Virtual element for the buckling problem of Kirchhoff–Love plates

Abstract

In this paper, we develop a virtual element method (VEM) of high order to solve the fourth order plate buckling eigenvalue problem on polygonal meshes. We write a variational formulation based on the Kirchhoff-Love model depending on the transverse displacement of the plate. We propose a C 1 conforming virtual element discretization of arbitrary order k ≥ 2 and we use the socalled Babuška-Osborn abstract spectral approximation theory to show that the resulting scheme provides a correct approximation of the spectrum and prove optimal order error estimates for the buckling modes (eigenfunctions) and a double order for the buckling coefficients (eigenvalues). Finally, we report some numerical experiments illustrating the behaviour of the proposed scheme and confirming our theoretical results on different families of meshes.

A virtual element method for the von Kármán equations

“…Proposition 3.3 has been proved in Proposition 4.2 of [16]. Proposition 3.4 and Lemma 3.4 can be seen for instance in Section 4 of [44] and Lemma 3.5 of [45], respectively. Proposition 3.2.…”

“…Remark 3.1. In (3.7) the vector function Π 1 ∇ ℎ is fully computable from the degrees of freedom given by the sets D 1 and D 2 (see for instance [44], Sect. 3 for further details).…”

“…. (see [17,43,44]) be the smallest eigenvalue of the following buckling spectral problem (cf. (2.5)):…”

“…Several virtual element methods based on conforming and non-conforming schemes have been developed to solve a wide variety of problems in Solid and Fluid Mechanics, for example [4][5][6]9,11,12,14,19,25,27,30,42,46,47]. Moreover, the VEM for thin structures has been developed in [16,24,29,30,44,45], whereas VEM for nonlinear problems have been introduced in [3,15,26,35,36,50] In this paper, we analyze a conforming 1 Virtual Element Method to approximate the isolated solutions of the von Kármán equations. We consider a variational formulation in terms of the transverse displacement and the Airy stress function, which contains bilinear and trilinear forms.…”

First-order VEM for Reissner–Mindlin plates

“…Due to its appealing features, VEM has been used to solve a wide range of prob-lems in Mechanics, e.g. plane elasticity [11][12][13], contact mechanics [14,15], fracture mechanics [16,17], solid nonlinear mechanics [18,19], Kirchhoff-Love plates [20][21][22].…”