Hostname: page-component-cd9895bd7-8ctnn Total loading time: 0 Render date: 2024-12-23T14:29:38.724Z Has data issue: false hasContentIssue false
  • English
  • Français

A Domain Decomposition Algorithm for Contact Problems:Analysis and Implementation

Published online by Cambridge University Press:  27 January 2009

J. Haslinger ,
R. Kučera  and
T. Sassi
J. Haslinger
Affiliation:
Department of Numerical Mathematics, Charles University Prague, 186 75 Prague, CZ
R. Kučera*
Affiliation:
Department of Mathematics and Descriptive Geometry, VŠB-TU Ostrava, 708 33 Ostrava, CZ
T. Sassi
Affiliation:
Department of Mathematics, University of Basse-Normandie, 14032 Caen, France
*
[email protected]
Get access

Abstract

The paper deals with an iterative method for numerical solving frictionlesscontact problems for two elastic bodies. Each iterative step consists of aDirichlet problem for the one body, a contact problem for the other one and twoNeumann problems to coordinate contact stresses. Convergence is proved by theBanach fixed point theorem in both continuous and discrete case. Numericalexperiments indicate scalability of the algorithm for some choices of therelaxation parameter.

Keywords

contact problemdomain decomposition method
Type
Research Article
Information
Mathematical Modelling of Natural Phenomena , Volume 4 , Issue 1: Modelling and numerical methods in contact mechanics , 2009 , pp. 123 - 146
Copyright
© EDP Sciences, 2009

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

L. Baillet, T. Sassi. Simulations numériques de différentes méthodes d'éments finis pour les problémes contact avec frottement. C. R. Acad. Sci, Paris, Ser. IIB, 331 (2003),789–796.
G. Bayada, J. Sabil, T. Sassi. Algorithme de décomposition de domaine pour un probléme de Signorini sans frottement. C. R. Acad. Sci. Paris, Ser. I335 (2002), 381–386.
Bayada, G., Sabil, J., Sassi, T.. A Neumann-Neumann domain decomposition algorithm for the Signorini problem. Appl. Math. Letters, 17 (2004), 1153–1159. CrossRef
Bjorstad, P. E., Widlund, O. B.. Iterative methods for the solution of elliptic problems on regions partitioned into substructures. SIAM J. Numerical Analysis, 23 (1986), No. 6, 1097–1120. CrossRef
Christensen, P. W., Klarbring, A., Pang, J. S., Strömberg, N.. Formulation and comparison of algorithms for frictional contact problems. Internat. J. Numer. Methods Engrg., 42 (1998), No. 1, 145–173. 3.0.CO;2-L>CrossRef
Dostál, Z., Schöberl, J.. Minimizing quadratic functions over non-negative cone with the rate of convergence and finite termination. Comput. Optim. Appl., 30 (2005), No. 1, 23–44. CrossRef
Eck, C., Wohlmuth, B.. Convergence of a Contact-Neumann iteration for the solution of two-body contact problems. Mathematical Models and Methods in Applied Sciences, 13 (2003), No. 8, 1103-1118. CrossRef
R. Glowinski, J. L. Lions, R. Trémoliére. Numerical analysis of variational inequalities. Studies in Mathematics and its Applications, Volume VIII, North-Holland, Amsterdam, 1981.
G. H. Golub, C. F. Van Loan. Matrix computation. The Johns Hopkins University Press, Baltimore, 1996.
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. Engrg., 191 (2002), No. 21-22, 2261–2281. CrossRef
J. Haslinger, I. Hlaváček, J. Nečas. Numerical methods for unilateral problems in solid mechanics. Handbook of Numerical Analysis, Volume IV, Part 2, North Holland, Amsterdam, 1996.
M. A. Ipopa. Algorithmes de Décomposition de Domaine pour les problémes de Contact: Convergence et simulations numériques. Thesis, Université de Caen, 2008.
N. Kikuchi, J. T. Oden. Contact problems in elasticity: A study of variational inequalities and finite element methods. SIAM, Philadelphia, 1988.
Kornhuber, R., Krause, R.. Adaptive multigrid methods for Signorini's problem in linear elasticity. Comput. Vis. Sci., 4 (2001), No. 1, 9–20., CrossRef
Krause, R., Wohlmuth, B.. A Dirichlet-Neumann type algorithm for contact problems with friction. Comput. Vis. Sci., 5 (2002), No. 3, 139–148. CrossRef
Le Tallec, P.. Domain decomposition methods in computational mechanics. Comput. Mech. Adv., 1 (1994), No. 2, 121–220.
J. Sabil. Modélisation et méthodes de décomposition de domaine pour des problémes de contact. Thesis, INSA de Lyon, 2004.