# A Time Integration Method Based on Galerkin Weak Form for Nonlinear Structural Dynamics
## 2 Basic Principle
### Galerkin Finite Element Method  
To solve the linear IVP of ODEs in Equation (5), a time-domain FEM based on the Galerkin weak form [25; 34; 42] was used in the present paper. Let \(\mathbf{v}\in H^{1}_{\text{\tiny{$\mathrm{\mathrm{\mathrm{\mathrm{\mathrm{ \mathrm{\mathrm{\mathrm{\mathrm{\mathrm{\mathrm{\mathrm{\mathrm{\mathrm{ \mathrm{\mathrm{ \mathrm{    }}}}}}}}}$}}}}}} \mathbf{u}^{h}\in H^{1}_{\text{\tiny{$\mathrm{\mathrm{\mathrm{\mathrm{ \mathrm{\mathrm{\mathrm{\mathrm{\mathrm{\mathrm{\mathrm{\mathrm{\mathrm{\mathrm{\mathrm{\mathrm{\mathrm{\mathrm{\mathrmmathrm{\mathrm \mathrm{\mathrm{\mathrm{\mathrm{\mathrmmathrm{  
\[\mathbf{u}^{h}=\mathbf{N}\mathbf{d}=\sum_{i=1}^{m+1}N_{i}(t)\mathbf{d}_{i} \tag{7a}\] \[\mathbf{v}^{h}=\mathbf{N}\mathbf{d}_{v}=\sum_{i=1}^{m+1}N_{i}(t)\mathbf{v}_{i} \tag{7b}\]  
where the shape functions \(N_{i}(t)\) (\(i=1,\,2,\,\ldots,\,m+1\)) are Lagrange polynomials, the shape function matrix \(\mathbf{N}=\left[N_{1}\mathbf{I},N_{2}\mathbf{I},\cdots,N_{m+1}\mathbf{I}\right]\), and \(\mathbf{I}\) is the unit matrix, \(\mathbf{d}=\left\{\mathbf{d}_{1}^{T}\mathbf{d}_{2}^{T}\cdots\mathbf{d}_{m+1} ^{T}\right\}\), and \(\mathbf{d}_{i}\) with \(\mathbf{d}_{1}=\mathbf{u}_{n}\), \(\mathbf{d}_{m+1}=\mathbf{u}_{n+1}\) is the nodal displacement vector at time \(\tilde{t}_{i}\), and \(\mathbf{d}_{v}=\left\{\mathbf{v}_{1}^{T}\mathbf{v}_{2}^{T}\cdots\mathbf{v}_{ m+1}^{T}\right\}^{T}\) is the vector of nodal test displacements. Substituting Equation (7b) into Equation (6), because of the arbitrariness of test displacement vector \(\mathbf{d}_{v}\), the following equation can be derived:  
\[\int\limits_{t_{n}}^{t_{n+1}}\mathbf{N}^{T}\mathbf{M}\ddot{\mathbf{u}}^{h} \mathrm{d}t+\int\limits_{t_{n}}^{t_{n+1}}\mathbf{N}^{T}\mathbf{\bar{C}}\ddot{ \mathbf{u}}^{h}\mathrm{d}t+\int\limits_{t_{n}}^{t_{n+1}}\mathbf{N}^{T}\mathbf{ \bar{K}}\mathbf{u}^{h}\mathrm{d}t=\int\limits_{t_{n}}^{t_{n+1}}\mathbf{N}^{T} \mathbf{\bar{f}}\mathrm{d}t. \tag{8}\]  
After implementing integration by parts for the first term of Equation (8), which is:  
\[\int\limits_{t_{n}}^{t_{n+1}}\mathbf{N}^{T}\mathbf{M}\ddot{\mathbf{u}}^{h} \mathrm{d}t=\left(\mathbf{N}^{T}\mathbf{M}\dot{\mathbf{u}}^{h}\right)_{t_{n}} ^{t_{n+1}}-\int\limits_{t_{n}}^{t_{n+1}}\dot{\mathbf{N}}^{T}\mathbf{M}\dot{ \mathbf{u}}^{h}\mathrm{d}t=\mathbf{N}(t_{n+1})^{T}\mathbf{M}\dot{\mathbf{u}}_{ n+1}-\mathbf{N}(t_{n})^{T}\mathbf{M}\dot{\mathbf{u}}_{n}-\int\limits_{t_{n}}^{t_{n+1}} \mathbf{N}^{T}\mathbf{M}\dot{\mathbf{u}}^{h}\mathrm{d}t \tag{9}\]  
where \(\dot{\mathbf{u}}_{n+1}\) and \(\dot{\mathbf{u}}_{n}\) represent the velocity at \(t_{n+1}\) and \(t_{n}\), respectively. Substituting Equation (7a) into Equation (8), and then taking a time coordinate mapping from \(t\in[t_{n},t_{n+1}]\) to \(\zeta\in[-1,1]\), the following algebraic equations are yielded:  
\[(\frac{-4}{\Delta t^{2}}m\mathbf{K}+\frac{2}{\Delta t}{}^{c}\mathbf{K}+{}^{k} \mathbf{K})\mathbf{d}+\frac{2\mathbf{M}}{\Delta t}(\mathbf{N}(1)^{T}\dot{ \mathbf{u}}_{n+1}-\mathbf{N}(-1)^{T}\dot{\mathbf{u}}_{n})=\mathbf{P} \tag{10}\]  
with  
\[{}^{m}\mathbf{K}=\int_{-1}^{1}\dot{\mathbf{N}}^{T}\mathbf{M}\dot{\mathbf{N}} \mathrm{d}t,{}^{c}\mathbf{K}=\int_{-1}^{1}\mathbf{N}^{T}\mathbf{\bar{C}}( \zeta)\dot{\mathbf{N}}\mathrm{d}t \tag{11}\]  
\[{}^{k}\mathbf{K}=\int_{-1}^{1}\mathbf{N}^{T}\mathbf{\bar{K}}(\zeta)\mathbf{N} \mathrm{d}t,\mathbf{P}=\int_{-1}^{1}\mathbf{N}^{T}\mathbf{d}t \tag{12}\]  
denoting  
\[\mathbf{K}=\frac{-4}{\Delta t^{2}}m\mathbf{K}+\frac{2}{\Delta t}{}^{c}\mathbf{ K}+{}^{k}\mathbf{K}. \tag{13}\]  
Equation (10) is equivalent to the equation:  
\[\mathbf{K}\mathbf{d}+\frac{2\mathbf{M}}{\Delta t}(\mathbf{N}(1)^{T}\dot{ \mathbf{u}}_{n+1}-\mathbf{N}(-1)^{T}\dot{\mathbf{u}}_{n})=\mathbf{P} \tag{14}\]  
which could further be written into the following tensor notation form:  
\[\mathbf{u}^{h}=\mathbf{N}\mathbf{d}=\sum_{i=1}^{m+1}N_{i}(t)\mathbf{d}_{i} \tag{7a}\]0  
\[\mathbf{u}^{h}=\mathbf{N}\mathbf{d}=\sum_{i=1}^{m+1}N_{i}(t)\mathbf{d}_{i} \tag{7a}\]1 A time element of degree \(m\).