Non-linear optimization of the shape functions when applying the finite element method to vibration problems

The finite element method constitutes, undoubtedly, one of the most powerful tools of computational mechanics. Minimizing the discretization error, on the other hand, is a question of basic concern in order to insure maximum reliability without increase in computer memory and/or CPU time. The present paper introduces the concept of the “ k-optimization parameter” contained in the shape functions. Once the Rayleigh-Ritz formulation of the finite element algorithm is performed and the eigenvalues are determined in the case of vibrations or elastic stability problems one is able to minimize them by optimizing with respect to k. The proposed method is applied to a classical vibrations problem—the vibrating string—and it is concluded that the main advantage of the method is the fact that excellent accuracy can be achieved with a small number of elements. In other words, the requirement of computer memory is considerably lower for a pre-assigned relative error.