Some energy conservative schemes for vibro-impacts of a beam on rigid obstacles
ESAIM: Mathematical Modelling and Numerical Analysis , Tome 45 (2011) no. 6, pp. 1163-1192.

Caused by the problem of unilateral contact during vibrations of satellite solar arrays, the aim of this paper is to better understand such a phenomenon. Therefore, it is studied here a simplified model composed by a beam moving between rigid obstacles. Our purpose is to describe and compare some families of fully discretized approximations and their properties, in the case of non-penetration Signorini's conditions. For this, starting from the works of Dumont and Paoli, we adapt to our beam model the singular dynamic method introduced by Renard. A particular emphasis is given in the use of a restitution coefficient in the impact law. Finally, various numerical results are presented and energy conservation capabilities of the schemes are investigated.

DOI : 10.1051/m2an/2011008
Classification : 35L85, 65M12, 74H15, 74H45
Mots-clés : variational inequalities, finite element method, elastic beam, dynamics, unilateral constraints, restitution coefficient 
@article{M2AN_2011__45_6_1163_0,
     author = {Pozzolini, C. and Salaun, M.},
     title = {Some energy conservative schemes for vibro-impacts of a beam on rigid obstacles},
     journal = {ESAIM: Mathematical Modelling and Numerical Analysis },
     pages = {1163--1192},
     publisher = {EDP-Sciences},
     volume = {45},
     number = {6},
     year = {2011},
     doi = {10.1051/m2an/2011008},
     mrnumber = {2833177},
     language = {en},
     url = {http://archive.numdam.org/articles/10.1051/m2an/2011008/}
}
TY  - JOUR
AU  - Pozzolini, C.
AU  - Salaun, M.
TI  - Some energy conservative schemes for vibro-impacts of a beam on rigid obstacles
JO  - ESAIM: Mathematical Modelling and Numerical Analysis 
PY  - 2011
SP  - 1163
EP  - 1192
VL  - 45
IS  - 6
PB  - EDP-Sciences
UR  - http://archive.numdam.org/articles/10.1051/m2an/2011008/
DO  - 10.1051/m2an/2011008
LA  - en
ID  - M2AN_2011__45_6_1163_0
ER  - 
%0 Journal Article
%A Pozzolini, C.
%A Salaun, M.
%T Some energy conservative schemes for vibro-impacts of a beam on rigid obstacles
%J ESAIM: Mathematical Modelling and Numerical Analysis 
%D 2011
%P 1163-1192
%V 45
%N 6
%I EDP-Sciences
%U http://archive.numdam.org/articles/10.1051/m2an/2011008/
%R 10.1051/m2an/2011008
%G en
%F M2AN_2011__45_6_1163_0
Pozzolini, C.; Salaun, M. Some energy conservative schemes for vibro-impacts of a beam on rigid obstacles. ESAIM: Mathematical Modelling and Numerical Analysis , Tome 45 (2011) no. 6, pp. 1163-1192. doi : 10.1051/m2an/2011008. http://archive.numdam.org/articles/10.1051/m2an/2011008/

[1] J. Ahn and D.E. Stewart, An Euler-Bernoulli beam with dynamic contact: discretization, convergence and numerical results. SIAM J. Numer. Anal. 43 (2005) 1455-1480. | MR | Zbl

[2] N.J. Carpenter, Lagrange constraints for transient finite element surface contact. Internat. J. Numer. Methods Engrg. 32 (1991) 103-128. | Zbl

[3] P. Deuflhard, R. Krause and S. Ertel, A contact-stabilized Newmark method for dynamical contact problems. Internat. J. Numer. Methods Engrg. 73 (2007) 1274-1290. | MR | Zbl

[4] Y. Dumont and L. Paoli, Vibrations of a beam between obstacles: convergence of a fully discretized approximation. ESAIM: M2AN 40 (2006) 705-734. | Numdam | MR | Zbl

[5] Y. Dumont and L. Paoli, Numerical simulation of a model of vibrations with joint clearance. Int. J. Comput. Appl. Technol. 33 (2008) 41-53.

[6] P. Hauret and P. Le Tallec, Energy controlling time integration methods for nonlinear elastodynamics and low-velocity impact. Comput. Methods Appl. Mech. Eng. 195 (2006) 4890-4916. | MR | Zbl

[7] H.B. Khenous, P. Laborde and Y. Renard, Mass redistribution method for finite element contact problems in elastodynamics. Eur. J. Mech. A. Solids 27 (2008) 918-932. | MR | Zbl

[8] K. Kuttler and M. Shillor, Vibrations of a beam between two stops, Dynamics of Continuous, Discrete and Impulsive Systems, Series B. Applications and Algorithms 8 (2001) 93-110. | MR | Zbl

[9] T.A. Laursen and V. Chawla, Design of energy conserving algorithms for frictionless dynamic contact problems. Internat. J. Numer. Methods Engrg. 40 (1997) 863-886. | MR | Zbl

[10] T.A. Laursen and G.R. Love, Improved implicit integrators for transient impact problems-geometric admissibility within the conserving framework. Internat. J. Numer. Methods Engrg. 53 (2002) 245-274. | MR | Zbl

[11] L. Paoli, Time discretization of vibro-impact. Philos. Trans. Roy. Soc. London A 359 (2001) 2405-2428. | MR | Zbl

[12] L. Paoli and M. Schatzman, A numerical scheme for impact problems. I. The one-dimensional case. SIAM J. Numer. Anal. 40 (2002) 702-733. | MR | Zbl

[13] L. Paoli and M. Schatzman, Numerical simulation of the dynamics of an impacting bar. Comput. Methods Appl. Mech. Eng. 196 (2007) 2839-2851. | MR | Zbl

[14] A. Petrov and M. Schatzman, Viscolastodynamique monodimensionnelle avec conditions de Signorini. C. R. Acad. Sci. Paris, I 334 (2002) 983-988. | MR | Zbl

[15] A. Petrov and M. Schatzman, A pseudodifferential linear complementarity problem related to a one dimensional viscoelastic model with Signorini condition. Arch. Rational Mech. Anal., to appear.

[16] Y. Renard, The singular dynamic method for constrained second order hyperbolic equations. Application to dynamic contact problems. J. Comput. Appl. Math. 234 (2010) 906-923. | MR

[17] R.L. Taylor and P. Papadopoulos, On a finite element method for dynamic contact-impact problems. Internat. J. Numer. Methods Engrg. 36 (1993) 2123-2140. | Zbl

Cité par Sources :