A polygonal Reissner-Mindlin plate element based on the scaled boundary finite element method

TL;DR

This work develops a polygonal Reissner–Mindlin plate element within a fully discretized scaled boundary finite element method to handle arbitrary polygonal meshes, including non-star-convex shapes. It introduces linear shape functions in both the scaling and radial directions and a dedicated assumed natural strain (ANS) formulation to alleviate transverse shear locking in the thin-plate limit. A two-field variational framework is then used to incorporate three-dimensional material laws while enforcing plane-stress conditions on the weak form, enabling 3D constitutive modeling without Poisson’s thickness locking. Numerical validations across multiple geometries demonstrate robust locking alleviation, mesh-flexibility, and effective integration of 3D material behavior, indicating practical applicability to complex plate analyses.

Abstract

In this work, a polygonal Reissner-Mindlin plate element is presented. The formulation is based on a scaled boundary finite element method, where in contrast to the original semi-analytical approach, linear shape functions are introduced for the parametrization of the scaling and the radial direction. This yields a fully discretized formulation, which enables the use of non-star-convex-polygonal elements with an arbitrary number of edges, simplifying the meshing process. To address the common effect of transverse shear locking for low-order Reissner-Mindlin elements in the thin-plate limit, an assumed natural strain approach for application on the polygonal scaled boundary finite elements is derived. Further, a two-field variational formulation is introduced to incorporate three-dimensional material laws. Here the plane stress assumptions are enforced on the weak formulation, facilitating the use of material models defined in three-dimensional continuum while considering the effect of Poisson's thickness locking. The effectiveness of the proposed formulation is demonstrated in various numerical examples.

Figures (24)

• Figure 1: Visualization of the degrees of freedom of the plate formulation with $3D$-material laws.
• Figure 2: Illustration of a polygonal element domain $\Omega_{e}$ (a) consisting of six sections, the scaling center with its position vector $\mathbf{X}_{0}$ and the element boundary $\partial\Omega_{e}$ and illustration of a section $\Omega_{s}$ (b) consisting of the scaling center with its position vector $\mathbf{X}_{0}$, the section boundary $\partial\Omega_{s}$, the boundary nodes $\tilde{\mathbf{X}}_{i}$, the position vector $\bar{\mathbf{X}}$ of a node at the boundary and the parameterization ($\xi,\eta$).
• Figure 3: Positioning of the tying points $A$, $B$ and $C$ for the assumed natural strain method against transverse shear locking in a scaled boundary finite element section.
• Figure 4: Element shapes used for the zero-energy mode test.
• Figure 5: Illustration of the load cases (a) moment loading $m$ and (b) uniformly distributed load $q$ acting on the cantilever plate.