## 4 Time Domain Spectral Finite Element Method (TDSFEM)  
The spectral finite element method is a relatively new computational technique that combines the properties of approximating polynomials of spectral methods and the approach to discretising the analysed area particular to the finite element method [82]. The finite element properties of the method are represented by the fact that every simple geometrical object is defined particular points (called nodes) with certain approximating functions (called shape functions or node functions). Those functions describe the distribution of the analysed physical properties inside the element and on its boundaries. The non uniform distribution of nodes within a single element (Figure 2, the abbreviation explanation has been supported at the article end) results from the distance between zeros of certain polynomials. This allows for avoiding the Runge phenomenon i.e., large oscillations of approximating polynomials near the area edges [128]. This fact is the background of employing the high-order polynomials in this method, which is impossible with equally distributed nodes in classical finite element methods.  
The process of defining and solving the mathematical model by the use of TDSFEM is similar to FEM and may be shortly represented by the following steps:  
Figure 2: Exemplary node distribution in TDSFEM element formulation.  
Divide the analysed structure into a finite number of geometrically simple elements, called spectral finite elements with a certain number of characteristic points called nodes. The spectral finite elements are connected together in a finite number of nodes located at their edges. The number of nodes in the element indicates a selection of the function used for description of the distribution of the physical properties inside the spectral finite elements, depending on their node values. These functions are called node functions or shape functions--Lobatto, Chebyshev or Laguerre polynomials.
2. Transform the ordinary or differential equations describing the analysed physical phenomenon to the equation of the spectral finite element method. This transformation may be a weak formulation of the method where the weighted residual method is applied or the strong formulation where the method of minimising the variation functional of the phenomenon is applied. The aforementioned equations being the problem description are composed at the level of individual elements and are called local equations, whereas the transformations mentioned correspond to the characteristic matrices of elements that are derived. At this step, the element matrices are aggregated, leading to global characteristic matrices.
3. Implement the boundary conditions.
4. Start the solution process with an appropriate numerical method leading to obtaining values of the sought physical properties in the nodes of individual elements. In the case of non-stationary problems, steps of matrices aggregation until the solution is obtained are repeated until the relevant completion condition is met.  
For more detailed information about the TDSFEM procedures, the reader is encouraged to get familiarised with [82]. An implementation of the TDSFEM with numerous examples of elaborated models applied to different structural elements have been presented by [129], where numerical models for wave propagation with dispersion effect have been described. The models have been experimentally verified and applied for damage detection analysis.