Nonconforming finite element analysis for a plate contact problem

SIAM Journal on Numerical Analysis

Volumn 40, Issue 5, 2002, Pages 1683-1697

  Han, Weimin ^a   Wang, Lie Heng ^b  

^a UNIVERSITY OF IOWA   (United States)

^b INSTITUTE OF GEOLOGY AND GEOPHYSICS   (China)

Author keywords

Elliptic variational inequality of fourth order; Nonconforming finite element method; Optimal order error estimate; Plate frictional contact problem

Indexed keywords

CONVERGENCE OF NUMERICAL METHODS; ERROR ANALYSIS; MATHEMATICAL MODELS; NON NEWTONIAN FLOW; NONLINEAR EQUATIONS; OPTIMAL CONTROL SYSTEMS; VARIATIONAL TECHNIQUES;

FRICTIONAL CONTACT;

FINITE ELEMENT METHOD;

EID: 0344464853     PISSN: 00361429     EISSN: None     Source Type: Journal    
DOI: 10.1137/S0036142901390731     Document Type: Article

References (30)

1. C. BAIOCCHI AND A. CAPELO, Variational and Quasivariational Inequalities: Applications to Free-Boundary Problems, John Wiley, New York, 1984.
2. G.P. BAZELEY, Y.K. CHEUNG, B.M. IRONS, AND O.C. ZIENKIEWIOZ, Triangular elements in bending - conforming and non - conforming solutions, in Proceedings of the Conference on Matrix Methods in Structural Mechanics, Air Force Institute of Technology, Wright-Patterson Air Force Base, OH, 1965.
3. S.C. BRENNER AND L.R. SCOTT, The Mathematical Theory of Finite Element Methods, Springer-Verlag, New York, 1994.
4. F. BREZZI, W.W. HAGER, AND P.A. RAVIART, Error estimates for the finite element solution of variational inequalities, Part I. Primal theory, Numer. Math., 28 (1977), pp. 431-443.
5. P.G. CIARLET, The Finite Element Method for Elliptic Problems, North-Holland, Amsterdam, 1978.
6. P.G. CIARLET, Basic error estimates for elliptic problems, in Handbook of Numerical Analysis, Vol. II, P.G. Ciarlet and J.-L. Lions, eds., North-Holland, Amsterdam, 1991, pp. 17-351.
7. I. DOGHRI, Mechanics of Deformable Solids, Springer-Verlag, Berlin, 2000.
8. G. DUVAUT AND J.-L. LIONS, Inequalities in Mechanics and Physics, Springer-Verlag, Berlin, 1976.
9. R. FALK, Error estimates for the approximation of a class of variational inequalities, Math. Comput., 28 (1974), pp. 963-971.
10. A. FRIEDMAN, Variational Principles and Free-boundary Problems, John Wiley, New York, 1982.
11. V. GIRAULT AND P.A. RAVIART, Finite Element Methods for Navier-Stokes Equations, Theory and Algorithms, Springer-Verlag, New York, 1986.
12. R. GLOWINSKI, Numerical Methods for Nonlinear Variational Problems, Springer-Verlag, New York, 1984.
13. R. GLOWINSKI, J.-L. LIONS, AND R. TRÉMOLIÈRES, Numerical Analysis of Variational Inequalities, North-Holland, Amsterdam, 1981.
14. W. HAN AND B.D. REDDY, Plasticity: Mathematical Theory and Numerical Analysis, Springer-Verlag, New York, 1999.
15. W. HAN AND M. SOFONEA, Quasiatatic Contact Problems in Viscoelasticity and Viscoplasticity, AMS and International Press, 2002.
16. J. HASLINGER, I. HLAVÁČEK, AND J. NEČAS, Numerical methods for unilateral problems in solid mechanics, in Handbook of Numerical Analysis, Vol. IV, P.G. Ciarlet and J.-L. Lions, eds., North-Holland, Amsterdam, 1996, pp. 313-485.
17. I. HLAVÁČEK, J. HASLINGER, J. NEČAS, AND J. LOVÍŠEK, Solution of Variational Inequalities in Mechanics, Springer-Verlag, New York, 1988.
18. B.M. IRONS AND A. RAZZAQUE, Experience with the patch test for convergence of finite element methods, in Mathematical Foundations of the Finite Element Method with Applications to Partial Differential Equations, A.R. Aziz, ed., Academic Press, New York, 1972, pp. 557-587.
19. N. KIKUCHI AND J.T. ODEN, Contact Problems in Elasticity: A Study of Variational Inequalities and Finite Element Methods, SIAM Stud. Appl. Math. 8, SIAM, Philadelphia, 1988.
20. D. KINDERLEHRER AND G. STAMPAOCHIA, An Introduction to Variational Inequalities and Their Applications, Academic Press, New York, 1980.
21. P. LASCAUX AND P. LESAINT, Some nonconforming finite elements for the plate bending problem, Rev. Française Automat. Informat. Recherche Opérationnelle Sér. Rouge Anal. Numér., 9 (1975), pp. 9-53.
22. P.D. PANAGIOTOPOULOS, Inequality Problems in Mechanics and Applications, Birkhäuser, Boston, 1985.
23. J.F. RODRIGUES, Obstacle Problems in Mathematical Physics, North-Holland, Amsterdam, 1987.
24. Z.C. SHI, The generalized patch test for Zienkiewicz's triangles, J. Comput. Math., 2 (1984), pp. 279-286.
25. Z.C. SHI, The F-E-M-Test for nonconforming finite elements, Math. Comp., 49 (1987), pp. 391-405.
26. F. STUMMEL, The generalized patch test, SIAM J. Numer. Anal., 16 (1979), pp. 449-471.
27. L.H. WANG, Morley's element approximation to a fourth order variational inequality with curvature obstacle, Math. Numer. Sinica, 12 (1990), pp. 279-284.
28. L.H. WANG, Some nonconforming finite element approximations of a fourth order Variational inequality with displacement obstacle, Math. Numer. Sinica, 12 (1990), pp. 352-356.
29. L.H. WANG, Some strongly discontinuous nonconforming finite element approximations for a fourth order variational inequality with displacement obstacle, Math. Numer. Sinica, 14 (1992), pp. 98-101.
30. M. WANG, On the necessity and sufficiency of the patch test for convergence of nonconforming finite elements, SIAM J. Numer. Anal., 39 (2001), pp. 363-384.