# Three-Dimensional Elastodynamic Analysis Employing Partially Discontinuous Boundary Elements
## 2 Time-Domain Boundary Integral Equation of Elastodynamics and Its Numerical Implementation
### Numerical Integration Scheme
The physical variable at any point in the element can be calculated by the standard shape function as  
\[u(\xi,\eta)=\sum\limits_{k=1}^{4}N_{k}(\xi  
when \(r<c_{2}\Delta t\), in which \(c_{2}\) represents the shear wave velocity and \(\Delta t\) represents the time step. The smaller the time step \(\Delta t\), the more drastically the function value changes.  
Therefore, this paper proposes an improved element subdivision scheme related to the time step as follows:  
1. Create a square with the source point \(\mathbf{p}\) as the center and \(2\)\(c_{2}\Delta t\) as the side length. If the square exceeds the boundary of the element (as shown by the red dashed lines in Figure 6), the boundary of the element is taken as the boundary of the square;
2. According to the position of the source point \(\mathbf{p}\) and the square area established, the element is subdivided into several triangular patches containing \(\mathbf{p}\) and quadrilateral patches without \(\mathbf{p}\), as shown by the black dashed lines in Figure 6;
3. The \(\alpha-\beta\) transformation method is used for integrals on the triangular patches, and the standard Gauss-Legendre quadrature method is used on the quadrilateral patches.  
Compared with the traditional element subdivision method, the improved element subdivision method related to the time step proposed in this paper can cause the Gauss points to be mainly distributed in the region of \(r<c_{2}\Delta t\) (the region where the kernel function value changes drastically). Therefore, in the case of the same number of Gauss points, the improved subdivision method can significantly improve the accuracy of the singular integral. To verify the computational accuracy of the improved element subdivision method, a numerical example is given next to compare the computational accuracy before and after the improvement.  
As shown in Figure 7, the coordinate of the source point \(\mathbf{p}\) is \((0,0)\), and the coordinates of the quadrilateral element are shown in the figure. The dotted circles represent the  
Figure 5: The curve graph of the displacement kernel function under different time steps.  
Figure 6: The improved element subdivision method: (**a**) \(\mathbf{p}\) is located at the corner; (**b**) \(\mathbf{p}\) is located on the edge; (**c**) \(\mathbf{p}\) is located in the element.  
boundaries of the wavefronts. The radius of the oscillation area changes with the time step \(\Delta t\). Consider the following integral function:  
\[I_{ij}=\int\limits_{S}U_{ij}^{MM}(\mathbf{p},\mathbf{q})dS(\mathbf{q}) \tag{22}\]  
where \(U_{ij}^{MM}(\mathbf{p},\mathbf{q})\) is the displacement kernel function after time integration. The material parameters in the function \(U_{ij}^{MM}(\mathbf{p},\mathbf{q})\) are defined as follows: [PERSON]'s modulus \(E=1.1\times 10^{5}\) N/m\({}^{2}\), density \(\rho=2.0\) kg/m\({}^{3}\), and [PERSON]'s ratio \(\
u=0.0\). The average relative error of the integration result is defined as follows:  
\[\text{error}=\frac{1}{n}\sum\limits_{i=1}^{i=3}\sum\limits_{j=1}^{i=3}\left| \frac{I_{ij}-I_{ij}^{*}}{I_{ij}^{*}}\right|_{I_{ij}^{*}\
eq 0} \tag{23}\]  
in which \(I_{ij}^{*}\) represents the reference result. In this example, the integration results of 50 * 50 Gauss points for each triangle patch and 25 * 25 Gauss points for each quadrilateral patch are used as reference results. With different time steps and different numbers of Gauss points, the errors of the integral results before and after the improvement are shown in Table 1. It can be seen that when the number of Gauss points used is the same, the computation error of the traditional element subdivision method becomes larger and larger as the time step \(\Delta t\) decreases. However, the computation error of the improved method can be kept at the same order of magnitude. In the case of the same time step, whether it is improved or not, the computation error will decrease as the number of Gauss points increases.