Locking free quadrilateral continuous/discontinuous finite element methods for the Reissner–Mindlin plate

Abstract

We develop a finite element method with continuous displacements and discontinuous rotations for the Mindlin-Reissner plate model on quadrilateral elements. To avoid shear locking, the rotations must have the same polynomial degree in the parametric reference plane as the parametric derivatives of the displacements, and obey the same transformation law to the physical plane as the gradient of displacements. We prove optimal convergence, uniformly in the plate thickness, and provide numerical results that confirm our estimates.

Developments of Mindlin-Reissner Plate Elements

“…Recently, some scholars, mostly among mathematicians, began to employ the discontinuous Galerkin (DG) finite element methods [17] to design plate elements [120][121][122][123][124][125][126]. This DG method admitted the discontinuities in the element discrete space, leading to new types of conforming or nonconforming elements.…”

“…Arnold et al [120] developed two DG Mindlin-Reissner plate element families: one was the fully discontinuous case and the other was the case of continuous rotations and nonconforming deflections. Hansbo et al [126,127] also proposed a quadrilateral element model with continuous displacements and discontinuous rotations. In this model, the rotations had the same order as the parametric derivatives of the displacements in the parametric reference plane, and they obeyed the same transformation law.…”

Two generalized conforming quadrilateral Mindlin–Reissner plate elements based on the displacement function

“…Playing a central role in practical engineering analysis, the MindlinReissner plate element is continuously being improved for better performance and to meet specific requirement. Among others, some of the latest developments include the NIPE formulations derived from the assumed-strain technique [16], the models with continuous displacement and discontinuous rotations [17,18], the element based on the high-order linked interpolation [19], the alternative alpha finite element method with discrete shear gap technique [20], and so on. However, two major challenges remain outstanding: (1) the performance of the element cannot be guaranteed when the mesh is severely distorted; and (2) the precisions for stress/resultant results are relatively lower than those for displacements.…”

Stabilised finite element methods for a bending moment formulation of the Reissner-Mindlin plate model

“…Hence, after integrating by parts in (15), it follows that ξ · ν, ω ∂Ω = 0 on ∂Ω, for all ξ ∈ H(div, Ω). The surjectivity of the normal trace implies ω = 0 on ∂Ω and thus ω ∈ H 1 0 (Ω).…”

“…In Figure 5 we depict the behaviour of this error as h tends to 0 and for different values of t. We observe that this norm tends to zero as predicted by the theory, and moreover it presents a very robust behaviour with respect to the value of t, showing results which, as in the errors for method (33), are virtually t-independent. Finally, to assess the presence, or lack of, locking in our method, we have repeated the experience carried out in the recent work [15], Section 5.3. For this, we have fixed a mesh from the sequence displayed in Figure 1, containing approximately 500 elements, solved the problem on it for different values of t, and have measured the maximum discrete displacement and compared it to the maximum exact displacement (measured in all of the nodes of the mesh).…”