On the numerical solution of plate bending problems by hybrid methods

 Access to full text

 Full (PDF)

1. [1] A. ADINI and R. W. CLOUGH, Analysis of plate bending by the finite element method, NSF Report G. 7337, 1961. 
2. [2] F. K. BOGNER, R. L. FOX and L. A. SCHMIT, The generation of interelement compatible stiffness and mass matrices by the use of interpolation formulas, Conference on Matric Methods in Structural Mechanics, Wright Patterson A.F.B., Ohio, 1965. 
3. [3] J. H. BRAMBLE and R. S. HILBERT, Estimation of linear functionals on Sobolev Spaces with applications to Fourier transforms andspline interpolation, SIAM J. Numer. Anal. 7, pp. 112-124, 1970. Zbl0201.07803MR263214
4. [4] J. H. BRAMBLE and R. S. HILBERT, Bounds for a class of linear functionals with applications to Hermite interpolation, Numer. Math., 16, pp.362-369, 1971. Zbl0214.41405MR290524
5. [5] J. H. BRAMBLE and M. ZLAMAL, Triangular Elements in the finite element method, Math. Comp. 24, pp. 809-820, 1970. Zbl0226.65073MR282540
6. [6] F. BREZZI, Sur une méthode hybride pour l'approximation du problème de la torsion d'une barre élastique Rend. Ist. Lombardo Sci. Lett. A, 108 (1974), 274-300. Zbl0351.73081MR378556
7. [7] F. BREZZI, Sur la méthode des éléments finis hybrides pour le problème biharmonique. Num. Math. 24(1975) 103-131. Zbl0316.65029MR391538
8. [8] F. BREZZI, On the existence uniqueness and approximation of saddle point problems arising from Lagrangian multipliers. R.A.I.R.O. 8-R2 (1974) 129-151. Zbl0338.90047MR365287
9. [9] P. G. CIARLET, Quelques méthodes d'éléments finis pour le problème d'une plaque encastrée, Colloque IRIA sur « Méthodes de calcul scientifique et technique », Roquencourt (Paris) 17-21 déc. 1973. (To appear in Springer Verlag). Zbl0285.65042MR440954
10. [10] P. G. CIARLET, Sur l'élément de Clough et Tocher. (To appear). Zbl0306.65070
11. [11] P. G. CIARLET and P. A. RAVIART, La méthode des éléments finis pour les problèmes aux limites elliptiques. (To appear). 
12. [12] P. G. CIARLET and P. A. RAVIART, A mixed finite element method for the biharmonic equation, Symposium on Mathematical Aspect of Finite Elements in P.D.E., University of Wisconsin, Madison, April 01-03, 1974. Zbl0337.65058
13. [13] P. G. CIARLET and P. A. RAVIART, A nonconforming method for the plate problem. (To appear). Zbl0337.65058
14. [14] P. G. CIARLET and P. A. RAVIART, General Lagrange and Hermite interpolation in $R^n$ with applications to finite element method, Arch. Rat. Mech. Anal. 46,pp. 177-189, 1972. Zbl0243.41004MR336957
15. [15] J. F. CIAVALDINI and J. C. NEDELEC. (To appear). 
16. [16] R. W. CLOUGH and J. L. TOCHER, Finite element stiffness matrices for analysis of plate in bending. Conference on Matrix Methods in Structural Mechanics, Wright Patterson A.F.B., Ohio, 1965. 
17. [17] G. FICHERA, Esistenza del minimo in un classico problema di calcolo delle variazioni, Rend. Acc. Nazionale Lincei, s. VIII, v. XI, fasc. 1-2, pp. 34-39, 1951. Zbl0054.04303MR46588
18. [18] G. FICHERA, Linear elliptic differential systems and eigenvalue problems, Lecture Notes, Springer Verlag, Berlin. 1965. Zbl0138.36104MR209639
19. [19] B. FRAEIJS DE VEUBEKE, Displacements and equilibrium models in the finite element method, Stress Analysis (éd. by O. C. Zienkiewicz and G. S. Holister), chap. 9, Wiley, 1964. Zbl0131.22903
20. [20] B. FRAEIJS DE VEUBEKE, Bending and stretching of plates, Conference on Matrix Methods in Structural Mechanics, Wright Patterson A.F.B., Ohio, 1965. Zbl0168.22602
21. [21] B. FRAEIJS DE VEUBEKE, Variational principles and the patch test. (To appear on Internat. J. Numer. Methods Engrg., 8, 783-801, 1974. Zbl0284.73043MR375911
22. [22] B. FRAEIJS DE VEUBEKE and O. C. ZIENKIEWICZ, Strain-energy bounds in finite-element analysis by slab analogy, Journal of strain analysis, Vol. 2 N. 4, 1967. 
23. [23] R. GLOWINSKI, Approximations externes, par éléments finis de Lagrange d'ordre un et deux, du problème de Dirichlet pour l'opérateur biharmonique. Méthode itérative de résolution des problèmes approchés, Conference on Numerical Analysis, Royal Irish Academy, 1972. Zbl0277.35003
