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A fictitious domain method for frictionless contact problems in elasticity using Nitsche’s method
Mathieu Fabre1; Jérôme Pousin1; Yves Renard2
1 Univ Lyon, INSA Lyon, CNRS UMR 5208, ICJ, F-69621, Villeurbanne, France
2 Univ Lyon, INSA Lyon, CNRS UMR 5208, ICJ, UMR 5259, LaMCoS, F-69621, Villeurbanne, France
The SMAI Journal of computational mathematics, Volume 2 (2016), pp. 19-50.

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.

Published online:
DOI: 10.5802/smai-jcm.8
Classification: 65N85, 35M85, 74M15
Keywords: fictitious domain method, Signorini’s problem, unilateral contact, finite element method, Nitsche’s method, a priori analysis

Mathieu Fabre 1; Jérôme Pousin 1; Yves Renard 2

1 Univ Lyon, INSA Lyon, CNRS UMR 5208, ICJ, F-69621, Villeurbanne, France
2 Univ Lyon, INSA Lyon, CNRS UMR 5208, ICJ, UMR 5259, LaMCoS, F-69621, Villeurbanne, France 
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     journal = {The SMAI Journal of computational mathematics},
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Mathieu Fabre; Jérôme Pousin; Yves Renard. A fictitious domain method for frictionless contact problems in elasticity using Nitsche’s method. The SMAI Journal of computational mathematics, Volume 2 (2016), pp. 19-50. doi : 10.5802/smai-jcm.8. https://smai-jcm.centre-mersenne.org/articles/10.5802/smai-jcm.8/

[1] C. Annavarapu; M. Hautefeuille; J. E. Dolbow A robust Nitsche’s formulation for interface problems, Comput. Methods Appl. Mech. Engrg., Volume 225-228 (2012), pp. 44-54 | DOI | MR | Zbl

[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, Volume 40 (2005), pp. 513-520 | DOI | MR | Zbl

[3] H. Brézis Équations et inéquations non linéaires dans les espaces vectoriels en dualité, Ann. Inst. Fourier, Volume 18 (1968) no. 1, pp. 115-175 | DOI | Numdam | Zbl

[4] E. Burman; P. Hansbo Fictitious domain finite element methods using cut elements: II. A stabilized Nitsche’s method, Comput. Methods Appl. Mech. Engrg., Volume 62 (2012) no. 4, pp. 328-341 | MR | Zbl

[5] F. Chouly An adaptation of Nitsche’s method to the Tresca friction problem, J. Math. Anal. Appl., Volume 411 (2014), pp. 329-339 | DOI | MR | Zbl

[6] F. Chouly; P. Hild A Nitsche-based method for unilateral contact problems: numerical analysis, SIAM J. Numer. Anal., Volume 51 (2013) no. 2, pp. 1295-1307 | DOI | MR | Zbl

[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., Volume 84 (2015), pp. 1089-1112 | DOI | MR | Zbl

[8] P. G. Ciarlet The Finite Element Method for Elliptic Problems, North-Holland, Amsterdam, 1978 | Zbl

[9] G. Duvaut; J. L. Lions Les inéquations en mécanique et en physique, Dunod, Paris, 1972 | Zbl

[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., Volume 8 (1963), pp. 91-140 | Zbl

[11] A. Fritz; S. Hüeber; B.I. Wohlmuth A comparison of mortar and Nitsche techniques for linear elasticity, Calcolo, Volume 41 (2004) no. 3, pp. 115-137 | DOI | MR | Zbl

[12] V. Girault; R. Glowinski Error analysis of a fictitious domain method applied to a Dirichlet problem, Japan J. Indust. Appl. Math., Volume 12 (1995) no. 3, pp. 487-514 | DOI | MR | Zbl

[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., Volume 327 (1998) no. 7, pp. 693-698 | DOI | MR

[14] A. Hansbo; P. Hansbo An unfitted finite element method, based on Nitsche’s method, for elliptic interface problems, Comput. Methods Appl. Mech. Engrg., Volume 191 (2002), pp. 5537-5552 | DOI | MR | Zbl

[15] J. Haslinger; I. Hlaváček; J. Nečas Numerical methods for unilateral problems in solid mechanics, Handbook of Numerical Analysis, Vol. IV (P.G. Ciarlet; J.L. Lions, eds.), North Holland, 1996, pp. 313-385 | DOI | Zbl

[16] J. Haslinger; Y. Renard A new fictitious domain approach inspired by the extended finite element method, SIAM J. Numer. Anal., Volume 47 (2009) no. 2, pp. 1474-1499 | DOI | MR | Zbl

[17] M. Juntunen; R. Stenberg Nitsche’s method for general boundary conditions, Math. Comp., Volume 78 (2009) no. 267, pp. 1353-1374 | DOI | MR | Zbl

[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 | Zbl

[19] G. I. Marchuk Methods of numerical mathematics, Applications of Mathematics, vol. 2, Springer-Verlag, New York, 1982

[20] N. Möes; E. Béchet; M. Tourbier Imposing Dirichlet boundary conditions in the extended finite element method, Internat. J. Numer. Methods Engrg., Volume 67 (2006) no. 12, pp. 1641-1669 | DOI | MR | Zbl

[21] N. Möes; J. Dolbow; T. Belytschko A finite element method for cracked growth without remeshing, Internat. J. Numer. Methods Engrg., Volume 46 (1999), pp. 131-150 | DOI | Zbl

[22] M. Moussaoui; K. Khodja Régularité des solutions d’un problème mêlé Dirichlet-Signorini dans un domaine polygonal plan, Commun. Partial Differential Equations, Volume 17 (1992), pp. 805-826 | DOI | Zbl

[23] S. Nicaise; Y. Renard; E. Chahine Optimal convergence analysis for the eXtended Finite Element, Internat. J. Numer. Methods Engrg., Volume 86 (2011), pp. 528-548 | DOI | MR | Zbl

[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, Volume 36 (1971) no. 1, pp. 9-15 | DOI | Zbl

[25] C. S. Peskin The immersed boundary method, Acta Numerica, Volume 11 (2002), pp. 479-517 | DOI | MR | Zbl

[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., Volume 199 (2010), pp. 1165-1177 | DOI | Zbl

[27] Y. Renard Generalized Newton’s methods for the approximation and resolution of frictional contact problems in elasticity, Comput. Methods Appl. Mech. Engrg., Volume 256 (2013), pp. 38-55 | DOI | MR | Zbl

[28] B. Wohlmuth Variationally consistent discretization schemes and numerical algorithms for contact problems, Acta Numerica, Volume 20 (2011), pp. 569-734 | DOI | MR | Zbl

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