## Basic Principle
### Improvement of Algorithm  
In order to introduce numerical dissipation into the GGW method, Ref. [31] proposes a feasible time integration scheme. A parameter \(\alpha\) is introduced to control the numerical damping by changing the form of the test function as Equation (16). At the same time, to calculate the integral terms in Equations (12) and (13), a specific numerical integration rule as Equation (17) is constructed using Equation (16).  
\[\widetilde{N}_{\alpha 1}=-\frac{(\zeta-1-\alpha)}{2},\ \widetilde{N}_{\alpha 2 }=\frac{(\zeta+1-\alpha)}{2} \tag{16}\]  
\[\int\limits_{-1}^{1}f(\zeta)\mathrm{d}\zeta=2f(\frac{\alpha}{2}) \tag{17}\]  
In this way, the time integration scheme with numerical dissipation for linear time elements is established based on the Galerkin weak form. However, the parameter \(\alpha\) in Equations (16) and (17) turns out to be too sensitive under certain circumstances, which will result in too fast decay (this is demonstrated in Section 4.3).  
To overcome this shortcoming of the GGW method, here we present an improved method in which the time step is divided into two subsections and another parameter \(\beta\) (\(0<\beta<1\)) is introduced in addition to \(\alpha\).  
In the first step, we take the step length to be \(\beta\Delta t\), and by using the GGW method, we obtain the solutions \(u_{n+\beta},\ \dot{u}_{n+\beta}\) and \(\ddot{u}_{n+\beta}\) at (\(t_{n}+\beta\Delta t\)). In the second step, we use \(u_{n+\beta},\ \dot{u}_{n+\beta}\) and \(\ddot{u}_{n+\beta}\) to compute the solutions \(u_{n+1},\ \dot{u}_{n+1}\) and \(\ddot{u}_{n+1}\) at \(t_{n+1}\) (\(t_{n+1}=t_{n}+\Delta t\)). The detailed computational strategy of the second sub-step is as follows:  
Considering the definition of integral as Equation (18) shown as follows:  
\[\left\{\begin{array}{l}u_{n+1}-u_{n}=\int\limits_{t_{n}}^{t_{n+1}}\dot{u}(t) \mathrm{d}t=\int\limits_{t_{n}}^{t_{n+1}+\beta\Delta t}\dot{u}(t)\mathrm{d}t+ \int\limits_{t_{n}+\beta\Delta t}^{t_{n+1}}\dot{u}(t)\mathrm{d}t\\ \dot{u}_{n+1}-\dot{u}_{n}=\int\limits_{t_{n}}^{t_{n+1}}\ddot{u}(t)\mathrm{d}t= \int\limits_{t_{n}}^{t_{n+1}+\beta\Delta t}\ddot{u}(t)\mathrm{d}t+\int\limits _{t_{n}+\beta\Delta t}^{t_{n+1}}\ddot{u}(t)\mathrm{d}t\end{array}\right. \tag{18}\]  
and using the algorithm of trapezoidal integral, Equation (18) can be rewritten into Equation (19).  
\[\left\{\begin{array}{l}u_{n+1}-u_{n}=\frac{\Delta t}{2}\big{[}\big{(}\dot{u} _{n}+\dot{u}_{n+\beta}\big{)}\beta+\big{(}\dot{u}_{n+\beta}+\dot{u}_{n+1}\big{)} (1-\beta)\big{]}\\ =\Big{(}\frac{\beta}{2}\dot{u}_{n}+\frac{1}{2}\dot{u}_{n+\beta}+\frac{1-\beta }{2}\dot{u}_{n+1}\Big{)}\Delta t\\ \dot{u}_{n+1}-\dot{u}_{n}=\frac{\Delta t}{2}\big{[}\big{(}\ddot{u}_{n}+\ddot{u }_{n+\beta})\beta+\big{(}\ddot{u}_{n+\beta}+\ddot{u}_{n+1}\big{)}(1-\beta) \big{]}\\ =\Big{(}\frac{\beta}{2}\ddot{u}_{n}+\frac{1}{2}\ddot{u}_{n+\beta}+\frac{1-\beta }{2}\ddot{u}_{n+1}\Big{)}\Delta t\end{array}\right. \tag{19}\]  
where \(\dot{u}_{n+\beta}\) and \(\ddot{u}_{n+\beta}\) are solutions solved from the first sub-step. Furthermore, Equation (19) can be simplified into the following:  
\[\begin{array}{l}u_{n+1}=u_{n}+\big{(}c_{1}\dot{u}_{n}+c_{2}\dot{u}_{n+\beta} +c_{3}\dot{u}_{n+1}\big{)}\Delta t\\ \dot{u}_{n+1}=\dot{u}_{n}+\big{(}c_{1}\ddot{u}_{n}+c_{2}\ddot{u}_{n+\beta}+c_{3} \ddot{u}_{n+1}\big{)}\Delta t\end{array} \tag{20}\]  
where \(c_{1}\), \(c_{2}\) and \(c_{3}\) are parameters, and \(c_{1}=\frac{\beta}{2},\ c_{2}=\frac{1}{2},\ c_{3}=\frac{1-\beta}{2}\). Generally, when \(\beta\) is \(0.5\), \(c_{1}=c_{3}=1/4\), \(c_{2}=1/2\). In addition, \(c_{1}+c_{2}+c_{3}=1\).  
It is obvious that, when \(\beta=1\), the second sub-step vanishes and the proposed method degenerates to the GGW method; when \(\beta=0\), the first sub-step vanishes and the proposed method degenerates to the Newmark method.