Existence of solutions to discrete semicoercive frictional contact problems

SIAM Journal on Optimization

Volumn 8, Issue 2, 1998, Pages 414-442

  Klarbring, Anders ^a   Pang, Jong Shi ^b  

^a LINKÖPING UNIVERSITY   (Sweden)

^b JOHNS HOPKINS UNIVERSITY   (United States)

Author keywords

Complementarity problems; Contact problems; Coulomb friction; Quasi variational inequality; Solution existence

Indexed keywords

EID: 0032222082     PISSN: 10526234     EISSN: None     Source Type: Journal    
DOI: 10.1137/S105262349629784X     Document Type: Article

References (55)

1. n dans luimême; application a la mécanique du contact, Math. Modelling Numer. Anal., 27 (1993), pp. 203-222.
2. A. ALART, F. LEBON, F. QUITTAU, AND K. REY, Frictional contact problem in elastostatics: Revisiting the uniqueness condition, in Contact Mechanics, M. Raous, M. Jean, and J. J. Moreau, eds., Plenum Press, New York, 1995, pp. 63-70.
3. A. M. AL-FAHED AND P. D. PANAGIOTOPOULOS, Multifingered frictional robot grippers: A new type of numerical implementation, Comput. & Structures, 42 (1992), pp. 555-562.
4. A. M. AL-FAHED, G. E. STAVROLAKIS, AND P. D. PANAGIOTOPOULOS, Hard and soft fingered robot grippers. The linear complementarity approach, Z. Angew. Math. Mech., 71 (1991), pp. 257-265.
5. L.-E. ANDERSSON, A quasistatic frictional problem with normal compliance, Nonlinear Anal., 16 (1991), pp. 347-369.
6. L.-E. ANDERSSON, A global existence result for a quasistatic contact problem with friction, Adv. Math. Sci. Appl., 5 (1995), pp. 249-286.
7. G. BONFANTI, A noncoercive friction problem with tangential applied forces in three dimensions, Boll. Un. Mat. Ital. B (7), 7 (1993), pp. 149-165.
8. D. CHAN AND J. S. PANG, The generalized quasi-variational inequality problem, Math. Oper. Res., 7 (1982), pp. 211-222.
9. P. W. CHRISTENSEN, A. KLARBRING, J. S. PANG, AND N. STROMBERG, Formulation and comparison of algorithms for fractional contact problems, Internat. J. Num. Methods Engrg., to appear.
10. P. W. CHRISTENSEN AND J. S. PANG, Fractional contact algorithms based on semismooth Newton methods, in Nonsmooth, Piecewise Smooth, Semismooth and Smoothing Method, M. Fukushina and L. Qi, eds., Kluwer Academic Publishers, Norwell, MA, to appear.
11. M. COCU, Existence of solutions of Signorini problems with friction, Internat. J. Engrg. Sci., 22 (1984), pp. 567-575.
12. R. W. COTTLE, J. S. PANG, AND R. E. STONE, The Linear Complementarity Problem, Academic Press, Boston, MA, 1992.
13. L. DEMKOWICZ AND J. T. ODEN, On some existence and uniqueness results in contact problems with nonlocal friction, Nonlinear Anal., 6 (1982), pp. 1075-1093.
14. I. DOUDOUMIS, E. MITSOPOULOU, AND N. CHARALAMBAKIS, The influence of the friction coefficients on the uniqueness of the solution of the unilateral contact problem, in Contact Mechanics, M. Raous, M. Jean, and J. J. Moreau, eds., Plenum Press, New York, 1995, pp. 63-70.
15. G. DUVAUT, Equilibre d'un solide élastique avec contact unilatéral et frottement de Coulomb, Comptes rendues Academie Sciences, Paris/A, 290 (1980), pp. 263-265.
16. G. DUVAUT AND J. L. LIONS, Inequalities in Mechanics and Physics, Springer, Berlin, 1976.
