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.
DOI: 10.5802/smai-jcm.8
Keywords: fictitious domain method, Signorini’s problem, unilateral contact, finite element method, Nitsche’s method, a priori analysis
@article{SMAI-JCM_2016__2__19_0,
author = {Mathieu Fabre and J\'er\^ome Pousin and Yves Renard},
title = {A fictitious domain method for frictionless contact problems in elasticity using {Nitsche{\textquoteright}s} method},
journal = {The SMAI Journal of computational mathematics},
pages = {19--50},
publisher = {Soci\'et\'e de Math\'ematiques Appliqu\'ees et Industrielles},
volume = {2},
year = {2016},
doi = {10.5802/smai-jcm.8},
zbl = {1416.74082},
mrnumber = {3633544},
language = {en},
url = {https://smai-jcm.centre-mersenne.org/articles/10.5802/smai-jcm.8/}
}
TY - JOUR AU - Mathieu Fabre AU - Jérôme Pousin AU - Yves Renard TI - A fictitious domain method for frictionless contact problems in elasticity using Nitsche’s method JO - The SMAI Journal of computational mathematics PY - 2016 SP - 19 EP - 50 VL - 2 PB - Société de Mathématiques Appliquées et Industrielles UR - https://smai-jcm.centre-mersenne.org/articles/10.5802/smai-jcm.8/ DO - 10.5802/smai-jcm.8 LA - en ID - SMAI-JCM_2016__2__19_0 ER -
%0 Journal Article %A Mathieu Fabre %A Jérôme Pousin %A Yves Renard %T A fictitious domain method for frictionless contact problems in elasticity using Nitsche’s method %J The SMAI Journal of computational mathematics %D 2016 %P 19-50 %V 2 %I Société de Mathématiques Appliquées et Industrielles %U https://smai-jcm.centre-mersenne.org/articles/10.5802/smai-jcm.8/ %R 10.5802/smai-jcm.8 %G en %F SMAI-JCM_2016__2__19_0
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] A robust Nitsche’s formulation for interface problems, Comput. Methods Appl. Mech. Engrg., Volume 225-228 (2012), pp. 44-54 | DOI | MR | Zbl
[2] 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] 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] 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] 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] 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] 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] The Finite Element Method for Elliptic Problems, North-Holland, Amsterdam, 1978 | Zbl
[9] Les inéquations en mécanique et en physique, Dunod, Paris, 1972 | Zbl
[10] 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 comparison of mortar and Nitsche techniques for linear elasticity, Calcolo, Volume 41 (2004) no. 3, pp. 115-137 | DOI | MR | Zbl
[12] 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] 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] 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] 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] 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] Nitsche’s method for general boundary conditions, Math. Comp., Volume 78 (2009) no. 267, pp. 1353-1374 | DOI | MR | Zbl
[18] Contact problems in elasticity: a study of variational inequalities and finite element methods, Studies in Applied Mathematics, vol. 8, SIAM, Philadelphia, 1988 | Zbl
[19] Methods of numerical mathematics, Applications of Mathematics, vol. 2, Springer-Verlag, New York, 1982
[20] 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] A finite element method for cracked growth without remeshing, Internat. J. Numer. Methods Engrg., Volume 46 (1999), pp. 131-150 | DOI | Zbl
[22] 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] Optimal convergence analysis for the eXtended Finite Element, Internat. J. Numer. Methods Engrg., Volume 86 (2011), pp. 528-548 | DOI | MR | Zbl
[24] 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] The immersed boundary method, Acta Numerica, Volume 11 (2002), pp. 479-517 | DOI | MR | Zbl
[26] 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] 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] Variationally consistent discretization schemes and numerical algorithms for contact problems, Acta Numerica, Volume 20 (2011), pp. 569-734 | DOI | MR | Zbl
Cited by Sources: