Keywords: recovery, patch, smoothing, superconvergent, Reissner-Mindlin plate, finite element method, error estimate.

Calculation of smoothed stress resultants for a Reissner-Mindlin plate is introduced. Smoothed stress resultants are obtained by post-processing finite element results with a new and simple patch recovery method. The patch recovery method, which uses local or patchwise polynomial representations for the basic unknowns and the corresponding derivative quantities of the boundary value problem was presented in [1]. In this method, local polynomial representations are formed such that they contain "built-in" information from the field equations and boundary conditions of the problem. In this paper, the patch recovery method, which is based on ideas shown in [1], is developed for the Reissner-Mindlin plate. In this case, basic unknowns are the deflection and the rotations. Derivative quantities are stress resultants of the plate: moments and shear forces. Unknown parameters of local polynomial representations are determined by fitting deflection and rotations to the corresponding finite element values at the nodes of the patch by least squares procedure. At boundary patches information from the boundary conditions is included into the least squares procedure by Lagrange multipliers method.

In the numerical examples, two low-order four noded plate element types are used. The first plate element is a conventional selectively integrated four node element and second element is stabilized form [2] of four noded MITC plate element [3]. Comparison study between the results obtained by the present and the well-known SPR (Superconvergent Patch Recovery) [4] methods is made.

With the help of patch recovery methods Zienkiewich-Zhu error estimates [5] are calculated. Numerical examples include thin and thick square and circular plates. In a numerical convergence study the energy norm is used as an error measure and the effectivity index is used as a measure of the quality of a posteriori error estimate. Numerical study shows that presented method works well.

1

J. Aalto and M. Perälä, "Two robust patch recovery methods with built-in field equations and boundary conditions", Finite element methods, superconvergence, post-processing and a-posteriori estimates, (edited by M. Krizek, P. Neittaanmäki, R. Stenberg), Lecture notes in pure and applied mathematics, Marcel Dekker, Inc.c, Vol. 196, 1-17, 1998.

2

M. Lyly, R. Stenberg and T. Vihinen, "A stable bilinear element for the Reissner-Mindlin plate problem", Computer Methods in Applied Mechanics and Engineering, 110, 343-357, 1993. doi:10.1016/0045-7825(93)90214-I

3

K.J. Bathe and E.N. Dvorkin, "A four-node plate bending element based on Mindlin/Reissner plate theory and mixed interpolation", International Journal for Numerical Methods in Engineering, 21, 367-383, 1985. doi:10.1002/nme.1620210213

4

O.C. Zienkiewich and J.Z. Zhu, "The superconvergent patch recovery and a posteriori error estimates. Part 1: The recovery technique", International Journal for Numerical Methods in Engineering, 33, 1331-1364, 1992. doi:10.1002/nme.1620330702

5

O.C. Zienkiewich and J.Z. Zhu, "A simple error estimator and adaptive procedures for practical engineering analysis", International Journal for Numerical Methods in Engineering, 24, 337-357, 1987. doi:10.1002/nme.1620240206

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