## 2 Simulation Model Establishment
### Finite Element Model of SICL  
In this article, a finite element model of a four-stage SICL is established. In the external circuit simulation model, a pulse power supply system model is established, and ANSYS Maxwell is used for coupled field-circuit simulation analysis [14]. Taking advantage of the strengths of both the field model and the circuit model, the transient launching process of the induction coil device is simulated [15; 16]. During the process of electromagnetic coil launch, the armature is suspended and accelerated without contact with the launch pipeline. Therefore, it is assumed that the central axis of the armature coincides with the central axis of the launching tube during the whole launching process, that is, the axis of the armature does not deviate from the axis of the drive coil. Under this condition, the three-dimensional launching system model is simplified into a two-dimensional axisymmetric simulation model, which is modeled in cylindrical coordinates. The two-dimensional finite element simulation model of the four-stage SICL established in Maxwell is shown in Figure 1. This diagram displays the key components of the drive coils and armature used in our ANSYS simulations, essential for understanding the impact of different energization methods analyzed in the study. The parameters of the simulation model for the launching device are shown in Table 1.  
This table lists essential specifications such as dimensions and materials for the drive coils and armature. Studying the parameter characteristics of the induction coil launch device is the basis for conducting research on coil launch devices. For a multi-stage induction coil launch device, as the driving coil accelerates the armature stage by stage, the speed at which the armature enters the driving coil increases. To improve the launch efficiency of the device, selecting appropriate device parameters involves analyzing the characteristics of each parameter as the armature entry speed increases. The two-dimensional schematic diagram of the coil launching device in this study is shown in Figure 2. The armature is launched in the direction of the arrow in the diagram, with a distance D = 5 mm between each stage of driving coils. The armature is progressively accelerated by four stages of coils. \(P_{1}\) represents the triggering position of the armature in the first-stage driving coil, while \(T_{i}\) (where \(i=2\), 3, and 4) represents the triggering time of the armature in the second- to fourth-stage driving coils. The driving coils are powered at the appropriate triggering times to accelerate the armature forward. Not all structural and electrical parameters of multi-stage induction coil devices need to be optimized. In this study, the armature to be launched is predetermined, and the armature parameters cannot be adjusted as the armature entry speed changes because the same armature is accelerated by different stages. As for the parameters of the driving coil, its inner diameter can usually be directly determined based on the known outer diameter of the armature, and the aspect ratio of the driving coil should gradually transition from short and thick coils to long and thin coils as the number of device stages increases, with the number of turns gradually decreasing. The coil launching devices of the PP and PN configurations studied in this article are shown in Figure 3, and the drive coils are arranged coaxially with the armature. Figure 3a shows the PP configuration launching device with same-direction current applied to the drive coils of the four stages, while Figure 3b shows the PN configuration launching device with alternating reverse-direction current applied to the drive coils of the four stages. In the process of Maxwell simulation, the properties of the drive coils in the PP configuration are set as positive for stages 1 to 4, while in the PN configuration, the properties of the drive coils in stages 1 and 3 are set as positive, and the properties of the drive coils in stages 1 and 3 are set as positive.