## 5 Conclusions  
This paper lays a solid foundation for further research on the effect of tank sloshing on the dynamic stability of ships. The phenomena of liquid sloshing in a ballast tank under  
Figure 11: Comparison of dynamic pressure at different rolling angles in case 1 and case 3 at two probes. (**a**) \(P_{1}\); (**b**) \(P_{2}\).  
Figure 12: Comparison of dynamic force at two rolling periods in case 3 and case 4. (**a**) case 3; (**b**) case 4.  
realistic ship rolling motion are numerically simulated by using the MPS method. After that, sloshing-induced force on sidewall of the tank is obtained and analyzed.  
Conclusions can be summarized as follows:  
(1) The phenomena of large deformation and nonlinear fragmentation of free surface, e.g., accumulation, bending and slamming, etc., can be found in liquid sloshing, which proves that a traditional quasi-static method cannot reflect the influence of the violent free surface on ship dynamic stability. In this case, meshfree methods should be used to get more realistic results.  
(2) The maximum sloshing-induced force is much bigger than the corresponding static one, e.g., 20% bigger in some cases. Meanwhile, both the rolling angle and period have significant effects on liquid sloshing. The sloshing intensity of the liquid in sloshing tanks is not positively correlated with the ship's undamped natural rolling frequency.  
(3) A large number of numerical simulations of liquid sloshing through a series of different ships' undamped natural periods can be carried out. The relationship between the rolling periods and the liquid filling rates of the most violent liquid sloshing will be investigated. By establishing the database of this relationship, it can provide effective suggestions for ship operators, such as changing the ballast water to adjust the dynamic stability of the ship.  
Although the MPS method has its own advantages, it also has the limitation of computing ability in dealing with large-scale violent free surface flow. This is the reason why this paper uses a scaled tank model to perform the simulation. Therefore, a more effective MPS method should be further investigated to simulate the liquid sloshing in all ship tanks of real size simultaneously.  
Conceptualization, X.S., Y.Z. and C.L.; methodology, X.S., Y.Z. and F.B.; Visualization, Y.Z.; writing--original draft preparation, X.S. and Y.Z.; software, Y.Z. and F.B.; data curation, C.L.; writing--review and editing, X.S., Y.Z. and Y.Y. All authors have read and agreed to the published version of the manuscript.  
The work was supported by the National Key Research and Development Program of China (No. 2019 YFE0111600), the National Natural Science Foundation of China (No. 52071049, No. 61971083, and No. 51939001), the LiaoNing Revitalization Talents Program (No. XLYC2002078), the Dalian Science and Technology Innovation Fund (No. 2019J11 CY015), and the project of intelligent ship testing and verification from the Ministry of Industry and Information Technology of the People's Republic of China (No. 2018/473).  
Not applicable.  
Readers can access our data by sending an email to the corresponding author [PERSON].  
The 3D ship design data from the Shanghai Merchant Ship Design and Research Institute (SDARI) and the validation data from the Computational Marine Hydrodynamics Lab (CMHL) of Shanghai Jiao Tong University are gratefully acknowledged. Meanwhile, the kind help of [PERSON] from Shenzhen Pilot Station in verifying the simulation results is greatly appreciated.  
The authors declare no conflict of interest.