The Galerkin element method for non-uniform frames

Abstract

The effectiveness of the Galerkin method (or the Rayleigh-Ritz method) is well known for its simplicity and fast convergent property when using a complete set of orthogonal functions (Galerkin functions) for given boundary conditions. We present a new method to form the element matrices by the Galerkin method when the boundary conditions are not known beforehand. The resulting element matrix converges to the exact dynamic stiffness matrix in the limit and eliminates the numerical instability problems of the latter when the vibration frequency is small or large. The present method can improve the finite element matrices in a numerically well-conditioned manner. The vibration of frames with non-uniform members is discussed.