## 1 Introduction  
Damage detection is the key aspect of Structural Health Monitoring (SHM), being defined as the acquisition, validation and analysis of technical data to facilitate life-cycle cost management decisions [1]. Numerous SHM techniques have become the subject of extensive scientific investigations [2; 3]. As one of the greatest challenges of the SHM system has been recognised, the knowledge of what physical parameters observe and how to identify their 'changes' [4]. There have been defined various techniques of efficient damage detection based on modal analysis [5; 6; 7; 8; 9; 10; 11], wave propagation [12; 13; 14; 15; 16; 17; 18; 19; 20; 21; 22; 23; 24; 25; 26; 27; 28; 29; 30], impedance changes [31; 32], ultrasonic inspection [33; 34; 35; 36; 37; 38], and many others [39; 40; 41; 42; 43; 44]. Modern engineering is based on numerical modelling of designed structures, and it has been a standard procedure for any technical design starting from a project through a thorough state analysis until the damage prognosis operations. Almost every reliable damage detection procedure requires dynamic analyses typically distinguished according to the range of excitation frequency. It is usually categorized as structural dynamics for low frequencies (up to few hundred hertz) and wave propagation for high frequencies (the order of kilohertz) [45]. Dynamic analyses are usually performed using the conventional Finite Element Method (FEM) [46; 47; 48], but, for wave propagation, many higher order vibrational modes participate in the motion and the classical FEM approach may be numerically inefficient. This fact originates from the very dense FE meshes resulting from very small wavelengths participating in vibrational movements. It is known that the size of FEs should be of the size comparable to the wavelength of the shortest signal component. Therefore, numerous attempts have been addressed in order to improve other numerical methods for simulation of wave propagation in structures, especially because some important phenomena like wave dispersion or attenuation [49] should be addressed. A process of numerical modelling of the designed structure is based on a solution of a set of partial differential or integral equations being an approximation of the problem [50; 51; 52; 53; 54; 55; 56; 57; 58; 59]. There are various numerical methods to solve partial differential equations, which have been roughly gathered in Table 1.  
The crucial point in the process of numerical modelling of structures is typically the balance between the time of calculation and the level of complication of the solution method chosen. As the number of SHM systems based on changes in wave propagation in structures increases every year, therefore it is important to use reliable solution methods for this type of physical phenomena [61]. For several decades, the number of applications of spectral methods for solution of wave problems has been constantly growing. According to the literature [62], the name for \"spectral method\" begins probably with the usage of Fourier series of sines and cosines as basis functions especially in reference to a time series evaluation and the basic frequencies of a process, i.e., the \"spectrum\". However, in the most general definition, the spectral methods involve defining the solution to a problem as an abridged series of known functions of the separate variables [63]. The choice of an appropriate spectral method should be defined by the two main aspects: the accuracy and the efficiency. In order to be useful, a spectral method should be performed to provide results of greater accuracy than conventional difference methods of a similar spatial resolution. The choice of convenient spectral representation depends also on the kind of boundary conditions in the problem. The calculation results obtained by the use of spectral methods should be more accurate while compared with the results produced with conventional methods.