## 5 Fast Solver Technology
### Complex Damped Spectral Solver  
The frequency-domain damped spectral Eigen element approach, which has already been developed for vibrational analysis [23; 24; 25; 26; 27; 28], benefits from the speed and accuracy of fast Fourier transform (FFT) and the fractional derivative damping models. The technique has a fundamental difference from FE method, where exact wave shape functions are used for element formulation instead of simple approximate shape functions in the FE method. These exact shape functions are used to formulate spectral elements, which can also be used to find the natural frequencies and mode shapes of structures. Although many parts of the technique have been presented extensively in previous publications [23; 24; 25; 26; 27; 28; 29; 30], some extended parts are represented here to elaborate on the efficiency of the method for damped vibrating systems. Consider a simple case of a beam with pinned joints, as shown in Figure 6.  
The spectral solution for the flexural vibration of a beam can be assumed as [23; 24; 25; 26]  
\[\hat{u}_{n}(x)=\ A_{n}e^{-i\kappa_{n}x}+B_{n}e^{-\kappa_{n}x}+C_{n}e^{-i\kappa _{n}(L-x)}+D_{n}e^{-\kappa_{n}(L-x)} \tag{1}\]  
where \(\kappa\) is the wavenumber. The boundary conditions at each frequency steps are  
\[x=0,\ L\ \ \rightarrow\ \hat{u}=0 \tag{2}\]  
\[x=0,\ L\ \ \rightarrow\ EI\frac{d^{2}\hat{u}}{dx^{2}}\hat{u}=0 \tag{3}\]  
which gives the four conditions in sine and cosine format as  
Figure 5: Zonal concepts for (**a**) partial and (**b**) full overlapping grids. Figure 6: Simply supported beam. \[\left\{ \begin{array}{c}A+C=0\\ A\cos\kappa L+B\sin\kappa L+C\cosh\kappa L+D\sinh\kappa L=0\\ A-C=0\\ -A\cos\kappa L-B\sin\kappa L+C\cosh\kappa L+D\sinh\kappa L=0\end{array} \right\} \tag{4}\]  
It can be seen that \(A=C=0\), and  
\[\left[ \begin{array}{cc}\sin\kappa L&\sinh\kappa L\\ -\sin\kappa L&\sinh\kappa L\end{array} \right]\left[ \begin{array}{c}B\\ D\end{array} \right]=0 \tag{5}\]  
As the hyperbolic term is zero, only when \(\kappa\)\(L=0\), then  
\[\sin\kappa L=0\rightarrow\kappa L=n\pi\ \rightarrow\ \sqrt{\omega}\left[ \frac{\rho A}{EI}\right]^{\frac{1}{4}} \tag{6}\]  
\[L=n\pi\ \rightarrow\ \omega=\frac{n^{2}\pi^{2}}{L^{2}}\left(\frac{EI}{ \rho A}\right)^{\frac{1}{2}} \tag{7}\]  
which gives the exact solution for the spectral Eigen frequencies of the beam element. An interesting point here is that, as the spectral solution uses an exact stiffness formulation, there is no limit on the frequency resolution. This exact solution of the governing differential equation of motion gives an infinite number of discrete resonant frequencies (\(n=1,\,2,\,3\,\dots\) ) using a single spectral element. To assess the efficiency of the spectral solution, consider a conventional FE numerical model of a beam based on an approximate interpolation shape function. The displacement, stiffness, and consistent mass matrices can be written as  
\[[u]=\left[ \begin{array}{c}\theta_{1}\\ \theta_{2}\end{array} \right];\ \ [K]=\frac{EI}{L^{3}}\left[ \begin{array}{cc}4L^{2}&2L^{2} \\ 2L&4L^{2}\end{array} \right] \tag{8}\]  
\[[M]=\frac{\rho AL}{420}\left[ \begin{array}{cc}4L^{2}&-3L^{2}\\ -3L^{2}&4L^{2}\end{array} \right] \tag{9}\]  
and for free vibration, the conventional FE Eigen equation of the beam can be written as  
\[\left\{\frac{EI}{L^{3}}\left[ \begin{array}{cc}4&2\\ 2&4\end{array} \right]-\omega^{2}\frac{\rho AL}{420}\left[ \begin{array}{cc}4&- 3\\ -3&4\end{array} \right]\right\}\left[ \begin{array}{c}\theta_{1}\\ \theta_{2}\end{array} \right]=\left[ \begin{array}{c}0\\ 0\end{array}