## 2 Basic Formulations
\right] \tag{12}\]  
\[\overline{\mathbf{M}}=\left(\mathbf{A}^{\rm T}\mathbf{C}_{\rm G}\mathbf{A}\right)\otimes\mathbf{M},\ \ \overline{\mathbf{K}}=\mathbf{C}_{\rm G}\otimes\mathbf{K},\ \ \ \overline{\mathbf{C}}=\left(\mathbf{A}^{\rm T}\mathbf{C}_{\rm G}\right)\otimes\mathbf{C} \tag{13}\]  
\[\mathbf{C}_{\rm G}=\mathrm{diag}\big{(}C_{j}\big{)},\ \ \ \overline{\mathbf{E}}=\mathbf{C}_{\rm G} \otimes\mathbf{E}, \tag{14}\]  
where '\(\mathrm{cs}\)' indicates column expansion of a matrix, '\(\otimes\)' is the Kronecker product, \(\mathbf{E}\) is a \(n\times n\) identity matrix, and \(\mathbf{C}_{j}\) is the coefficient of Gauss-Lobatto quadrature. Diagonal matrix \(\mathbf{C}_{\rm G}=\mathrm{diag}(\mathbf{C}_{j})\). Appendix A presents the derivations of Equations (8) to (10). Equation (13) shows that the expanded mass matrix \(\overline{\mathbf{M}}\) and stiffness \(\overline{\mathbf{K}}\) are symmetrical, but \(\overline{\mathbf{C}}\) is not, even though \(\mathbf{C}\) is symmetrical. Substituting Equations (8) to (10) into Equation (1), and noting that \(\delta\overline{\mathbf{U}}\) is arbitrary, we have the dynamic equilibrium equation in algebraic form as  
\[\overline{\mathbf{U}}^{\rm T}[\overline{\mathbf{K}}-\overline{\mathbf{M}}-\overline{\mathbf{C} }]=\overline{\mathbf{F}}^{\rm T}\overline{\mathbf{E}}, \tag{15}\]or in a compact and standard form as  
\[\mathbf{G}\overline{\mathbf{U}}=\mathbf{Q}, \tag{16}\]  
where  
\[\mathbf{G}=\overline{\mathbf{K}}-\overline{\mathbf{M}}-(\overline{\mathbf{C}})^{\mathsf{T}},\quad \mathbf{Q}=\overline{\mathbf{E}}\overline{\mathbf{F}}. \tag{17}\]  
Equation (16) can also be written in a different form as  
\[\mathbf{KUC}_{\mathrm{G}}-\mathbf{MUA}^{\mathsf{T}}\mathbf{C}_{\mathrm{G}}\mathbf{A}-\mathbf{CUC} _{\mathrm{G}}\mathbf{A}=\mathbf{FC}_{\mathrm{G}} \tag{18}\]  
which can be reduced to a Sylvester equation whose solutions can be found by the method of [PERSON] and [PERSON] [51] if we neglect damping. Then it has no row or column expansion of matrix in Equation (18), or it has no additional storage needed except for \(\mathbf{A}\) and the diagonal matrix \(\mathbf{C}_{\mathrm{G}}\) when compared with a step-by-step method. After deriving Equation (15), a DQTFEM has been formulated, which is different from classical TFEMs where Hermite interpolations are used and displacement and velocity are involved for all nodes. From the above derivation, one can see that, in DQTFEM, displacements are the only nodal parameter at all time sampling points since Lagrange interpolations are used; refer to Equations (11) and (12). In order to impose initial conditions and satisfy \(\mathrm{C}^{1}\)-compatible conditions in the present method, the nodal parameter vector \(\overline{\mathbf{U}}\) should be modified to include velocities for all spatial DOFs at the first time sampling point. To achieve this, an accurate and convenient method is to modify \(\overline{\mathbf{U}}\) as  
\[\widetilde{\mathbf{U}}^{\mathsf{T}}=\left[ \begin{array}{cccccccccccc}u_{11}& \cdots&u_{n1},&u_{12}&\cdots&u_{n2},&\cdots&,u_{1(N-1)}&\cdots&u_{n(N-1)},& \dot{u}_{11}&\cdots&\dot{u}_{n1}\end{array} \right], \tag{19}\]  
which is formulated by substituting the displacements \(u_{1N},\ \ \cdots,u_{nN}\) of the \(N\)th time sampling point by the velocities \(\dot{u}_{11},\ \ \cdots,\dot{u}_{n1}\) of the first time sampling point. The relation between \(\widetilde{\mathbf{U}}\) and \(\overline{\mathbf{U}}\) can be readily obtained by DQ rule as  
\[\widetilde{\mathbf{U}}=\mathbf{H}\overline{\mathbf{U}} \tag{20}\]  
\[\mathbf{H}=\mathbf{T}\otimes\mathbf{E} \tag{21}\]  
where \(\mathbf{T}\) is formed by substituting the first row of \(\mathbf{A}\) for the \(N\)th row of an unit matrix, that is  
\[\mathbf{T}=\left[ \begin{array}{cccccccc}1&0&0&\cdots&0&0&0\\ 0&1&0&\cdots&0&0&0\\ 0&0&1&\cdots&0&0&0\\ \vdots&\vdots&\vdots&\ddots&\vdots&\vdots&\vdots\\ 0&0&0&\cdots&1&0&0\\ 0&0&0&\cdots&0&1&0\\ A_{11}&A_{12}&A_{13}&\cdots&A_{1(N-2)}&A_{1(N-1)}&A_{1N}\end{array} \right].