The Generalized Finite Element Method Applied to Free Vibration of Framed Structures

Abstract

The vibration analysis is an important stage in the design of mechanical systems and buildings subject to dynamic loads like wind and earthquake. The dynamic characteristics of these structures are obtained by the free vibration analysis. The Finite Element Method (FEM) is commonly used in vibration analysis and its approximated solution can be improved using two refinement techniques: h and p-versions. The h-version consists of the refinement of element mesh; the p-version may be understood as the increase in the number of shape functions in the element domain without any change in the mesh. The conventional p-version of FEM consists of increasing the polynomial degree in the solution. The h-version of FEM gives good results for the lowest frequencies but demands great computational cost to work up the accuracy for the higher frequencies. The accuracy of the FEM can be improved applying the polynomial p refinement. Some enriched methods based on the FEM have been developed in last 20 years seeking to increase the accuracy of the solutions for the higher frequencies with lower computational cost. Engels (1992) and Ganesan & Engels (1992) present the Assumed Mode Method (AMM) which is obtained adding to the FEM shape functions set some interface restrained assumed modes. The Composite Element Method (CEM) (Zeng, 1998a and 1998b) is obtained by enrichment of the conventional FEM local solution space with non-polynomial functions obtained from analytical solutions of simple vibration problems. A modified CEM applied to analysis of beams is proposed by Lu & Law (2007). The use of products between polynomials and Fourier series instead of polynomials alone in the element shape functions is recommended by Leung & Chan (1998). They develop the Fourier p-element applied to the vibration analysis of bars, beams and plates. These three methods have the same characteristics and they will be called enriched methods in this chapter. The main features of the enriched methods are: (a) the introduction of boundary conditions follows the standard finite element procedure; (b) hierarchical p refinements are easily implemented and (c) they are more accurate than conventional h version of FEM. At the same time, the Generalized Finite Element Method (GFEM) was independently proposed by Babuska and colleagues (Melenk & Babuska, 1996; Babuska et al., 2004; Duarte et al., 2000) and by Duarte & Oden (Duarte & Oden, 1996; Oden et al., 1998) under the following names: Special Finite Element Method, Generalized Finite Element Method, Finite Element Partition of Unity Method, hp Clouds and Cloud-Based hp Finite Element Method.