Stabilized finite elements for Tresca friction problem. (English) Zbl 1497.65227
Summary: We formulate and analyze a Nitsche-type algorithm for frictional contact problems. The method is derived from, and analyzed as, a stabilized finite element method and shown to be quasi-optimal, as well as suitable as an adaptive scheme through an a posteriori error analysis. The a posteriori error indicators are validated in a numerical experiment.
MSC:
| 65N30 | Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs |
| 65N15 | Error bounds for boundary value problems involving PDEs |
| 74M10 | Friction in solid mechanics |
| 74M15 | Contact in solid mechanics |
| 74S05 | Finite element methods applied to problems in solid mechanics |
References:
| [1] | L. Baillet and T. Sassi, Mixed finite element methods for the Signorini problem with friction. Numer. Methods Part. Differ. Equ. 22 (2006) 1489-1508. · Zbl 1105.74041 · doi:10.1002/num.20147 |
| [2] | F. Chouly, An adaptation of Nitsche’s method to the Tresca friction problem. J. Math. Anal. App. 411 (2014) 329-339. · Zbl 1311.74112 · doi:10.1016/j.jmaa.2013.09.019 |
| [3] | F. Chouly, M. Fabre, P. Hild, R. Mlika, J. Pousin and Y. Renard, An overview of recent results on Nitsche’s method for contact problems. In: Geometrically Unfitted Finite Element Methods and Applications, edited by S.P.A. Bordas, E.N. Burman, M.G. Larson and M.A. Olshanskii. Vol. 121 of Lecture Notes in Computational Science and Engineering. Springer (2017). · Zbl 1390.74003 |
| [4] | F. Chouly, M. Fabre, P. Hild, J. Pousin and Y. Renard, Residual-based a posteriori error estimation for contact problems approximated by Nitsche’s method. IMA J. Numer. Anal. 38 (2018) 921-954. · Zbl 1477.65204 · doi:10.1093/imanum/drx024 |
| [5] | F. Chouly, R. Mlika and Y. Renard, An unbiased Nitsche’s approximation of the frictional contact between two elastic structures. Numer. Math. 139 (2018) 593-631. · Zbl 1391.74169 |
| [6] | F. Chouly, A. Ern and N. Pignet, A hybrid high-order discretization combined with Nitsche’s method for contact and Tresca friction in small strain elasticity. SIAM J. Sci. Comput. 42 (2020) A2300-A2324. · Zbl 1452.65328 · doi:10.1137/19M1286499 |
| [7] | P. Dörsek and J.M. Melenk, Adaptive hp-FEM for the contact problem with Tresca friction in linear elasticity: the primal-dual formulation and a posteriori error estimation. Appl. Numer. Math. 60 (2010) 689-704. · Zbl 1306.74036 · doi:10.1016/j.apnum.2010.03.011 |
| [8] | G. Duvaut and J.L. Lions, Inequalities in Mechanics and Physics. Springer, Berlin (1976). · Zbl 0331.35002 · doi:10.1007/978-3-642-66165-5 |
| [9] | T. Gustafsson, Source code for the numerical experiment: kinnala/paper-fricnitsche. Preprint DOI: 10.5281/zenodo.5851711 (2022). |
| [10] | T. Gustafsson and G.D. McBain, scikit-fem: a Python package for finite element assembly. J. Open Source Softw. 5 (2020) 2369. |
| [11] | T. Gustafsson, R. Stenberg and J. Videman, Mixed and stabilized finite element methods for the obstacle problem. SIAM J. Numer. Anal. 55 (2017) 2718-2744. · Zbl 1378.65135 · doi:10.1137/16M1065422 |
| [12] | T. Gustafsson, R. Stenberg and J. Videman, On Nitsche’s method for elastic contact problems. SIAM J. Sci. Comput. 42 (2020) B425-B446. · Zbl 1447.65143 · doi:10.1137/19M1246869 |
