Analysis of some low order quadrilateral Reissner-Mindlin plate elements

Abstract

Abstract. Four quadrilateral elements for the Reissner-Mindlin plate model are considered. The elements are the stabilized MITC4 element of Lyly, Stenberg and Vihinen (1933), the MIN4 element of Tessler and Hughes (1983), the Q4BL element of Zienkiewicz et al. (1993) and the FMIN4 element of Kikuchi and Ishii (1999). For all elements except the Q4BL element, a unifying variational formulation is introduced, and optimal H 1 and L 2 error bounds uniform in the plate thickness are proven. Moreover, we propose a modified Q4BL element and show that it admits the optimal H 1 and L 2 error bounds uniform in the plate thickness. In particular, we study the convergence behavior of all elements regarding the mesh distortion.

H$^2-$ Korn's Inequality and the Nonconforming Elements for The Strain Gradient Elastic Model

“…Moreover, the technicalities may also be used to derive shaper error bounds for the elements in [35,42,24]. Guided by the broken H 2 −Korn's inequality, we can design robust elements for the nonlinear strain gradient elastic models, thin beam and thin plate with strain gradient effect in [20,17] by combining the tricks in [8,33] and the machinery developed in this article, which will be left for further pursuit.…”

Developments of Mindlin-Reissner Plate Elements

“…Maunder and de Almeida [170] proposed equilibrium models in which the stress fields were divided into hyperstatic part and spurious kinematic part. Ming and Shi [173] provided a unified variational formulation for three different elements, MIN4 [42], Q9 [183], and FMIN4 [184], and proved their optimal error bounds. Castellazzi and Krysl [164] proposed a new assumed-strain technique with use of the nodal integration, in which the weighted residual method was employed to weakly enforce the equilibrium equation and the kinematic equation.…”

Analysis of a 3‐node mixed‐shear‐projected triangular element method for Reissner–Mindlin plates

“…To overcome the shear locking difficulty and derive 'locking-free' or robust plate bending elements that are valid for the analysis of thick and thin plates, significant efforts are devoted to the development of simple and efficient triangular and quadrilateral finite elements in the past few decades. The most common approach is to modify the variational formulation with some reduction operator so as to weaken the Kirchhoff constraint (see [2,3,4,5,6,7,8,9,10,11,12,15,17,19,20,21,22,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38] and the references therein).…”