# Three-Dimensional Elastodynamic Analysis Employing Partially Discontinuous Boundary Elements
## 2 Time-Domain Boundary Integral Equation of Elastodynamics and Its Numerical Implementation
### Time-Domain Boundary Integral Equation  
The governing equation of the elastodynamic problem for homogeneous, isotropic, linear elastic materials [23] is  
\[(\lambda+G)u_{jjj}(\mathbf{q},t)+Gu_{i,jj}(\mathbf{q},t)+\rho b_{i}(\mathbf{q},t )=\rho\ddot{u}_{i}(\mathbf{q},t) \tag{1}\]  
with boundary conditions  
\[\begin{array}{l}u_{i}(\mathbf{q},t)=\overline{u}_{i}(\mathbf{q},t),\;\forall \mathbf{q}\in S^{ui}\\ p_{i}(\mathbf{q},t)=\overline{p}_{i}(\mathbf{q},t),\;\forall\mathbf{q}\in S^{ pi}\end{array} \tag{2}\]  
and initial conditions  
\[\begin{array}{l}u_{i}(\mathbf{q},0)=u_{i}^{0}(\mathbf{q}),\;\forall\mathbf{q }\in V\\ u_{i}(\mathbf{q},0)=v_{i}^{0}(\mathbf{q}),\;\forall\mathbf{q}\in V\end{array} \tag{3}\]  
in which the symbols used are as follows:  
* \(i\) and \(j\) represent the three directions of the 3D space;
* \(u_{i}\) represents the displacement component;
* \(p_{i}\) represents the traction component in the \(i\) direction;
* \(b_{i}\) represents the body force per mass;
* \(\dot{u}_{i}\) is the first-order derivative of displacement \(u_{i}\) versus time \(t\);
* \(\ddot{u}_{i}\) is the second-order derivative of displacement \(u_{i}\) versus time \(t\);
* \(\overline{u}_{i}\) and \(\overline{p}_{i}\) represent the boundary displacement on \(S^{ui}\) and the boundary traction on \(S^{pi}\), respectively, where the whole boundary \(S=S^{ui}\cup S^{pi}\);
* \(V\) represents the whole domain of the analysis object;
* \(\rho\) is the medium density;
* \(\lambda\) and \(G\) are the Lame constants given by \[\lambda=\frac{\
u E}{(1+\
u)(1-2\
u)},\;G=\frac{E}{2(1+\
u)}\] (4) where \(E\) and \(\
u\) represent [PERSON]'s modulus and [PERSON]'s ratio, respectively. The time-domain boundary integral equation (BIE) corresponding to the governing Equation (1) can be expressed as \[\begin{array}{l}C_{ij}(\mathbf{p})u_{j}(\mathbf{p},t)=\int\limits_{s} \int\limits_{0}^{t}u_{ij}^{s}(\mathbf{p},\mathbf{q};t-\tau)p_{j}(\mathbf{q}, \tau)\mathrm{d}\tau\mathrm{d}S(\mathbf{q})-\int\limits_{s}\int\limits_{0}^{t} p_{ij}^{s}(\mathbf{p},\mathbf{q};t-\tau)u_{j}(\mathbf{q},\tau)\mathrm{d}\tau \mathrm{d}S(\mathbf{q})\\ +\rho\int\limits_{V}\int\limits_{0}^{t}u_{ij}^{s}(\mathbf{p},\mathbf{q};t- \tau)b_{j}(\mathbf{q},\tau)\mathrm{d}\tau\mathrm{d}V(\mathbf{q})+\rho\int \limits_{V}u_{ij}^{s}(\mathbf{p},\mathbf{q};t)v_{i}^{0}(\mathbf{q})\mathrm{d}V (\mathbf{q})\\ +\rho\int\limits_{V}\dot{u}_{ij}^{s}(\mathbf{p},\mathbf{q};t)u_{i}^{0}( \mathbf{q})\mathrm{d}V(\mathbf{q})\end{array}\] (5) in which the meanings of the symbols used are as follows:
* \(C_{ij}=\delta_{ij}/2\) when the source point \(\mathbf{p}\) is located on smooth surfaces, and \(C_{ij}=\delta_{ij}\) when the source point \(\mathbf{p}\) is located in the domain (\(\delta_{ij}=1\) when \(i=j\), otherwise \(\delta_{ij}=0\));
* \(u_{ij}^{s}(\mathbf{p},\mathbf{q};t-\tau)\) and \(p_{ij}^{s}(\mathbf{p},\mathbf{q};t-\tau)\) are the time-dependent displacement fundamental solution and the traction fundamental solution, respectively, whose expressions can be found in [23];
* \(\dot{u}_{ij}^{s}(\mathbf{p},\mathbf{q};t)\) represents the velocity fundamental solution, which is the first-order derivative of \(u_{ij}^{s}(\mathbf{p},\mathbf{q};t-\tau)\) versus time. Assuming that the body force and initial conditions are both zero, Equation (5) can be simplified to \[C_{ij}(\mathbf{p})u_{j}(\mathbf{p},t)=\int\limits_{s}\int\limits_{0}^{t}u_{j}^ {s}(\mathbf{p},\mathbf{q};t-\tau)p_{j}(\mathbf{q},\tau)\mathrm{d}\tau\mathrm{d }S(\mathbf{q})-\int\limits_{s}\int\limits_{0}^{t}p_{ij}^{s}(\mathbf{p}, \mathbf{q};t-\tau)u_{j}(\mathbf{q},\tau)\mathrm{d}\tau\mathrm{d}S(\mathbf{q})\] (6)