## 3 Space--Time Dynamic Analysis Method Based on DCM
### Space-Time Discretization and Numerical Scheme of DC Dynamic Analysis  
For the above one-dimensional transient problem, a two-dimensional DC grid is needed for discretization. The orthogonal rectangular grids shown in Figure 1 are selected as the space-time discretization scheme, and the Equation (4) is discretized as the following form  
\[\frac{EI}{L^{4}}\sum_{k=1}^{N}c_{ik}^{(2^{4})}y_{k}+\frac{\rho}{T^{2}}\sum_{k =1}^{N}c_{ik}^{(\tau^{2})}y_{k}-\frac{\rho\zeta}{T^{2}L^{2}}\sum_{k=1}^{N}c_{ ik}^{(2^{4}\tau^{2})}y_{k}=F(L\xi_{i},T\tau_{i}) \tag{9}\]  
where \(c_{ik}^{(\xi^{4})}\) and \(c_{ik}^{(\tau^{2})}\) are the DC weight coefficients of the fourth order partial derivatives of \(\xi\) and the second order partial derivatives of \(\tau\), \(c_{ik}^{(2^{4}\tau^{2})}\) is the fourth order mixed partial derivatives of \(\xi\) and \(\tau\), and \(N\) is the total number of discrete nodes. If only one spatio-temporal grid is substituted into the calculation of the whole-time length \(T\), in order to achieve the ideal accuracy, a considerable number of nodes need to be used, which will cause the matrix dimension to be too large and ill-conditioned. Therefore, in order to achieve long-term dynamic analysis, the whole space-time domain can be equally divided into (\(n\!-\!1\)) small space-time segments by referring to the progressive integration method. The length of each time step is \(\Delta T\), and the equipartition process is shown in Figure 2.  
For each time step \(\Delta T\), its spatial-temporal discrete form is shown in Figure 3. Its initial time connects to the end of the previous time step. The initial displacement and initial velocity correspond to the displacement and velocity at the end of the previous time step. For convenience, group the nodes in the grid. Let the nodes at the initial time \(\tau=0\) form \(B_{3}\), the nodes satisfying \(\xi=0\) and \(\tau\
eq 0\) form \(B_{1}\), the nodes satisfying \(\xi=1\) and \(\tau\
eq 0\) form \(B_{2}\), all the remaining nodes form \(B_{i}\), and the nodes satisfying \(\tau=1\) in \(B_{i}\) form \(B_{4}\). In particular, the case where any two combination symbols correspond to a combination of the underlying markers represents the union of the two node groups, such as \(B_{12}\) being a combination of \(B_{1}\) and \(B_{2}\). The gradual calculation method needs to extract the initial conditions (displacement and velocity) of each time step. At the same time, in order to substitute the boundary conditions, the discrete control Equation (9) needs to be reconstructed. Each partial derivative term in Equation (9) is decomposed as follows:  
Figure 1: Orthogonal rectangular grids. Figure 2: Division of space–time domain. \[ \begin{array}{ll}\sum\limits_{k=1}^{N}c_{ik}^{(\xi^{4})}y_{k}&=\sum\limits_{k= B_{12}}\sum\limits_{j=B_{N}}c_{ik}^{(\xi^{2})}c_{kj}^{(\xi^{2})}y_{j}+\sum\limits_{k= B_{N}}\sum\limits_{j=B_{N}}c_{ik}^{(\xi^{2})}c_{kj}^{(\xi^{2})}y_{j}\\ &=L^{2}\sum\limits_{k=B_{12}}c_{ik}^{(\xi^{2})}\frac{\partial^{2}y}{\partial \xi^{2}}\Big{|}_{k}+\sum\limits_{k=B_{N}}\sum\limits_{j=B_{N}}\sum\limits_{l= B_{12}}c_{ik}^{(\xi^{2})}c_{kj}^{(\xi^{2})}c_{jl}^{(\xi^{2})}y_{l}+\sum \limits_{k=B_{N}}\sum\limits_{j=B_{N}}\sum\limits_{l=B_{3}}c_{ik}^{(\xi^{2})}c _{kj}^{(\xi^{2})}c_{jl}^{(\xi^{2})}y_{l}\\ &+\sum\limits_{k=B_{N}}\sum\limits_{j=B_{N}}\sum\limits_{l=B_{i}}c_{ik}^{(\xi^ {2})}c_{kj}^{(\xi^{2})}c_{jl}^{(\xi^{2})}y_{l}\\ \end{array}  \tag{10}\]  
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