# An Improved Differential Quadrature Time Element Method
## 2 Basic Rules and Formulations
### The Formulae of the Improved DQTEM  
The ordinary differential equation in structural dynamics has the form as  
\[\mathbf{M}\ddot{\mathbf{x}}+\mathbf{C}\dot{\mathbf{x}}+\mathbf{K}\mathbf{x}=\mathbf{f} \tag{7}\]  
where \(\mathbf{M}=[M_{ij}]_{M\times M}\), \(\mathbf{C}=[C_{ij}]_{M\times M}\) and \(\mathbf{K}=[K_{ij}]_{M\times M}\) are the mass, damping and stiffness matrices which are formulated explicitly by using spatial DQFEM, and will be indicated below. The displacement vector \(\mathbf{x}\) and load vector \(\mathbf{f}\) are defined as  
\[\left\{\begin{array}{l}\mathbf{x}^{\mathrm{T}}=[x_{1}(t)\ x_{2}(t)\ \cdots\ x_{M-1}(t)\ x_{M}(t)]\\ \mathbf{f}^{\mathrm{T}}=[f_{1}(t)\ f_{2}(t)\ \cdots\ f_{M-1}(t)\ f_{M}(t)]\end{array}\right. \tag{8}\]By using the DQ rule, the velocities at sampling points can be obtained  
\[\dot{x}_{ik}=\sum_{j=1}^{n}A_{kj}^{(1)}x_{ij}\quad(i=1,2,\cdots,M\text{ and }k=1,2,\cdots,n) \tag{9}\]  
where \(n\) is the number of DOFs of time element, a dot over the letter denotes the time derivative; the first subscript \(i\) of \(x\) or \(\dot{x}\) stands for the spatial DOF number, the second subscripts \(j\) and \(k\) stand for the time point number; and \(\dot{x}_{ik}\) and \(x_{ij}\) represent the values of \(\dot{x}_{i}\) at time point \(k\) and \(x_{i}\) at time point \(j\), respectively. Note that Gauss-Lobatto-Chebyshev points [3] are employed as time sampling points in present work. Let \(\mathbf{x}_{i}=[x_{i1},\ x_{i2},\ \cdots\ x_{in}]\), based on the DQ rule, the velocity vector \(\dot{\mathbf{x}}_{i}\) and acceleration vector \(\ddot{\mathbf{x}}_{i}\) can be written in a compact form as  
\[\dot{\mathbf{x}}_{i}=\mathbf{A}_{t}\mathbf{x}_{i} \tag{10}\]  
\[\ddot{\mathbf{x}}_{i}=\mathbf{A}_{t}\dot{\mathbf{x}}_{i}=(\mathbf{A}_{t})^{2}\mathbf{x}_{i}=B_{t} \mathbf{x}_{i} \tag{11}\]  
where \(\mathbf{A}_{t}\) and \(\mathbf{B}_{t}\) are the first and second order weighting coefficient matrices with respect to time, respectively. Substituting Equations (10) and (11) into Equation (7) yields  
\[\mathbf{M}\mathbf{X}\mathbf{B}_{t}^{\text{T}}+\mathbf{C}\mathbf{X}\mathbf{A}_{t}^{\text{T}}+\mathbf{K}\mathbf{ X}=\mathbf{F} \tag{12}\]  
and  
\[\left\{\begin{array}{l}\mathbf{X}^{\text{T}}=[\mathbf{x}_{1}\ \mathbf{x}_{2}\ \cdots\ \mathbf{x}_{M-1}\ \mathbf{x}_{M}]\\ \mathbf{F}^{\text{T}}=[\mathbf{f}_{1}\mathbf{f}_{2}\ \cdots\ \mathbf{f}_{M-1}\mathbf{f}_{M}]\end{array}\right. \tag{13}\]  
where \(\mathbf{f}_{i}=[f_{i1},\ f_{i2},\ \cdots\ f_{in}]\) has the similar definition as \(\mathbf{x}_{i}\). According to matrix theory, Equation (12) can be transformed into another form  
\[\mathbf{G}\mathbf{Z}=\mathbf{R} \tag{14}\]  
where \(\mathbf{Z}=\text{cs}\mathbf{X})\), \(\mathbf{R}=\text{cs}(\mathbf{F})\), \"\(\text{cs}\)\" indicates column expansion of a matrix, and  
\[\left\{\begin{array}{l}\mathbf{Z}^{\text{T}}=[x_{11}\cdots\ x_{M1}\ x_{12}\ \cdots\ x_{M2}\ x_{1n}\ \cdots\ x_{Mn}]\\ \mathbf{R}^{\text{T}}=[f_{11}\cdots\ f_{M1}\ f_{12}\ \cdots\ f_{M2}\ f_{1n}\ \cdots\ f_{Mn}] \end{array}\right. \tag{15}\]  
\[\mathbf{G}=\mathbf{B}_{t}\otimes\mathbf{M}+\mathbf{A}_{t}\otimes\mathbf{C}+\mathbf{E}\otimes\mathbf{K} \tag{16}\]  
where \"\(\otimes\)\" is Kronecker product, and \(\mathbf{E}\) is an \(n\times n\) unit matrix. Introducing initial conditions into Equation (14), one can solve displacements \(\mathbf{X}\), and the velocities and accelerations can be achieved through Equations (10) and (11).  
For a system with Rayleigh damping, its damping matrix can be represented by  
\[\mathbf{C}=\alpha\mathbf{K}+\beta\mathbf{M} \tag{17}\]  
where \(\alpha\) and \(\beta\) are the pre-determined constants [43; 44]. When \(\mathbf{C}=\mathbf{0}\), Equation (12) can be solved accurately and efficiently by the Bartels and Stewart method [45].