# A Time Integration Method Based on Galerkin Weak Form for Nonlinear Structural Dynamics
## 4 A Feasible Scheme with Numerical Dissipation  
To be sufficiently effective in solutions of structural dynamics, the time integration scheme should possess controllable numerical dissipation, such that the false vibration components of higher orders could be removed and the accuracy of results could be guaranteed. However, as discussed in Section 3, the integral scheme of Equation (15) does not have such numerical dissipation. In order to get over this difficulty, a feasible time integration scheme with numerical dissipation is proposed using the Galerkin weak form method. The detailed construction procedure is as follows. Considering the Galerkin weak form shown in Equation (6) with linear time elements, let the trial function remain as the common interpolation of linear polynomials while the test function is as follows:  
\[\mathbf{\tilde{v}}^{h}=\mathbf{N}_{a}\mathbf{d}_{\sigma}=\sum_{i=1}^{2}N_{ai} (\zeta)\mathbf{v}_{i} \tag{30}\]  
where \(N_{ai}\) (\(i\) = 1,2) is the following constructed linear polynomial with a parameter '\(\alpha\)' being introduced, i.e.,:  
\[N_{a1}=-\frac{(\zeta-1-\alpha)}{2},N_{a2}=\frac{(\zeta+1-\alpha)}{2}. \tag{31}\]  
Figure 2: Percentage period elongation due to the proposed algorithm with \(m\) = 1,2,3: (**a**) Percentage period elongation with \(m\) = 1–3 comparing with that of trapezoidal rule and Bathe method. (**b**) Percentage period elongation with \(m\) = 2. (**c**) Percentage period elongation with \(m\) = 3. That is to say, the test function in Equation (32) doesn't satisfy the C\({}_{0}\) continuity between elements any more. Substituting such a trial function \(\mathbf{u}^{h}\) and test function \(\mathbf{\bar{v}}^{h}\) into Equation (6), in consideration of the arbitrariness of \(\mathbf{d}_{v}\), the following solution formula for Equation (5) may be obtained:  
\[\mathbf{K}_{\alpha}\mathbf{d}+\frac{2\mathbf{M}}{\Delta t}(\mathbf{N}(1)^{T} \dot{\mathbf{u}}_{n+1}-\mathbf{N}(-1)^{T}\dot{\mathbf{u}}_{n})=\mathbf{P}_{ \alpha}. \tag{32}\]  
In Equation (32), there are many integration terms in both \(\mathbf{K}_{\alpha}\) and \(\mathbf{P}_{\alpha}\) which are also with respect to the parameter \(\alpha\). Again, for the target to introduce numerical damping, the following integral formulae are used instead of the common Gauss integral in calculations of all the integral terms, i.e.,:  
\[\int\limits_{-1}^{1}f(\zeta)\mathrm{d}\zeta=2f(\frac{\alpha}{2}). \tag{33}\]  
Finally the following Newton-Raphson iterative scheme is yielded:  
\[ \begin{array}{l}(\frac{2}{\Delta t^{2}}\mathbf{M}+\frac{1}{\Delta t}(1+ \frac{\alpha}{2})\bar{\mathbf{C}}_{\alpha/2}^{(k)}+\frac{1}{2}(1+\frac{\alpha }{2})^{2}\bar{\mathbf{K}}_{\alpha/2}^{(k)})\mathbf{u}_{n+1}^{(k+1)}=(1+\frac{ \alpha}{2})\mathbf{f}_{\alpha/2}^{(k)}\\ \qquad+(\frac{2}{\Delta t^{2}}\mathbf{M}+\frac{1}{\Delta t}(1+\frac{\alpha}{2 })\bar{\mathbf{C}}_{\alpha/2}^{(k)}-\frac{1}{2}(1-\frac{\alpha^{2}}{4})\bar{ \mathbf{K}}_{\alpha/2}^{(k)})\mathbf{u}_{n}+\frac{2\mathbf{M}}{\Delta t}\dot{ \mathbf{u}}_{n}\\ \\ (-\frac{2}{\Delta t^{2}}\mathbf{M}+\frac{1}{\Delta t}(1-\frac{\alpha}{2})\bar{ \mathbf{C}}_{\alpha/2}^{(k)}+\frac{1}{2}(1-\frac{\alpha^{2}}{4})\bar{\mathbf{ K}}_{\alpha/2}^{(k)})\mathbf{u}_{n+1}^{(k+1)}+\frac{2\mathbf{M}}{\Delta t}\dot{ \mathbf{u}}_{n+1}^{(k+1)}=\\ (1-\frac{\alpha}{2})\mathbf{f}_{\alpha/2}^{(k)}+(-\frac{2}{\Delta t^{2}} \mathbf{M}+\frac{1}{\Delta t}(1-\frac{\alpha}{2})\bar{\mathbf{C}}_{\alpha/2} ^{(k)}-\frac{1}{2}(1-\frac{\alpha}{2})^{2}\bar{\mathbf{K}}_{\alpha/2}^{(k)}) \mathbf{u}_{n}\end{array}