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Quadratic finite elements with non-matching grids for theunilateral boundary contact

Published online by Cambridge University Press:  17 June 2013

S. Auliac ,
Z. Belhachmi ,
F. Ben Belgacem  and
F. Hecht
S. Auliac
Affiliation:
LJLL, C.N.R.S. Université Pierre et Marie Curie, B.C. 187, 4 place Jussieu, 75252 Paris Cedex 05, France.. [email protected]
Z. Belhachmi
Affiliation:
LMIA, EA CNRS, Université deq Haute Alsace, Rue des Frères Lumière, 68096 Mulhouse, France.; [email protected]
F. Ben Belgacem
Affiliation:
LMAC, EA 2222, Université de Technologie de Compiègne, BP 20529, 60205 Compiègne Cedex, France.; [email protected] I2M (UMR CNRS 5295), Site ENSCBP, 16 Avenue Pey Berland, 33607 Pessac Cedex, France.
F. Hecht
Affiliation:
LJLL, C.N.R.S. Université Pierre et Marie Curie, B.C. 187, 4 place Jussieu, 75252 Paris Cedex 05, France.; [email protected]
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Abstract

We analyze a numerical model for the Signorini unilateral contact, based on the mortarmethod, in the quadratic finite element context. The mortar frame enables one to usenon-matching grids and brings facilities in the mesh generation of different components ofa complex system. The convergence rates we state here are similar to those alreadyobtained for the Signorini problem when discretized on conforming meshes. The matching forthe unilateral contact driven by mortars preserves then the proper accuracy of thequadratic finite elements. This approach has already been used and proved to be reliablefor the unilateral contact problems even for large deformations. We provide however somenumerical examples to support the theoretical predictions.

Keywords

Unilateral contact conditionsquadratic finite elementsnon-matching gridsmortar matching
Type
Research Article
Information
ESAIM: Mathematical Modelling and Numerical Analysis , Volume 47 , Issue 4: Direct and inverse modeling of the cardiovascular and respiratory systems , July 2013 , pp. 1185 - 1205
Copyright
© EDP Sciences, SMAI, 2013

