## 5 Fast Solver Technology
### Numerical Damping Models  
This part of the paper presents a methodology to incorporate damping into the exact spectral-Eigen method. The fractional derivatives damping models have already been developed [31] to model the material intrinsic damping as well as damping devices in engineering systems. One of the simplest fractional derivates damping models [26] has been incorporated into the spectral element technique for transient analyses herein. Consider the governing differential equation of motion for a simple element as  
\[\frac{d^{2}\hat{u}}{dx^{2}}+\omega^{2}\lambda\hat{u}=0 \tag{27}\]  
where \(\lambda=\rho/E\). The damped form of the equation can be written as [28]  
\[\frac{d^{2}\hat{u}}{dx^{2}}+\omega^{2}\lambda(i\omega)\hat{u}=0 \tag{28}\]  
where, in this case, \(\lambda(i\omega)=\frac{\rho}{E(i\omega)}\). The only change that occurs in the spectral relation for the damped element is  
\[\kappa_{c}=\pm\omega(\lambda)^{\frac{1}{2}}\hskip 14.226378 pt-\hskip 14.226378 pt \rightarrow\kappa_{f}=\pm\omega[\lambda(i\omega)]^{\frac{1}{2}} \tag{29}\]  
where subscripts \(c\) and \(f\) represent the conventional and fractional derivative module cases, respectively. The following spectral relation is given when the fractional derivative constitutive relation is employed:  
\[\kappa_{f}=\pm\omega\Bigg{[}\frac{\Big{(}1+a(i\omega)^{b}\Big{)}\rho}{e+f(i \omega)^{c}}\Bigg{]}^{\frac{1}{2}} \tag{30}\]and the fractional derivative element shape function can be written as [23; 28]  
\[\hat{u}(x)=\left(\frac{\sin\kappa_{f}(L-x)}{\sin\kappa_{f}L}\right)\hat{u}_{1}\ + \left(\frac{\sin\kappa_{f}x}{\sin\kappa_{f}L}\right)\hat{u}_{2} \tag{31}\]  
Thus, the dynamic stiffness matrix, which includes the fractional derivative damping modulus effects, is  
\[\left[\hat{K}_{f}\right]=\left[ \begin{array}{cc}\frac{\kappa_{f}L}{\tan \left(\kappa_{f}L\right)}&\frac{-\kappa_{f}L}{\sin\left(\kappa_{f}L\right)}\\ \frac{-\kappa_{f}L}{\sin\left(\kappa_{f}L\right)}&\frac{\kappa_{f}L}{\tan \left(\kappa_{f}L\right)}\end{array} \right] \tag{32}\]  
An important point here is that the dynamic stiffness matrix includes the damping effects; therefore, there is no need to assemble an individual numerical damping matrix for the structure. For the more general case of viscoelastic material damping modelling, a simplest conventional constitutive equation that adequately describes the infinitesimal deformation of a viscoelastic body can be written as [27]  
\[\sum_{n=0}^{\infty}u_{n}\frac{d^{m}\sigma(t)}{dt^{n}}=\sum_{m=0}^{\infty}q_{m }\frac{d^{m}\varepsilon(t)}{dt^{m}} \tag{33}\]  
where \(u_{n}\) and \(q_{m}\) are constant coefficients. The stress and strain in this form are related through multiple derivatives in time. One of the advantages of the spectral method is that time-dependent effects can be easily incorporated into the formulation. The spectral form of Equation (23) is [30]  
\[\left[\sum_{n=0}^{\infty}u_{n}(i\omega)^{n}\right]\hat{\sigma}=\left[\sum_{m=0 }^{\infty}q_{m}(i\omega)^{m}\ \ \ \right]\hat{\varepsilon} \tag{34}\]  
or it can be written in simple form as  
\[\hat{\sigma}=\frac{\sum_{m=0}^{\infty}q_{m}(i\omega)^{m}}{\sum_{n=0}^{\infty} u_{n}(i\omega)^{n}}\hat{\varepsilon} \tag{35}\]  
which is the viscoelastic form of the linear-elastic constitutive relation. The more familiar form of the viscoelastic behavior that adequately describes the infinitesimal deformation of a viscoelastic engineering systems has been proposed as [27]  
\[\sigma+a\frac{d\sigma}{dt}=b\varepsilon+c\frac{d\varepsilon}{dt} \tag{36}\]  
where \(a\), \(b\), and \(c\) are constants. This linear differential equation can be expressed in fractional derivative form as  
\[\sigma(t)+\sum_{i=1}^{M}a_{i}D^{b_{i}}[\sigma(t)]=e_{0}\varepsilon(t)+\sum_{j =1}^{N}f_{j}D^{c_{j}}[\varepsilon(t)] \tag{37}\]  
where \(a_{i}\), \(b_{i}\), \(c_{j}\), \(e_{0}\), and \(f_{j}\) are model parameters.