Generalized Newton methods for the 2D-Signorini contact problem with friction in function space
ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Volume 39 (2005) no. 4, p. 827-854

The 2D-Signorini contact problem with Tresca and Coulomb friction is discussed in infinite-dimensional Hilbert spaces. First, the problem with given friction (Tresca friction) is considered. It leads to a constraint non-differentiable minimization problem. By means of the Fenchel duality theorem this problem can be transformed into a constrained minimization involving a smooth functional. A regularization technique for the dual problem motivated by augmented lagrangians allows to apply an infinite-dimensional semi-smooth Newton method for the solution of the problem with given friction. The resulting algorithm is locally superlinearly convergent and can be interpreted as active set strategy. Combining the method with an augmented lagrangian method leads to convergence of the iterates to the solution of the original problem. Comprehensive numerical tests discuss, among others, the dependence of the algorithm's performance on material and regularization parameters and on the mesh. The remarkable efficiency of the method carries over to the Signorini problem with Coulomb friction by means of fixed point ideas.

DOI : https://doi.org/10.1051/m2an:2005036
Classification:  49M05,  49M29,  74M10,  74M15,  74B05
Keywords: Signorini contact problems, Coulomb and Tresca friction, linear elasticity, semi-smooth Newton method, Fenchel dual, augmented lagrangians, complementarity system, active sets
@article{M2AN_2005__39_4_827_0,
author = {Kunisch, Karl and Stadler, Georg},
title = {Generalized Newton methods for the 2D-Signorini contact problem with friction in function space},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis - Mod\'elisation Math\'ematique et Analyse Num\'erique},
publisher = {EDP-Sciences},
volume = {39},
number = {4},
year = {2005},
pages = {827-854},
doi = {10.1051/m2an:2005036},
zbl = {pre02213941},
mrnumber = {2165681},
language = {en},
url = {http://www.numdam.org/item/M2AN_2005__39_4_827_0}
}

Kunisch, Karl; Stadler, Georg. Generalized Newton methods for the 2D-Signorini contact problem with friction in function space. ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Volume 39 (2005) no. 4, pp. 827-854. doi : 10.1051/m2an:2005036. http://www.numdam.org/item/M2AN_2005__39_4_827_0/

[1] P. Alart and A. Curnier, A mixed formulation for frictional contact problems prone to Newton like solution methods. Comput Methods Appl. Mech. Engrg. 92 (1991) 353-375. | Zbl 0825.76353

[2] J. Alberty, C. Carstensen, S.A. Funken and R. Klose, Matlab implementation of the finite element method in elasticity. Computing 69 (2002) 239-263.

[3] A. Bensoussan and J. Frehse, Regularity results for nonlinear elliptic systems and applications, Springer-Verlag, Berlin. Appl. Math. Sci. 151 (2002). | MR 1917320 | Zbl 1055.35002

[4] M. Bergounioux, M. Haddou, M. Hintermüller and K. Kunisch, A comparison of a Moreau-Yosida-based active set strategy and interior point methods for constrained optimal control problems. SIAM J. Optim. 11 (2000) 495-521. | Zbl 1001.49034

[5] X. Chen, Z. Nashed and L. Qi, Smoothing methods and semismooth methods for nondifferentiable operator equations. SIAM J. Numer. Anal. 38 (2000) 1200-1216. | Zbl 0979.65046

[6] P.W. Christensen and J.S. Pang, Frictional contact algorithms based on semismooth Newton methods, in Reformulation: nonsmooth, piecewise smooth, semismooth and smoothing methods, Kluwer Acad. Publ., Dordrecht. Appl. Optim. 22 (1999) 81-116. | Zbl 0937.74078

[7] P.W. Christensen, A. Klarbring, J.S. Pang and N. Strömberg, Formulation and comparison of algorithms for frictional contact problems. Internat. J. Numer. Methods Engrg. 42 (1998) 145-173. | Zbl 0917.73063

[8] Z. Dostál, J. Haslinger and R. Kučera, Implementation of the fixed point method in contact problems with Coulomb friction based on a dual splitting type technique. J. Comput. Appl. Math. 140 (2002) 245-256. | Zbl 1134.74418

[9] C. Eck and J. Jarušek, Existence results for the static contact problem with Coulomb friction. Math. Models Methods Appl. Sci. 8 (1998) 445-468. | Zbl 0907.73052

[10] I. Ekeland and R. Témam, Convex analysis and variational problems, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA. Classics in Applied Mathematics 28 (1999). | MR 1727362 | Zbl 0939.49002

[11] D. Gilbarg and N.S. Trudinger, Elliptic partial differential equations of second order, Springer-Verlag, Berlin, second edition. Grundlehren der Mathematischen Wissenschaften 224 (1983). | MR 737190 | Zbl 0562.35001

[12] R. Glowinski, Numerical methods for nonlinear variational problems. Springer Series in Computational Physics. Springer-Verlag, New York (1984). | MR 737005 | Zbl 0536.65054

