## 4 Time Domain Spectral Finite Element Method (TDSFEM)
### Wave Propagation in 2D Elements  
The previous subsection shows that the TDSFEM can be used to efficiently model the phenomenon of mechanical wave propagation in one-dimensional elements. The changes in wave propagation may be successfully used to diagnose the analysed components. However, the excellent majority of technical devices requires more complex geometries to be used in the modelling process. The examplesof modelling two-dimensional structures will be discussed here, in particular shell and plate elements designed and analysed with TDSFEM. Multi-mode formulation of a higher order transversely deformable shell-type spectral finite element (SFE) for dynamic analysis of isotropic structures is presented and analysed in [135]. A method to generate the dispersion curves associated with the displacement fields obtained is presented by the authors, which utilises [PERSON]'s principle and which leads to a set of associated equations of motion. These equations have been presented in the paper for the case of the analysed shell SFE as well as the dispersion curves resulting from them. The accuracy of the formulation proposed is assessed by comparison of dispersion curves obtained for the current model with dispersion curves obtained for exact solutions of the problem as well as comparison with shell theories well-known from the literature. Finally, in order to demonstrate the effectiveness of the current formulation of a higher order transversely deformable shell-type SFE, a series of numerical tests were performed. These comprised the analysis of natural frequencies and modes of vibration of an isotropic spherical shell as well as the wave propagation analysis in the case of the same spherical shell and a half-pipe shell-like structure. Combined application of TD-SFEM and the higher order transversely deformable shell SFE developed by the authors helps to reduce significantly modelling and discretisation errors of numerical investigation providing excellent accuracy, especially in comparison to the classical FEM and typical FEs commonly used for that purpose. A TDSFEM deformed thin-walled spherical shell element for modal analysis and for an effective structural health monitoring technique based on propagation of guided elastic waves has been given in [87; 136]. The first paper deals with certain problems related to static and modal analysis of isotropic shell structures representing a pressure vessel. The robustness and effectiveness of the spectral approach have been successfully demonstrated. Static and modal responses of the structure have been investigated by the use of transversally deformable shell-type spectral finite elements and the results of this investigation have been compared to known analytical solutions as well as those obtained by the use of commercially available software for the finite element method. The second paper takes advantage of the TDSFEM modelling technique in application to certain structural health monitoring (SHM) strategy. This has been realized based on results of numerical calculation and experimental investigations obtained with laser scanning vibrometry (LSV) and subsequent application of two different integral-based indices for damage quantification, these being the integral mean value (IMV) and the root mean square (RMS). Numerical tests by SFEM have been carried out for a square plate with through-hole damage, a cracked fuselage section with two stiffeners and a cracked wing section skin, all made out of aluminium. Experimental measurements by LSV included tests on a square aluminium plate, composite plate and a composite stabilizer of a PZL W-3A helicopter, all with simulated damage. During numerical and experimental investigations, damage sensitivity of both IMV and RMS indices was tested by considering various displacement components, signal time-scales or weighting factors. A certain practical method for quick construction of RMS damage maps based on experimental measurements by LSV has been also presented.