## 3 Frequency Domain Spectral Finite Element Method (FDSFEM)
### Wave Propagation in 1D Elements
This subject has been analysed by [116], where the FDSFEM has been employed to model the behaviour of wave propagating in a functionally graded (FG) beam subjected to high frequency impulse loading, which can be either thermal or mechanical. The element, based on the first order shear deformation theory, has exact dynamic stiffness matrix and it takes into account depthwise variation of material properties. Structures, subjected to temperature burst kind of loading, can also be analysed using this element, with great ease. A continuation of the research may be found in [117], where a new higher-order spectral element (SE) has been developed for wave propagation analysis of a functionally graded material (FGM) beam in the presence of thermal and mechanical loading. The element has also been based on first order shear deformation theory (FSDT) and takes into account the depthwise contraction due to [PERSON]'s ratio. The formulation of the SE has been performed using a general procedure based upon the advantages of linear algebra with ascertained effectiveness. Material properties have varied along the thickness of the beam. The effect of different temperature fields at the top and bottom layer of the beam has also been taken into account by solving the heat conduction equation and considering the strain due to the computed thermal field. The extra propagating mode due to the presence of [PERSON]'s contraction has been captured successfully by the element. The effect of the contraction on the spectrum and dispersion relation has been shown in detail. FDSFEM has been a base for useful modifications, as presented in [118], describing a wavelet spectral finite element. It is an analytical-numerical method, based on the use of wavelet spectral finite elements (WSFE), and has been presented for studying the nonlinear interaction of the flexural wave switch of a breathing crack present in a slender beam. The cracked beam is discretized using a wavelet spectral finite element switch, while also using compactly supported Daubechies scaling functions for approximating the temporal dependence of the transverse displacement. A rotational spring has been used to model the open crack condition, whereas the behaviour of the beam in a closed-crack condition has been assumed to be similar to that of an intact beam. Composite materials due to their specific physical properties are very often used in different technical applications. Despite their significant advantages, they are also subject to destruction processes, and analysis of these phenomena by the use of FDSFEM has been a subject of interest of some research groups. As it can be found in [119], utilisation of FDSFEM is suitable to quantify the defect due to porosity and also to locate the porous regions in a composite beam-type structure. The actual measured response from a structure and the numerically obtained time responses are used for the estimation of porosity, by solving a nonlinear optimization problem. The effect of the length of the porous region (in the propagation direction), on the time responses, is studied. The damage force indicator technique is used to locate the porous region in a beam and also to find its length, using the measured wave propagation responses. The concept of complex modes of dynamic structures has been analysed here [120]. It shows how complex modes can be interpreted in terms of wave propagation phenomena caused by either localized damping or propagation to the surrounding media. Numerical simulation results have been presented for different kinds of structures exhibiting modal and wave propagation characteristics: straight beams, an L-shaped beam, and a three-dimensional frame structure. With the FDSFEM, it is straightforward to use infinite elements, usually known as throw-off elements, to represent the propagation to infinity, which is a possible cause of modal complexity. With the FDSFEM model, the exact dynamic behaviour of structures can be investigated.