[13] P. Grisvard, Elliptic problems in nonsmooth domains, Pitman (Advanced Publishing Program), Boston, MA. Monographs Stud. Math. 24 (1985). | MR 775683 | Zbl 0695.35060

[14] W. Han and M. Sofonea, Quasistatic contact problems in viscoelasticity and viscoplasticity, American Mathematical Society, Providence, RI. AMS/IP Studies in Advanced Mathematics 30 (2002). | MR 1935666 | Zbl 1013.74001

[15] J. Haslinger, Approximation of the Signorini problem with friction, obeying the Coulomb law. Math. Methods Appl. Sci. 5 (1983) 422-437. | Zbl 0525.73130

[16] J. Haslinger, Z. Dostál and R. Kučera, On a splitting type algorithm for the numerical realization of contact problems with Coulomb friction. Comput. Methods Appl. Mech. Engrg. 191 (2002) 2261-2281. | Zbl 1131.74344

[17] M. Hintermüller and M. Ulbrich, A mesh-independence result for semismooth Newton methods. Math. Prog., Ser. B 101 (2004) 151-184. | Zbl 1079.65065

[18] M. Hintermüller, K. Ito and K. Kunisch, The primal-dual active set strategy as a semismooth Newton method. SIAM J. Optim. 13 (2003) 865-888. | Zbl 1080.90074

[19] M. Hintermüller, V. Kovtunenko and K. Kunisch, Semismooth Newton methods for a class of unilaterally constrained variational inequalities. Adv. Math. Sci. Appl. 14 (2004) 513-535. | Zbl 1083.49023

[20] I. Hlaváček, J. Haslinger, J. Nečas and J. Lovíšek, Solution of Variational Inequalities in Mechanics. Springer, New York. Appl. Math. Sci. 66 (1988). | MR 952855 | Zbl 0654.73019

[21] S. Hüeber and B. Wohlmuth, A primal-dual active strategy for non-linear multibody contact problems. Comput. Methods Appl. Mech. Engrg. 194 (2005) 3147-3166. | Zbl 1093.74056

[22] K. Ito and K. Kunisch, Augmented Lagrangian methods for nonsmooth, convex optimization in Hilbert spaces. Nonlinear Anal. 41 (2000) 591-616. | Zbl 0971.49014

[23] K. Ito and K. Kunisch, Semi-smooth Newton methods for variational inequalities of the first kind. ESAIM: M2AN 37 (2003) 41-62. | Numdam | Zbl 1027.49007

[24] N. Kikuchi and J.T. Oden, Contact problems in elasticity: a study of variational inequalities and finite element methods, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA. SIAM Stud. Appl. Math. 8 (1988). | MR 961258 | Zbl 0685.73002

[25] A. Klarbring, Mathematical programming and augmented Lagrangian methods for frictional contact problems, A. Curnier, Ed. Proc. Contact Mechanics Int. Symp. (1992).

[26] R. Krause, Monotone Multigrid Methods for Signorini's Problem with Friction. Ph.D. Thesis, FU Berlin (2001).

[27] P. Laborde and Y. Renard, Fixed point strategies for elastostatic frictional contact problems. Rapport Interne 03-27, MIP Laboratory, Université Paul Sabatier, Toulouse (2003).

[28] A.Y.T. Leung, Guoqing Chen and Wanji Chen, Smoothing Newton method for solving two- and three-dimensional frictional contact problems. Internat. J. Numer. Methods Engrg. 41 (1998) 1001-1027. | Zbl 0905.73079

[29] C. Licht, E. Pratt and M. Raous, Remarks on a numerical method for unilateral contact including friction, in Unilateral problems in structural analysis, IV (Capri, 1989), Birkhäuser, Basel. Internat. Ser. Numer. Math. 101 (1991) 129-144. | Zbl 0762.73076

[30] J. Nečas, J. Jarušek and J. Haslinger, On the solution of the variational inequality to the Signorini problem with small friction. Boll. Unione Math. Ital. 5 (1980) 796-811. | Zbl 0445.49011

[31] C.A. Radoslovescu and M. Cocu, Internal approximation of quasi-variational inequalities. Numer. Math. 59 (1991) 385-398. | Zbl 0742.65055

[32] M. Raous, Quasistatic Signorini problem with Coulomb friction and coupling to adhesion, in New developments in contact problems, P. Wriggers and Panagiotopoulos, Eds., Springer Verlag. CISM Courses and Lectures 384 (1999) 101-178. | Zbl 0942.74052

[33] G. Stadler, Infinite-Dimensional Semi-Smooth Newton and Augmented Lagrangian Methods for Friction and Contact Problems in Elasticity. Ph.D. Thesis, University of Graz (2004).

[34] G. Stadler, Semismooth Newton and augmented Lagrangian methods for a simplified friction problem. SIAM J. Optim. 15 (2004) 39-62. | Zbl 1106.90078

[35] M. Ulbrich, Semismooth Newton methods for operator equations in function spaces. SIAM J. Optim. 13 (2003) 805-842. | Zbl 1033.49039