With cedram.org   version française
Search for an article
Table of contents for this volume | Previous article | Next article
Mathieu Fabre; Jérôme Pousin; Yves Renard
A fictitious domain method for frictionless contact problems in elasticity using Nitsche’s method
SMAI-Journal of computational mathematics, 2 (2016), p. 19-50, doi: 10.5802/smai-jcm.8
Article PDF
Class. Math.: 65N85, 35M85, 74M15
Keywords: fictitious domain method, Signorini’s problem, unilateral contact, finite element method, Nitsche’s method, a priori analysis

Abstract

In this paper, we develop and analyze a finite element fictitious domain approach based on Nitsche’s method for the approximation of frictionless contact problems of two deformable elastic bodies. In the proposed method, the geometry of the bodies and the boundary conditions, including the contact condition between the two bodies, are described independently of the mesh of the fictitious domain. We prove that the optimal convergence is preserved. Numerical experiments are provided which confirm the correct behavior of the proposed method.

Bibliography

[1] C. Annavarapu, M. Hautefeuille & J. E. Dolbow, “A robust Nitsche’s formulation for interface problems”, Comput. Methods Appl. Mech. Engrg. 225-228 (2012), p. 44-54 Article |  MR 2917495 |  Zbl 1253.74096
[2] S. Bertoluzza, M. Ismail & B. Maury, “The fat boundary method: Semi-discrete scheme and some numerical experiments”, Domain Decomposition Methods in Science and Engineering, Lecture Notes in Computational Science and Engineering 40 (2005), p. 513-520 Article |  MR 2235779 |  Zbl 1067.65121
[3] H. Brézis, “Équations et inéquations non linéaires dans les espaces vectoriels en dualité”, Ann. Inst. Fourier 18 (1968) no. 1, p. 115-175 Cedram |  MR 270222 |  Zbl 0169.18602
[4] E. Burman & P. Hansbo, “Fictitious domain finite element methods using cut elements: II. A stabilized Nitsche’s method”, Comput. Methods Appl. Mech. Engrg. 62 (2012) no. 4, p. 328-341  MR 2899249
[5] F. Chouly, “An adaptation of Nitsche’s method to the Tresca friction problem”, J. Math. Anal. Appl. 411 (2014), p. 329-339 Article |  MR 3118488
[6] F. Chouly & P. Hild, “A Nitsche-based method for unilateral contact problems: numerical analysis”, SIAM J. Numer. Anal. 51 (2013) no. 2, p. 1295-1307 Article |  MR 3045657 |  Zbl 1268.74033
[7] F. Chouly, P. Hild & Y. Renard, “Symmetric and non-symmetric variants of Nitsche’s method for contact problems in elasticity: theory and numerical experiments”, Math. Comp. 84 (2015), p. 1089-1112 Article |  MR 3315501
[8] P. G. Ciarlet, The Finite Element Method for Elliptic Problems, North-Holland, Amsterdam, 1978  MR 520174 |  Zbl 0511.65078
[9] G. Duvaut & J. L. Lions, Les inéquations en mécanique et en physique, Dunod, Paris, 1972  MR 464857 |  Zbl 0298.73001
[10] G. Fichera, “Problemi elastostatici con vincoli unilaterali: Il problema di Signorini con ambigue condizioni al contorno”, Atti Accad. Naz. Lincei Mem. Cl. Sci. Fis. Mat. Natur. 8 (1963), p. 91-140  MR 178631 |  Zbl 0146.21204
[11] A. Fritz, S. Hüeber & B.I. Wohlmuth, “A comparison of mortar and Nitsche techniques for linear elasticity”, Calcolo 41 (2004) no. 3, p. 115-137 Article |  MR 2199546 |  Zbl 1099.65123
