An adaptive generalized finite element method applied to free vibration analysis of straight bars and trusses

In this paper, a first application of an adaptive Generalized Finite Element Method to free longitudinal vibration analysis of straight bars and trusses is presented. 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 variational problem of free vibration is formulated and 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 application of this technique in a truss is also presented. The frequencies obtained by the adaptive Generalized Finite Element Method are compared with those obtained by the analytical solution, the Composite Element Method and the h-version of Finite Element Method.