# Three-Dimensional Elastodynamic Analysis Employing Partially Discontinuous Boundary Elements
## 4 Numerical Examples
### Longitudinal Forced Vibration of a Cantilevered Beam
An inappropriate node inward shrink ratio can easily cause unstable results, so it is not suitable for the time-domain BEM. In order to verify the effectiveness of the improved singular element subdivision method proposed in Section 3.2, we compared the accuracy of the results before and after the improvement. It should be mentioned that the partially discontinuous element is adopted in the following study. Since the singular integral only appears in the first analysis step, we select the results of the first ten analysis steps to clearly observe the difference between the results before and after the improvement. The comparison results under the time steps \(\Delta t=0.00356\) s(\(\beta\approx 0.8\)), \(\Delta t=0.00178\) s(\(\beta\approx 0.4\)), and \(\Delta t=0.00089\) s(\(\beta\approx 0.2\)) are shown in Figure 14a-c, respectively. In the figures, the \"Exact\" curve represents the exact solution, the \"Traditional\" curve represents the results before the improvement, and the \"Improved\" curve represents the results after the improvement. It can be seen that the improved results are in better agreement with the exact solutions. The results before the improvement become unstable as the time step decreases. This demonstrates that the improved singular element subdivision method can reduce the side effect of the time step on the stability of the result and effectively improve the accuracy of the singular integration. However, it should be noted that although the proposed subdivision method has improved the result accuracy in the first ten steps, the computational error becomes larger and larger with the accumulation of time steps, and instability results may still appear. The subdivision method proposed in this paper cannot completely solve the stability problem of the time-domain BEM but can improve the stability of the result. Figure 12: Displacement response at the free end computed by using completely discontinuous elements in terms of (**a**) the cases of unstable results and (**b**) the cases of stable results. Figure 13: Traction response at the fixed end computed by using completely discontinuous elements in terms of (**a**) the cases of unstable results and (**b**) the cases of stable results. Finally, the condition number of matrix \(\mathbf{A}\) in Equation (21) with a different inward shrink ratio \(\lambda\) is studied. The condition number of matrix \(\mathbf{A}\) is computed by  
\[cond(\mathbf{A})=\left\|\mathbf{A}\right\|_{2}\cdot\left\|\mathbf{A}^{-1} \right\|_{2} \tag{26}\]  
in which \(\left\|\mathbf{A}\right\|_{2}\) represents the two-norm of matrix \(\mathbf{A}\). Three common types of discontinuous element are considered, as shown in Figure 15. In the figure, case 1 shows that four nodes are shrunk into the element, case 2 shows that three nodes are shrunk into the element, and case 3 shows that two nodes are shrunk into the element. Case 1 corresponds to the completely discontinuous element, and cases 2 and 3 correspond to the partially discontinuous element. Through computing the condition number of matrix \(\mathbf{A}\) in Equation (21) with a different inward shrink ratio \(\lambda\), Figure 16 is plotted for a results comparison. From Figure 16, the following can be seen:  
(1) When the \(\lambda\) = 0, the condition number is equal to 1. The condition number of \(\mathbf{A}\) becomes larger as the value of \(\lambda\) increases. In case 1, the matrix \(\mathbf{A}\) becomes singular when \(\lambda\) = 0.5. The result verifies the conclusion that the value of \(\lambda\) should not be too large. (2) The condition numbers of \(\mathbf{A}\) in case 1 are larger than those in the other two cases. This verifies the conclusion that the result stability of the completely discontinuous element (corresponding to case 1) is poorer than that of the partially discontinuous element (corresponding to case 2 and case 3).