Our purpose is to describe and compare some families of fully discretized approximations and their properties, in the case of vibro-impact of plates between rigid obstacles with non-penetration Signorini’s conditions. To this aim, the dynamical Kirchhoff–Love plate model is considered and an extension to plates of the singular dynamic method, introduced by Renard and previously adapted to beams by Pozzolini and Salaün, is described. A particular emphasis is given in the use of an adapted Newmark scheme in which intervene a discrete restitution coefficient. Finally, various numerical results are presented and energy conservation capabilities of several numerical schemes are investigated and discussed.
Accepté le :
DOI : 10.1051/m2an/2015094
Mots clés : Variational inequalities, finite element method, elastic plates, dynamics, unilateral constraints
@article{M2AN_2016__50_6_1585_0,
author = {Pozzolini, C\'edric and Renard, Yves and Sala\"un, Michel},
title = {Energy conservative finite element semi-discretization for vibro-impacts of plates on rigid obstacles},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis },
pages = {1585--1613},
publisher = {EDP-Sciences},
volume = {50},
number = {6},
year = {2016},
doi = {10.1051/m2an/2015094},
mrnumber = {3580115},
zbl = {1388.74107},
language = {en},
url = {http://archive.numdam.org/articles/10.1051/m2an/2015094/}
}
TY - JOUR AU - Pozzolini, Cédric AU - Renard, Yves AU - Salaün, Michel TI - Energy conservative finite element semi-discretization for vibro-impacts of plates on rigid obstacles JO - ESAIM: Mathematical Modelling and Numerical Analysis PY - 2016 SP - 1585 EP - 1613 VL - 50 IS - 6 PB - EDP-Sciences UR - http://archive.numdam.org/articles/10.1051/m2an/2015094/ DO - 10.1051/m2an/2015094 LA - en ID - M2AN_2016__50_6_1585_0 ER -
%0 Journal Article %A Pozzolini, Cédric %A Renard, Yves %A Salaün, Michel %T Energy conservative finite element semi-discretization for vibro-impacts of plates on rigid obstacles %J ESAIM: Mathematical Modelling and Numerical Analysis %D 2016 %P 1585-1613 %V 50 %N 6 %I EDP-Sciences %U http://archive.numdam.org/articles/10.1051/m2an/2015094/ %R 10.1051/m2an/2015094 %G en %F M2AN_2016__50_6_1585_0
Pozzolini, Cédric; Renard, Yves; Salaün, Michel. Energy conservative finite element semi-discretization for vibro-impacts of plates on rigid obstacles. ESAIM: Mathematical Modelling and Numerical Analysis , Tome 50 (2016) no. 6, pp. 1585-1613. doi : 10.1051/m2an/2015094. http://archive.numdam.org/articles/10.1051/m2an/2015094/
R.A. Adams, Sobolev spaces. Academic Press (1975). | MR | Zbl
and , An Euler-Bernoulli beam with dynamic contact: Discretization, convergence and numerical results. SIAM J. Numer. Anal. 43 (2005) 1455–1480. | DOI | MR | Zbl
and , A mixed formulation for frictional contact problems prone to Newton like solution methods. Comput. Methods Appl. Mech. Eng. 92 (1991) 353–375. | DOI | MR | Zbl
B. Brogliato, Nonsmooth Mechanics, edited by E.D. Sontag, M. Thoma. Springer, London (1999). | Zbl
, Lagrange constraints for transient finite element surface contact. Int. J. Numer. Methods Eng. 32 (1991) 103–128. | DOI | Zbl
P.G. Ciarlet, The finite element method for elliptic problems. North-Holland (1978). | MR | Zbl
P.G. Ciarlet, Basic error estimates for elliptic problems, in Vol. II of Handbook of Numerical Analysis. North-Holland (1991) 17–351. | MR | Zbl
, , and , Convergence of mass redistribution method for the wave equation with a unilateral constraint at the boundary ESAIM: M2AN 48 (2014) 1147–1169. | DOI | Numdam | MR | Zbl
, and , A contact-stabilized Newmark method for dynamical contact problems. Int. J. Numer. Methods Eng. 73 (2007) 1274–1290. | DOI | MR | Zbl
and , Vibrations of a beam between obstacles: Convergence of a fully discretized approximation. ESAIM: M2AN 40 (2006) 705–734. | DOI | Numdam | MR | Zbl
and , Numerical simulation of a model of vibrations with joint clearance. Int. J. Comput. Appl. Technol. 33 (2008) 41–53. | DOI
D. Doyen, Méthodes numériques pour des problèmes dynamiques de contact et de fissuration. Thèse de l’Université Paris-Est (2010).
