## 5 Conclusions  
In this study, we investigated the elastic-plastic ejection from grooved aluminum surfaces under double supported shocks using the SPH method. First, the present method was validated by the pressure-temperature relationship of the material during the loading and unloading of the shock wave and the 2D high-velocity impact. Then, the mechanism of the second ejection, and the effects of the second shock pressure and time interval on the second ejection were analyzed. The main conclusions are summarized as follows:  
(1) The spallation region, formed at the bottom of the bubble during the first ejection due to rarefaction wave interactions, initially expanded fan-shaped into the sample interior, followed by widening and eventual stabilization. The second ejecta can be classified into high-, medium-, and low-velocity categories with clear boundaries. The total mass of the second ejection increased significantly compared to the first ejection. High- and medium-velocity ejecta were primarily generated by the recompression of spallation material, while the majority of the total mass of the second ejecta arose from low-velocity ejecta due to the significant increase in the defect area of the bubble and the convergence of the first ejecta.  
(2) The shock pressure threshold required for the second ejection was significantly reduced compared with the first ejection. A distinct second ejection with a total mass of 8.793 mg/cm\({}^{2}\) was observed at a second shock pressure of 16.9 GPa, surpassing the total mass of the first ejection (3.889 mg/cm\({}^{2}\)). The maximum velocity of the second ejecta increased linearly with the surface jump velocity. The total mass of the second ejecta increased with the surface jump velocity. However, the rate of increase gradually decreased, primarily affecting the low-velocity ejecta, whereas the total mass of the high- and medium-velocity ejecta remained approximately constant.  
(3) The second ejection was significantly affected by the time interval between shocks, primarily due to the evolution of the spallation region at the bottom of the bubble in the first ejection and the convergence of the first ejecta. The structures of the high- and medium-velocity ejecta stabilized owing to the stabilization of the spallation region, while the low-velocity ejecta continued to evolve due to the interaction with the first ejecta. The maximum velocity of the second ejection decreased and converged to a constant value as the time interval increased. Furthermore, the total mass of the high- and medium-velocity ejecta remained approximately constant, while the total mass of the low-velocity ejecta increased progressively, stabilizing at a constant value over time.  
**Author Contributions:** Conceptualization, W.L.; methodology, W.L.; software, W.L.; validation, W.L. and H.X.; formal analysis, W.L.; investigation, W.L. and H.X.; resources, W.L.; data curation, W.L. and H.X.; writing--original draft preparation, W.L.; writing--review and editing, W.L. and H.X.; visualization, W.L.; supervision, W.L.; project administration, W.L.; funding acquisition, W.L. All authors have read and agreed to the published version of the manuscript.  
**Funding:** This research was funded by China Postdoctoral Science Foundation, grant number 2023M740256.  
**Institutional Review Board Statement:** Not applicable.  
**Informed Consent Statement:** Not applicable.  
**Data Availability Statement:** The original contributions presented in this study are included in the article. Further inquiries can be directed to the corresponding author.  
**Conflicts of Interest:** The authors declare no conflicts of interest.