## 2 Materials and Methods
First, the velocity at the half time step \(n+1/2\) is as follows:  
\[\mathbf{v}_{i}^{n+\frac{1}{2}}=\mathbf{v}_{i}^{n}+\frac{1}{2}\Delta t\bigg{(}\frac{ \mathrm{d}v_{i}}{\mathrm{d}t}\bigg{)}^{n}, \tag{20}\]  
where \(n\) denotes the present time step. Subsequently, the density, specific internal energy, and position are solved using the updated velocity by Equation (20). The updated formulas for the density, specific internal energy, and position at the new time step \(n+1\) are as follows:  
\[\rho_{i}^{n+1}=\rho_{i}^{n}+\Delta t\bigg{(}\frac{\mathrm{d}\rho_{i}}{\mathrm{ d}t}\bigg{)}^{n+\frac{1}{2}}, \tag{21}\]  
\[e_{i}^{n+1}=e_{i}^{n}+\Delta t\bigg{(}\frac{\mathrm{d}e_{i}}{\mathrm{d}t} \bigg{)}^{n+\frac{1}{2}}, \tag{22}\]  
\[\mathbf{x}_{i}^{n+1}=\mathbf{x}_{i}^{n}+\Delta t\bigg{(}\frac{\mathrm{d}\mathbf{x}_{i}}{ \mathrm{d}t}\bigg{)}^{n+\frac{1}{2}}. \tag{23}\]  
Finally, the velocity at the new time step is updated. \[\mathbf{v}_{i}^{n+1}=\mathbf{v}_{i}^{n+\frac{1}{2}}+\frac{1}{2}\Delta t\bigg{(}\frac{ \mathrm{d}\mathbf{v}_{i}}{\mathrm{d}t}\bigg{)}^{n+1}. \tag{24}\]  
The pressure of the material in the ejection is up to tens of GPa or even hundreds of GPa, so the Mie-Gruneisen equation of state [46] is employed as follows:  
\[p=\bigg{(}1-\frac{1}{2}\gamma_{p}\eta\bigg{)}p_{H}(\rho)+\gamma_{p}\rho e, \tag{25}\]  
\[p_{H}(\rho)=\left\{ \begin{array}{ll}a_{0}\eta+b_{0}\eta^{2}+c_{0}\eta^{3},& \eta\geqslant 0\\ a_{0},&\eta<0\end{array} \right., \tag{26}\]  
where \(\eta\) is the compression ratio, \(\gamma_{p}\) is the Gruneisen constant about pressure, and the parameters \(a_{0}\), \(b_{0}\), and \(c_{0}\) are given by  
\[\left\{ \begin{array}{ll}a_{0}=\rho_{0}c^{2}\\ b_{0}=a_{0}[1+2(s-1)]\\ c_{0}=a_{0}\Big{[}2(s-1)+3(s-1)^{2}\Big{]}\end{array} \right.^{,} \tag{27}\]  
where \(s\) is the material parameter. The deviatoric stress tensor component is determined by applying the [PERSON] stress rate, as expressed below:  
\[\dot{S}^{\alpha\beta}=2G\bigg{(}\dot{\varepsilon}^{\alpha\beta}-\frac{1}{3}\delta ^{\alpha\beta}\dot{\varepsilon}^{\gamma\gamma}\bigg{)}+S^{\alpha\gamma}\dot{R}^ {\beta\gamma}+S^{\beta\gamma}\dot{R}^{\alpha\gamma}, \tag{28}\]  
where \(G\) denotes the shear modulus. The strain rate tensor component \(\dot{\varepsilon}^{\alpha\beta}\) and the rotation rate tensor component \(\dot{R}^{\alpha\beta}\) are expressed as  
\[\dot{\varepsilon}^{\alpha\beta}=\frac{1}{2}\bigg{(}\frac{\partial\
u^{\alpha} }{\partial x^{\beta}}+\frac{\partial\
u^{\beta}}{\partial x^{\alpha}}\bigg{)}, \tag{29}\]  
\[\dot{R}^{\alpha\beta}=\frac{1}{2}\bigg{(}\frac{\partial\
u^{\alpha}}{\partial x ^{\beta}}-\frac{\partial\
u^{\beta}}{\partial x^{\alpha}}\bigg{)}. \tag{30}\]  
The effective stress \(S_{eff}\) is calculated based on the following equation:  
\[S_{eff}=\sqrt{\frac{3}{2}S^{\alpha\beta}S^{\alpha\beta}}. \tag{31}\]  
The plastic flow is determined using the [PERSON] yield criterion, as follows:  
\[S^{\alpha\beta}=\zeta S^{\alpha\beta}, \tag{32}\]  
\[\zeta= \begin{cases}\sqrt{\frac{2Y^{2}}{3\delta^{\alpha\beta}S^{\alpha\beta}}},&S_{eff}>Y\\ 1,&S_{eff}\leqslant Y\end{cases} , \tag{33}\]  
where \(Y\) denotes the yield strength. The Steinberg-Guinan model [51] is employed as follows:  
\[G=G_{0}\bigg{[}1+\bigg{(}\frac{G_{p}^{\prime}}{G_{0}}\bigg{)}\frac{p}{\eta^{1/ 3}}-\bigg{(}\frac{G_{T}^{\prime}}{G_{0}}\bigg{)}(T-300)\bigg{]}, \tag{34}\]  
\[Y=Y_{0}(1+\beta\epsilon^{p})^{n}\bigg{[}1+\bigg{(}\frac{Y_{p}^{\prime}}{Y_{0} }\bigg{)}\frac{p}{\eta^{1/3}}-\bigg{(}\frac{Y_{T}^{\prime}}{Y_{0}}\bigg{)}(T- 300)\bigg{]}, \tag{35}\]  
and must satisfy the following condition:  
\[Y_{0}(1+\beta\epsilon^{p})^{n}\leq Y_{\max}, \tag{36}\]  
where \(T\) denotes the temperature; \(\epsilon^{p}\) corresponds to the equivalent plastic strain; \(\beta\) and \(n\) are the work-hardening parameters; and \(G_{p}^{\prime}/G_{0}\), \(G_{T}^{\prime}/G_{0}\), \(Y_{p}^{\prime}/Y_{0}\), \(Y_{T}^{\prime}/Y_{0}\), and \(Y_{\max}\) are material constants. The subscript \(0\) refers to the reference state (\(T=300\) K, \(p=0\)).