Comparative Analysis of the Adaptive Generalized Finite Element Method and the Finite Element Method for Free Vibration of Bars

Abstract

In this paper, the results of the adaptive Generalized Finite Element Method to free longitudinal vibration analysis of straight bars are compared with the results of the h-versions and the hierarchical p-version of the Finite Element Method. The Generalized Finite Element Method is developed by enriching the standard Finite Element Method space, whose basis performs a partition of unity, with knowledge about the differential equation being solved. The enrichment functions used are dependent on the geometric and mechanical properties of the element. The proposed approach converges very fast and is able to approximate the frequency related to any vibration mode. The main aspects of the adaptive Generalized Finite Element Method are presented and discussed. The efficiency and convergence of the proposed method in vibration analysis of uniform and non-uniform straight bars are checked. The frequencies obtained by the adaptive Generalized Finite Element Method are also compared with those obtained by the analytical solution.