## 2 Materials and Methods
### Simulation of Frost Heave in the Dynamic Models
#### 2.2.2 A Novel FEM Model Based on Explicit Algorithm  
After the occurrence of uneven subgrade frost heave, the slab track was lifted by the surface of the roadbed. Therefore, in addition to the track irregularities and the contact loss area considered in the above two models, the deformations and the constraints of the slab track itself were changed. The change would also alter the dynamic properties of the track structure, and the two existing models cannot simulate this very well. Thus, a novel dynamic model was built to analyze the effects on the dynamic response of the whole vehicle-track system when the track was lifted up by the subgrade frost heave. For convenience, the novel model was called Model-C.  
Model-C uses the central difference method to perform explicit time integration on the kinetic equation throughout the whole time period [46], and the dynamic state of the previous increment is applied to calculate the dynamic state of the next increment. At the initial increment, the relation expressed by Equation (8) is used to solve the dynamic equilibrium equation in the solution program. \[M\text{:}\tilde{u}=P-I \tag{8}\]  
In Equation (8), \(M\) is the mass matrix of nodes, \(\tilde{u}\) is the initial nodal acceleration, \(P\) is the initial external force, and \(I\) is the initial internal force of elements. The nodal acceleration at time of \(t\) is calculated as:  
\[\left.\ddot{u}\right|_{(t)}=M^{-1}\left(P-I\right)\right|_{(t)} \tag{9}\]  
In Equation (9), \(\left.\left(P-I\right)\right|_{(t)}\) is the difference value between the external force and the internal force of elements at time of \(t\). The central difference method is used to perform time integration on the acceleration, and the acceleration is assumed to be constant when computing the change in velocity. The speed at the  
Figure 5: Theoretical topology models of different simulation methods of subgrade frost heave. (**a**) Theoretical model of Model-A; (**b**) Theoretical model of Model-B; (**c**) Theoretical model of Model-C.  
midpoint of the next increment is determined by the speed of the current increment and the midpoint of the previous increment, as shown in Equation (10). \[\dot{u}\big{|}_{(t+\frac{\Delta t}{2})}=\ \dot{u}\big{|}_{(t+\frac{\Delta t}{2})}+ \frac{\Delta t|_{(t+\Delta t)}+\Delta t|_{(t)}}{2}\ \ddot{u}\big{|}_{(t)} \tag{10}\]  
In Equation (10), \(\Delta t|\) refers to the time interval between two increments. Moreover, the displacement of the next increment is determined by the displacement at the beginning of the increment and the integration of the velocity, as shown in Equation (11). \[u|_{(t+\Delta t)}=u_{(t)}+\Delta t|_{(t+\Delta t)}\ \dot{u}|_{(t+\frac{\Delta t}{2})} \tag{11}\]  
After the responses of nodes are calculated, the strain increments of elements are further calculated according to the strain rate shown in Equation (12), and the strain increment is substituted into the constitutive relation to calculate the stress, as shown in Equation (13). \[d\varepsilon_{(t)}=\ \dot{\varepsilon}_{(t)}=f\Big{(}\dot{u}\big{|}_{(t+ \Delta t)}-\dot{u}_{t}\Big{)} \tag{12}\]  
In Equation (12), \(\dot{\varepsilon}_{(t)}\) refers to the strain rate; \(\dot{f}\Big{(}\dot{u}\big{|}_{(t+\Delta t)}-\dot{u}_{t}\Big{)}\) is the computational formula of the strain rate, which is the function of \(\dot{u}_{t}\) and \(\ \dot{u}\big{|}_{(t+\Delta t)}\). \[\sigma_{(t+\Delta t)}=f\Big{(}\sigma_{(t)},\ \ dc\Big{)} \tag{13}\]  
In Equation (13), \(f\Big{(}\sigma_{(t)},\ \ dc\Big{)}\) is the constitutive relation described by \(\sigma_{(t)}\) and \(d\varepsilon\). The explicit algorithm in Model-C based on the center of the central difference method always adopts a diagonal or centralized mass matrix, so it is not necessary to solve simultaneous equations when solving acceleration. The end state of the increment only depends on the initial state at the beginning of the increment, which greatly reduces the calculation cost. We next present the analytical concept of Model-C (shown in Figure 6).