## 2 Validation of Simulation Model  
Details of hypervelocity impact tests conducted by NASA are confidential, so only part of the test phenomenon can be found in public reports. There are no quantitative test data to validate the simulation model. At present, the material parameters, boundary conditions, and contact definitions in a high-speed impact simulation model have mainly been validated based on the Varas test [9]. In the test, the projectile is spherical with a diameter of 1.25 cm and an impact velocity of 900 m/s. The tank is rectangular with dimensions of 75 cm, 15 cm, 15 cm, and 0.25 cm in length, width, height and wall thickness, respectively. The test device is displayed in Figure 1.  
The simulation model consisted of the projectile, tank, water, and air. The tank walls were divided into the entry wall, exit wall, and lateral walls. Lagrange grids were constructed for the projectile and tank, whilst Euler grids were constructed for the water and air. These two grids were coupled through the ALE algorithm [14]. The interaction between the projectile and tank wall was achieved through the eroding surface-to-surface contact, and the interaction between the projectile and water was achieved through the automatic surface-to-surface contact. The non-reflecting boundary conditions were applied to the air boundary interfaces to prevent the shock wave from reentering the air field. Water was allowed to flow into the air region by sharing nodes on their interfaces. The mesh sizes of the projectile, tank walls, water, and air were 0.1 cm, 0.2 cm, 0.4 cm, and 2 cm, respectively. The whole model consisted of 377,605 elements. SOLID 164 elements with eight nodes were adopted, and each node had three DOFs. The computing time step was 0.6 \(\upmu\)s. The simulation model and its local magnification diagram are exhibited in Figure 2.  
Figure 1: Schematic diagram of the Varas test device. The air was modeled using the constitutive equation of MAT_NULL and the state equation of EOS_LINEAR_POLYNOMIAL [14]. The linear polynomial equation of state is linear in internal energy. The pressure \(P\) is given as follows:  
\[P=C_{0}+C_{1}\mu+C_{2}\mu^{2}+C_{3}\mu^{3}+(C_{4}+C_{5}\mu+C_{6}\mu^{2})E \tag{1}\]  
where \(C_{0}\)-\(C_{6}\) are the polynomial equation coefficients and \(\mu=(\rho/\rho_{0})-1\), \(\rho/\rho_{0}\) is the ratio of the current density to initial density. The air is considered as an ideal gas by setting \(C_{0}=C_{1}=C_{2}=C_{3}=C_{6}\) = 0. The water was modeled using the constitutive equation of MAT_NULL and the state equation of EOS_GRUNEISEN [14]. The Gruneisen equation with cubic shock velocity-particle velocity (\(v_{\mathrm{s}}\)-\(v_{\mathrm{p}}\)) defines pressure \(P\) as follows:  
\[P=\frac{\rho_{0}C^{2}\mu\left[1+\left(1-\frac{\gamma_{0}}{2}\right)\mu-\frac{ \gamma_{0}}{2}\mu^{2}\right]}{\left[1-(S_{1}-1)\mu-S_{2}\frac{\mu^{2}}{\mu+1} -S_{3}\frac{\mu^{3}}{\left(\mu+1\right)^{2}}\right]^{2}}+(\gamma_{0}+a\mu)E \tag{2}\]  
where \(S_{1}\)-\(S_{3}\) are the slope coefficients of the \(v_{\mathrm{s}}\)-\(v_{\mathrm{p}}\) curve and \(\gamma_{0}\) is the Gruneisen gamma. The properties and parameters for the water and air are listed in Table 1[14]. The steel projectile and aluminum tank were modeled using the constitutive equation of MAT_JOHNSON_COOK and the state equation of EOS_GRUNEISEN. [PERSON] and [PERSON] express the flow stress \(\sigma_{y}\) as follows:  
\[\sigma_{y}=(A+B(\overline{\epsilon^{p}})^{n}(1+C\ln\epsilon^{*})\left[1-(T^{* })^{m}\right] \tag{3}\]  
where \(A\) is the quasi-static yield stress; \(B\) is the strain hardening modulus; and \(C\), \(n\), \(m\) are the material constants. These equations are widely used for metallic materials because numerous efforts have been exerted to determine the parameters. The material properties and parameters used for aluminum [14] and steel [19] are summarized in Table 2.