Application of the recursive finite element approach on 2D periodic structures under harmonic vibrations

Abstract

The frequency response function is a quantitative measure used in structural analysis and engineering design; hence, it is targeted for accuracy. For a large structure, a high number of substructures, also called cells, must be considered, which will lead to a high amount of computational time. In this paper, the recursive method, a finite element method, is used for computing the frequency response function, independent of the number of cells with much lesser time costs. The fundamental principle is eliminating the internal degrees of freedom that are at the interface between a cell and its succeeding one. The method is applied solely for free (no load) nodes. Based on the boundary and interior degrees of freedom, the global dynamic stiffness matrix is computed by means of products and inverses resulting with a dimension the same as that for one cell. The recursive method is demonstrated on periodic structures (cranes and buildings) under harmonic vibrations. The method yielded a satisfying time decrease with a maximum time ratio of 1/18 and a percentage difference of 19%, in comparison with the conventional finite element method. Close values were attained at low and very high frequencies; the analysis is supported for two types of materials (steel and plastic). The method maintained its efficiency with a high number of forces, excluding the case when all of the nodes are under loads.