Quadratic finite elements with non-matching grids for the unilateral boundary contact

 Access to full text

1. [1] R.A. Adams, Sobolev Spaces. Academic Press (1975). Zbl1098.46001MR450957
2. [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). Zbl0259.00014MR347104
3. [3] L. Baillet and T. Sassi, Mixed finite element formulation in large deformation frictional contact problem. European J. Comput. Mech.14 (2005) 287–304. Zbl1226.74027
4. [4] Z. Belhachmi and F. Ben Belgacem, Quadratic finite element for Signorini problem. Math. Comput.72 (2003) 83–104. Zbl1112.74446MR1933319
5. [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. Zbl0940.74056MR1674556
6. [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. Zbl1086.74037MR2134092
7. [7] F. Ben Belgacem and Y. Renard, Hybrid finite element methods for the Signorini problem. Math. Comput.72 (2003) 1117–1145. Zbl1023.74043MR1972730
8. [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. Zbl0797.65094MR1268898
9. [9] S.C. Brenner and L.R. Scott, Mathematical Theory of Finite Element Methods. Texts Appl. Math. Springer Verlag, New-York 15 (1994). Zbl0804.65101MR1278258
10. [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). Zbl1108.65060MR2265882
11. [11] F. Brezzi and M. Fortin, Mixed and Hybrid Finite Element Methods, Springer Series Comput. Math., vol. 15. Springer Verlag, New York (1991). Zbl0788.73002MR1115205
12. [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. Zbl0369.65030MR448949
13. [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. Zbl0911.65117
14. [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. Zbl1149.74051MR2457761
15. [15] P.-G. Ciarlet, The Finite Element Method for Elliptic Problems. North Holland (1978). Zbl0511.65078MR520174
16. [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. Zbl0637.41034MR878688
17. [17] G. Duvaut and J.-P. Lions, Les inéquations en mécanique et en physique. Dunod, Paris (1972). Zbl0298.73001
18. [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. MR2334147
19. [19] R.S. Falk, Error Estimates for the Approximation of a Class of Variational Inequalities. Math. Comput.28963–971 (1974). Zbl0297.65061MR391502
20. [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. Zbl1102.74033
21. [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. Zbl1084.74050MR2144916
22. [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). Zbl0873.73079MR1422506
23. [23] F. Hecht, Freefem++. Third Edition, Version 3.11-1 http://www.freefem.org/ff++http://www.freefem.org/ff++. 
24. [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). Zbl1260.74027MR3022224
25. [25] P. Hild, Problèmes de contact unilatéral et maillages éléments finis incompatibles. Thèse de l’Université Paul Sabatier, Toulouse 3 (1998). 
26. [26] P. Hild, Numerical implementation of two nonconforming finite element methods for unilateral contact. Comput. Methods Appl. Mech. Eng.184 (2000) 99–123. Zbl1009.74062MR1752624
27. [27] P. Hild, P. Laborde. Quadratic finite element methods for unilateral contact problems. Appl. Numer. Math.41 (2002) 401–421. Zbl1062.74050MR1903172
28. [28] D. Hua, L. Wang. A mixed finite element method for the unilateral contact problem in elasticity. Sci. China Ser. A49 (2006) 513–524. Zbl1104.74059MR2250480
29. [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. Zbl1083.74047MR2177139
30. [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. Zbl1114.74058MR2149369
31. [31] N. Kikuchi and J.T. Oden, Contact Problems in Elasticity: A Study of Variational Inequalities and Finite Element Methods. SIAM (1988). Zbl0685.73002MR961258
32. [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. Zbl1178.74169
33. [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. Zbl1239.74070MR2872022
34. [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. MR2432699
35. [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. Zbl1148.74044MR2474630
36. [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. Zbl0806.35049MR1177293
37. [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. Zbl1169.74627MR2396042
38. [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. Zbl1060.74636MR2097761
39. [39] P. Seshaiyer and M. Suri, Uniform h − p Convergence Results for the Mortar Finite Element Method. Math. Comput. 69 521–546 (2000). Zbl0944.65113MR1642762
40. [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. [41] B.I. Wohlmuth, A Mortar Finite Element Method Using Dual Spaces for the Lagrange Multiplier. SIAM J. Numer. Anal. 38 (2001) 989–1012,. Zbl0974.65105MR1781212
42. [42] B. Wohlmuth, R. Krause. Monotone multigrid methods on nonmatching grids for nonlinear multibody contact problems. SIAM J. Sci. Comput.25 (2003) 324–347. Zbl1163.65334MR2047208
43. [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. Zbl1161.74497MR2120292
44. [44] Z.-H. Zhong, Finite Element Procedures for Contact-Impact Problems. Oxford University Press (1993). 

1. Shougui Zhang, Xiaolin Li, Boundary augmented Lagrangian method for the Signorini problem