In a previous paper [F. Chouly, P. Hild and Y. Renard, A Nitsche finite element method for dynamic contact. 1. Space semi-discretization and time-marching schemes. ESAIM: M2AN 49 (2015) 481–502.], we adapted Nitsche’s method to the approximation of the linear elastodynamic unilateral contact problem. The space semi-discrete problem was analyzed and some schemes (-scheme, Newmark and a new hybrid scheme) were proposed and proved to be well-posed under appropriate CFL conditions. In the present paper we look at the stability properties of the above-mentioned schemes and we proceed to the corresponding numerical experiments. In particular we prove and illustrate numerically some interesting stability and (almost) energy conservation properties of Nitsche’s semi-discretization combined to the new hybrid scheme.
DOI : 10.1051/m2an/2014046
Keywords: Unilateral contact, elastodynamics, Nitsche’s method, time-marching schemes, stability
Chouly, Franz 1 ; Hild, Patrick 2 ; Renard, Yves 3
@article{M2AN_2015__49_2_503_0,
author = {Chouly, Franz and Hild, Patrick and Renard, Yves},
title = {A {Nitsche} finite element method for dynamic contact: 2. {Stability} of the schemes and numerical experiments },
journal = {ESAIM: Mathematical Modelling and Numerical Analysis },
pages = {503--528},
year = {2015},
publisher = {EDP Sciences},
volume = {49},
number = {2},
doi = {10.1051/m2an/2014046},
zbl = {1311.74114},
language = {en},
url = {https://www.numdam.org/articles/10.1051/m2an/2014046/}
}
TY - JOUR AU - Chouly, Franz AU - Hild, Patrick AU - Renard, Yves TI - A Nitsche finite element method for dynamic contact: 2. Stability of the schemes and numerical experiments JO - ESAIM: Mathematical Modelling and Numerical Analysis PY - 2015 SP - 503 EP - 528 VL - 49 IS - 2 PB - EDP Sciences UR - https://www.numdam.org/articles/10.1051/m2an/2014046/ DO - 10.1051/m2an/2014046 LA - en ID - M2AN_2015__49_2_503_0 ER -
%0 Journal Article %A Chouly, Franz %A Hild, Patrick %A Renard, Yves %T A Nitsche finite element method for dynamic contact: 2. Stability of the schemes and numerical experiments %J ESAIM: Mathematical Modelling and Numerical Analysis %D 2015 %P 503-528 %V 49 %N 2 %I EDP Sciences %U https://www.numdam.org/articles/10.1051/m2an/2014046/ %R 10.1051/m2an/2014046 %G en %F M2AN_2015__49_2_503_0
Chouly, Franz; Hild, Patrick; Renard, Yves. A Nitsche finite element method for dynamic contact: 2. Stability of the schemes and numerical experiments. ESAIM: Mathematical Modelling and Numerical Analysis , Tome 49 (2015) no. 2, pp. 503-528. doi: 10.1051/m2an/2014046
R.A. Adams, Sobolev spaces. Vol. 65 of Pure Appl. Math. Academic Press, New York, London (1975). | Zbl
and , Formulation and analysis of conserving algorithms for frictionless dynamic contact/impact problems. Comput. Methods Appl. Mech. Engrg. 158 (1998) 269–300. | Zbl | DOI
, and , A frictionless viscoelastodynamic contact problem with energy consistent properties: numerical analysis and computational aspects. Comput. Methods Appl. Mech. Engrg. 198 (2009) 669–679. | Zbl | DOI
, and , Lagrange constraints for transient finite element surface contact. Int. J. Numer. Methods Engrg. 32 (1991) 103–128. | Zbl | DOI
, and , A Nitsche finite element method for dynamic contact. 1. Space semi-discretization and time-marching schemes. ESAIM: M2AN 49 (2015) 481–502. | Zbl | Numdam | DOI
P.G. Ciarlet, Handbook of Numerical Analysis, The finite element method for elliptic problems. Edited by P.G. Ciarlet and J.L. Lions, vol. II, Chap. 1. North Holland (1991) 17–352. | Zbl
