×

Path-following and augmented Lagrangian methods for contact problems in linear elasticity. (English) Zbl 1119.49028

Summary: A certain regularization technique for contact problems leads to a family of problems that can be solved efficiently using infinite-dimensional semismooth Newton methods, or in this case equivalently, primal-dual active set strategies. We present two procedures that use a sequence of regularized problems to obtain the solution of the original contact problem: first-order augmented Lagrangian, and path-following methods. The first strategy is based on a multiplier-update, while path-following with respect to the regularization parameter uses theoretical results about the path-value function to increase the regularization parameter appropriately. Comprehensive numerical tests investigate the performance of the proposed strategies for both a 2D as well as a 3D contact problem.

MSC:

49M15 Newton-type methods
74M15 Contact in solid mechanics
49M37 Numerical methods based on nonlinear programming
65K05 Numerical mathematical programming methods
49S05 Variational principles of physics

Software:

FEMLAB
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Alart, P.; Curnier, A., A mixed formulation for frictional contact problems prone to Newton like solution methods, Comput. Methods Appl. Mech. Eng., 92, 353-375 (1991) · Zbl 0825.76353
[2] Alberty, J.; Carstensen, C.; Funken, S. A.; Klose, R., Matlab implementation of the finite element method in elasticity, Computing, 69, 3, 239-263 (2002) · Zbl 1239.74092
[3] Bertsekas, D. P., Constrained optimization and Lagrange multiplier methods. Computer Science and Applied Mathematics (1982), Academic Press Inc.: Academic Press Inc. New York
[4] Carstensen, C.; Funken, S. A., Averaging technique for FE—a posteriori error control in elasticity. I. Conforming FEM, Comput. Methods Appl. Mech. Eng., 190, 18-19, 2483-2498 (2001) · Zbl 0981.74063
[5] COMSOL Group. FEMLAB: Multiphysics Modeling. \( \langle;\) http://www.femlab.com \(\rangle;\); COMSOL Group. FEMLAB: Multiphysics Modeling. \( \langle;\) http://www.femlab.com \(\rangle;\)
[6] Forsgren, A.; Gill, P. E.; Wright, M. H., Interior methods for nonlinear optimization, SIAM Rev., 44, 4, 525-597 (2003) · Zbl 1028.90060
[7] Glowinski, R., Numerical Methods for Nonlinear Variational Problems, (Springer Series in Computational Physics (1984), Springer: Springer New York) · Zbl 0456.65035
[8] Haslinger, J.; Dostál, Z.; Kučera, R., On a splitting type algorithm for the numerical realization of contact problems with Coulomb friction, Comput. Methods Appl. Mech. Eng., 191, 21-22, 2261-2281 (2002) · Zbl 1131.74344
[9] Hintermüller, M.; Ito, K.; Kunisch, K., The primal-dual active set strategy as a semismooth Newton method, SIAM J. Optim., 13, 3, 865-888 (2003) · Zbl 1080.90074
[10] Hintermüller, M.; Kovtunenko, V.; Kunisch, K., Semismooth Newton methods for a class of unilaterally constrained variational inequalities, Adv. Math. Sci. Appl., 14, 2, 513-535 (2004) · Zbl 1083.49023
[11] M. Hintermüller, K. Kunisch, Path-following methods for a class of constrained minimization problems in function space, Technical Report TR04-11, CAAM, Rice University, 2004, SIAM J. Optim. 17 (1) (2006) 159-187.; M. Hintermüller, K. Kunisch, Path-following methods for a class of constrained minimization problems in function space, Technical Report TR04-11, CAAM, Rice University, 2004, SIAM J. Optim. 17 (1) (2006) 159-187.
[12] Huang, H. C.; Han, W.; Zhou, J. S., The regularization method for an obstacle problem, Numer. Math., 69, 2, 155-166 (1994) · Zbl 0817.65050
[13] Hüeber, S.; Wohlmuth, B., A primal-dual active set strategy for non-linear multibody contact problems, Comput. Methods Appl. Mech. Eng., 194, 3147-3166 (2005) · Zbl 1093.74056
[14] Ito, K.; Kunisch, K., Augmented Lagrangian methods for nonsmooth, convex optimization in Hilbert spaces, Nonlinear Anal., Ser. A: Theory Methods, 41, 5-6, 591-616 (2000) · Zbl 0971.49014
[15] Ito, K.; Kunisch, K., Semi-smooth Newton methods for variational inequalities of the first kind, M2AN Math. Model. Numer. Anal., 37, 1, 41-62 (2003) · Zbl 1027.49007
[16] N. Kikuchi, J.T. Oden, Contact problems in elasticity: a study of variational inequalities and finite element methods, SIAM Studies in Applied Mathematics, vol. 8, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 1988.; N. Kikuchi, J.T. Oden, Contact problems in elasticity: a study of variational inequalities and finite element methods, SIAM Studies in Applied Mathematics, vol. 8, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 1988. · Zbl 0685.73002
[17] Kinderlehrer, D., Remarks about Signorini’s problem in linear elasticity, Ann. Scuola Norm. Sup. Pisa, 4, 8, 605-645 (1981) · Zbl 0482.73017
[18] Kornhuber, R.; Krause, R., Adaptive multigrid methods for Signorini’s problem in linear elasticity, Comp. Vis. Sci., 4, 1, 9-20 (2001) · Zbl 1051.74045
[19] R. Krause, Monotone multigrid methods for Signorini’s problem with friction, Ph.D. Thesis, FU Berlin, 2001.; R. Krause, Monotone multigrid methods for Signorini’s problem with friction, Ph.D. Thesis, FU Berlin, 2001.
[20] Schöberl, J., Efficient contact solvers based on domain decomposition techniques, Comput. Math. Appl., 42, 8-9, 1217-1228 (2001), (Numerical methods and computational mechanics (Miskolc, 1998)) · Zbl 1048.74052
[21] 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.; 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.
[22] Ulbrich, M., Semismooth Newton methods for operator equations in function spaces, SIAM J. Optim., 13, 3, 805-842 (2003) · Zbl 1033.49039
[23] M. Weiser, Function space complementarity methods for optimal control problems, Ph.D. Thesis, Freie Universität Berlin, 2001.; M. Weiser, Function space complementarity methods for optimal control problems, Ph.D. Thesis, Freie Universität Berlin, 2001. · Zbl 1014.49022
[24] Weiser, M., Interior point methods in function space, SIAM J. Control Optim., 44, 5, 1766-1786 (2005) · Zbl 1132.49024
[25] Wriggers, P., Computational Contact Mechanics (2002), Wiley: Wiley New York
[26] Ye, Y., Interior Point Algorithms, (Wiley-Interscience Series in Discrete Mathematics and Optimization (1997), Wiley: Wiley New York) · Zbl 0814.90096
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.