A Nitsche finite element method for dynamic contact: 1. Space semi-discretization and time-marching schemes

Chouly, Franz; Hild, Patrick; Renard, Yves

ESAIM: Mathematical Modelling and Numerical Analysis , Tome 49 (2015) no. 2, pp. 481-502.

This paper presents a new approximation of elastodynamic frictionless contact problems based both on the finite element method and on an adaptation of Nitsche’s method which was initially designed for Dirichlet’s condition. A main interesting characteristic is that this approximation produces well-posed space semi-discretizations contrary to standard finite element discretizations. This paper is then mainly devoted to present an analysis of the space semi-discretization in terms of consistency, well-posedness and energy conservation, and also to study the well-posedness of some time-marching schemes ($\theta$-scheme, Newmark and a new hybrid scheme). The stability properties of the schemes and the corresponding numerical experiments can be found in a second paper [F. Chouly, P. Hild and Y. Renard, A Nitsche finite element method for dynamic contact. 2. Stability analysis and numerical experiments. ESAIM: M2AN 49 (2015) 503–528.].

Classification : 65N12, 65N30, 74M15
Mots clés : Unilateral contact, elastodynamics, finite elements, Nitsche’s method, time-marching schemes, stability

R.A. Adams, Sobolev spaces. Pure Appl. Math., vol. 65. Academic Press, New York, London (1975). | Zbl

J. Ahn and D.E. Stewart, Existence of solutions for a class of impact problems without viscosity. SIAM J. Math. Anal. 38 (2006) 37–63. | DOI | Zbl

P. Alart and A. Curnier, A generalized newton method for contact problems with friction. J. Mech. Theor. Appl. 7 (1988) 67–82. | Zbl

F. Armero and E. Petőcz, Formulation and analysis of conserving algorithms for frictionless dynamic contact/impact problems. Comput. Methods Appl. Mech. Eng. 158 (1998) 269–300. | DOI | Zbl

M. Astorino, F. Chouly, and M.A. Fernández, An added-mass free semi-implicit coupling scheme for fluid-structure interaction. C. R. Math. Acad. Sci. Paris 347 (2009) 99–104. | DOI | Zbl

J.-P. Aubin and A. Cellina, Differential inclusions, vol. 264 of Grundlehren der Mathematischen Wissenschaften. Springer-Verlag, Berlin (1984). | Zbl

R. Becker, P. Hansbo, and R. Stenberg, A finite element method for domain decomposition with non-matching grids. ESAIM: M2AN 37 (2003) 209–225. | DOI | Numdam | Zbl

S.-C. Brenner and L.-R. Scott, The Mathematical Theory of Finite Element Methods, vol. 15 of Texts Appl. Math. Springer-Verlag, New York, 2007. | Zbl

H. Brezis, Équations et inéquations non linéaires dans les espaces vectoriels en dualité. Ann. Inst. Fourier 18 (1968) 115–175. | DOI | Numdam | Zbl

E. Burman and M.A. Fernández, Stabilization of explicit coupling in fluid-structure interaction involving fluid incompressibility. Comput. Methods Appl. Mech. Eng. 198 (2009) 766–784. | DOI | Zbl

G. Choudury and I. Lasiecka, Optimal convergence rates for semidiscrete approximations of parabolic problems with nonsmooth boundary data. Numer. Funct. Anal. Optim. 12 (1991) 469–485 (1992). | DOI | Zbl

F. Chouly, An adaptation of Nitsche’s method to the Tresca friction problem. J. Math. Anal. Appl. 411 (2014) 329–339. | DOI | Zbl

F. Chouly and P. Hild, A Nitsche-based method for unilateral contact problems: numerical analysis. SIAM J. Numer. Anal. 51 (2013) 1295–1307. | DOI | Zbl

F. Chouly, P. Hild, and Y. Renard, Symmetric and non-symmetric variants of Nitsche’s method for contact problems in elasticity: theory and numerical experiments. Math. Comp. (2014). DOI:. | DOI

F. Chouly, P. Hild and Y. Renard, A Nitsche finite element method for dynamic contact. 2. Stability of the schemes and numerical experiments. ESAIM: M2AN 49 (2015) 503–528. | DOI | Numdam | Zbl

P.G. Ciarlet, Handbook of Numerical Analysis. The finite element method for elliptic problems. Edited by P.G. Ciarlet and J.L. Lions. In vol II, chap. 1. North Holland (1991) 17–352. | Zbl

F. Dabaghi, A. Petrov, J. Pousin, and Y. Renard, Convergence of mass redistribution method for the one-dimensional wave equation with a unilateral constraint at the boundary. ESAIM: M2AN 48 (2014) 1147–1169. | DOI | Numdam | Zbl

C. D’Angelo and P. Zunino, Numerical approximation with Nitsche’s coupling of transient Stokes’/Darcy’s flow problems applied to hemodynamics. Appl. Numer. Math. 62 (2012) 378–395. | DOI | Zbl

R. Dautray and J.-L. Lions, Analyse mathématique et calcul numérique pour les sciences et les techniques. Évolution: semi-groupe, variationnel. Vol. 8. Masson, Paris (1988). | Zbl

