×

Fixed point strategies for elastostatic frictional contact problems. (English) Zbl 1132.74032

Summary: Several fixed point strategies and Uzawa algorithms (for classical and augmented Lagrangian formulations) are presented to solve the unilateral contact problem with Coulomb friction. These methods are analysed, without introducing any regularization, and a theoretical comparison is performed. Thanks to a formalism coming from convex analysis, some new fixed point strategies are presented and compared with known methods. The analysis is first performed on continuous Tresca problem, and then on the finite dimensional Coulomb problem derived from an arbitrary finite element method.

MSC:

74M15 Contact in solid mechanics
74M10 Friction in solid mechanics
74S05 Finite element methods applied to problems in solid mechanics
74G25 Global existence of solutions for equilibrium problems in solid mechanics (MSC2010)
PDF BibTeX XML Cite
Full Text: DOI Link

References:

[1] . Les Inéquations en Mécanique et en Physique. Dunod: Paris, 1972.
[2] Boundary value problems of elasticity. Handbuch des Physik VI a/2. Springer, Dunod: Paris, 1972; 391–424.
[3] Renard, SIAM Journal on Mathematical Analysis 38 pp 452– (2006)
[4] Nečas, Bolletino U.M.I. 5 pp 796– (1980)
[5] , . Numerical methods for unilateral problems in solids mechanics. Handbook of Numerical Analysis, vol. IV. Elsevier Science: Amsterdam, 1996; 313–485.
[6] . Contact Problems in Elasticity. SIAM: Philadelphia, PA, 1988.
[7] Sobolev Spaces. Academic Press: New York, 1975. · Zbl 0314.46030
[8] Andersson, Applied Mathematics and Optimisation 42 pp 169– (2000)
[9] Jarušek, Czechoslovak Mathematical Journal 33 pp 237– (1983)
[10] Jarušek, Czechoslovak Mathematical Journal 34 pp 619– (1984)
[11] Eck, Mathematical Models and Methods in Applied Sciences 8 pp 445– (1998)
[12] Hild, Comptes Rendus de l Academie des Sciences, Paris 337 pp 685– (2003)
[13] Hild, Quarterly Journal of Mechanics and Applied Mathematics 57 pp 225– (2004)
[14] Alart, Computer Methods in Applied Mechanics and Engineering 92 pp 353– (1991)
[15] . Variational Analysis. Springer: Berlin, 1998. · Zbl 0888.49001
[16] Programmation Mathématique, Théorie et Algorithmes. Dunod: Paris, 1983.
[17] Infinite-dimensional semi-smooth newton and augmented Lagrangian methods for friction and contact problems in elasticity. Ph.D. Thesis, University of Graz, 2004.
[18] Stadler, Journal of Computational and Applied Mathematics 203 pp 533– (2007)
[19] Khenous, Applied Numerical Mathematics 56 pp 163– (2006)
[20] Opérateurs Maximaux Monotones et Semi-groupes de Contractions Dans les Espaces de Hilbert. North-Holland: Amsterdam, 1973. · Zbl 0252.47055
[21] Rockafellar, SIAM Journal on Control and Optimization 14 pp 877– (1976)
[22] Ben Belgacem, Mathematics of Computation 72 pp 1117– (2003)
[23] , . Annals of University of Craiova. Mathematical Computations Science Series, vol. 30(1), 2003; 45–52.
[24] Ben Belgacem, Applied Numerical Mathematics 54 pp 1– (2005)
[25] , . Finite Element Method for Hemivariational Inequalities, Theory, Methods and Applications. Kluwer Academic Publishers: Dordrecht, 1999. · Zbl 0949.65069
[26] Hild, Applied Numerical Mathematics 41 pp 401– (2002)
[27] Bernardi, SIAM Journal on Numerical Analysis 35 pp 1893– (1998)
[28] The Finite Element Method for Elliptic Problems, Studies in Mathematics and its Applications. North-Holland: Amsterdam, 1978.
[29] De Saxcé, Comptes Rendus de l Acadmie des Sciences Série II 314 pp 125– (1992)
[30] Bousshine, Mechanical Sciences 44 pp 2189– (2002)
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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.