## 4 Validation of the Proposed Numerical Algorithm for the Inversion of Laplace Transformation
### Investigation of Stress Wave Propagation in a Semi-Infinite Rod at \(x=0.5\) m  
This case investigates stress wave propagation at the position \(x=0.5\) m on the semi-infinite rod with different impedances, as shown in Figure 11 and Table 5. All cases initially generate compressive stress, with notable variations in the second cycle. Specifically, the \(Z_{1}=0.5Z_{2}\) case alternates between tensile stress and compressive stress, approaching zero; the \(Z_{1}=0.5Z_{2}\) case sees stress oscillating around zero; and \(Z_{1}=2Z_{2}\) consistently shows decreasing compressive stress towards zero. Particularly, when \(Z_{1}=Z_{2}\), the stress wave appears in only one stress compression cycle, after which the stress returns to oscillate around zero. Each cycle's duration remains constant at \(T_{i}=2L/c_{1}=69.17\) us in all cases. To further validate the accuracy of the method we propose, the results obtained are compared to finite element method (FEM) results. This 3D finite element analysis is a kind of numerical experiment to simulate the real performance of the system. The 3D FEM model is constructed as shown in Figure 12. The parameters for material properties and the geometries of the striker rod and the finite rod used in this model are detailed in Table 1; the length of the striker is 0.2 m, and that of the finite rod is 2.5 m. The approximate element size is 0.01 m, with a maximum deviation factor of 0.05 and a minimum element size of 0.001 m. The boundary conditions for the finite rod are set with one end free at \(x=0\) and fixed at \(x=L\). In the Abaqus finite element analysis code, the solid element type C3D8R was employed and had 11,193 nodes and 8672 elements in the model, as shown in Figure 12b. In the simulation conducted using the Abaqus/Explicit [20], the system of equations was solved through an explicit time integration method. The time increment selected for the simulation is \(\Delta t=0.5\times 10^{-7}\) (s). This setting guarantees that the condition \(\Delta t\leq L_{\text{min}}/c=1.729\times 10^{-7}\) (s), as recommended for stable explicit dynamic \begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline \multirow{3}{*}{**Cycle**} & \multicolumn{3}{c}{\(Z_{1}=0.5Z_{2}\)} & \multicolumn{3}{c}{\(Z_{1}=Z_{2}\)} & \multicolumn{3}{c}{\(Z_{1}=2Z_{2}\)} \\ \cline{2-9}  & \multirow{2}{*}{**Type**} & **Time (\(\mu\)s)** & **\(\sigma_{1}\)** & **Time (MPa)** & **\(\sigma_{1}\)** & **Time (MPa)** & **\(\sigma_{1}\)** \\  & & **(\(\mu\)s)** & **(MPa)** & **(\(\mu\)s)** & **(MPa)** & **(\(\mu\)s)** & **(MPa)** \\ \hline \multirow{3}{*}{1} & 1/2 & 17.29–51.88 & \(-\)351.05 & 17.29–51.88 & \(-\)263.286 & 17.29–51.88 & \(-\)175.52 \\  & 1/2 & 51.88–86.46 & 0 & 51.88–86.46 & 0 & 51.88–86.46 & 0 \\ \hline \multirow{3}{*}{2} & 1/2 & 86.46–121.05 & 117.02 & 86.46–121.05 & 0 & 86.46–121.05 & \(-\)58.51 \\  & 1/2 & 121.05–155.64 & 0 & 121.05–155.64 & 0 & 121.05–155.64 & 0 \\ \hline \multirow{3}{*}{3} & 1/2 & 155.64–190.22 & \(-\)39.01 & 155.64–190.22 & 0 & 155.64–190.22 & \(-\)19.5 \\  & 1/2 & 190.22–224.81 & 0 & 190.22–224.81 & 0 & 190.22–224.81 & 0 \\ \hline \multirow{3}{*}{4} & 1/2 & 224.81–259.39 & 13.00 & 224.81–259.39 & 0 & 224.81–259.39 & \(-\)6.50 \\  & 1/2 & 259.39–293.98 & 0 & 259.39–293.98 & 0 & 259.39–293.98 & 0 \\ \hline \multirow{3}{*}{5} & 1/2 & 293.98–328.57 & \(-\)4.33 & 293.98–328.57 & 0 & 293.98–328.57 & \(-\)2.17 \\  & 1/2 & 328.57–363.15 & 0 & 328.57–363.15 & 0 & 328.57–363.15 & 0 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Stress variation in rods with \(v=0.3\) and the solution without [PERSON]’s effect (Equation (48)) over the cycles at \(x=-0.1\) m with different impedances, under the initial impact velocity \(2v_{0}=11.6\) m/s, \(Z_{2}=32\),087.204 kg/s.