## 4 Discussion and Conclusions  
In this study an ad-hoc developed constant strain triangular (CST) finite element has been presented for calculating the three-dimensional time-transient response in a viscoelastic octagonal rod. The [PERSON]'s principle has been used to formulate the weak form of the wave equation that is next transformed in the frequency domain. The major benefit of the frequency transformation comes from  
Figure 11: Response for the \(y\)-dof of node 13 at different distances source-receiver \(d\)=\(\mathrm{z_{R}}\)-\(\mathrm{z_{S}}\). (a) absolute value of the scaled frequency response function, (b) time-transient response.  
the possibility to easily account for linear viscoelastic constitutive relations. By solving a complex eigenvalue problem for a given wave frequency, wave equation roots are obtained in terms of complex wavenumbers, describing the spatial frequency and attenuation of the guided waves. A modal summation procedure and the application of the inverse time Fourier transform lead to the time-transient response of the problem for an applied time-dependent load.  
In the computation of the time-transient response, the proposed SFE formulation is reliable, capturing mode separation as well as pulse dispersion, stable, showing no or limited activity before the first arrival, and fast. The computational time mainly depends on the maximum excited frequency that sets the minimum number of finite elements in the mesh and the sampling frequency for which the waveguide frequency response [via Equation (29)] must be calculated to guarantee time-transient solution accuracy. For the proposed formulation the distance source-receiver only marginally affects the computational time and memory requirements, resulting advantageous if compared to a time transient Finite Element simulation for wave problems where an increased propagation distance involves a massive increase of dofs. This is particular true for high frequency computations, where the wavelengths are small and an accurate prediction by FE procedures involves an intensive three-dimensional discretization.  
For example, the computation of each FRF and the corresponding time response required approximately 260 seconds using a Pentium 4 processor on a PC with 2 GB of RAM. This computational time is negligible if compared to the time needed by an equivalent full 3D Finite Element simulation to simulate waves at 10 meters distance.  
In addition, SFE formulations allow for a very efficient multiple time-transient response calculation by means of an optimized post-processing procedure. In the time-transient response calculation, in fact, the majority of the computational CPU time is required to compute the waveguide eigenvalues and eigenvectors at several different frequencies solving the eigenvalue problem of Equation (18). The required time increases substantially for an increased number of dofs of the considered waveguide mesh. For a given waveguide, and especially for large dofs problems, wavenumbers and wavevectors can be calculated at the several different frequencies of interest only once and stored. Next, these data can be used to quickly calculate the time-transient response of the waveguide at the different dofs of the mesh, considering different distances source-receiver and also different spatial and temporal load distribution.