Quadratic finite elements with non-matching grids for the unilateral boundary contact
ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Volume 47 (2013) no. 4, p. 1185-1205

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

DOI : https://doi.org/10.1051/m2an/2012064
Classification:  35J85,  65N30,  74M15
Keywords: unilateral contact conditions, quadratic finite elements, non-matching grids, mortar matching
@article{M2AN_2013__47_4_1185_0,
     author = {Auliac, S. and Belhachmi, Z. and Ben Belgacem, F. and Hecht, F.},
     title = {Quadratic finite elements with non-matching grids for the unilateral boundary contact},
     journal = {ESAIM: Mathematical Modelling and Numerical Analysis - Mod\'elisation Math\'ematique et Analyse Num\'erique},
     publisher = {EDP-Sciences},
     volume = {47},
     number = {4},
     year = {2013},
     pages = {1185-1205},
     doi = {10.1051/m2an/2012064},
     mrnumber = {3082294},
     language = {en},
     url = {http://www.numdam.org/item/M2AN_2013__47_4_1185_0}
}
Auliac, S.; Belhachmi, Z.; Ben Belgacem, F.; Hecht, F. Quadratic finite elements with non-matching grids for the unilateral boundary contact. ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Volume 47 (2013) no. 4, pp. 1185-1205. doi : 10.1051/m2an/2012064. http://www.numdam.org/item/M2AN_2013__47_4_1185_0/

[1] R.A. Adams, Sobolev Spaces. Academic Press (1975). | MR 450957 | Zbl 1098.46001

[2] 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). | MR 347104 | Zbl 0259.00014

[3] L. Baillet and T. Sassi, Mixed finite element formulation in large deformation frictional contact problem. European J. Comput. Mech. 14 (2005) 287-304. | Zbl 1226.74027

[4] Z. Belhachmi and F. Ben Belgacem, Quadratic finite element for Signorini problem. Math. Comput. 72 (2003) 83-104. | MR 1933319 | Zbl 1112.74446

[5] F. Ben Belgacem, P. Hild and P. Laborde, Extension of the mortar finite element method to a variational inequality modeling unilateral contact. Math. Models Methods Appl. Sci. 9 (1999) 287-303. | MR 1674556 | Zbl 0940.74056

[6] F. Ben Belgacem, Y. Renard and L. Slimane, A Mixed Formulation for the Signorini Problem in nearly Incompressible Elasticity. Appl. Numer. Math. 54 (2005) 1-22. | MR 2134092 | Zbl 1086.74037

[7] F. Ben Belgacem and Y. Renard, Hybrid finite element methods for the Signorini problem. Math. Comput. 72 (2003) 1117-1145. | MR 1972730 | Zbl 1023.74043

[8] 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. | MR 1268898 | Zbl 0797.65094

[9] S.C. Brenner and L.R. Scott, Mathematical Theory of Finite Element Methods. Texts Appl. Math. Springer Verlag, New-York 15 (1994). | MR 1278258 | Zbl 0804.65101

[10] 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). | MR 2265882 | Zbl 1108.65060

[11] F. Brezzi and M. Fortin, Mixed and Hybrid Finite Element Methods, Springer Series Comput. Math., vol. 15. Springer Verlag, New York (1991). | MR 1115205 | Zbl 0788.73002

[12] F. Brezzi, W.W. Hager and P.A. Raviart, Error estimates for the finite element solution of variational inequalities. Numer. Math. 28 (1977) 431-443. | MR 448949 | Zbl 0369.65030

[13] 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. | Zbl 0911.65117

[14] A. Chernov, M. Maischak and E.P. Stephan, hp-mortar boundary element method for two-body contact problems with friction. Math. Methods Appl. Sci. 31 (2008) 2029-2054. | MR 2457761 | Zbl 1149.74051

[15] P.-G. Ciarlet, The Finite Element Method for Elliptic Problems. North Holland (1978). | MR 520174 | Zbl 0511.65078

[16] M. Crouzeix and V. Thomée, The Stability in Lp and W1,p of the L2-Projection on Finite Element Function Spaces. Mathods Comput. 48 (1987) 521-532. | MR 878688 | Zbl 0637.41034

[17] G. Duvaut and J.-P. Lions, Les inéquations en mécanique et en physique. Dunod, Paris (1972). | Zbl 0298.73001

[18] S. Faletta, 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. | MR 2334147

[19] R.S. Falk, Error Estimates for the Approximation of a Class of Variational Inequalities. Math. Comput. 28 963-971 (1974). | MR 391502 | Zbl 0297.65061

[20] K.A. Fischer and P. Wriggers, Frictionless 2D contact formulations for finite deformations based on the mortar method. Comput. Mech. 36 (2005) 226-244. | Zbl 1102.74033

[21] B. Flemisch, M.A. Puso and B.I. Wohlmuth, A new dual mortar method for curved interfaces: 2D elasticity. Internat. J. Numer. Methods Eng. 63 (2005) 813-832. | MR 2144916 | Zbl 1084.74050

[22] 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). | MR 1422506 | Zbl 0873.73079

[23] F. Hecht, Freefem++. Third Edition, Version 3.11-1 http://www.freefem.org/ff++http://www.freefem.org/ff++.

