## 1 Introduction
Similarly, the efficiency of the calculation process for a comparable number of degrees of freedom should be analogous [63]. Despite the fact that there are many possibilities of computational techniques employed to solve various types of problems considered in SHM applications [61], it is very difficult to indicate one universal numerical method for such problems. Therefore, it is important to choose a method that is suitable and efficient. As presented in Table 1, there are various numerical methods of modelling of structures successfully used in [64; 65; 66; 67; 68]; however, for the problems of wave propagation modelling, the most popular are spectral methods. Two basic approaches may be distinguished. The first one has been popularised by [PERSON] and is named Frequency Domain Spectral Element Method (FDSEM) [79]. This method is a semi-analytical technique commonly used for modelling of guided waves propagating in 1D and 2D structural elements [26]. The governing partial differential equations (PDEs) describing the wave problem are converted into a frequency domain by the use of Fourier transform of the displacement variables. Then, it is possible to operate on a set of ordinary differential equations (ODEs) for each discretized frequency instead of on PDEs. The obtained ODEs solutions are then used as shape functions to formulate the elemental dynamic stiffness matrix. The procedure relates the transformed nodal forces to the corresponding transformed nodal displacements. Such approach, similar to the Finite Element Method (FEM), can be used to model structures with more complicated geometries. In comparison with FEM, the FDSFEM leads to significant computational efficiency, especially in modelling of anisotropic and inhomogeneous structures with damages [45]. The limitation of FDSFEM is the difficulty of mathematical modelling of finite length structures due to the problem of Fourier transform expansion into more complicated geometries. The second approach is named the Time Domain Spectral Finite Element Method (TDSFEM) and it has been found to be extremely efficient in simulating guided wave propagation in computational solid and fluid mechanics. This scheme \begin{table}
\begin{tabular}{c l} \hline \hline
**Method** & **Example** \\ \hline finite difference & e.g., parabolic, hyperbolic, finite-difference time-domain \\ finite volume & e.g., high-resolution \\ finite element & e.g., finite element, spectral element \\ meshless (meshfree) & e.g., material point \\ domain decompositions & e.g., fictitious domain \\ others & e.g., methods of lines, boundary element, wavelet method \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Solution methods of numerical partial differential equations [60]. adjusts the disadvantage of FE in terms of large computational time resulting from dense spatial and temporal discretization. The main difference between TDSFEM and FEM is the use of other types of polynomials as shape functions in the former method. The integration points in TDSFEM are usually connected with the nodes resulting in consistent diagonal mass matrix formulation helping in faster convergence [80]. TDSFEM has been successfully implemented for simulation of wave propagation in 1D, 2D and 3D waveguides [81; 82; 83; 84], which includes modelling of piezoelectrically (PZT) induced wave propagation [83; 84; 85], and wave interaction with damages [86; 87; 88]. In many of these works, a 2D model with an appropriate plate theory has been adopted to reduce the number of degrees of freedom (DOFs) associated with 3D models [81; 85; 89]. As stated in [46], the spectral method [63], the spectral element method [65; 90; 91; 92; 93; 94; 95; 96] and the spectral finite element method [97; 98; 99; 45; 97; 99; 46] and each of the following references have been developed to solve wave propagation problems. Analysing the literature resources, it is sometimes difficult to recognise whether a method belongs to one or another of these methods.