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.
@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},
volume = {39},
year = {2005},
pages = {827-854},
doi = {10.1051/m2an:2005036},
mrnumber = {2165681},
zbl = {pre02213941},
language = {en},
url = {http://dml.mathdoc.fr/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, Tome 39 (2005) pp. 827-854. doi : 10.1051/m2an:2005036. http://gdmltest.u-ga.fr/item/M2AN_2005__39_4_827_0/
[1] and, 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] ,, and, Matlab implementation of the finite element method in elasticity. Computing 69 (2002) 239-263.
[3] and, Regularity results for nonlinear elliptic systems and applications, Springer-Verlag, Berlin. Appl. Math. Sci. 151 (2002). | MR 1917320 | Zbl 1055.35002
[4] ,, and, 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] , and, Smoothing methods and semismooth methods for nondifferentiable operator equations. SIAM J. Numer. Anal. 38 (2000) 1200-1216. | Zbl 0979.65046
[6] and, 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] ,, and, Formulation and comparison of algorithms for frictional contact problems. Internat. J. Numer. Methods Engrg. 42 (1998) 145-173. | Zbl 0917.73063
[8] , and, 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] and, Existence results for the static contact problem with Coulomb friction. Math. Models Methods Appl. Sci. 8 (1998) 445-468. | Zbl 0907.73052
[10] and, 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] and, Elliptic partial differential equations of second order, Springer-Verlag, Berlin, second edition. Grundlehren der Mathematischen Wissenschaften 224 (1983). | MR 737190 | Zbl 0562.35001
[12] , Numerical methods for nonlinear variational problems. Springer Series in Computational Physics. Springer-Verlag, New York (1984). | MR 737005 | Zbl 0536.65054
[13] , Elliptic problems in nonsmooth domains, Pitman (Advanced Publishing Program), Boston, MA. Monographs Stud. Math. 24 (1985). | MR 775683 | Zbl 0695.35060
[14] and, 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] , Approximation of the Signorini problem with friction, obeying the Coulomb law. Math. Methods Appl. Sci. 5 (1983) 422-437. | Zbl 0525.73130
[16] , and, 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] and, A mesh-independence result for semismooth Newton methods. Math. Prog., Ser. B 101 (2004) 151-184. | Zbl 1079.65065
[18] , and, The primal-dual active set strategy as a semismooth Newton method. SIAM J. Optim. 13 (2003) 865-888. | Zbl 1080.90074
[19] , and, Semismooth Newton methods for a class of unilaterally constrained variational inequalities. Adv. Math. Sci. Appl. 14 (2004) 513-535. | Zbl 1083.49023
[20] ,, and, Solution of Variational Inequalities in Mechanics. Springer, New York. Appl. Math. Sci. 66 (1988). | MR 952855 | Zbl 0654.73019
[21] and, A primal-dual active strategy for non-linear multibody contact problems. Comput. Methods Appl. Mech. Engrg. 194 (2005) 3147-3166. | Zbl 1093.74056
[22] and, Augmented Lagrangian methods for nonsmooth, convex optimization in Hilbert spaces. Nonlinear Anal. 41 (2000) 591-616. | Zbl 0971.49014
[23] and, Semi-smooth Newton methods for variational inequalities of the first kind. ESAIM: M2AN 37 (2003) 41-62. | Numdam | Zbl 1027.49007
[24] and, 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] , Mathematical programming and augmented Lagrangian methods for frictional contact problems, A. Curnier, Ed. Proc. Contact Mechanics Int. Symp. (1992).
[26] , Monotone Multigrid Methods for Signorini's Problem with Friction. Ph.D. Thesis, FU Berlin (2001).
[27] and, Fixed point strategies for elastostatic frictional contact problems. Rapport Interne 03-27, MIP Laboratory, Université Paul Sabatier, Toulouse (2003).
[28] , 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] , and, 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] , and, 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] and, Internal approximation of quasi-variational inequalities. Numer. Math. 59 (1991) 385-398. | Zbl 0742.65055
[32] , 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] , Infinite-Dimensional Semi-Smooth Newton and Augmented Lagrangian Methods for Friction and Contact Problems in Elasticity. Ph.D. Thesis, University of Graz (2004).
[34] , Semismooth Newton and augmented Lagrangian methods for a simplified friction problem. SIAM J. Optim. 15 (2004) 39-62. | Zbl 1106.90078
[35] , Semismooth Newton methods for operator equations in function spaces. SIAM J. Optim. 13 (2003) 805-842. | Zbl 1033.49039