First locking-free plane-elastic finite element: Historia mathematica

Computer Methods in Applied Mechanics and Engineering

Volumn 190, Issue 11-12, 2000, Pages 1323-1366

  Pitkäranta, Juhani ^a  

^a AALTO UNIVERSITY   (Finland)

Author keywords

[No Author keywords available]

Indexed keywords

APPROXIMATION THEORY; ELASTICITY; ERROR ANALYSIS; STRAIN;

CONFORMING BILINEAR QUADRILATERAL FINITE ELEMENT FORMULATION;

FINITE ELEMENT METHOD;

EID: 0342995757     PISSN: 00457825     EISSN: None     Source Type: Journal    
DOI: 10.1016/S0045-7825(00)00163-8     Document Type: Article

References (32)

1. I. Babuŝka, A.K. Aziz, On the angle condition in the finite element method, SIAM J. Numer. Anal. 13 (1976) 214-226.
2. T. Belytschko, W.E. Bachrach, Efficient implementation of quadrilaterals with high coarse-mesh accuracy, Comput. Meth. Appl. Mech. Engrg. 54 (1986) 279-301.
3. T. Belytschko, W.K. Liu, B.E. Engelman, The gamma-elements and related developments, in: T.J.R. Hughes, E. Hinton (Eds.), Finite Element Methods for Plate and Shell Problems, Pineridge, Swansea, vol. 2, 1986, pp. 316-347.
4. D. Braess, Finite Elements, Cambridge University Press, Cambridge, 1997.
5. S. Brenner, L.R. Scott, The Mathematical Theory of Finite Element Methods, Springer, Berlin, 1994.
6. P.G. Ciarlet, The Finite Element Method for Elliptic Problems, North-Holland, Amsterdam, 1978.
7. R.W. Clough, Original formulation of the finite element method, Finite Elements Anal. Design 7 (1990) 89-101.
8. R. Courant, Variational methods for the solution of problems of equilibrium and vibration, Bull. Amer. Math. Soc. 49 (1943) 1-43.
9. M. Fröier, L. Nilsson, A. Samuelsson, The rectangular plane stress element by Turner Pian and Wilson, Int. J. Numer. Meth. Engrg. 8 (1974) 433-437.
10. R.H. Gallagher, Finite Element Analysis Fundamentals, Prentice-Hall, Englewood Cliffs, NJ, 1975.
11. T.J.R. Hughes, The Finite Element Method: Linear Static and Dynamic Finite Element Analysis, Prentice-Hall, Englewood Cliffs, NJ, 1987.
12. B.M. Irons, A. Razzaque, Experience with patch test for convergence of finite elements, in: A.K. Aziz (Ed.), The Mathematical Foundations of the Finite Element Method with Applications to Partial Differential Equations, Academic Press, New York, 1972, pp. 557-587.
13. C. Johnson, J. Pitkäranta, Analysis of some mixed finite element methods related to reduced integration, Math. Comp. 38 (1982) 375-400.
14. P. Lesaint, On the convergence of Wilson's nonconforming element for solving the elastic problem, Comput. Meth. Appl. Mech. Engrg. 7 (1976) 1-16.
15. P. Lesaint, M. Zlámal, Convergence of the nonconforming Wilson element for arbitrary quadrilateral meshes, Numer. Math. 36 (1980) 33-52.
16. R.H. MacNeal, Finite Elements: Their Design and Performance, Marcel Dekker, New York, 1994.
17. R.H. MacNeal, A theorem regarding the locking of tapered four-node membrane elements, Int. J. Numer. Meth. Engrg. 24 (1987) 1793-1799.
18. O. Ovaskainen, J. Pitkäranta, An energy method approach to the problem of elastic strip, SIAM J. Appl. Math. 58 (1998) 999-1021.
19. T.H.H. Pian, Derivation of element stiffnes matrices by assumed stress distributions, AIAA J. 2 (1964) 1333-1336.
20. J. Pitkäranta, The problem of membrane locking in finite element analysis of cylindrical shells, Numer. Math. 61 (1992) 523-542.
21. J. Pitkäranta, Y. Leino, O. Ovaskainen, J. Piila, Shell deformation states and the finite element method: A benchmark study of cylindrical shells, Comput. Meth. Appl. Mech. Engrg. 133 (1996) 157-182.
22. J. Pitkäranta, R. Stenberg, Error bounds for the approximation of the Stokes problem using bilinear/constant elements on irregular quadrilateral meshes, in: J.R. Whiteman (Ed.), The Mathematics of Finite Elements and Applications V MAFELAP 1984, Academic Press, New York, 1985, pp. 325-334.
23. J. Pitkäranta, M. Suri, Design principles and error analysis for reduced-shear plate-bending finite elements, Numer. Math. 75 (1996) 223-266.
24. J. Pitkäranta, M. Suri, Upper and lower error bounds for some plate-bending finite elements, Numer. Math. 84 (2000) 611-648.
25. J. Robinson, The Life And Work of Early FEM Pioneers, Robinson and Associates, 1985.
26. Z.C. Shi, A convergence condition for the quadrilateral Wilson element, Numer. Math. 44 (1984) 349-361.
27. Z.C. Shi, B. Jiang, W. Xue, A new superconvergence property of Wilson nonconforming finite element, Numer. Math. 78 (1997) 259-268.
28. G. Strang, Variational crimes in the finite element method, in: A.K. Aziz (Ed.), The Mathematical Foundations of the Finite Element Method with Applications to Partial Differential Equations, Academic Press, New York, 1972, pp. 689-710.
29. G. Strang, G.J. Fix, An Analysis of the Finite Element Method, Prentice-Hall, Englewood Cliffs, NJ, 1973.
30. R.L. Taylor, P.J. Beresford, E.L. Wilson, A nonconforming element for stress analysis, Int. J. Numer. Meth. Engrg. 10 (1976) 1211-1219.
31. M.J. Turner, R.W. Clough, H.C. Martin, L.J. Topp, Stiffness and deflection analysis of complex structures, J. Aeronaut. Sci. 23 (1956) 805-823.
32. E.L. Wilson, R.L. Taylor, W. Doherty, J. Ghaboussi, Incompatible displacement models, in: S.J. Fenves et al. (Eds.), Numerical and Computer Methods in Structural Mechanics, Academic Press, New york, 1973, pp. 43-57.