## 3 Numerical Simulation and Comparison of the Results  
Based on the Equations (10-13), the jet and slug velocity of typical LSCs were generated and compared to numerical simulation results created from a commercial hydrocode Autodyntm. Because there is no direct relation between the detonation and LSCs liner velocity in the equations above, the following Gurney analysis of an asymmetric open-face sandwich configuration was added in the calculation [4]. \[\mathrm{V_{p}}=\sqrt{2E}\left(\frac{\left(1+2\frac{M}{c}\right)^{3}+1}{6\left( 1+\frac{M}{c}\right)}+\frac{M}{c}\right)^{-0.5} \tag{14}\]  
where \(V_{p}\) is the liner velocity, \(\sqrt{2E}\) is Gurney velocity, \(M\) and \(C\) are the unit mass of liner and explosives in \(\rho_{0}\)y\({}_{0}\) (density and thickness). For the numerical simulation, the copper liner thickness was varied from 0.5, 1 and 1.5 mm, and the thickness of C4 explosives were varied to maintain the M/C values in the range of 1 to 5 with an asymmetric square shaped liner and explosives representing a reasonable size of LSCs. The material properties of the liner and explosives were determined to be identical in both theoretical and numerical calculations as shown in Table 1.  
\begin{table}
\begin{tabular}{l c c c} \hline \hline \multicolumn{2}{c}{_Material: Copper (liner)_[5]} & \multicolumn{2}{c}{_Material: C4 (explosives)_[6]} \\ \hline \multicolumn{2}{c}{_Equation of State_} & \multicolumn{2}{c}{_Shock_} & \multicolumn{2}{c}{_Equation of State_} & \multicolumn{1}{c}{_JWL_} \\ \hline _Reference density_ & _8.96000_ \(\times\) _10\({}^{-1}\) (kg/cm3)_ & _Reference density_ & _1.60100_ \(\times\) _10\({}^{-3}\) (kg/cm3)_ \\ _Parameter C1_ & _3.89000_ (_m/ns_)_ & _Parameter A_ & _6.09770_ \(\times\) _10\({}^{-1}\) (TPa)_ \\ _Parameter S1_ & _1.8900_ (_non_)_ & _Parameter B_ & _6.09770_ \(\times\) _10\({}^{-1}\) (TPa)_ \\ _Parameter C2_ & _3.89200_ (_m/ns_)_ & _Parameter R1_ & _4.50000_ (_none_)_ \\ _Parameter S2_ & _1.48900_ (_none_)_ & _Parameter R2_ & _1.40000_ (_none_)_ \\ _Reference Temperature_ & _2.53000_ \(\times\) _10\({}^{-1}\) (_R_)_ & _Parameter W_ & _2.50000_ \(\times\) _10\({}^{-1}\) (none)_ \\ _Strength_ & _Joinoin_ & _C-J Detonation velocity_ & _8.19300_ (_m/ns_)_ \\ _Shear Modulus_ & _4.60000_ \(\times\) _10\({}^{-2}\) (TPa)_ & _C-J Energy / unit volume_ & _9.00000_ \(\times\) _10\({}^{-3}\) (L/mm3)_ \\ _Yield Stress_ & _9.00000_ \(\times\) _10\({}^{-5}\) (TPa)_ & _C-J Pressure_ & _2.80000_ \(\times\) _10\({}^{-2}\) (TPa)_ \\ _Haltering Constant_ & _2.92000_ \(\times\) _10\({}^{-4}\) (TPa)_ & _Burn on compression_ & _0.0000_ (_none_)_ \\ _Hardening Exponent_ & _3.10000_ \(\times\) _10\({}^{-1}\) (none)_ & _Pre-burn bulk modulus_ & _0.00000_ (_TPa_)_ \\ _Strain Rate Constant_ & _2.50000_ \(\times\) _10\({}^{-2}\) (none)_ & _Adiabatic constant_ & _0.00000_ (_none_)_ \\ _Thermal Softening Exponent_ & _1.09000_ (_none_)_ & _Auto-convert to Ideal Gas_ & _Yes_ \\ _Melting Temperature_ & _1.35600_ \(\times\) _10\({}^{1}\) (K)_ & _Additional Options (Beta)_ & _None_ \\ _Ref. Strain Rate (s)_ & _1.00000_ (_none_)_ & _Strength_ & _None_ \\ _Strain Rate Correction_ & _1 st Order_ & _Failure_ & _None_ \\ _Failure_ & _Principal Strain_ & _Erosion_ & _None_ \\ _Principal Tensile Failure_ & _2.50000_ \(\times\) _10\({}^{-1}\) (none)_ & & \\ _Strain_ & _1.01000_ \(\times\) _10\({}^{20}\) (none)_ & & \\ _Surface Pressure_ & _1. To compare the results between the analytical calculations of the steady-state modified Birkhoff theory and the numerical simulation results, the jet velocity in the numerical simulation was measured in three different liner collapse completion rates to achieve a general tendency of the jet velocity. The jet velocity in the numerical simulation was measure at 25%, 50% and 75% of liner collapse completion stages and tabulated in Table 2, and the average values are included in Figure 7. The entire LSCs liner was designed with \(\alpha=40^{\circ}\) and \(l=25\) mm.