24. [24] B. M. IRONS and A. RAZZAQUE, Experience with the patch test for convergence of finite elements, The Mathematical Foundations of the Finite Element Method with Applications to Partial Differential Equations (A. K. Aziz, Editor), pp. 557-587, Academic Press, New York, 1972. Zbl0279.65087MR423839
25. [25] C. JOHNSON, On the convergence ofsome mixed finite element methods inplate bending problems, Num. Math, 21 (1),43-62 (1973). Zbl0264.65070MR388807
26. [26] C. JOHNSON, Convergence of another mixed finite element method for plate bending problems, Report N. 27, Department of Mathematics, Chalmers Institute of Technology and the University of Goteborg, 1972. 
27. [27] P. LASCAUX and P. LESAINT, Convergence de certains éléments finis non conformes pour le problème de la flexion des plaques minces. (To appear). 
28. [28] J. L. LIONS and E. MAGENES, Non homogeneous boundary value problems and applications, I, II, Grund. B. 181-182, Springer-Berlin, 1970. Zbl0227.35001
29. [29] B. MERCIER, Numerical solution of the biharmonic problem by mixed finite elements of class C°. Boll U.M.I., 4, 10, 1974. Zbl0332.65058MR378442
30. [30] J. T. ODEN, Some contributions to the mathematical theory of mixed finite element approximations, Tokyo Seminar on Finite Elements, 1973. Zbl0374.65060
31. [31] T. H. H. PIAN, Hybrid Models, Int. Symp. on Numerical and Computer Methods in Structural Mechanics, Urbana, Illinois, Sept. 1971. 
32. [32] T. H. H. PIAN, Finite element formulation by variational principles with relaxed continuity requirements, The Mathematical Foundations of the Finite Element Method with Applications to Partial Differential Equations (A. K. Aziz, Editor), Academic Press, New York, 1972. Zbl0274.65033MR413552
33. [33] T. H. H. PIAN and P. TONG, Basis of finite element methods for solid continua, Internat. J. Numer. Methods Engrg, Vol. I, 3-28, 1969. Zbl0252.73052
34. [34] T. H. H. PIAN and P. TONG, A variational principle and the convergence of a finite element method based on assumed stress distribution, Int. J. Solids & Struct. 5, pp. 463-472, 1969. Zbl0167.52805
35. [35] T. H. H. PIAN and P. TONG, Rationalization in Deriving element stiffness Matrix by assumed stress approach, Proceedings Second Conf. on Matrix Methods in Structural Mechanics, FFDL TR-68-150. Wright Patterson, AFB, Ohio, pp. 441-469, 1968. 
36. [36] P. A. RAVIART, Cours de troisième cycle, de l'Université de Paris VI, 1971-1972. 
37. [37] P. A. RAVIART, Hybrid finite element methods for solving 2nd order elliptic equations. Conference on Numerical Analysis, Royal Irish Academy, Dublin, July 29-August 2, 1974. Zbl0339.65061
38. [38] P. A. RAVIART and J. M. THOMAS, Hybrid finite element methods and 2nd order elliptic problems. (To appear.) Zbl0364.65082
39. [39] G. SANDER, Applications de la méthode des éléments finis à la flexion des plaques, Université de Liège, Fac. de Sci. Appl., Pubblic. N. 15, 1969. 
40. [40] G. STRANG, Approximation in the finite element method, Numer. Math., 19, pp. 81-98, 1972. Zbl0221.65174MR305547
41. [41] G. STRANG, Variational Crimes in the finite element method, The Mathematical Foundations of the Finite Element Method with Applications to Partial Differential Equations (A. K. Aziz, Editor), pp. 689-710, Academic Press, NewYork, 1972. Zbl0264.65068MR413554
42. [42] G. STRANG and G. FIX, An Analysis of the Finite Element Method, Prentice Hall Englewood Cliffs, 1973. Zbl0356.65096MR443377
43. [43] L. TARTAR. (To appear.) 
44. [44] S. TIMOSCHENKO, Theory of Plates and Shells, McGraw-Hill, New York, 1959. JFM66.1049.02
45. [45] J. M. THOMAS. (To appear.) 
46. [46] O. C. ZIENKIEWICZ, The finite element method in engineering Science, McGraw-Hill, London, 1971. Zbl0237.73071MR315970
47. [47] M. ZLAMAL, On the finite element method, Numer. Math., 12, pp. 394-409, 1968. Zbl0176.16001MR243753

1. Alfio Quarteroni, Error estimates for the assumed stresses hybrid methods in the approximation of 4th order elliptic equations
2. Pierre Lesaint, Milos Zlamal, Superconvergence of the gradient of finite element solutions
3. T. Scapolla, A mixed finite element method for the biharmonic problem
4. Claudio Canuto, A hybrid finite element method to compute the free vibration frequencies of a clamped plate