Design of Galerkin Generalized Least Squares methods for Timoshenko beams

A class of finite element methods, the Galerkin Generalized Least Squares methods, are developed and applied to model the steady-state response of Timoshenko beams. An optimal method is designed using a linear interpolation of the response such that there is zero finite element dispersion error. The classical method of selective reduced integration and a modified version of selective reduced integration with mass lumping are shown to fall under the Galerkin Generalized Least Squares framework. Numerical experiments in wave propagation demonstrate the dramatic superiority of the new optimal method over the standard approaches. The goal of the new methods is to decrease the computational burden required to achieve a desired accurcy level at a particular frequency thereby enabling larger scale, higher frequency computations for a given platform.