# Three-Dimensional Elastodynamic Analysis Employing Partially Discontinuous Boundary Elements
## 4 Numerical Examples
### Longitudinal Forced Vibration of a Cantilevered Beam  
The geometric model used in this example is shown in Figure 8. The left end of the beam is fixed, and the right end is subjected to a Heaviside-type load \(p(t)=1000\) H(\(t\)) Pa. The material parameters of the beam are [PERSON]'s modulus \(E=1.1\times 10^{5}\) N/m\({}^{2}\), density \(\rho=2.0\) kg/m\({}^{3}\), and [PERSON]'s ratio \(\
u=0.0\). By setting \(\
u=0.0\), the problem in this example can be regarded as the longitudinal forced vibration of a one-dimensional rod. The analytical solutions to the problem are available. The displacement solution and the internal force solution can be calculated by the formulas  
\[u(x,t)=\frac{8 pL}{\pi^{2}c^{2}\rho}\sum_{n=1,3,5,...}^{\infty}\frac{(-1)^{(n-1 )/2}}{n^{2}}(1-\cos\frac{n\pi ct}{2L})\sin\frac{n\pi x}{2L} \tag{24}\]  
and  
\[p(x,t)=-\frac{4 pE}{\pi c^{2}\rho}\sum_{n=1,3,5,...}^{\infty}\frac{(-1)^{(n-1)/2 }}{n}(1-\cos\frac{n\pi ct}{2L})\cos\frac{n\pi x}{2L} \tag{25}\]  
respectively, where \(c=\sqrt{E/\rho}\) is the wave speed. The discretized mesh model is shown in Figure 9a, which includes 72 linear quadrilateral elements and 126 nodes. In this model, the nodes located at the corner points are shrunk to the surfaces in proportion \(\lambda=a/b(0<\lambda<0.5)\) (as shown in Figure 9b), where \(a\) represents the moving distance of the node and \(b\) represents the edge length of the element. The computation results of the time-domain BEM are sensitive to the time step. An improper time step length will cause the results to diverge, which is unstable. Therefore, in many papers [24; 25; 26], the time step is selected according to the dimensionless parameters \(\beta=c_{1}\Delta t/d\), where \(c_{1}\) is the pressure wave velocity, \(\Delta t\) is the time step, and \(d\) is the element edge length. The smaller the value of \(\beta\) is, the more unstable the result will be. The larger the value of \(\beta\) is, the lower the computational accuracy. In general, acceptable results can be obtained when \(\beta\approx 1\). In this experiment, we choose the time step that can obtain stable results for computation. On this basis, the influence of the inward shrink ratio of the element node on the computation result is investigated. Figure 8: The geometric model and the boundary conditions of the beam: (**a**) geometric model; (**b**) load curve. When \(\Delta t=0.00445\,\mathrm{s}(\beta\approx 1)\), the displacement response at the free end and the traction response at the fixed end under different \(\lambda\) values are shown in Figures 10 and 11, respectively. In these two figures, the left picture and the right picture show unstable results and stable results, respectively. It can be seen that the results are unstable when \(\lambda=0.05\) and \(\lambda=0.45\). The results are stable when \(0.1\leq\lambda\leq 0.4\), and the result curves overlap each other in these cases. Therefore, the value of \(\lambda\) should avoid being too large or too small, and the computational accuracy is insensitive to the value of \(\lambda\) if the results converge. Under the same conditions, if completely discontinuous elements are used for discretization, the mesh model contains 72 linear quadrilateral elements and 288 nodes, which are greater values than those using partially discontinuous elements. The displacement response at the free end and the traction response at the fixed end under different \(\lambda\) values  
Figure 11: Traction response at the fixed end computed by using partially discontinuous elements in terms of (**a**) the cases of unstable results and (**b**) the cases of stable results. Figure 9: The mesh model of the beam: (**a**) mesh model; (**b**) inward shrink ratio \(\lambda\). are shown in Figures 12 and 13, respectively. It can be seen that the results tend to be stable only when \(\lambda=0.1\), and the results are unstable in other cases. This demonstrates that the performance of the completely discontinuous element is very sensitive to the position of the element node.