Publications Meetings The Profession Membership Programs Math Samplings Policy & Advocacy In the News About the AMS

   
Remote Access
Green Open Access
Mathematics of Computation
Mathematics of Computation
ISSN 1088-6842(online) ISSN 0025-5718(print)

 

Triangular elements in the finite element method


Authors: James H. Bramble and Miloš Zlámal
Journal: Math. Comp. 24 (1970), 809-820
MSC: Primary 65.66
MathSciNet review: 0282540
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: For a plane polygonal domain $ \Omega $ and a corresponding (general) triangulation we define classes of functions $ {p_m}(x,y)$ which are polynomials on each triangle and which are in $ {C^{(m)}}(\Omega )$ and also belong to the Sobolev space $ W_2^{(m + 1)}(\Omega )$. Approximation theoretic properties are proved concerning these functions. These results are then applied to the approximate solution of arbitrary-order elliptic boundary value problems by the Galerkin method. Estimates for the error are given. The case of second-order problems is discussed in conjunction with special choices of approximating polynomials.


References [Enhancements On Off] (What's this?)

  • [1] Shmuel Agmon, Lectures on elliptic boundary value problems, Prepared for publication by B. Frank Jones, Jr. with the assistance of George W. Batten, Jr. Van Nostrand Mathematical Studies, No. 2, D. Van Nostrand Co., Inc., Princeton, N.J.-Toronto-London, 1965. MR 0178246 (31 #2504)
  • [2] J. H. Argyris, I. Fried & D. W. Scharpf, "The tuba family of plate elements for the matrix displacement method," Aeronautical J. Roy. Aeronautical Soc., v. 72, 1968, pp. 618-623.
  • [3] K. Bell, Analysis of Thin Plates in Bending Using Triangular Finite Elements, The Technical University of Norway, Trondheim, 1968.
  • [4] K. Bell, "A refined triangular plate bending finite element," Internat. J. Numer. Methods in Engrg., v. 1, 1969, pp. 101-122.
  • [5] I. S. Berezin & N. P. Židkov, Computing Methods. Vol. I, 2nd ed., Fizmatgiz, Moscow, 1962; English transl., of 1st ed., Pergamon Press, New York, 1965. MR 30 #4372; MR 31 #1756.
  • [6] W. Bosshard, "Ein neues, vollverträgliches endliches Element für Plattenbiegung," Abh. Int. Verein. Brückenbau and Hochbau, Zürich, v. 28, 1968, pp. 27-40.
  • [7] J. H. Bramble and S. R. Hilbert, Estimation of linear functionals on Sobolev spaces with application to Fourier transforms and spline interpolation, SIAM J. Numer. Anal. 7 (1970), 112–124. MR 0263214 (41 #7819)
  • [8] Jean Céa, Approximation variationnelle des problèmes aux limites, Ann. Inst. Fourier (Grenoble) 14 (1964), no. fasc. 2, 345–444 (French). MR 0174846 (30 #5037)
  • [9] R. Courant, Variational methods for the solution of problems of equilibrium and vibrations, Bull. Amer. Math. Soc. 49 (1943), 1–23. MR 0007838 (4,200e), http://dx.doi.org/10.1090/S0002-9904-1943-07818-4
  • [10] J. J. Goël, List of Basic Functions for Numerical Utilisation of Ritz's Method. Application to the Problem of the Plate, École Polytechnique Féderale, Lausanne, 1969.
  • [11] Jan Kadlec, The regularity of the solution of the Poisson problem in a domain whose boundary is similar to that of a convex domain, Czechoslovak Math. J. 14 (89) (1964), 386–393 (Russian, with English summary). MR 0170088 (30 #329)
  • [12] J. Kratochvíl, A. Ženíšek & M. Zlámal, "A simple algorithm for the stiffness matrix of triangular plate bending finite elements," Numer. Methods in Engineering. (To appear.)
  • [13] J.-L. Lions and E. Magenes, Problèmes aux limites non homogènes et applications. Vol. 1, Travaux et Recherches Mathématiques, No. 17, Dunod, Paris, 1968 (French). MR 0247243 (40 #512)
  • [14] J. Nitsche, Lineare Spline-Funktionen und die Methoden von Ritz für elliptische Randwertprobleme, Arch. Rational Mech. Anal. 36 (1970), 348–355 (German). MR 0255043 (40 #8250)
  • [15] M. J. Turner, R. W. Clough, H. C. Martin & L. J. Topp, "Stiffness and deflection analysis of complex structures," J. Aeronautical Sci., v. 23, 1956, pp. 805-823.
  • [16] M. Visser, The Finite Element Method in Deformation and Heat Conduction Problems, Delft, 1968.
  • [17] Alexander Ženíšek, Interpolation polynomials on the triangle, Numer. Math. 15 (1970), 283–296. MR 0275014 (43 #772)
  • [18] O. C. Zienkiewicz, The Finite Element Method in Structural and Continuum Mechanics, McGraw-Hill, New York, 1967.
  • [19] Miloš Zlámal, On the finite element method, Numer. Math. 12 (1968), 394–409. MR 0243753 (39 #5074)
  • [20] M. Zlámal, "A finite element procedure of the second order of accuracy," Numer. Math., v. 16, 1970, pp. 394-402.
  • [21] A. C. Felippa, Refined Finite Element Analysis of Linear and Nonlinear TwoDimensional Structures, SESM Report No. 66-22, University of California, Berkeley, Calif., 1967.
  • [22] E. Anderheggen, Programme zur Methode der finiten Elemente, Institut für Baustatik, Eidgenössische Technische Hochschule, Zürich, 1969.
  • [23] G. R. Cowper, E. Kosko, G. M. Lindberg & M. D. Olson, "Formulation of a new triangular plate bending element," C.A.S.I. Trans., v. 1, 1968, pp. 86-90.
  • [24] G. R. Cowper, E. Kosko, G. M. Lindberg & M. D. Olson, "Static and dynamic applications of a high-precision triangular plate bending element," AIAA J., v. 7, 1969, pp. 1957-1965.

Similar Articles

Retrieve articles in Mathematics of Computation with MSC: 65.66

Retrieve articles in all journals with MSC: 65.66


Additional Information

DOI: http://dx.doi.org/10.1090/S0025-5718-1970-0282540-0
PII: S 0025-5718(1970)0282540-0
Keywords: Finite element method, Ritz method, Galerkin method, piecewise polynomial subspaces, approximation of solution, elliptic boundary problems
Article copyright: © Copyright 1970 American Mathematical Society