Numerical issues concerning the wave and finite element method for free and forced vibrations of waveguides

The wave and finite element (WFE) method is a numerical approach to the calculation of the wave properties of structures of arbitrary complexity. The method starts from a finite element (FE) model of only a short segment of the structure, typically by using existing element libraries and commercial FE packages. The dynamic stiffness matrix of the segment is obtained, a periodicity condition applied and an eigenvalue problem formed whose solution gives the dispersion equations and wave mode shapes. These define a wave basis from which the forced response can be found straightforwardly. Although straightforward in application, the WFE method is prone to numerical difficulties. These are discussed in this paper and methods to avoid or remove them described. Attention is focused on 1-dimensional waveguide structures, for which numerical problems are most severe. Three ways of phrasing the eigenvalue problem for free wave propagation are presented and a method based on singular value decomposition is proposed to determine eigenvectors at low frequencies. Discretisation errors are seen to occur if the segment is too large, while round-off errors occur if the segment is too small. This can be overcome by forming a super-segment from the concatenation of two or more segments. The forced response is then considered. The use of a reduced wave basis removes many problems. Direct calculation of the waves excited by a point force is very prone to poor numerical conditioning but can be circumvented by exploiting the orthogonality of the left and right eigenvectors. Numerical examples are presented.