17. G. FICHERA, Boundary value problems in elasticity with unilateral constraints, in Encyclopedia of Physics, Vol. VI a/2, S. Flügge, ed., Springer-Verlag, Berlin, 1972.
18. F. GASTALDI, Remarks on a Noncoercive Contact Problem with Friction in Elastostatics, Tech. report 649, Istituto di analisi numerica, University of Pavia, Pavia, 1988.
19. F. GASTALDI, M. D. P. MONTEIRO MARQUES, AND J. A. C. MARTINS, A two degree-of-freedom quasistatic contact problem with viscous damping, Adv. Math. Sci. Appl., submitted.
20. F. GASTALDI, M. D. P. MONTEIRO MARQUES, AND J. A. C. MARTINS, Mathematical analysis of a two degree-of-freedom frictional contact problem with discontinuous solutions, Nonlinear Differential Equations Appl., submitted.
21. I. HLAVÁĆEK, J. HASLINGER, J. NEČAS, AND J. LOVÍČEK, Numerical Solution of Variational Inequalities, Springer Series in Applied Mathematical Sciences 66, Springer-Verlag, New York, 1988.
22. V. JANOVSKÝ, Catastrophic Features of Coulomb Friction Model, Tech. report KNM-0105044/80, Department of Numerical Analysis, Charles University, Prague, Czech Republic, 1980.
23. V. JANOVSKÝ, Catastrophic features of Coulomb friction model, in The Mathematics of Finite Elements and Applications, J. R. Whiteman, ed., Academic Press, London, 1981, pp. 259-264.
24. J. JARUŠEK, Contact problems with friction. Coercive case, Czechoslovak Math. J., 33 (1983), pp. 237-261.
25. J. JARUŠEK, Contact problems with friction. Semicoercive case, Czechoslovak Math. J., 34 (1984), pp. 619-629.
26. J. H. JELLET, A Treatise on the Theory of Friction, Macmillan and Co., London, 1872.
27. J. J. KALKER, Three-dimensional elastic bodies in rolling contact, Kluwer, Dordrecht, 1990.
28. Y. KATO, Signorini's problem with friction in linear elasticity, Japan J. Appl. Math., 4 (1987), pp. 237-268.
29. N. KIKUCHI AND J. T. ODEN, Contact Problems in Elasticity: A Study of Variational Inequalities and Finite Element Methods, SIAM Studies Appl. Math., Philadelphia, PA, 1988.
30. A. KLARBRING, Contact Problems with Friction - Using a Finite Dimensional Description and the Theory of Linear Complementarity, Thesis no. 20, LIU-TEK-LIC-1984:3, Linköping Studies Sci. Tech., Linköping Institute of Technology, Linköping, Sweden, 1984.
31. A. KLARBRING, A mathematical programming approach to three-dimensional contact problems with friction, Comput. Methods Appl. Mech. Engrg, 58 (1986), pp. 175-200.
32. A. KLARBRING, Contact problems with friction by linear complementarity, in Unilateral Problems in Structural Analysis, Vol. 2 (CISM Courses and Lectures, No. 304), G. Del Piero and F. Maceri, eds., Springer, Wien, 1987, pp. 197-219.
33. A. KLARBRING, Derivation and analysis of rate boundary-value problems with friction, European J. Mech. A Solids, 1 (1990), pp. 211-226.
34. A. KLARBRING, Examples of non-uniqueness and non-existence of solutions to quasistatic contact problems with friction, Ingenieur-Archiv, 60 (1990), pp. 529-541.
35. A. KLARBRING, Mathematical programming and augmented Lagrangian methods for frictional contact problems, in Proc. Contact Mechanics International Symposium, A. Curnier, ed., Presses Polytechniques et Universitaires Romandes, Lausanne, Switzerland, 1992, pp. 409-422.
