## 3 Frequency Domain Spectral Finite Element Method (FDSFEM)
### Wave Propagation in 2D Elements  
A plate is a body where one of the dimensions is substantially smaller than the other two. Fundamentally, plate theory is an approximate structural theory and therefore it is best to approach it by way of variational principle [79]. Of the numerous plate theories that have been developed in the late 19 th century, two are widely accepted and used in engineering. These are the Kirchhoff-Love theory of plates and the Mindlin-Reissner theory of plates. Plates in flexure are the two-dimensional equivalent of beams, and classical plate theory (Kirchhoff-Love) is its equivalent of the Bernoulli-Euler beam theory, whereas the in-plane or membrane behaviour of plates is analogous to that of rods. The two-dimensionality of plates affects the wave propagation modelling as the wave is no longer confined to follow a 1D guide [79]. The formulation of spectral elements for 2D structural waveguides poses extra complexity [121]. The reduced equation in the frequency domain is no longer an ODE but remains a PDE in terms of the space variables. This PDE is not readily solvable and another transform is necessary to reduce the equation to one spatial dimension. A possibility consists in moving to the frequency/wavenumber domain, through a 2D Fourier transformation. Thus, the unknown variable is further decomposed, normally using a Fourier series representation as:  
\[\hat{\mathbf{u}}\left(x,y,\omega_{n}\right)=\sum_{m=0}^{M-1}\hat{\mathbf{u}}\left(x, \eta_{m},\omega_{n}\right)\left\{sin(\eta_{m}y)cos(\eta_{m}y)\right\}, \tag{30}\]where \(M\) is the number of Fourier series points and \(\eta_{m}\) is the discrete wavenumber related to the spatial window (see Equation (25) for time space relation) \(Y\) by  
\[\eta_{m}=m\Delta\eta=\frac{m\eta_{f}}{M}=\frac{m}{M\Delta y}=\frac{m}{Y}, \tag{31}\]  
with \(\Delta y\) denoting the spatial sampling rate and \(\eta_{f}\) being the highest wavenumber captured by \(\Delta y\). The spatial variation of the load determines \(M\). Using this representation, the governing equation becomes an ordinary differential equation in \(x\) and again can be solved exactly for some cases. This exact solution is again used as the interpolation function for the unknown in the spectral element formulation. Thus, for each frequency \(\omega_{n}\) and wavenumber \(\eta_{m}\), the dynamic stiffness matrix is formed and assembled and the unknown variable is solved for its force amplitude \(\tilde{\mathbf{u}}_{n,m}\) as  
\[\tilde{\mathbf{K}}_{n,m}\tilde{\mathbf{u}}_{n,m}=\tilde{\mathbf{f}}_{n,m}, \tag{32}\]  
where \(\tilde{\mathbf{f}}_{n,m}\) is the force amplitude of applied load. First, \(\tilde{\mathbf{u}}_{n,m}=\tilde{\mathbf{u}}\left(x,\eta_{m},\omega_{n}\right)\), and \(\hat{\mathbf{u}}\left(x,y,\omega_{n}\right)\) is recovered by the Fourier series and \(\mathbf{u}\left(x,y,t\right)\) is recovered by the Inverse Fast Fourier Transformation (IFFT) algorithm [121]. To obtain the exact solution of the ODE in frequency domain (for 1D analysis) or FWD (for 2D analysis), it is assumed that the solution of \(\hat{\mathbf{u}}\left(x,\omega_{n}\right)\) or \(\hat{\mathbf{u}}\left(x,\eta_{m},\omega_{n}\right)\) is in the form \(u_{0}e^{-j\mathrm{i}x}\), where \(u_{0}\) is an unknown constant and \(k\) is the unknown wavenumber in the assumed direction of propagation. This formulation is valid only for a constant coefficient ODE [121]. For this reason, careful attention should be paid while considering problems of modelling of wave propagation in 2D structural elements. One of the popular modifications of FDSFEM is its combination with wavelet analysis (Wavelet Spectral Finite Element Method WSFEM). In such a formulation, the dependent variable can be approximated by a certain scaling function. Further analysis steps have been described in detail by [122; 123; 124; 147; 148], where successful analysis of wave propagating in different plate type elements have been shown.