A virtual element method for the vibration problem of Kirchhoff plates
Provides a theoretically grounded and practical numerical method for engineers analyzing plate vibrations on complex polygonal meshes.
This paper develops a virtual element method for computing vibration modes of thin Kirchhoff plates on polygonal meshes, proving optimal error estimates for eigenfunctions and double order for eigenvalues, with numerical experiments confirming the theory.
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 $H^2(Ω)$ 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 schemeprovides a correct approximation of the spectrum and prove optimal order error estimates for the eigenfunctions and a double order for the eigenvalues. The analysis restricts to simply connected polygonal clamped plates, not necessarily convex. 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.