K. Deimling, Multivalued differential equations. In vol. 1 of de Gruyter Series Nonlin. Anal. Appl. Walter de Gruyter & Co., Berlin (1992). | Zbl

C. Eck, J. Jarušek, and M. Krbec, Unilateral contact problems. In vol. 270 of Pure Appl. Math. Chapman & Hall/CRC, Boca Raton, FL (2005). | Zbl

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

R. Glowinski and P. Le Tallec, Augmented Lagrangian and operator-splitting methods in nonlinear mechanics. In vol. 9 of SIAM Studies Appl. Math. Society for Industrial and Applied Mathematics, Philadelphia, PA (1989). | Zbl

O. Gonzalez, Exact energy and momentum conserving algorithms for general models in nonlinear elasticity. Comput. Methods Appl. Mech. Eng. 190 (2000) 1763–1783. | DOI | Zbl

W. Han and M. Sofonea, Quasistatic contact problems in viscoelasticity and viscoplasticity. In vol. 30 of AMS/IP Stud. Adv. Math. American Mathematical Society, Providence, RI (2002). | Zbl

A. Hansbo and P. Hansbo, A finite element method for the simulation of strong and weak discontinuities in solid mechanics. Comput. Methods Appl. Mech. Eng. 193 (2004) 3523–3540. | DOI | Zbl

P. Hansbo, Nitsche’s method for interface problems in computational mechanics. GAMM-Mitt. 28 (2005) 183–206. | DOI | Zbl

P. Hansbo, J. Hermansson, and T. Svedberg, Nitsche’s method combined with space-time finite elements for ALE fluid-structure interaction problems. Comput. Methods Appl. Mech. Eng. 193 (2004) 4195–4206. | DOI | Zbl

J. Haslinger, I. Hlavcáˇek, and J. Nečas, Handbook of Numerical Analysis. Numerical methods for unilateral problems in solid mechanics. Edited by P.G. Ciarlet and J.L. Lions. In Vol. IV, chap. 2. North Holland (1996) 313–385. | Zbl

P. Hauret and P. Le Tallec, Energy-controlling time integration methods for nonlinear elastodynamics and low-velocity impact. Comput. Methods Appl. Mech. Engrg. 195 (2006) 4890–4916. | DOI | Zbl

B. Heinrich and B. Jung, Nitsche mortaring for parabolic initial-boundary value problems. Electron. Trans. Numer. Anal. 32 (2008) 190–209. | Zbl

P. Heintz and P. Hansbo, Stabilized Lagrange multiplier methods for bilateral elastic contact with friction. Comput. Methods Appl. Mech. Eng. 195 (2006) 4323–4333. | DOI | Zbl

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).

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. | DOI | Zbl

N. Kikuchi and J.T. Oden, Contact problems in elasticity: a study of variational inequalities and finite element methods. In vol. 8 of SIAM Stud. Appl. Math. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA (1988). | Zbl

J.U. Kim, A boundary thin obstacle problem for a wave equation. Comm. Partial Differ. Equ. 14 (1989) 1011–1026. | DOI | Zbl

T.A. Laursen, Computational contact and impact mechanics. Springer-Verlag, Berlin (2002). | Zbl

T.A. Laursen and V. Chawla, Design of energy conserving algorithms for frictionless dynamic contact problems. Int. J. Numer. Methods Eng. 40 (1997) 863–886. | DOI | Zbl

G. Lebeau and M. Schatzman, A wave problem in a half-space with a unilateral constraint at the boundary. J. Differ. Equ. 53 (1984) 309–361. | DOI | Zbl

J. Nitsche, Über ein Variationsprinzip zur Lösung von Dirichlet-Problemen bei Verwendung von Teilräumen, die keinen Randbedingungen unterworfen sind. Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg 36 (1971) 9–15. | DOI | Zbl

C. Pozzolini, Y. Renard, and M. Salaün, 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 | Zbl

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. | DOI | MR | Zbl

Y. Renard, Generalized Newton’s methods for the approximation and resolution of frictional contact problems in elasticity. Comput. Meth. Appl. Mech. Engrg. 256 (2013) 38–55. | DOI | MR | Zbl

R. Stenberg, On some techniques for approximating boundary conditions in the finite element method. In proc. of International Symposium on Mathematical Modelling and Computational Methods Modelling 94 (Prague, 1994). J. Comput. Appl. Math. 63 (1995) 139–148. | DOI | Zbl

V. Thomée, Galerkin finite element methods for parabolic problems. In vol. 25 of Springer Ser. Comput. Math. Springer-Verlag, Berlin (1997). | Zbl

B. Wohlmuth, Variationally consistent discretization schemes and numerical algorithms for contact problems. Acta Numer. (2011) 569–734. | Zbl

P. Wriggers, Computational Contact Mechanics. Wiley (2002).

P. Wriggers and G. Zavarise, A formulation for frictionless contact problems using a weak form introduced by Nitsche. Comput. Mech. 41 (2008) 407–420. | DOI | Zbl