## 4 Time Domain Spectral Finite Element Method (TDSFEM)
### Wave Propagation in 2D Elements
A novel formulation of a spectral plate element for wave propagation in isotropic structures is presented in [137]. In this formulation of the spectral plate finite element as approximation functions, Chebyshev polynomials of the first kind have been employed. The element makes use of on an extended form of the displacement field that enables one to investigate selectively or simultaneously both symmetric and anti-symmetric modes of Lamb waves propagating in plate structures. In addition, the dispersion relations associated with the extended displacement field have been presented and discussed in the paper. The effectiveness of the new spectral plate finite element has been illustrated on examples of propagation of elastic waves in a flat aluminium panel in the case of the fundamental symmetric and anti-symmetric modes of Lamb waves. A pseudo-spectral Mindlin plate element with Chebyshev polynomials chosen as the elemental interpolating functions as they are proper for wave propagation modelling has been proposed by [138]. A diagonal system matrix has been obtained and therefore the equations of motion could be easily solved in the time domain by applying the central difference method of reducing the computational cost. Static, dynamic and elastic wave propagation problems have been analysed to obtain the efficiency of the proposed pseudo-spectral Mindlin plate element using two integration schemes for elemental stiffness matrix. In [139], the authors have presented an interesting node distribution (Figure 3) for the proposed plate element model based on TDSFEM. The intention of these investigations has been the effective, computing time reducing simulation of Lamb wave propagation using methods for the numerical calculation. For this purpose, high-order shape functions and the Gauss-Lobatto-Legendre integration scheme has been introduced. Using the example of 2D plane strain plate elements, it has been shown that this method is able to provide an efficient analysis for the wave propagation in damaged and undamaged structures. The presented procedure has been verified by analytical computations as well as by alternatively obtained references. Numerical analysis of physical phenomena occurring in elements of machines and devices requires a certain level of awareness of the differences between real structures and their numerical equivalents. For this reason, it is worth mentioning here the work on estimating numerical errors of specific modelling methods. Some research results concerned on this subject has been performed by [140]. In this work, the Legendre spectral finite elements have been examined through numerical experiments for static and dynamic Reissner-Mindlin plate bending and a mixed-quadrature scheme. Legendre spectral finite elements are high-order Lagrangian-interpolant finite elements with nodes located at the Gauss-Lobatto-Legendre quadrature points. Solutions on unstructured meshes have been examined in terms of accuracy as a function of the number of model nodes and total operations. While nodal-quadrature Legendre spectral finite elements have been shown elsewhere to be free of shear locking on structured grids, locking has been demonstrated on unstructured grids. Legendre spectral finite elements with mixed quadrature have been, however, locking free and are significantly more accurate than low-order finite-elements for a given model size or total computation time. An example of similar elements but regarding composite plates for static and dynamic deformation on unstructured grids has been analysed by [141]. The authors have analysed Legendre spectral finite elements application to Reissner-Mindlin composite plates. Nodal quadrature has been employed for mass-matrix calculations, and that has yield diagonal mass matrices. Full quadrature or mixed-reduced quadrature has been used for stiffness-matrix calculations.