Virtual Element for the Buckling Problem of Kirchhoff-Love plates
This work provides a theoretically sound and numerically verified method for plate buckling analysis, which is important for structural engineering applications.
The paper develops a high-order virtual element method for solving the buckling eigenvalue problem of Kirchhoff-Love plates on polygonal meshes, proving optimal error estimates and double order for eigenvalues, with numerical confirmation.
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 $k\ge2$ and we use the so-called Babuska--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.