[24] 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). | MR 3022224 | Zbl 1260.74027

[25] P. Hild, Problèmes de contact unilatéral et maillages éléments finis incompatibles. Thèse de l'Université Paul Sabatier, Toulouse 3 (1998).

[26] P. Hild, Numerical implementation of two nonconforming finite element methods for unilateral contact. Comput. Methods Appl. Mech. Eng. 184 (2000) 99-123. | MR 1752624 | Zbl 1009.74062

[27] P. Hild, P. Laborde. Quadratic finite element methods for unilateral contact problems. Appl. Numer. Math. 41 (2002) 401-421. | MR 1903172 | Zbl 1062.74050

[28] D. Hua, L. Wang. A mixed finite element method for the unilateral contact problem in elasticity. Sci. China Ser. A 49 (2006) 513-524. | MR 2250480 | Zbl 1104.74059

[29] S. Hüeber, B.I. Wohlmuth. An optimal a priori error estimate for nonlinear multibody contact problems. SIAM J. Numer. Anal. 43 (2005) 156-173. | MR 2177139 | Zbl 1083.74047

[30] S. Hüeber, M. Mair and B.I. Wohlmuth, 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. | MR 2149369 | Zbl 1114.74058

[31] N. Kikuchi and J.T. Oden, Contact Problems in Elasticity: A Study of Variational Inequalities and Finite Element Methods. SIAM (1988). | MR 961258 | Zbl 0685.73002

[32] T.Y. Kim, J.E. Dolbow and T.A. Laursen, A Mortared Finite Element Method for Frictional Contact on Arbitrary Surfaces. Comput. Mech. 39 (2007) 223-235. | Zbl 1178.74169

[33] T.A. Laursen, M.A. Puso and J. Sandersc, 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. | MR 2872022 | Zbl 1239.74070

[34] T.A. Laursen and B. Yang, New Developments in Surface-to-Surface Discretization Strategies for Analysis of Interface Mechanics. Computational Plasticity. Comput. Methods Appl. Sci. 7 (2010) 67-86. | MR 2432699

[35] M.-X. Li, Q. Lin and S.-H. Zhang, Superconvergence of finite element method for the Signorini problem. J. Comput. Appl. Math. 222 (2008) 284-292. | MR 2474630 | Zbl 1148.74044

[36] M. Moussaoui and K. Khodja, Régularité des solutions d'un problème mêlé Dirichlet-Signorini dans un domaine polygonal plan. Commun. Part. Differ. Equ. 17 (1992) 805-826. | MR 1177293 | Zbl 0806.35049

[37] M.A. Puso, T.A. Laursen and J. Solberg, A segment-to-segment mortar contact method for quadratic elements and large deformations. Comput. Meth. Appl. Mech. and Eng. 197 (2008) 555-566. | MR 2396042 | Zbl 1169.74627

[38] M.A. Puso and T.A. Laursen, A Mortar Segment-to-Segment Contact Method for Large Deformation Solid Mechanics. Comput. Methods Appl. Mech. Eng. 193 (2004) 601-629. | MR 2097761 | Zbl 1060.74636

[39] P. Seshaiyer and M. Suri, Uniform h − p Convergence Results for the Mortar Finite Element Method. Math. Comput. 69 521-546 (2000). | MR 1642762 | Zbl 0944.65113

[40] 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).

[41] B.I. Wohlmuth, A Mortar Finite Element Method Using Dual Spaces for the Lagrange Multiplier. SIAM J. Numer. Anal. 38 (2001) 989-1012,. | MR 1781212 | Zbl 0974.65105

[42] B. Wohlmuth, R. Krause. Monotone multigrid methods on nonmatching grids for nonlinear multibody contact problems. SIAM J. Sci. Comput. 25 (2003) 324-347. | MR 2047208 | Zbl 1163.65334

[43] B. Yang, T.A. Laursen, X. Meng. Two dimensional mortar contact methods for large deformation frictional sliding. Internat. J. Numer. Methods Eng. 62 (2005) 1183-1225. | MR 2120292 | Zbl 1161.74497

[44] Z.-H. Zhong, Finite Element Procedures for Contact-Impact Problems. Oxford University Press (1993).