## 1 Introduction  
The transition section, which connects soil subgrade and rigid structures (bridges, tunnels and culverts, etc. ), is one of the weakest structures in the long-term operation of a high-speed railway [1]. Studying dynamic characteristics of the system is of great theoretical significance for the long-term stability and safe operation of the existing high-speed railway, and also provides technical reserves for future high-speed railway construction [2]. A field test is a direct way to study the dynamic characteristics of the transition section. However, it is expensive, labor intensive, and it is still challenging to cover various conditions. Alternatively, numerical simulation methods have been used to overcome the disadvantages of the field test method to study the dynamic characteristics of the transition section under different conditions [3]. There are three kinds of models:  
(1) Train-track coupling model: This is mainly used to study the influence of the stiffness and deformation on the dynamic response and comfort of the train by simplifying the subgrade as a mass layer, spring and damper, even without considering the vibration effect of the subgrade. This is a straightforward method adopted by many scholars. For example, this simple coupling model was employed to study the response of a track when the train ran on a transition zone [4]. The similar methodology was also used to establish a simplified three-dimentional (3D) train-track model to conduct sensitivity analysis on the vehicle speed, vehicle load, auxiliary track number, and track stiffness [5]. More recently, the dynamic responses of track in the marine environment were also modelled by the train-track coupling model [6]. (2) Track-subgrade coupling model: This is mainly used to study the characteristics of subgrade filling and has made a significant contribution to the research of subgrade dynamic characteristics by simplifying the trainload as moving load [7]. For example, the coupling model was adopted to study the dynamic response of the culvert embankment transition section, and also the effect of strengthening the foundation on reducing the uneven settlement of the transition zone was analyzed [8]. In addition, this coupling model could be combined with wave analysis technology to show that the gradual change of the stiffness of the transition section is conducive to the dynamic stability of track subgrade, as confirmed both by [9; 10]. (3) Vehicle-track-subgrade model: The interaction between train, track and subgrade is considered in this type of model. For example, a bridge-embankment transition model was established by [11], where the vehicle was simplified as a multi-body system, and the vehicle-track-subgrade system was coupled by the wheel-rail contact force. The wheel-rail contact force serves an essential feature to couple the vehicle, track and subgrade. The similar methodology relying on the contact force was also found in [12; 13], where [12] focused on the effect of filling materials on the dynamic properties of transition zones and [13] on the influence of bridge-pier settlement on the high-speed railway's dynamic response. In addition to the use of the wheel-rail contact force, the vehicle, track and subgrade could also be coupled using a fastener connection that couples the vehicle-track and track-subgrade subsystems, as illustrated in [14]. The above overview shows that the first and second modeling approaches oversimplify or ignore the dynamic response of either the subgrade or trains running on the track. While the third modeling approach acknowledges all contribution from the vehicle, track and subgrade, the modeling of the transition structure and material composition becomes very complex and a large number of system freedom degrees are generated. Therefore, the calculation efficiency is not satisfactory, and the complex modeling is not convenient for practical engineering application. The authors have also been studying the dynamic responses of the transition section in recent years.