Linked interpolation for Reissner‐Mindlin plate elements: Part I—A simple quadrilateral

Abstract

SUMMARYIn most plate elements using the Reissner-Mindlin assumptions, the interpolations used for the lateral displacements (w) and the rotation (8) involve the independent representation of each variable by its nodal values, usually with identical interpolations. To ensure a higher order of expansion for displacement MI its representation is linked in the present paper with both sets of nodal variables.Conditions necessary for the use of such expansions are established here and the paper shows the development of a linear quadrilateral element (Q4BL) whose performance and robustness are good (although it possesses one singularity if only three degrees of freedom are prescribed).In Part 11 we apply the identical formulation to develop a triangular element (T3BL) which performs equally well and is fully robust.

The weak patch test for nonhomogeneous materials modeled with graded finite elements

“…In addition, Belytschko and Lasry (1988) have studied the behavior of a distorted element with a fractal patch test, which is also valid in the weak patch test sense. The patch test has been applied to many problem-types including, for example, mixed displacement-pressure finite element formulations Zienkiewicz et al, 1986;Razzaque, 1986;Wu and Chen, 1997;Zienkiewicz and Taylor, 1997), three-dimensional (3-D) solid elements (Loikkanen and Irons, 1984), plate bending elements (Samuelsson et al, 1987;Zienkiewicz and Lefebvre, 1988;Zhifei, 1993;Zienkiewicz et al, 1993;Auricchio and Taylor, 1993;Aricchio and Taylor, 1994;Martins and Sabino, 1997;Park and Choi, 1997), and shell elements (Herrmann, 1989). The patch test has also been used as a fundamental tool to create new elements or to improve existing ones (Ju and Sin, 1996;Cheung et al, 2002;Piltner and Taylor, 2000).…”

Analysis of some low order quadrilateral Reissner-Mindlin plate elements

“…However, there is still a gap between plate elements commonly used in engineering and those for which a sound mathematical theory exists. Even worse, the existing analyses are mostly confined to either triangular elements or rectangular elements, but exclude most widely used quadrilateral elements, like the stabilized MITC4 element of Lyly, Stenberg and Vihinen [27], the aniso-parametrically interpolated MIN4 element of Tessler and Hughes [35], the linked interpolated Q4BL element of Zienkiewicz et al [39], as well as an element proposed recently by Kikuchi and Ishii [21] (FMIN4). These elements are constructed in different settings and perform extremely well in the benchmark computation, however, a rigorous convergence analysis for them seems lacking except the stabilized MITC4 element [25,26,30], and the connections between them are unclear.…”

“…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.…”

A locking-free Reissner-Mindlin quadrilateral element

“…Our argument benefits from a local linked interpolation ( [25], [26], mathematically analyzed by [38], [39]) which helps us to establish a special Helmholtz-decomposition. Consequently, we can bound the inconsistency term (q,β − Π hβ ) by O(h 2 ).…”