A virtual element method for the vibration problem of Kirchhoff plates

¹ GIMNAP, Departamento de Matemática, Universidad del Bío-Bío, Casilla 5-C, Concepción, Chile, and Centro de Investigación en Ingeniería Matemática (CI 2MA), Universidad de Concepción, Concepción, Chile
² Departamento de Ciencias Exactas, Universidad de Los Lagos, Casilla 933, Osorno, Chile
³ Departamento de Ingeniería Matemática, Universidad de Concepción, Casilla 160-C, Concepción, Chile, and Centro de Investigación en Ingeniería Matemática (CI 2MA), Universidad de Concepción, Concepción, Chile

^* Corresponding author: dmora@ubiobio.cl

Received: 13 January 2017
Accepted: 21 August 2017

Abstract

The aim of this paper is to develop a virtual element method (VEM) for the vibration problem of thin plates on polygonal meshes. We consider a variational formulation relying only on the transverse displacement of the plate and propose an H²(Ω) conforming discretization by means of the VEM which is simple in terms of degrees of freedom and coding aspects. Under standard assumptions on the computational domain, we establish that the resulting scheme provides a correct approximation of the spectrum and prove optimal order error estimates for the eigenfunctions and a double order for the eigenvalues. Finally, we report several numerical experiments illustrating the behaviour of the proposed scheme and confirming our theoretical results on different families of meshes. Additional examples of cases not covered by our theory are also presented.