Approximation of the vibration modes of a plate by Reissner-Mindlin equations

Abstract:

This paper deals with the approximation of the vibration modes of a plate modelled by the Reissner-Mindlin equations. It is well known that, in order to avoid locking, some kind of reduced integration or mixed interpolation has to be used when solving these equations by finite element methods. In particular, one of the most widely used procedures is the mixed interpolation tensorial components, based on the family of elements called MITC. We use the lowest order method of this family. Applying a general approximation theory for spectral problems, we obtain optimal order error estimates for the eigenvectors and the eigenvalues. Under mild assumptions, these estimates are valid with constants independent of the plate thickness. The optimal double order for the eigenvalues is derived from a corresponding $L^2$-estimate for a load problem which is proven here. This optimal order $L^2$-estimate is of interest in itself. Finally, we present several numerical examples showing the very good behavior of the numerical procedure even in some cases not covered by our theory.