[12] V. Girault & R. Glowinski, “Error analysis of a fictitious domain method applied to a Dirichlet problem”, Japan J. Indust. Appl. Math. 12 (1995) no. 3, p. 487-514 Article |  MR 1356667 |  Zbl 0843.65076
[13] R. Glowinski & Y. Kuznetsov, “On the solution of the Dirichlet problem for linear elliptic operators by a distributed Lagrange multiplier method”, C. R. Acad. Sci. Paris Sér. I Math. 327 (1998) no. 7, p. 693-698 Article |  MR 1652674 |  Zbl 1005.65127
[14] A. Hansbo & P. Hansbo, “An unfitted finite element method, based on Nitsche’s method, for elliptic interface problems”, Comput. Methods Appl. Mech. Engrg. 191 (2002), p. 5537-5552 Article |  Zbl 1035.65125
[15] J. Haslinger, I. Hlaváček & J. Nečas, Numerical methods for unilateral problems in solid mechanics, in P.G. Ciarlet, J.L. Lions, ed., Handbook of Numerical Analysis, Vol. IV, North Holland, 1996, p. 313-385  MR 1422506 |  Zbl 0873.73079
[16] J. Haslinger & Y. Renard, “A new fictitious domain approach inspired by the extended finite element method”, SIAM J. Numer. Anal. 47 (2009) no. 2, p. 1474-1499 Article |  MR 2497337 |  Zbl 1205.65322
[17] M. Juntunen & R. Stenberg, “Nitsche’s method for general boundary conditions”, Math. Comp. 78 (2009) no. 267, p. 1353-1374 Article |  MR 2501054 |  Zbl 1198.65223
[18] N. Kikuchi & J. T. Oden, Contact problems in elasticity: a study of variational inequalities and finite element methods, Studies in Applied Mathematics, vol. 8, SIAM, Philadelphia, 1988  MR 961258 |  Zbl 0685.73002
[19] G. I. Marchuk, Methods of numerical mathematics, Applications of Mathematics, vol. 2, Springer-Verlag, New York, 1982  MR 661258 |  Zbl 0485.65003
[20] N. Möes, E. Béchet & M. Tourbier, “Imposing Dirichlet boundary conditions in the extended finite element method”, Internat. J. Numer. Methods Engrg. 67 (2006) no. 12, p. 1641-1669 Article |  MR 2258515 |  Zbl 1113.74072
[21] N. Möes, J. Dolbow & T. Belytschko, “A finite element method for cracked growth without remeshing”, Internat. J. Numer. Methods Engrg. 46 (1999), p. 131-150 Article |  Zbl 0955.74066
[22] M. Moussaoui & K. Khodja, “Régularité des solutions d’un problème mêlé Dirichlet-Signorini dans un domaine polygonal plan”, Commun. Partial Differential Equations 17 (1992), p. 805-826 Article |  MR 1177293 |  Zbl 0806.35049
[23] S. Nicaise, Y. Renard & E. Chahine, “Optimal convergence analysis for the eXtended Finite Element”, Internat. J. Numer. Methods Engrg. 86 (2011), p. 528-548 Article |  MR 2815989 |  Zbl 1216.74029
[24] 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) no. 1, p. 9-15 Article |  MR 341903 |  Zbl 0229.65079
[25] C. S. Peskin, “The immersed boundary method”, Acta Numerica 11 (2002), p. 479-517 Article |  MR 2009378 |  Zbl 1123.74309
[26] E. Pierres, M.C. Baietto & A. Gravouil, “A two-scale extended finite element method for modeling 3D crack growth with interfacial contact”, Comput. Methods Appl. Mech. Engrg. 199 (2010), p. 1165-1177 Article |  MR 2594832 |  Zbl 1227.74088
[27] Y. Renard, “Generalized Newton’s methods for the approximation and resolution of frictional contact problems in elasticity”, Comput. Methods Appl. Mech. Engrg. 256 (2013), p. 38-55 Article |  MR 3029045
[28] B. Wohlmuth, “Variationally consistent discretization schemes and numerical algorithms for contact problems”, Acta Numerica 20 (2011), p. 569-734 Article |  MR 2805157