| [13] | J. Haslinger and I. Hlaváçek, Approximation of the Signorini problem with friction by a mixed finite element method. J. Math. Anal. App. 86 (1982) 99-122. · Zbl 0486.73099 · doi:10.1016/0022-247X(82)90257-8 |
| [14] | J. Haslinger and T. Sassi, Mixed finite element approximation of 3D contact problems with given friction: error analysis and numerical realization. ESAIM: Math. Modell. Numer. Anal. 38 (2004) 563-578. · Zbl 1080.74046 · doi:10.1051/m2an:2004026 |
| [15] | J. Haslinger, I. Hlaváček and J. Nečas, Numerical methods for unilateral problems in solid mechanics. In: Finite Element Methods (Part 2), Numerical Methods for Solids (Part 2). Vol. 4 of Handbook of Numerical Analysis. Elsevier (1996) 313-485. · Zbl 0873.73079 · doi:10.1016/S1570-8659(96)80005-6 |
| [16] | P. Hild and V. Lleras, Residual error estimators for Coulomb friction. SIAM J. Numer. Anal. 47 (2009) 3550-3583. · Zbl 1410.74067 · doi:10.1137/070711554 |
| [17] | P. Hild and Y. Renard, An error estimate for the Signorini problem with Coulomb friction approximated by finite elements. SIAM J. Numer. Anal. 45 (2007) 2012-2031. · Zbl 1146.74050 · doi:10.1137/050645439 |
| [18] | P. Hild and Y. Renard, A stabilized Lagrange multiplier method for the finite element approximation of contact problems in elastostatics. Numer. Math. 115 (2010) 101-129. · Zbl 1194.74408 · doi:10.1007/s00211-009-0273-z |
| [19] | S. Hüeber, G. Stadler and B.I. Wohlmuth, A primal-dual active set algorithm for three-dimensional contact problems with Coulomb friction. SIAM J. Sci. Comput. 30 (2008) 572-596. · Zbl 1158.74045 · doi:10.1137/060671061 |
| [20] | J.D. Hunter, Matplotlib: A 2d graphics environment. Comput. Sci. Eng. 9 (2007) 90-95. |
| [21] | N. Kikuchi and J.T. Oden, Contact Problems in Elasticity: A Study of Variational Inequalities and Finite Element Methods. SIAM, Philadelphia (1988). · Zbl 0685.73002 · doi:10.1137/1.9781611970845 |
| [22] | P. Laborde and Y. Renard, Fixed point strategies for elastostatic frictional contact problems. Math. Methods Appl. Sci. 31 (2008) 415-441. · Zbl 1132.74032 · doi:10.1002/mma.921 |
| [23] | T.A. Laursen, Computational Contact and Impact Mechanics, corr. 2nd printing. Springer, Heidelberg (2003). · doi:10.1007/978-3-662-04864-1 |
| [24] | R. Verfürth, A Posteriori Error Estimation Techniques for Finite Element Methods. Oxford University Press, Oxford (2013). · Zbl 1279.65127 · doi:10.1093/acprof:oso/9780199679423.001.0001 |
| [25] | P. Virtanen, R. Gommers, T.E. Oliphant, M. Haberland, T. Reddy, D. Cournapeau, E. Burovski, P. Peterson, W. Weckesser, J. Bright, S.J. van der Walt, M. Brett, J. Wilson, K. Jarrod Millman, N. Mayorov, A.R.J. Nelson, E. Jones, R. Kern, E. Larson, C.J. Carey, İ. Polat, Y. Feng, E.W. Moore, J. VanderPlas, D. Laxalde, J. Perktold, R. Cimrman, I. Henriksen, E.A. Quintero, C.R. Harris, A.M. Archibald, A.H. Ribeiro, F. Pedregosa, P. van Mulbregt and SciPy 1.0 Contributors, SciPy 1.0: Fundamental algorithms for scientific computing in Python. Nat. Methods 17 (2020) 261-272. |
| [26] | B. Wohlmuth, Variationally consistent discretization schemes and numerical algorithms for contact problems. Acta Numer. 20 (2011) 569-734. · Zbl 1432.74176 · doi:10.1017/S0962492911000079 |
| [27] | P. Wriggers, Computational Contact Mechanics, 2nd edition, Springer-Verlag, Berlin Heidelberg (2006). · Zbl 1104.74002 · doi:10.1007/978-3-540-32609-0 |
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