## 4 Validation and Application  
### Validation Example 1: A Block Sliding along a Slope  
This example considers an initially static block sliding along a slope with an incline angle of 30\({}^{\circ}\) due to gravitational force. The block is a cube with side length of 1 m. The slope is a plate with a length of 10 m, width of 1.6 m and thickness of 0.4 m. The block and the slope are both elastic and have the same elastic constants. The material parameters are set as density \(\rho=7850\) kg/m\({}^{3}\), shear modulus \(G=76.9\) GPa and bulk modulus \(K=166.7\) GPa. The gravitational acceleration is set to \(9.8\) m/s\({}^{2}\) for \(z\)-direction and zero for other directions. Uniform particles with a spacing of 0.2 m are used to discretize the block and the slope. There are 216 particles for the block and 1377 particles for the slope. Figure 4 shows the initial SPH calculation model for the present problem. In this example, the time step is set to \(5\times 10^{-6}\) s, and the artificial viscosity constants \(C_{L}\) and \(C_{Q}\) are 0.25 and 1.0, respectively. The total simulation time is set to 1.8 s. Since the problem is a small deformation problem, surface reconstruction is only performed at the first time step. Figure 5 shows the sliding processes of the block simulated by the present contact algorithm for both frictional and frictionless cases. In the figure, the triangular meshes represent the reconstructed surfaces of the block and the slope, and the small spheres are boundary particles. The same display method is used in the following sections to show deformed shapes of contact objects. The arrows in the figure indicate the velocity vectors of particles. It can be seen that the shapes of the block and the slope are well reconstructed. The deformations of the block and the slope are negligible, which means they can be considered as rigid bodies. The velocity vectors of all particles are uniform and parallel to the slope surface. This indicates that the block slides on the slope smoothly, and there is no significant numerical oscillation excited by the contact calculation. By assuming the block and the slope are rigid bodies, analytical solutions to the displacements of the block can be easily obtained for a quantitative assessment of the accuracy of our numerical results. Figure 6 shows the displacements of the block in \(x\)- and  
Figure 4: Initial SPH calculation model for a block sliding along a slope. Figure 5: Calculation results for a block sliding along a slope: (**a**) Friction coefficient \(\mu=0\); (**b**) Friction coefficient \(\mu=0.3\). \(z\)-directions. The displacements of the block are represented by the displacements of the particle closest to the center of the top face of the block. The results calculated with the present contact algorithm are in excellent agreement with the corresponding analytical solutions. This demonstrates that the present contact algorithm can achieve very good accuracy for such small deformation problems. For comparison, the particle-to-particle contact algorithm [6,8] is also used to calculate the present problem. The algorithm detects contacts between SPH objects by the following criterion  
\[R_{i}+R_{j}-d_{ij}>0, \tag{26}\]  
where \(R_{i}=0.5h_{i}/\alpha_{\mathrm{h}}\) and \(R_{j}=0.5h_{j}/\alpha_{\mathrm{h}}\) are the radii of particles \(i\) and \(j\), respectively. It enforces the non-penetration contact condition with a penalty method. With this method, normal penalty forces are applied to particles in contact in order to prevent penetration between them.