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Studies in plate flexure. I: A mixed finite element formulation for Reissner-Mindlin plate theory: Uniform convergence of all higher-order spaces. (English) Zbl 0611.73077
A new mixed finite element formulation of Reissner-Mindlin theory is presented which improves upon the stability properties of the Galerkin formulation. General convergence theorems are proved which are uniformly valid for all values of the plate thickness, including the Poisson- Kirchhoff limit. As long as the dependent variables are interpolated with functions of sufficiently high order, the formulation is convergent. No special devices are required.

##### MSC:
 74S05 Finite element methods applied to problems in solid mechanics 74K20 Plates 74S30 Other numerical methods in solid mechanics (MSC2010)
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##### References:
 [1] Arnold, D.N., Discretization by finite elements of a model parameter dependent problem, Numer. math., 37, 405-421, (1981) · Zbl 0446.73066 [2] Arnold, D.N.; Falk, R.S., A uniformly accurate finite element method for the Mindlin-Reissner plate, () · Zbl 0696.73040 [3] Babuška, I., Error bounds for finite element method, Numer. math., 16, 322-333, (1971) · Zbl 0214.42001 [4] Bathe, K.J.; Brezzi, F., On the convergence of a four-node plate bending element based on Mindlin/Reissner plate theory and mixed interpolation, (), 491-503 · Zbl 0589.73068 [5] Bathe, K.J.; Dvorkin, E.N., A four-node plate bending element based on Mindlin/Reissner plate theory and a mixed interpolation, Internat. J. numer. meths. engrg., 21, 367-383, (1985) · Zbl 0551.73072 [6] Bercovier, M., On C0 beam elements with shear and their corresponding penalty function formulation, Comput. math. appl., 8, 4, 245-256, (1982) · Zbl 0486.73070 [7] Bernadou, M.; Boisserie, J.M., The finite element method in thin shell theory: application to arch dam simulations, (1982), Birkhäuser Boston · Zbl 0497.73069 [8] Brezzi, F., On the existence, uniqueness and approximation of saddle-point problems arising from Lagrange multipliers, RAIRO ser. rouge anal. numér., R-2, 129-151, (1974) · Zbl 0338.90047 [9] F. Brezzi, From Stokes to Mindlin, Private communication, 1986. [10] Brezzi, F.; Fortin, M., Numerical approximation of Mindlin-Reissner plates, Math. comp., 47, 175, 151-158, (1986) · Zbl 0596.73058 [11] Ciarlet, P.G., The finite element method for elliptic problems, (1978), North-Holland Amsterdam · Zbl 0445.73043 [12] Franca, L.P., New mixed finite element methods, () · Zbl 0651.65078 [13] Hughes, T.J.R., The finite element method: linear static and dynamic finite element analysis, (1987), Prentice-Hall Englewood Cliffs, NJ [14] Hughes, T.J.R.; Franca, L.P., A new finite element formulation for computational fluid dynamics: VII. the Stokes problem with various well-posed boundary conditions: symmetric formulations that converge for all velocity/pressure spaces, Comput. meths. appl. mech. engrg., 65, 85-96, (1987) · Zbl 0635.76067 [15] Hughes, T.J.R.; Franca, L.P., Convergence of transverse shear stresses in the finite element analysis of plates, (1987), Institute for Computer Methods in Applied Mechanics and Engineering, Division of Applied Mechanics, Stanford University Stanford, CA, Preprint [16] Hughes, T.J.R.; Franca, L.P.; Balestra, M., A new finite element formulation for computational fluid dynamics: V. circumventing the babuška-Brezzi condition: A stable Petrov-Galerkin formulation of the Stokes problem accommodating equal-order interpolations, Comput. meths. appl. mech. engrg., 59, 85-99, (1986) · Zbl 0622.76077 [17] Hughes, T.J.R.; Tezduyar, T.E., Finite elements based upon Mindlin plate theory with particular reference to the four-node bilinear isoparametric element, J. appl. mech., 48, 587-596, (1981) · Zbl 0459.73069 [18] Kikuchi, F., On a finite element scheme based on the discrete Kirchhoff assumption, Numer. math., 24, 211-231, (1975) · Zbl 0295.73067 [19] Kikuchi, F., On a mixed method related to the discrete Kirchhoff assumption, (), 137-154 [20] Loula, A.F.D.; Hughes, T.J.R.; Franca, L.P.; Miranda, I., Mixed Petrov-Galerkin method for the Timoshenko beam, Comput. meths. appl. mech. engrg., 63, 133-154, (1987) · Zbl 0607.73076 [21] MacNeal, R.H., Derivation of element stiffness matrices by assumed strain distribution, Nucl. engrg. design, 70, 3-12, (1982) [22] Pitkäranta, J., On simple finite element methods for Mindlin plates, (), 187-190 [23] Pitkäranta, J., Analysis of some low-order finite element schemes for Mindlin-Reissner and Kirchhoff plates, () · Zbl 0654.73043
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