36. A. KLARBRING, Mathematical programming in contact problems, in Computational Methods in Contact Mechanics, M. H. Aliabadi and C. A. Brebbia, eds., Computational Mechanics Publications, Southampton, UK, 1993, pp. 233-263.
37. A. KLARBRING AND G. BJÖRKMAN, A mathematical programming approach to contact problems with friction and varying contact surface, Comput. & Structures, 30 (1988), pp. 1185-1198.
38. A. KLARBRING, A. MIKELIĆ, AND M. SHILLOR, Frictional contact problems with normal compliance, Internat. J. Engrg. Sci., 26 (1988), pp. 811-832.
39. A. KLARBRING, A. MIKELIĆ, AND M. SHILLOR, On friction problems with normal compliance, Nonlinear Anal., 13 (1989), pp. 935-955.
40. A. KLARBRING, A. MIKELIĆ, AND M. SHILLOR, A global existence result for the quasistatic frictional contact problem with normal compliance, Internat. Ser. Numer. Math. 101, Birkhäuser, Basel, 1991, pp. 85-111.
41. J. KYPARISIS, On uniqueness of Kuhn-Tucker multipliers in nonlinear programming, Math. Programming, 32 (1985), pp. 242-246.
42. O. L. MANGASARIAN, A simple characterization of solution sets of convex programs, Oper. Res. Lett., 7 (1988), pp. 21-26.
43. J. A. C. MARTINS, M. D. P. MONTEIRO MARQUES, AND F. GASTALDI, On an example of non-existence of solutions to a quasistatic frictional contact problem, European J. Mech. A Solids, 13 (1994), pp. 113-133.
44. J. A. C. MARTINS AND J. T. ODEN, Existence and uniqueness results for dynamic contact problems with nonlinear normal and friction interface laws, Nonlinear Anal., 11 (1987), pp. 407-428.
45. J. A. C. MARTINS, F. M. F. SIMÕES, F. GASTALDI, AND M. D. P. MONTEIRO MARQUES, Dissipative graph solutions for a 2 degree-of-freedom quasistatic frictional contact problem, Internat. J. Engrg. Sci., 33 (1995), pp. 1959-1986.
46. M. D. P. MONTEIRO MARQUES, Differential Inclusions in Nonsmooth Mechanical Problems: Shocks and Dry Friction, Birkhäuser, Basel, 1993.
47. J. J. MOREAU, Unilateral contact and dry friction in finite freedom dynamics, in Nonsmooth Mechanics and Applications, J. J. Moreau and P. D. Panagiotopoulos, eds., Springer-Verlag, Berlin, 1988, pp. 1-81.
48. J. NEC̃AS, J. JARUŠEK, AND J. HASLINGER, On the solution of the variational inequality to the Signorini problem with small friction, Boll. Un. Mat. Ital. B(5), 17 (1980), no. 2, pp. 796-811.
49. P. D. PANAGIOTOPOULOS AND A. M. AL-FAHED, Robot hand grasping and related problems: Optimal control and identification, Internat. J. Robotics Res., 13 (1994), pp. 127-136.
50. J. S. PANG AND J. C. TRINKLE, Complementarity formulations and existence of solutions of multi-rigid-body contact problems with Coulomb friction, Math. Programming, 73 (1996), pp. 199-226.
51. J. S. PANG, J. C. TRINKLE, AND G. LO, A complementarity approach to a quasi-static multi-rigid-body contact problem, Comput. Optim. Appl., 5 (1996), pp. 139-154.
52. M. RAOUS, M. JEAN, AND J. J. MOREAU, EDS., Contact Mechanics, Plenum Press, New York, 1995.
53. S. M. ROBINSON, Some continuity properties of polyhedral multifunctions, Math. Programming Study, 14 (1981), pp. 206-214.
54. J. C. TRINKLE, J. S. PANG, S. SUDARSKY, AND G. LO, On dynamic multi-rigid-body contact problems with Coulomb friction, Z. Angew. Math. Mech., 77 (1997), pp. 267-279.
55. L. K. VY AND D. D. ANG, Frictional contact of an elastic body with a rigid support, Nonlinear Anal., 25 (1995), pp. 339-343.