# A Time Integration Method Based on Galerkin Weak Form for Nonlinear Structural Dynamics
## 2 Basic Principle  
### Governing Equations and Iterative Schemes  
For nonlinear dynamic problems in solid mechanics and structural engineering, discretization in the space dimension generally results in the following governing equations in the time domain, which are named dynamic equations hereinafter:  
\[\left\{\begin{array}{l}\mathbf{M}\ddot{\mathbf{u}}+\mathbf{f_{d}}(\dot{ \mathbf{u}})+\mathbf{f_{s}}(\mathbf{u})=\mathbf{f_{ext}}\,,\,0<t\leq t_{f}\\ \mathbf{u}(0)=\mathbf{u_{0}}\,,\,\dot{\mathbf{u}}(0)=\dot{\mathbf{u}}_{0}\\ \end{array}\right. \tag{1}\]  
where \(\mathbf{u}\equiv\mathbf{u}(t)\) is the unknown vector of vibration displacements/rotations, \(\mathbf{M}\) is the mass matrix, \(\mathbf{f_{d}}\) is the vector of damping forces depending on velocity \(\dot{\mathbf{u}}\equiv\mathbf{d}(t)/\mathrm{d}t\), \(\mathbf{f_{s}}\) is the vector of resilience depending on displacement, and \(\mathbf{f_{ext}}\) represents the vector of externally applied forces. The solution domain is supposed to be \((0,t_{f}]\), with \(\mathbf{u_{0}}\) and \(\dot{\mathbf{u}_{0}}\) being the given initial condition vectors.  
When \(\mathbf{f_{d}}\) or \(\mathbf{f_{s}}\) is the nonlinear function of \(\dot{\mathbf{u}}\) or \(\mathbf{u}\), Equation (1) is a typical IVP of nonlinear ordinary differential equations (ODEs). A common strategy for solving such ODEs is to transform them into linear equations and then perform iteration. Thus the Newton-Raphson method is first applied to Equation (1) which results in the following linearized ODEs:  
\[\left\{\begin{array}{l}\mathbf{M}\ddot{\mathbf{u}}^{(k+1)}+\mathbf{\tilde{C }}^{(k)}\dot{\mathbf{u}}^{(k+1)}+\mathbf{\tilde{K}}^{(k)}\mathbf{u}^{(k+1)} \,=\mathbf{f_{ext}}-\mathbf{f_{int}}^{(k)}\,,\,t_{n}\leq t\leq t_{n+1}\\ \mathbf{u}^{(k+1)}(t_{n})=\mathbf{u}_{n},\quad\dot{\mathbf{u}}^{(k+1)}(t_{n}) =\dot{\mathbf{u}}_{n}\\ \end{array}\right. \tag{2}\]  
where  
\[\mathbf{\tilde{C}}^{(k)}=(\left.\frac{\partial\mathbf{f_{d}}}{\partial\dot{ \mathbf{u}}}\right|_{\dot{\mathbf{u}}^{(k)}}\mathbf{\tilde{K}}^{(k)}=(\left. \frac{\partial\mathbf{f_{s}}}{\partial\mathbf{u}}\right|_{\mathbf{u}^{(k)}}) \tag{3}\]  
\[\mathbf{f_{int}}^{(k)}=(\left.\mathbf{f_{d}}^{(k)}\right|_{\dot{\mathbf{u}}^{( k)}}-\mathbf{\tilde{C}}^{(k)}\dot{\mathbf{u}}^{(k)})+(\left.\mathbf{f_{s}}^{(k)} \right|_{\mathbf{u}^{(k)}}-\mathbf{\tilde{K}}^{(k)}\mathbf{u}^{(k)}). \tag{4}\]  
Here \(\mathbf{u}^{(k)}\), \(\dot{\mathbf{u}}^{(k)}\), and \(\ddot{\mathbf{u}}^{(k)}\) are quantities obtained in the \(k\)-th iteration step (\(k\) = 0,1,2,... ), the tangential damping matrix \(\mathbf{\tilde{C}}^{(k)}\), tangential stiffness matrix \(\mathbf{\tilde{K}}^{(k)}\) and internal force vector \(\mathbf{f}_{\text{int}}^{(k)}\) are functions of time. For convenience, we omit the superscript \({}^{\prime}\)\({}^{\prime}\) in Equation (2) temporarily and denote \(\mathbf{f}=\mathbf{f_{ext}}-\mathbf{f_{int}}\). Then, Equation (2) can be written into the following simplified form, which is the main solution target of the next subsection:  
\[\left\{\begin{array}{l}\mathbf{M}\ddot{\mathbf{u}}+\mathbf{\tilde{C}}\dot{ \mathbf{u}}+\mathbf{\tilde{K}}\mathbf{u}\,=\mathbf{f},\quad t_{n}\leq t\leq t _{n+1}\\ \mathbf{u}(t_{n})=\mathbf{u}_{n},\quad\dot{\mathbf{u}}(t_{n})=\dot{\mathbf{u} }_{n}\\ \end{array}\right.. \tag{5}\]