F. Dabaghi, A. Petrov, J. Pousin and Y. Renard, Numerical approximation of a one dimensional elastodynamic contact problem based on mass redistribution method. Submitted (2013). Available at . | HAL
, , and , Convergence of mass redistribution method for the one-dimensional wave equation with a unilateral constraint at the boundary. ESAIM: M2AN 48 (2014) 1147–1169. | Zbl | Numdam | DOI
, and , A contact-stabilized Newmark method for dynamical contact problems. Int. J. Numer. Methods Engrg. 73 (2008) 1274–1290. | Zbl | DOI
, and , Time-integration schemes for the finite element dynamic Signorini problem. SIAM J. Sci. Comput. 33 (2011) 223–249. | Zbl | DOI
and , Vibrations of a beam between obstacles. Convergence of a fully discretized approximation. ESAIM: M2AN 40 (2006) 705–734. | Zbl | Numdam | DOI
A. Ern and J.-L. Guermond, Theory and practice of finite elements. In vol. 159 of Appl. Math. Sci. Springer-Verlag, New York (2004). | Zbl
, Exact energy and momentum conserving algorithms for general models in nonlinear elasticity. Comput. Methods Appl. Mech. Engrg. 190 (2000) 1763–1783. | Zbl | DOI
, and , A stable energy-conserving approach for frictional contact problems based on quadrature formulas. Int. J. Numer. Methods Engrg. 73 (2008) 205–225. | Zbl | DOI
, Mixed interpretation and extensions of the equivalent mass matrix approach for elastodynamics with contact. Comput. Methods Appl. Mech. Engrg. 199 (2010) 2941–2957. | Zbl | DOI
and , Energy-controlling time integration methods for nonlinear elastodynamics and low-velocity impact. Comput. Methods Appl. Mech. Engrg. 195 (2006) 4890–4916. | MR | Zbl | DOI
, , and , Finite element analysis of nonsmooth contact. Comput. Methods Appl. Mech. Engrg. 180 (1999) 1–26. | MR | Zbl | DOI
H.B. Khenous, Problèmes de contact unilatéral avec frottement de Coulomb en élastostatique et élastodynamique. Etude mathématique et résolution numérique. Ph.D. thesis, INSA de Toulouse (2005).
, and , Mass redistribution method for finite element contact problems in elastodynamics. Eur. J. Mech. A Solids 27 (2008) 918–932. | MR | Zbl | DOI
and , Presentation and comparison of selected algorithms for dynamic contact based on the Newmark scheme. Appl. Numer. Math. 62 (2012) 1393–1410. | MR | Zbl | DOI
and , Design of energy conserving algorithms for frictionless dynamic contact problems. Int. J. Numer. Methods Engrg. 40 (1997) 863–886. | MR | Zbl | DOI
and , Improved implicit integrators for transient impact problems – geometric admissibility within the conserving framework. Int. J. Numer. Methods Engrg. 53 (2002) 245–274. | MR | Zbl | DOI
, Time discretization of vibro-impact. R. Soc. Lond. Philos. Trans. Ser. A Math. Phys. Eng. Sci. 359 (2001) 2405–2428. | MR | Zbl | DOI
and , A numerical scheme for impact problems. I. The one-dimensional case. SIAM J. Numer. Anal. 40 (2002) 702–733. | MR | Zbl | DOI
and , A numerical scheme for impact problems. II. The multidimensional case. SIAM J. Numer. Anal. 40 (2002) 734–768. | MR | Zbl | DOI
, The singular dynamic method for constrained second order hyperbolic equations: application to dynamic contact problems. J. Comput. Appl. Math. 234 (2010) 906–923. | MR | Zbl | DOI
, Generalized Newton’s methods for the approximation and resolution of frictional contact problems in elasticity. Comput. Meth. Appl. Mech. Engrg. 256 (2013) 38–55. | MR | Zbl | DOI
B. Wohlmuth, Variationally consistent discretization schemes and numerical algorithms for contact problems. Acta Numerica (2011) 569–734. | MR | Zbl
P. Wriggers, Computational Contact Mechanics. Wiley (2002).
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