# Three-Dimensional Elastodynamic Analysis Employing Partially Discontinuous Boundary Elements
## 2 Time-Domain Boundary Integral Equation of Elastodynamics and Its Numerical Implementation
### Numerical Implementation of the BIE  
Equation (6) can be solved numerically by using the time step method to deal with the time integral and the collocation point method to process the space integral. If we discretize the time interval [0, \(t\)] into \(M\) time steps of duration \(\Delta t\), the discretized time nodal \(t_{M}=M\Delta t\), where \(M=1,2,\,\ldots,\,N\). Lagrange interpolation is used to approximate the dynamic response at any time in the discrete time interval [\(t_{M-1}\), \(t_{M}\)]. In this paper, both displacement and traction adopt linear interpolation in the time interval, that is, for a specific time step \(M\), the time interpolation formulas of displacement and traction are as follows:  
\[ \begin{array}{l}u_{i}(\mathbf{q},\tau)=\sum\limits_{m=1}^{M}\left[\varphi^{m -1}u_{i}^{m-1}(\mathbf{q})+\varphi^{m}u_{i}^{m}(\mathbf{q})\right]\\ p_{i}(\mathbf{q},\tau)=\sum\limits_{m=1}^{M}\left[\varphi^{m-1}p_{i}^{m-1}( \mathbf{q})+\varphi^{m}p_{i}^{m}(\mathbf{q})\right]\end{array}  \tag{7}\]  
in which  
\[\varphi^{m-1}=\frac{m\Delta t-\tau}{\Delta t}\Phi_{m}(\tau),\,\,\varphi^{m}= \frac{\tau-(m-1)\Delta t}{\Delta t}\Phi_{m}(\tau) \tag{8}\]  
with  
\[\Phi_{m}(\tau)=H(\tau-(m-1)\Delta t)-H(\tau-m\Delta t) \tag{9}\]  
Substituting Equation (7) into Equation (6), the BIE of the time-discrete format can be obtained as follows:  
\[ \begin{array}{l}C_{ij}(\mathbf{p})u_{j}^{M}(\mathbf{p})=\sum\limits_{m=1}^{ M}\int_{S}\int_{(m-1)\Delta t}^{m\Delta t}u_{ij}^{s}(\mathbf{p},\mathbf{q};M \Delta t-\tau)(\frac{m\Delta t-\tau}{\Delta t}p_{j}^{m-1}(\mathbf{q})+\frac{ \tau-(m-1)\Delta t}{\Delta t}p_{j}^{m}(\mathbf{q}))\mathrm{d}\tau\mathrm{d}S( \mathbf{q})\\ -\sum\limits_{m=1}^{M}\int_{S}\int_{(m-1)\Delta t}^{m\Delta t}p_{j}^{s}( \mathbf{p},\mathbf{q};M\Delta t-\tau)(\frac{m\Delta t-\tau}{\Delta t}u_{j}^{m -1}(\mathbf{q})+\frac{\tau-(m-1)\Delta t}{\Delta t}u_{j}^{m}(\mathbf{q})) \mathrm{d}\tau\mathrm{d}S(\mathbf{q})\end{array}  \tag{10}\]  
The time integral in Equation (10) can be calculated analytically. After time integration, the BIE without the time variable \(\tau\) can be obtained, as shown below:  
\[ \begin{array}{l}C_{ij}(\mathbf{p})u_{j}^{M}(\mathbf{p})=\int_{S}U_{ij}^{MM }p_{j}^{M}(\mathbf{q})\mathrm{d}S(\mathbf{q})-\int_{S}P_{ij}^{MM}u_{j}^{M}( \mathbf{q})\mathrm{d}S(\mathbf{q})\\ \hskip 28.452756 pt+\sum\limits_{m=1}^{M-1}\int_{S}U_{ij}^{MM}p_{j}^{m}( \mathbf{q})\mathrm{d}S(\mathbf{q})-\sum\limits_{m=1}^{M-1}\int_{S}P_{ij}^{MM}u_ {j}^{m}(\mathbf{q})\mathrm{d}S(\mathbf{q})\\ \hskip 28.452756 pt+\int_{S}U_{ij}^{MM}p_{j}^{0}(\mathbf{q})\mathrm{d}S(\mathbf{ q})-\int_{S}P_{ij}^{MM}u_{j}^{0}(\mathbf{q})\mathrm{d}S(\mathbf{q})\end{array}  \tag{11}\]  
in which \(U_{ij}^{MM}\) and \(P_{ij}^{MM}\) represent the displacement and traction kernel functions after time integration, respectively, and \(u_{j}^{0}(\mathbf{q})\) and \(p_{j}^{0}(\mathbf{q})\) represent the displacement and traction at the initial moment, respectively. Under zero initial conditions, \(u_{j}^{0}(\mathbf{q})=p_{j}^{0}(\mathbf{q})=0\). After discretizing the boundary S into \(L\) elements, we use the \(K\)-th Lagrange interpolation function to formulate the BIE of the spatial discretization:  
\[ \begin{array}{l}C_{ij}(\mathbf{p})u_{j}^{M}(\mathbf{p})=\sum\limits_{l=1}^{ L}\sum\limits_{k=1}^{K}p_{j}^{Mk}\int_{S_{l}}U_{ij}^{MM}N_{k}(\mathbf{q}) \mathrm{d}S_{l}(\mathbf{q})-\sum\limits_{l=1}^{L}\sum\limits_{k=1}^{K}u_{j}^{ Mll}\int_{S_{l}}P_{ij}^{MM}N_{k}(\mathbf{q})\mathrm{d}S_{l}(\mathbf{q})\\ \hskip 28.452756 pt+\sum\limits_{m=1}^{M-1}\sum\limits_{l=1}^{L}\sum\limits_{k=1}^{ K}p_{j}^{mll}\int_{S_{l}}U_{ij}^{MM}N_{k}(\mathbf{q})\mathrm{d}S_{l}(\mathbf{q})- \sum\limits_{m=1}^{M-1}\sum\limits_{l=1}^{L}\sum\limits_{k=1}^{K}u_{j}^{mll}\int _{S_{l}}P_{ij}^{MM}N_{k}(\mathbf{q})\mathrm{d}S_{l}(\mathbf{q})\end{array}