## 2 SPH Methods
In the SPH numerical scheme, \(\Pi_{\mathrm{ab}}\) in Equation (4) is an artificial viscosity term that increases the stability properties of the algorithm. Here, the artificial viscosity has the following form:  
\[\Pi_{\mathrm{ab}}=\left\{ \begin{array}{ll}\frac{-\alpha\overline{\xi}_{ \mathrm{ab}}\Pi_{\mathrm{ab}}}{\overline{\xi}_{\mathrm{ab}}}&\mathbf{u}_{ \mathrm{ab}}\cdot\mathbf{r}_{\mathrm{ab}}<0\\ 0&\mathbf{u}_{\mathrm{ab}}\cdot\mathbf{r}_{\mathrm{ab}}\geq 0\end{array} \right. \tag{7}\]  
where \(\alpha\) is a dimensionless coefficient; \(\overline{\rho}_{\mathrm{ab}}\triangleq(\rho_{\mathrm{a}}+\rho_{\mathrm{b}})/2\) is the average value of the density of particles a and b; \(\overline{\xi}_{\mathrm{ab}}\triangleq(c_{\mathrm{a}}+c_{\mathrm{b}})/2\) is the average value of the sound speed for the two particles; \(\mathbf{r}_{\mathrm{ab}}\triangleq\mathbf{r}_{\mathrm{a}}-\mathbf{r}_{ \mathrm{b}}\) is the difference in spatial locations between the two particles; and \(\mu_{\mathrm{ab}}\) is calculated by a modified form of [19] as follows:  
\[\mu_{\mathrm{ab}}=\mathrm{h}\frac{\mathbf{u}_{\mathrm{ab}}\cdot\mathbf{r}_{ \mathrm{ab}}}{\overline{\rho}_{\mathrm{ab}}^{2}+e\mathrm{h}^{2}}\overline{k}_ {ab} \tag{8}\]  
where \(\epsilon\) is a small number to prevent singularity; \(\overline{k}_{\mathrm{ab}}\triangleq(k_{\mathrm{a}}+k_{\mathrm{b}})/2\), and \(k\) is defined as:  
\[k_{\mathrm{a}}=\frac{|\
abla\cdot\mathbf{u}_{\mathrm{a}}|}{|\
abla\cdot\mathbf{ u}_{\mathrm{a}}|+|\mathbf{S}_{\mathrm{a}}:\mathbf{S}_{\mathrm{a}}|+10^{-4}c_{ \mathrm{a}}/\mathrm{h}} \tag{9}\]  
where \(\mathbf{S}=\mathrm{sym}(\
abla\mathbf{u})\) is the rate\(-\)of\(-\)strain tensor and \(\mathbf{S}:\mathbf{S}\) is the tensor operation. Since SPH can be affected by a lack of stability, introducing artificial viscosity can help reduce this problem, considering an artificial viscosity coefficient term [20] as shown in Equation (10):  
\[\left\{ \begin{array}{ll}\Pi\,\mathrm{\SIUnitSymboli}=\left\{\begin{array}{ ll}\frac{\alpha c_{\mathrm{ij}}\phi_{\mathrm{ij}}+\beta\,\phi_{\mathrm{ij}}^{2}}{ \overline{\psi}_{\mathrm{ij}}}\cdot\overline{x}_{\mathrm{ij}}<0\\ 0&\overline{\psi}_{\mathrm{ij}}\cdot\overline{x}_{\mathrm{ij}}\geq 0\\ \phi_{\mathrm{ij}}=\frac{\mathrm{h}_{\mathrm{ii}}\overline{\psi}_{\mathrm{ii}} \cdot\overline{x}_{\mathrm{ij}}}{\overline{\left|x_{\mathrm{ij}}\right|}+( \epsilon\mathrm{h}_{\mathrm{ii}})^{2}}\end{array}\right.\end{array} \right. \tag{10}\]  
where c,\(\rho\) and h are sound speed, density, smoothing length, and kernel function, respectively, and the subscript \"\(\mathrm{ij}\)\" represents a mean value. A preliminary analysis was carried out to obtain the best set\(-\)up of parameters \(\alpha\), \(\beta\) and \(\varepsilon\). As regards artificial viscosity parameters, a series of calculations with different settings was carried out, covering the range of \(\alpha\) and \(\beta\) from 0.3 to 0.01; in particular, the values of the two parameters were always kept equal to each other, and it was decided to modify the value of \(\varepsilon\) by the same percentage, starting from a value equal to 0.1 in correspondence with the maximum value of \(\alpha\) and \(\beta\). In this case of sloshing calculations, satisfactory results were obtained in correspondence to values of \(\alpha\) and \(\beta\) lower than 0.1. In contrast, for higher values, calculation results were clearly dampened without capturing any impact phenomenon. In particular, it was found that the best results were obtained with values of \(\alpha\) and \(\beta\) set to 0.05 and 0.01, and it was chosen to adopt the latter, which corresponds to low artificial viscosity. A relatively large number of computations were performed using SPH to analyze the effect of some parameters, particularly artificial viscosity. The results of this analysis are not included in this paper, and only the final set\(-\)up results are presented (artificial viscosity parameters set to 0.01).