**2. Technical Background**

As mentioned in Introduction, proper numerical modelling of designed structures has become, nowadays, a key aspect of modern engineering, and it usually covers the total product life cycle starting from its design, current state analysis, and damage prognosis for life-time assessment. Changes in propagating waves have been one of the most popular physical measures used for damage detection purposes [38–44]. For their proper numerical analysis it is extremely important to employ reliable numerical tools [45–50]. For modelling of the wave propagation phenomenon, various commonly known computer methods have been in use. However, the classical finite element method (FEM) may be numerically inefficient, since in the process of wave propagation many higher order vibrational modes participate in the motion. For this reason very dense FE meshes, resulting from very small wavelengths participating in vibrational movements, should be applied because the size of FEs should be comparable to the wavelength of the shortest signal component. This procedure significantly increases the computational size of analysed problems, and directs researchers to other, more efficient

numerical methods [51–53]. At this point, we would like to strongly emphasise that in the current paper we do not discuss the numerical consequences of taking into account such physical phenomena as wave dispersion and/or attenuation, as we instead concentrate on numerical properties of particular computational approaches.

For several decades, for wave propagation modelling purposes, the time domain spectral finite element method (TDSFEM), proposed by Patera [54], has been successfully applied [36,38,45,55–61]. It is a computational technique, which combines the properties of approximating polynomials of spectral methods and the same approach as the FEM to divide the analysed area into finite elements. The finite element properties of the method are represented by the fact that for every simple geometrical object, particular points (called nodes) with certain approximating functions (called shape functions or node functions) are defined. These functions describe the distributions of analysed physical quantities within elements and along their boundaries. Non-uniform distribution of nodes within a single element is the result of the distance between zeros of certain polynomials. This allows one to avoid the Runge phenomenon; i.e., large oscillations of approximating polynomials near the structural/element edges [62]. This fact is central for employing high-order polynomials in this method, which is impossible in the case of equally distributed nodes in classical finite element methods [36].

The process of building and solving a numerical model by the use of TDSFEM is similar to FEM and consists of the following steps:


For more detailed information about TDSFEM, procedures can be found in [36]. Implementation of TDSFEM with numerous examples of sophisticated models applied to different structural elements have been presented by [42], wherein numerical models for wave propagation with dispersion effects have been described. The models have been experimentally verified and applied for damage detection analysis.
