Monotone multigrid methods on nonmatching grids for nonlinear multibody contact problems

SIAM Journal on Scientific Computing

Volumn 25, Issue 1, 2003, Pages 324-347

  Wohlmuth, Barbara I ^a   Krause, Rolf H ^b  

^a UNIVERSITY OF STUTTGART   (Germany)

^b FREIE UNIVERSITÄT BERLIN   (Germany)

Author keywords

Contact problems; Dual space; Linear elasticity; Monotone methods; Mortar finite elements; Multigrid methods; Nonmatching triangulations

Indexed keywords

APPROXIMATION THEORY; BOUNDARY VALUE PROBLEMS; COMPUTER SIMULATION; FINITE ELEMENT METHOD; ITERATIVE METHODS; LAGRANGE MULTIPLIERS; MATHEMATICAL TRANSFORMATIONS;

DUAL SPACE; GENERALIZED MORTAR METHOD; LINEAR ELASTICITY; MONOTONE MULTIGRID METHODS; MORTAR FINITE ELEMENTS; NONCONFORMING DOMAIN DECOMPOSITION TECHNIQUES; NONLINEAR MULTIBODY CONTACT PROBLEMS; NONMATCHING TRIANGULATIONS;

PARTIAL DIFFERENTIAL EQUATIONS;

EID: 1842564097     PISSN: 10648275     EISSN: None     Source Type: Journal    
DOI: 10.1137/S1064827502405318     Document Type: Article

References (49)

