## 3 Model Validation  
To validate the SPH numerical model, a benchmark test was carried out by using our self\(-\)developed SPH simulation code. The computational results were compared with the experimental data of [PERSON] et al. [20].  
In the experiments of [PERSON] et al. [20], the tank dimensions were 900 \(\times\) 508 \(\times\) 50 mm. Our simulation cases studied a longitudinal section of the tank. The computational water tank is shown in Figure 2. The tank was rolled along the rolling axis (184 mm over the bottom line of the tank) following a sinusoidal type rotating motion whose amplitude is \(\theta_{max}=4^{\circ}\) and with different excitation frequencies. Six sensors are located on the boundary of the tank as shown in Figure 2.  
Two water filling levels considered in the present analysis are shown in Table 1. For both water levels, the experiments of [PERSON] et al. (2018) were carried out with the tank rolling in the resonance period (cases A and B). For case A, the impact pressure was recorded on the sides for sensor 1. For case B, pressure sensors were located at positions 3 and 6.  
The SPH computational parameters were used as follows: equation of state coefficient \(\text{B}=2.8\times 10^{5}\), artificial viscosity coefficient \(\alpha=0.01\), and particle spacing \(\Delta x=2\) mm. A constant time step of \(\Delta\text{t}=1\times 10^{-5}s\) was used for the frog-leap time integration.  
For each of cases A and B, as the tank starts to roll, the main water wave generates gradually and travels from one side of the tank to the other. The water wave breaks during its traveling and impacts the wall of the tank. The experiment results show that the water motion in the tank was periodic despite the high dissipation due to wave breaking.  
Figure 3 compares some snapshots of water-free surfaces simulated by the SPH model with experimental pictures of case A. Figure 3a-c shows the initial stage of sloshing. SPH results show good agreement with the experimental pictures. Figure 3d-f show the first wave-breaking event. The jet forms and starts to break at 2.9 s. This plunge-type breaker entraps air and air-water mixture impacts on the wall subsequently. The shape and position of the breaker simulated by SPH are quite similar to the experimental snapshot. Figure 3g-i show the following three times the water wave impacted on the side wall of the tank. The shape and height of the water splash of the SPH numerical results accord quite well with the experimental results.  
\begin{table}
\begin{tabular}{c c c c} \hline \hline
**Case** & **Water Filling Level (mm)** & **Level/Tank Height (\%)** & **Period (s)** \\ \hline A & 93 & 18.3 & 1.91 \\ B & 222 & 43.7 & 1.32 \\ \hline \hline \end{tabular}
\end{table}
Table 1: Description of test cases.  
Figure 2: Tank geometry and position of the sensors (in millimeters).  
Figure 3: _Cont._  
To validate the accuracy of our SPH model, the SPH-computed pressure time series at the sensor points were compared with the experimental data of [PERSON] et al. (2019) in Figure 4 for both cases A and B. Figure 4 shows that the simulated pressure time series of our SPH model are in good agreement with experimental data in both amplitude and phase of pressure oscillation, even after a dozen sloshing periods.  
Figure 3: Comparisons of SPH–computed snapshots of water–free surface (**right**) with experimental pictures (**left**) for case A.  
Experimental and SPH results showed very good agreement in terms of free surface shape, the impact phenomenon, and impact pressure on the wall, which means that our self-developed SPH model is capable of simulating the sloshing problems and capturing the water impact pressure on the solid wall accurately.