## 4 Validation and Application
### Validation Example 2: Impact between Two Elastic-Plastic Blocks  
This example considers the impact of two blocks travelling with an initial velocity of 200 m/s in opposite directions. The two blocks are just in contact in the initial time. They have the same geometric dimensions, which are 10 mm in length and width and 4 mm in thickness. The materials of the blocks are the same and assumed to be elastic-perfectly plastic material. The material parameters are set as density \(\rho\) = 7850 kg/m\({}^{3}\), shear modulus \(G\) = 76.9 GPa, bulk modulus \(K\) = 166.7 GPa and yield strength \(\sigma_{0}\) = 600 MPa. Uniform particles with a spacing of 0.5 mm are used to discretized the blocks. There are 3969 particles for each block. Figure 10 shows the initial SPH calculation model for the present problem. For this example, the time step is set to 2.5 \(\times\) 10\({}^{-8}\) s, the artificial viscosity constants are the same as those used in the previous example, and surface reconstruction is triggered in every time step. Figure 11 shows the deformed shapes of the blocks at two different time instants. The deformation results calculated with the present contact algorithm are compared with those obtained using the FEM in LS-DYNA [27]. In the FEM model, the element size is the same as the particle spacing used in the SPH model. The scale factor for the time step is 0.1, and the contacts between the blocks are treated by the algorithm defined by the keyword \"CONTACT_AUTOMATIC_SURFACE_TO_SURFACE along with segments in contact regions of the two blocks specified as slave and master surfaces, so that an optimal result is obtained. From Figure 8, it can be seen that the complex shapes of the blocks are properly reproduced by the reconstruction approach employed in the present contact algorithm, and the shapes of the blocks calculated by SPH are in good agreement with those calculated by the FEM. Figure 10: Initial SPH calculation model for high-velocity impact of two blocks. Figure 9: Comparison of the velocities of the block calculated by the present algorithm and the particle-to-particle contact algorithm: (**a**) Velocity in \(x\)-direction; (**b**) Velocity in \(z\)-direction. Figure 12 gives a comparison of the deformed width and thickness of the upper block simulated by the SPH and the FEM. The width is measured as the distance between the midpoints of the opposite sides of the bottom surface, and the thickness is measured as the distance between the midpoints of the bottom and top surfaces. It can be seen that the variations of width and thickness calculated by SPH are in good agreement with those calculated using the FEM. The relative error is about 0.5% for the final width and 3.7% for the final thickness. Figure 13 gives a comparison for the global \(z\)-direction velocity histories of the two blocks simulated by the SPH and the FEM. Overall, the global \(z\)-direction velocity histories of SPH agree well with those of the FEM. Interestingly, the final velocities of the two blocks are in excellent agreement with those of the FEM. Figure 11: Deformed shapes of the blocks calculated by different methods (Side view): (**a**) Present algorithm; (**b**) FEM. Figure 12: Comparison of width and thickness of the upper block calculated by SPH and FEM: (**a**) Width; (**b**) Thickness. The results shown above demonstrate that the present contact algorithm is also applicable to impact contact problems with significant deformations and can achieve good accuracy for such problems. With this example, the computational cost of the whole SPH simulation and the contact calculation with the present contact algorithm is examined. The computational cost is the consumed CPU time on a personal computer with Intel(r) Core(tm) i7-8750H CPUs (Operating frequency: 2.2 GHz) and 16 GB RAM. The computations are conducted in serial mode. The total computational cost includes nearly 5 s of CPU time spent on the output of result files.