 80a] NOOR A., PETERS 1 ., "Reduced basis technique for nonlinear analysis of structures",

AIAA Journal- N4, vol. 18, 1980, p. 455-462.

[NOO SOb] NooR A., PETERS J., "Tracing Post-Limit-Point Paths with reduced Basis Technique",

Comput. Methods Appl. Mech. Engrg., vol. 28, 1980, p. 217-240.

[NOO 85] NOOR A., "Hybrid analytical technique for nonlinear analysis of structures", AIAA

Journal - N6, vol. 23, 1985, p. 938-946.

(POT 97) POTIER-FERRY M., DAMIL N., BRAIKAT B., DESCAMPS 1., CADOU 1., CAO H.,

ELHAGEHUSSEIN A., "Traitement des fortes non-linearite par Ia methode asymptotique

numerique", C. R. A cad. Sci. Paris, vol. t 314, 1997, p. 171-177, Serie II b.

[RIK 84] RIKS E., "Some Computational aspects of stability of nonlinear structures", Comput.

MethodsAppl. Mech. Engrg., vol. 47,1984, p. 219-259.

[SIM 96) SIMO 1., TARNOW N., "The discrete Energy-Momentum method. Conserving algorithms

for nonlinear elastodynamics", Int. J. Numer. Meth. Eng., vol. 136, 1996, p. 293-

[SWE 90) SWEMPLINSKA-STUPNICKA W., The Behavior of Nonlinear Vibrating Systems -

Vol. I and 2, Kluwer Academic Publisher, I st edition, 1990.

[THO 68) THOMPSON 1 ., WALKER A., 'The nonlinear perturbation analysis of discrete structural

systems", Int. J. Solids Structures, vol. 4, 1968, p. 757-758.

[WAL 69) WALKER A., "A nonlinear finite element analysis of shallow circular arches", Int.

J. Solids Structures, vol. 5, 1969, p. 97-107.

(ZAH 99) ZAHROUNI H., COCHELIN B., POTIER-FERRY M., "Asymptotic-numerical methods

for shells with finite rotations", Comput. Methods Appl. Mech. Engrg., vol. 175, 1999,

p. 71-85.

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Published

2000-03-27

How to Cite

Cochelin, B. ., & Compain, C. . (2000). An Asymptotic Numerical Method for non-linear transient dynamics. European Journal of Computational Mechanics, 9(1-3), 113–128. Retrieved from https://journals.riverpublishers.com/index.php/EJCM/article/view/2933

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