Y. Renard and J. Pommier, An open source generic C++ library for finite element methods. Available at http://home.gna.org/getfem/ (2016).
, and , A stable energy conserving approach for frictional contact problems based on quadrature formulas. Int. J. Numer. Methods Eng. 73 (2008) 205–225. | DOI | MR | Zbl
, and , Energy controlling time integration methods for nonlinear elastodynamics and low-velocity impact. Comput. Methods Appl. Mech. Eng. 195 (2006) 4890–4916. | DOI | MR | Zbl
, Mixed Interpretation and Extensions of the Equivalent Mass Matrix Approach for Elastodynamics with Contact. Comput. Methods Appl. Mech. Eng. 199 (2010) 2941–2957. | DOI | MR | Zbl
R.A. Ibrahim, V.I. Babitsky and M. Okuma, Vibro-Impact Dynamics of Ocean Systems and Related Problems. Vol. 44 of Lect. Notes Appl. Comput. Mech. Springer (2009).
, and , Mass redistribution method for finite element contact problems in elastodynamics. Eur. J. Mech., A/Solids 27 (2008) 918–932. | DOI | MR | Zbl
and , Vibrations of a beam between two stops, Dynamics of Continuous, Discrete and Impulsive Systems. Ser. B, Appl. Algorithms 8 (2001) 93–110. | MR | Zbl
and , Design of energy conserving algorithms for frictionless dynamic contact problems. Int. J. Numer. Methods Eng. 40 (1997) 863–886. | DOI | MR | Zbl
and , Improved implicit integrators for transient impact problems-geometric admissibility within the conserving framework. Int. J. Numer. Methods Eng. 53 (2002) 245–274. | DOI | MR | Zbl
, Time discretization of vibro-impact. Philos. Trans. R. Soc. Lond., A 359 (2001) 2405–2428. | DOI | MR | Zbl
and , A numerical scheme for impact problems. I. The one-dimensional case. SIAM J. Numer. Anal. 40 (2002) 702–733. | DOI | MR | Zbl
and , Numerical simulation of the dynamics of an impacting bar. Comput. Methods Appl. Mech. Eng. 196 (2007) 2839–2851. | DOI | MR | Zbl
, and , Viscoélastodynamique monodimensionnelle avec conditions de Signorini. C. R. Acad. Sci. Paris, I 334 (2002) 983–988. | DOI | MR | Zbl
A. Petrov and M. Schatzman, A pseudodifferential linear complementarity problem related to a one dimensional viscoelastic model with Signorini condition. Archive for Rational Mechanics and Analysis. Springer (2009).
and , Some energy conservative schemes for vibro-impacts of a beam on rigid obstacles. ESAIM: M2AN 45 (2011) 1163–1192. | DOI | Numdam | MR | Zbl
, and , Vibro-Impact of a plate on rigid obstacles: existence theorem, convergence of a scheme and numerical simulations. IMA J. Numer. Anal. 33 (2013) 261–294. | DOI | MR | Zbl
, The singular dynamic method for constrained second order hyperbolic equations. Application to dynamic contact problems. J. Comput. Appl. Math. 234 (2010) 906–923. | DOI | MR | Zbl
and , On a finite element method for dynamic contact-impact problems. Int. J. Numer. Methods Eng. 36 (1993) 2123–2140. | DOI | Zbl
Y. Mochida, Bounded Eigenvalues of Fully Clamped and Completely Free Rectangular Plates. Master thesis of the University of Waikato Hamilton, New Zealand (2007).
Cité par Sources :