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References

R.A. Adams, Sobolev Spaces. Academic Press (1975).
A.K. Aziz and I. Babuška, The mathematical foundations of the finite element method with applications to partial differential equations. Academic Press, New York (1972).
Baillet, L. and Sassi, T., Mixed finite element formulation in large deformation frictional contact problem. European J. Comput. Mech. 14 (2005) 287–304. Google Scholar
Belhachmi, Z. and Ben Belgacem, F., Quadratic finite element for Signorini problem. Math. Comput. 72 (2003) 83–104. Google Scholar
Ben Belgacem, F., Hild, P. and Laborde, P., Extension of the mortar finite element method to a variational inequality modeling unilateral contact. Math. Models Methods Appl. Sci. 9 (1999) 287–303. Google Scholar
Ben Belgacem, F., Renard, Y. and Slimane, L., A Mixed Formulation for the Signorini Problem in nearly Incompressible Elasticity. Appl. Numer. Math. 54 (2005) 1–22. Google Scholar
Ben Belgacem, F. and Renard, Y., Hybrid finite element methods for the Signorini problem. Math. Comput. 72 (2003) 1117–1145. Google Scholar
C. Bernardi, Y. Maday and A.T. Patera,A New Nonconforming Approach to Domain Decomposition: The Mortar Element Method, Collège de France seminar, edited by H. Brezis, J.-L. Lions. Pitman (1994) 13–51.
Brenner, S.C. and Scott, L.R., Mathematical Theory of Finite Element Methods. Texts Appl. Math. Springer Verlag, New-York 15 (1994). Google Scholar
J.-F. Bonnans, J. Ch. Gilbert, C. Lemaréchal and C.A. Sagastizábal, Numerical optimization: Theoretical and practical aspects. Universitext (Second revised ed. translation of 1997 French ed.). Springer-Verlag, Berlin (2006).
F. Brezzi and M. Fortin, Mixed and Hybrid Finite Element Methods, Springer Series Comput. Math., vol. 15. Springer Verlag, New York (1991).
Brezzi, F., Hager, W.W. and Raviart, P.A., Error estimates for the finite element solution of variational inequalities. Numer. Math. 28 (1977) 431–443. Google Scholar
L. Cazabeau, Y. Maday and C. Lacour, Numerical quadratures and mortar methods. In Computational Sciences for the 21-st Century, edited by Bristeau et al., Wiley and Sons (1997) 119–128.
Chernov, A., Maischak, M. and Stephan, E.P., hp-mortar boundary element method for two-body contact problems with friction. Math. Methods Appl. Sci. 31 (2008) 2029–2054. Google Scholar
P.-G. Ciarlet, The Finite Element Method for Elliptic Problems. North Holland (1978).
Crouzeix, M. and Thomée, V., The Stability in L p and W 1,p of the L 2-Projection on Finite Element Function Spaces. Mathods Comput. 48 (1987) 521–532. Google Scholar
G. Duvaut and J.-P. Lions, Les inéquations en mécanique et en physique. Dunod, Paris (1972).
Faletta, S., The Approximate Integration in the Mortar Method Constraint. Domain Decomposition Methods in Science and Engineering XVI. Lect. Notes Comput. Sci. Eng. Part III, 55 (2007) 555–563. Google Scholar
Falk, R.S., Error Estimates for the Approximation of a Class of Variational Inequalities. Math. Comput. 28 963–971 (1974). Google Scholar
Fischer, K.A. and Wriggers, P., Frictionless 2D contact formulations for finite deformations based on the mortar method. Comput. Mech. 36 (2005) 226–244. Google Scholar
Flemisch, B., Puso, M.A. and Wohlmuth, B.I., A new dual mortar method for curved interfaces: 2D elasticity. Internat. J. Numer. Methods Eng. 63 (2005) 813–832. Google Scholar
J. Haslinger, I. Hlavcáček and J. Nečas, Numerical Methods for Unilateral Problems in Solid Mechanics, in Handbook of Numerical Analysis, Volume IV, Part 2, edited by P.G. Ciarlet and J.L. Lions. North Holland (1996).
F. Hecht, Freefem++. Third Edition, Version 3.11-1 http://www.freefem.org/ff++http://www.freefem.org/ff++.
P. Hild and Y. Renard, An improved a priori error analysis for finite element approximations of Signorini’s problem. SIAM J. Numer. Analys. to appear (2012).
P. Hild, Problèmes de contact unilatéral et maillages éléments finis incompatibles. Thèse de l’Université Paul Sabatier, Toulouse 3 (1998).
Hild, P., Numerical implementation of two nonconforming finite element methods for unilateral contact. Comput. Methods Appl. Mech. Eng. 184 (2000) 99–123. Google Scholar
Hild, P., Laborde, P.. Quadratic finite element methods for unilateral contact problems. Appl. Numer. Math. 41 (2002) 401–421. Google Scholar
Hua, D., Wang, L.. A mixed finite element method for the unilateral contact problem in elasticity. Sci. China Ser. A 49 (2006) 513–524. Google Scholar
Hüeber, S., Wohlmuth, B.I.. An optimal a priori error estimate for nonlinear multibody contact problems. SIAM J. Numer. Anal. 43 (2005) 156–173. Google Scholar
Hüeber, S., Mair, M. and Wohlmuth, B.I., A priori error estimates and an inexact primal-dual active set strategy for linear and quadratic finite elements applied to multibody contact problems. Appl. Numer. Math. 54 (2005) 555–576. Google Scholar
N. Kikuchi and J.T. Oden, Contact Problems in Elasticity: A Study of Variational Inequalities and Finite Element Methods. SIAM (1988).
Kim, T.Y., Dolbow, J.E. and Laursen, T.A., A Mortared Finite Element Method for Frictional Contact on Arbitrary Surfaces. Comput. Mech. 39 (2007) 223–235. Google Scholar
Laursen, T.A., Puso, M.A. and Sandersc, J., Mortar contact formulations for deformable-deformable contact: Past contributions and new extensions for enriched and embedded interface formulations. Comput. Methods Appl. Mech. Eng. 205-208 (2012) 3–15. Google Scholar
Laursen, T.A. and Yang, B., New Developments in Surface-to-Surface Discretization Strategies for Analysis of Interface Mechanics. Computational Plasticity. Comput. Methods Appl. Sci. 7 (2010) 67–86. Google Scholar
Li, M.-X., Lin, Q. and Zhang, S.-H., Superconvergence of finite element method for the Signorini problem. J. Comput. Appl. Math. 222 (2008) 284–292. Google Scholar
Moussaoui, M. and Khodja, K., Régularité des solutions d’un problème mêlé Dirichlet–Signorini dans un domaine polygonal plan. Commun. Part. Differ. Equ. 17 (1992) 805–826. Google Scholar
Puso, M.A., Laursen, T.A. and Solberg, J., A segment-to-segment mortar contact method for quadratic elements and large deformations. Comput. Meth. Appl. Mech. and Eng. 197 (2008) 555–566. Google Scholar
Puso, M.A. and Laursen, T.A., A Mortar Segment-to-Segment Contact Method for Large Deformation Solid Mechanics. Comput. Methods Appl. Mech. Eng. 193 (2004) 601–629. Google Scholar
Seshaiyer, P. and Suri, M., Uniform h − p Convergence Results for the Mortar Finite Element Method. Math. Comput. 69 521–546 (2000). Google Scholar
L. Slimane, Méthodes mixtes et traitement du verrouillage numérique pour la résolution des inéquations variationnelles. Thèse l’Institut National des Sciences Appliquées de Toulouse (2001).
Wohlmuth, B.I., A Mortar Finite Element Method Using Dual Spaces for the Lagrange Multiplier. SIAM J. Numer. Anal. 38 (2001) 989–1012,. Google Scholar
Wohlmuth, B., Krause, R.. Monotone multigrid methods on nonmatching grids for nonlinear multibody contact problems. SIAM J. Sci. Comput. 25 (2003) 324–347. Google Scholar
Yang, B., Laursen, T.A., Meng, X.. Two dimensional mortar contact methods for large deformation frictional sliding. Internat. J. Numer. Methods Eng. 62 (2005) 1183–1225. Google Scholar
Z.-H. Zhong, Finite Element Procedures for Contact-Impact Problems. Oxford University Press (1993).