## 3 Frequency Domain Spectral Finite Element Method (FDSFEM)
### Wave Propagation in 1D Elements  
As it has been already stated, a numerical analysis of various components of technical structures requires appropriate structural theory and the best way to approach it is the use of variational principle [79]. The calculation of the physical parameters of interest usually requires certain stress-strain relation based on the assumed displacement field. In the case of wave propagation in rods, usually the elementary rod theory has been utilised. It assumes that the rod supports only 1D axial stress. It further assumes that the lateral contraction (or the Poisson's ratio effect) can be neglected:  
\[\epsilon_{xx}=\frac{\partial u}{\partial x}. \tag{27}\]  
Figure 1: Flow diagram for a signal reconstruction procedure with FDSFEM. However, for wave propagation analysis, the one mode rod theory is not sufficient; therefore, for this purpose, rod theories with a different number of modes have been considered. The most popular is [PERSON]'s rod theory, which, in a modified version, assumes that the transverse strain is related to the axial strain by \(\epsilon_{tr}=-\
u\epsilon\), therefore the transverse velocity is given by:  
\[\dot{u_{tr}}=r\dot{e_{tr}}=-\
u r\dot{e}=-\
u r\frac{\partial\dot{u}}{\partial \dot{x}}. \tag{28}\]  
Here, it is assumed that the transverse displacement is proportional to the distance \(r\) from the centroid of the cross section. Apart from a single-mode theory, there have also been multi-mode theories developed. For example, a two mode theory, as a modification of [PERSON]'s theory, takes into account the shearing deformation due to the transverse displacement but retains the Poisson's ratio relation between the axial and transverse strains [79]. An assumption of the the transverse deformation independent of the axial deformation leads to the so-called Mindlin-Herrmann rod theory. Expansion of the deformations in a Taylor series and retaining one additional term in each expansion leads to the three-mode theory displacement relation. The detailed description is given in [79; 82]. Numerous examples devoted to the modelling of the phenomenon of wave propagation in one-dimensional elements can be classified, for example, due to the type of rod theory used or the type of material modelled. In the group of articles devoted to the application of different rod theories, one should distinguish the works [101; 102; 103; 108], where different models derived with FDSFEM suitable for modelling of wave propagation in elements with various damages have been proposed. As damage, the authors have proposed a transverse, open and not propagating crack and an additional mass. Those particular papers present wave propagation in isotropic material. The theories used have been as follows: elementary rod theory, Love rod theory, two modes rod theory and three-modes rod theory. The authors have shown that the results obtained with the proposed numerical models coincide well with the results of the experimental analysis, therefore the utilisation of more advanced rod theories should be considered, especially for wave propagation analysis. One-dimensional elements have been used for modelling of a structural local regions of non-uniformity by [104] where an FDSFEM super-element have been proposed. Important aspects of the damage detection system is damage quantification, especially in a stochastic dynamic approach, which has been analysed by [105]. The results provided have shown that it is possible to identify parameter uncertainty (natural spatial distribution of elastic modulus, Poisson's ratio, thickness, density, etc.) in material properties and including these uncertainties in a structural model for numerical examples in an axially vibrating undamaged and damaged structure with distributed parameters. The conventional FDSFEM requires a throw-off element to be added to the structural boundaries to act as a conduit for energy to transmit out of the system. This makes the method sometimes difficult to model wave reflection at boundaries, especially for more complicated geometries.