Abstract
Finite element methods for solving biharmonic boundary value problems are considered. The particular problem discussed is that of a clamped thin plate. This problem is reformulated in a weak form in the Sobolev space \({\text{W}}_2^2 \). Techniques for setting up conforming trial functions for the weak problem are described, and these functions are utilized in a Galerkin technique to produce finite element solutions. The shortcomings of various trial function formulations are discussed, and a macro-element approach to local mesh refinement using rectangular elements is given.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Agmon, S.: Lectures on Elliptic Boundary Value Problems. Van Nostrand, Princeton, 1965.
Birkhoff, G., and Mansfield, Lois.: Compatible triangular finite elements, (to appear).
Birkhoff, G., Schultz, M.H., and Varga, R.S.: Piecewise Hermite interpolation in one and two variables with applications to partial differential equations. Numer. Math. 11, 232–256, 1968.
Bramble, J.H., and Hilbert, S.R., Bounds for a class of linear functionals with applications to Hermite interpolation. Numer.Math. 16, 362–369, 1971.
Bramble, J.H., and Zlamal, M.: Triangular elements in the finite element method. Math. Comp. 24, 809–820, 1970.
Ciariet, P.G.: Conforming and nonconforming finite element methods for solving the plate problem. pp.21–32 of G.A. Watson (ed.), Proc.Conf. Numerical Solution of Differential Equations. Lecture Notes in Mathematics, No.363, Springer-Verlag, Berlin, 1974.
Ciariet, P.G., and Raviart, P.-A.: General Lagrange and Hermite interpolation in Rn with applications to finite element methods. Arch. Rat. Mech. Anal. 46, 177–199, 1972.
Clough, R.W., and Tocher, J.L.: Finite element stiffness matrices for analysis of plate bending. Proc. 1st Conf. Matrix Methods in Structural Mechanics, Wright-Patterson A.F.B., Ohio, 1965.
Collatz, L.: Hermitean methods for initial value problems in partial differential equations. pp.41–61 of J.J.H. Miller (ed.), Topics in Numerical Analysis. Academic Press, London, 1973.
Gallagher, R.H.: Finite Element Analysis: Fundamentals, (to appear).
Gregory, J.A., and Whiteman, J.R.: Local mesh refinement with finite elements for elliptic problems. Technical Report TR/24, Department of Mathematics, Brunei University, 1974.
Kolar, V., Kratochvil, J., Zlamal, M., and Zenisek, A.: Technical, Physical and Mathematical Principles of the Finite Element Method. Academia, Prague, 1971.
Westbrook, D.R.: A variational principle with applications in finite elements. J.Inst.Maths. Applics. 14, 79–82, 1974.
Whiteman, J.R.: Lagrangian finite element and finite difference methods for Poisson problems. In L. Collatz (ed.), Numerische Behandlung von Differentialgleichungen. I.S.N.M., Birkhauser Verlag, Basel, 1974.
Zlamal, M.: On the finite element method. Numer.Math. 12, 394–409, 1968.
Author information
Authors and Affiliations
Editor information
Rights and permissions
Copyright information
© 1975 Springer Basel AG
About this chapter
Cite this chapter
Whiteman, J.R. (1975). Conforming Finite Element Methods for the Clamped Plate Problem. In: Albrecht, J., Collatz, L. (eds) Finite Elemente und Differenzenverfahren. International Series of Numerical Mathematics / Internationale Schriftenreihe zur Numerischen Mathematik / Série Internationale D’Analyse Numérique, vol 28. Birkhäuser, Basel. https://doi.org/10.1007/978-3-0348-5861-8_10
Download citation
DOI: https://doi.org/10.1007/978-3-0348-5861-8_10
Publisher Name: Birkhäuser, Basel
Print ISBN: 978-3-0348-5862-5
Online ISBN: 978-3-0348-5861-8
eBook Packages: Springer Book Archive