On two simple virtual Kirchhoff-Love plate elements for isotropic and anisotropic materials

P. Wriggers,O. Allix,B. Hudobivnik

doi:10.1007/s00466-021-02106-1

Abstract

The virtual element method allows to revisit the construction of Kirchhoff-Love elements because the C^1-continuity condition is much easier to handle in the VEM framework than in the traditional Finite Elements methodology. Here we study the two most simple VEM elements suitable for Kirchhoff-Love plates as stated in Brezzi and Marini (Comput Methods Appl Mech Eng 253:455–462, 2013). The formulation contains new ideas and different approaches for the stabilisation needed in a virtual element, including classic and energy stabilisations. An efficient stabilisation is crucial in the case of C^1-continuous elements because the rank deficiency of the stiffness matrix associated to the projected part of the ansatz function is larger than for C^0-continuous elements. This paper aims at providing engineering inside in how to construct simple and efficient virtual plate elements for isotropic and anisotropic materials and at comparing different possibilities for the stabilisation. Different examples and convergence studies discuss and demonstrate the accuracy of the resulting VEM elements. Finally, reduction of virtual plate elements to triangular and quadrilateral elements with 3 and 4 nodes, respectively, yields finite element like plate elements. It will be shown that these C^1-continuous elements can be easily incorporated in legacy codes and demonstrate an efficiency and accuracy that is much higher than provided by traditional finite elements for thin plates.

Highlights

The necessity to accurately and efficiently model plates has a long history based on the fact that plates are installed as structural members in almost every building, in many machines and in airplanes
This paper aims at providing engineering inside in how to construct simple and efficient virtual plate elements for isotropic and anisotropic materials and at comparing different possibilities for the stabilisation
First elements that met C 1 -continuity were developed within the TUBA series of elements in [4]

Summary

Introduction

The necessity to accurately and efficiently model plates has a long history based on the fact that plates are installed as structural members in almost every building, in many machines and in airplanes. This has led early on to the development of plate elements in most discretisation schemes. Composite elements, consisting of four triangles were provided by [12,18] which allowed to use such composite element as general quadrilateral other formulations for strictly rectangular elements can be found e.g. in [14] All these formulations have the disadvantage that besides deflection w and rotations w,x , w,y additional higher order kinematical quantities, e.g

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Conclusion