## 3 Frequency Domain Spectral Finite Element Method (FDSFEM)
### Wave Propagation in 1D Elements
To overcome this limitation, imaginary spectral elements have been proposed [106], which are combined with the real structural elements to model wave reflections at structural boundaries. The efficiency and accuracy of this proposed approach have been verified by comparing the numerical simulation results with measured results of one-dimensional stress wave propagation in a steel bar. The method has also been applied to model wave propagation in a steel bar with not only boundary reflection, but also reflections from single and multiple cracks (from 10 to 50% of the rod light). The reflection and transmission coefficients, which have been obtained from the discrete spring model, have been adopted to quantify the discontinuities. Experimental tests of wave propagation in a steel bar with one crack of different depths have also been carried out. Presented numerical simulations and experimental results have shown that the proposed method is effective and reliable in modelling wave propagation in one-dimensional waveguides with reflections from boundary and structural discontinuities in case of rod elements. The process of modelling of wave propagation in beam elements is most often related to the use of two beam theories, i.e., the Bernoulli-Euler and the Timoshenko theories. In the first one, the tensile stress experienced by the beam may be expressed as:  
\[\sigma=-zE\frac{d^{2}w}{dx^{2}}, \tag{29}\]  
where \(z\) is the distance from the neutral axis to a point of interest and \(M\) is the bending moment. The simplest approach is usually based on the force-displacement relation, the concept of equilibrium and compatibility with exact solutions to the governing equations of motion, which is the direct relationship between nodal forces and nodal displacements (Force-Displacement Relation Method). In the force-displacement relationship method, the spectral element matrices are formulated by the use of the exact dynamic shape functions derived from the exact wave solutions to the frequency-domain governing equations. Thus, the method can only be used when the exact wave solutions to the frequency-domain governing equations are available in the closed forms. However, there is also an alternative approach available, while the spectral element matrix has been derived from the transfer matrix formulated directly from the frequency-domain state-vector equation of motion [107]. This specific approach has been analysed for propagation phenomena in elements made out of different materials by [108; 109; 110; 111; 112; 113]. The authors have analysed the influence of different types of damages on elastic wave propagation in isotropic (aluminium, steel) and anisotropic (glass-epoxy composite) materials. An interaction of wave propagating in a steel-concrete rebar with damages has been analysed by the use of FDSFEM by [114]. The Timoshenko beam theory with FDSFEM has been recently successfully utilised by [115] for modelling the vibration parameters of a beam element with an open crack. In this particular formulation, an open crack has been modelled by the two rotational and extensional massless springs. The applied moving load functions with either a constant velocity or a constant acceleration have been transformed into frequency-domain by utilizing the dynamic shape functions, as calculated by the solution of governing equations of wave propagation in free vibrations. The authors state that, with the proposed FDSFEM formulation, one is allowed to obtain correct results in a much shorter time of calculation while compared to the classical FEM, especially in terms of analysis of moving load. From a scientific point of view, the models of beam elements made of homogeneous materials in which the elastic wave is propagated have been quite well developed. However, there is an area related to modelling of the phenomenon of elastic wave propagation in functional elements composed of several materials such as, for example, functionally graded material (FGM) elements.