An adaptive generalized finite element method applied to free vibration analysis of straight bars and trusses: Fid-Bau Portal

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

Arndt, M. / Machado, R.D. / Scremin, A.

The main contribution of the present study consisted in proposing an adaptive Generalized Finite Element Method for vibration analysis. This study performed a preliminary formulation of free vibration analysis of straight bars and trusses by the proposed method. The Generalized Finite Element Method results were compared with those obtained by the h-version of Finite Element Method and the c-version of the Composite Element Method. In this adaptive Generalized Finite Element Method, trigonometric enrichment functions depending on geometric and mechanical properties of the elements were added to the linear Finite Element Method shape functions by the partition of unity approach. This technique allows an accurate adaptive process that converges very fast and is able to refine the frequency related to a specific vibration mode. In addition the enrichment functions are easily obtained and the introduction of boundary conditions follows the standard finite element procedure. The results have shown that the adaptive Generalized Finite Element Method achieves narrower precision than the c-version of Composite Element Method and the h-version of Finite Element Method in free longitudinal vibration analysis of uniform and non-uniform straight bars for the same number of degrees of freedom. It has been observed that even for problems where the exact solutions are not represented by trigonometric functions, like non-uniform bars, the results from the adaptive method are accurate with relatively few degrees of freedom. This method has been applied in free vibration analysis of trusses showing results very close to those of the Composite Element Method. It is worth remarking that the adaptive Generalized Finite Element Method is an iterative process that requires less computational effort than it appears. Instead of dealing with a matrix of (n+2m) x (n+2m) dimension, the problem is divided in one of n x n, and two of 2m x 2m matrices, significantly reducing the amount of arithmetic. Since the adaptive approach requires much less degrees of freedom than the standard Finite Element Method, the adaptive process spends less computational effort in order to obtain similar accuracy. The adaptive Generalized Finite Element Method has shown to be efficient in the analysis of longitudinal vibration of bars and has indicated that it can be applied even for a coarse discretization scheme in complex practical problems. Future research will extend this adaptive method to other structural elements like beams, plates and shells.