1. H. Bader and R. Hoppe, Multigrid Solution of Signorini Type Problems in Contact Elastostatics, Tech. report TUM-M9304, Technische Universität München, München, 1993.
2. P. Bastian, K. Birken, K. Johannsen, S. Lang, N. Neuss, H. Rentz-Reichert, and C. Wieners, UG - a flexible software toolbox for solving partial differential equations, Computing and Visualization in Science, 1 (1997), pp. 27-40.
3. F. B. Belgacem and Y. Renard, Hybrid finite element methods for the signorini problem, Math. Comp., 72 (2003), pp. 1117-1145.
4. F. Ben Belgacem, The mortar finite element method with Lagrange multipliers, Numer. Math., 84 (1999), pp. 173-197.
5. F. Ben Belgacem, P. Hild, and P. Laborde, Approximation of the unilateral contact problem by the mortar finite element method, C. R. Acad. Sci. Paris Sér. I Math., 324 (1997), pp. 123-127.
6. F. Ben Belgacem, P. Hild, and P. Laborde, Extension of the mortar finite element method to a variational inequality modeling unilateral contact, Math. Models Methods Appl. Sci., 9 (1999), pp. 287-303.
7. C. Bernardi, Y. Maday, and A. Patera, Domain decomposition by the mortar element method, in Asymptotic and Numerical Methods for Partial Differential Equations with Critical Parameters, H. G. Kaper, M. Garbey, and G. W. Pieper, eds., Reidel, Dordrecht, 1993, pp. 269-286.
8. C. Bernardi, Y. Maday, and T. Patera, A new nonconforming approach to domain decomposition: The mortar element method, in Nonlinear Partial Differential Equations and Their Applications, H. Brezis and J. L. Lions, eds., Pitman, Boston, 1994, pp. 13-51.
9. P. Boieri, F. Gastaldi, and D. Kinderlehrer, Existence, uniqueness and regularity results for the two-body contact problem, Appl. Math. Optim., 15 (1987), pp. 251-227.
10. C. Carstensen, O. Scherf, and P. Wriggers, Adaptive finite elements for elastic bodies in contact, SIAM J. Sci. Comput., 20 (1999), pp. 1605-1626.
11. P. Coorevits, P. Hild, and J.-P. Pelle, A posteriori error estimation for unilateral contact with matching and non-matching meshes, Comput. Methods Appl. Mech. Engrg., 186 (2000), pp. 65-83.
12. Z. Dostál, Box constrained quadratic programming with proportioning and projections, SIAM J. Optim., 7 (1997), pp. 871-887.
13. P. Eberhard, Kontaktuntersuchungen durch hybride Mehrköorpersystem/Finite Elemente Simulationen, Shaker Verlag, Aachen, 2000.
14. C. Eck, O. Steinbach, and W. Wendland, A symmetric boundary element method for contact problems with friction, Math. Comput. Simulation, 50 (1999), pp. 43-61.
15. C. Eck and B. Wohlmuth, Convergence of a Contact-Neumann iteration for the solution of two-body contact problems, Math. Models Methods Appl. Sci., 13 (2003), pp. 1-16.
16. R. Glowinski, Numerical Methods for Nonlinear Variational Problems, Springer Series in Computational Physics, Springer-Verlag, New York, 1984.
17. W. Hackbursch and H. Mittelmann, On multigrid methods for variational inequalities, Numer. Math., 42 (1983), pp. 65-76.
18. J. Haslinger and I. Hlavácek, Contact between elastic bodies. I. Continuous problems, Appl. Math., 25 (1980), pp. 324-327.
19. J. Haslinger and I. Hlaváček, Contact between elastic bodies. II. Finite element analysis., Appl. Math., 26 (1981), pp. 263-290.
20. H. Hertz, Über die Berührung fester elastischer Körper, Journal für Mathematik, 92 (1882), pp. 156-171.
21. P. Hild, A propos d'approximation par éléments finis optimale pour les problèmes de contact unilatéral, C. R. Acad. Sci. Paris Sér. I Math., 330 (1998), pp. 1233-1236.
22. P. Hild, Numerical implementation of two nonconforming finite element methods for unilateral contact, Comput. Methods Appl. Mech. Engrg., 184 (2000), pp. 99-123.
23. P. Hild and P. Laborde, Quadratic finite element methods for unilateral contact problems, Appl. Numer. Math., 41 (2002), pp. 401-421.
24. R. H. W. Hoppe, Multigrid algorithms for variational inequalities, SIAM J. Numer. Anal., 24 (1987), pp. 1046-1065.
25. R. H. W. Hoppe and R. Kornhuber, Adaptive multilevel methods for obstacle problems, SIAM J. Numer. Anal., 31 (1994), pp. 301-323.
26. J. N. I. Hlaváček, J. Haslinger, and J. Jovíšek, Solution of Variational Inequalities in Mechanics, Springer, Berlin, 1988.
27. N. Kikuchi and J. T. Oden, Contact Problems in Elasticity: A Study of Variational Inequalities and Finite Element Methods, Studies in Applied Mathematics 8, SIAM, Philadelphia, 1988.
28. C. Kim, R. D. Lazarov, J. E. Pasciak, and P. S. Vassilevski, Multiplier spaces for the mortar finite element method in three dimensions, SIAM J. Numer. Anal, 39 (2001), pp. 519-538.
29. R. Kornhuber, Monotone multigrid methods for elliptic variational inequalities I, Numer. Math., 69 (1994), pp. 167-184.
30. R. Kornhuber, Monotone multigrid methods for elliptic variational inequalities II, Numer. Math., 72 (1996), pp. 481-499.
31. R. Kornhuber, Adaptive Monotone Multigrid Methods for Nonlinear Variational Problems, Teubner-Verlag, Stuttgart, 1997.
32. R. Kornhuber and R. Krause, Adaptive multigrid methods for Signorini's problem in linear elasticity, Computing and Visualization in Science, 4 (2001), pp. 9-20.
33. R. Krause, Monotone Multigrid Methods for Signorini's Problem with Friction, Ph.D. thesis, Freie Universität Berlin, Berlin, 2001.
34. R. Krause and B. Wohlmuth, A Dirichlet-Neumann type algorithm for contact problems with friction, Computing and Visualization in Science, 5 (2002), pp. 139-148.
35. P. Le Tallec, Numerical methods for nonlinear three-dimensional elasticity, in Handbook of Numerical Analysis III, P. G. Ciarlet and J. L. Lions, eds., North-Holland, Amsterdam, 1994, pp. 465-622.
36. K. Lhalouani and T. Sassi, The Mortar Finite Element Method for Some Problems of Unilateral Contact with Non-local Coulomb's Friction, Tech. report UMR 5585, Laboratoire de Mécanique des Contacts, Institute National des Sciences Appliqueés de Lyon, Lyon, France, 1999.
37. P. Oswald and B. Wohlmuth, On polynomial reproduction of dual FE bases, in Thirteenth International Conference on Domain Decomposition Methods, N. Debit, M. Garbey, R. Hoppe, J. Pèriaux, D. Keyes, and Y. Kuznetsov, eds., CIMNE, Barcelona, Spain, 2001, pp 85-96. Available online at http://www.ddm.org/DD13/Oswald.pdf
38. G. Pietrzak and A. Curnier, Large deformation frictional contact mechanics: Continuum formulation and augmented Lagrangian treatment, Comput. Methods Appl. Mech. Engrg., 177 (1999), pp. 351-381.
39. J. L. R. Glowinski and J. Trémolières, Numerical Analysis of Variational Inequalities, North-Holland, Amsterdam, 1981.
40. J. Schöberl, Efficient contact solvers based on domain decomposition techniques, Comput. Math. Appl., 42 (2001), pp. 1217-1228.
41. K. Willner and L. Gaul, Contact description by fem based on interface physics, in Proceedings of the 5th International Computational Plasticity Conference (Spain, 1997), Part I, D. R. J. Owen, E. Oñate, and E. Hinton, eds., CIMNE, "The International Center for Numerical Methods in Engineering," Barcelona, Spain, 1997, pp. 884-891.
42. B. Wohlmuth, A mortar finite element method using dual spaces for the Lagrange multiplier, SIAM J. Numer. Anal., 33 (2000), pp. 989-1014.
43. B. Wohlmuth, A Short Note on: An Optimal A Priori Estimate for Nonlinear Multibody Contact Problems, JANS report 2003/013, University of Stuttgart, Stuttgart, Germany, 2003.
44. B. Wohlmuth, A V-Cycle Multigrid Approach for Mortar Finite Elements, SFB report 2003/33, University of Stuttgart, Stuttgart, Germany, 2003.
45. B. Wohlmuth, Discretization Methods and Iterative Solvers Based on Domain Decomposition, Lecture Notes in Comput. Sci. Engrg. 17, Springer, Heidelberg, 2001.
46. B. Wohlmuth, A comparison of dual Lagrange multiplier spaces for mortar finite element discretizations, Math. Model. Numer. Anal., 36 (2002), pp. 995-1012.
47. P. Wriggers, Finite element algorithms for contact problems, Arch. Comput. Methods Engrg., 2 (1995), pp. 1-49.
48. P. Wriggers, Finite elements for the thermomechanical contact and adaptive finite element analysis of contact problems., in New developments in contact problems., CISM Courses Lect., Springer, 1999, pp. 179-246.
49. J. Zou, A new fast solver - monotone MG method (MMG), J. Comput. Math., 5 (1987), pp. 325-335.