# Power Dissipation and Wear Modeling in Wheel-Rail Contact
## 3 Numerical Solution Methods  
In the literature, there are many proposed numerical models and methods to implement wear laws as well as to calculate wheel-rail energy dissipation and wear (see the references in review papers [1, 8]). All these approaches are facing and trying to solve the same challenges consisting in the mutual dependence between the simulated dynamical model and rail or wheel profiles altered due to wear. Nowadays, most frequently, a two-step approach is used. First, for a given profile, a vehicle dynamic problem is solved and then updated due to the wear geometry of profiles being calculated. Next, for a given updated profile from the previous step, a dynamic problem is solved. In the rolling contact problem (4)-(8) and (10)-(12), the calculated contact traction and wear depth depend on the thermal distortion of the wheel and the rail strip. On the other hand, the heat flow as well as the temperature achieved by the strip directly depend on contact pressure and on the rail profile. So, it is a system of coupled equations governing the thermo-elasto-plastic contact problem. Numerically, either we can solve mechanical and thermal equations simultaneously, i.e., in the same time instant, or we can decouple this system of equations and solve them sequentially in time. Since the second strategy is easier to implement, it has been used. The wear evaluation algorithm in time instant \(t\) and the consecutive time instant \(t+dt\), where \(dt\) denotes the time increment, is displayed on Figure 2. Assume the temperature \(\theta^{t}\) at time \(t\) is given. For time \(t\), the mechanical subsystem (5)-(8) is solved and normal contact traction \(F_{2}^{t}\) and wear \(w^{t}\) are evaluated. Based on it, the computations in the next time instant \(t+dt\) are executed. In this step, the thermal system (9) and (10) is solved, and the temperature \(\theta^{t+dt}\) at time \(t+dt\) is calculated. The heat flux \(Q^{t+dt}\) at time step \(t+dt\) in (10) is calculated as a time instant from formula \(Q^{t+dt}=\mu VF_{2}^{t}\) using traction force calculated in the previous time step. The solution of the thermo-mechanical system is stopped if the difference in temperatures calculated in consecutive time instants satisfies:  
\[\mid\theta^{t+dt}-\theta^{t}\mid\textit{leq}, \tag{17}\]  
for a given suitable small \(\epsilon>0\). The stopping condition (17) means that the temperature has reached a stable state and is almost constant. In the case that this condition is not satisfied, for a new calculated temperature \(\theta^{t+dt}\) at time step \(t+dt\), the mechanical subsystem (5)-(8) is solved. When the thermo-mechanical system (5)-(10) is solved, the worn volume material \(Vol^{t}\) is evaluated based on (13)-(16) and an update of the rail profile is calculated. Due to nonsmoothness, the friction condition (8) requires regularization [26, 28] to numerically solve the thermo-mechanical system (1)-(16). We shall use the arctan function to calculate tangential traction on the contact interface based on normal traction:  
\[F_{1}=F_{1}(\epsilon,F_{2},u_{1})=-\mu\mid F_{2}\mid\ \ \arctan\ \ (\frac{du_{1}}{ \epsilon}). \tag{18}\]  
The finite element method [32, 33, 36] is used to replace the system (4)-(8) and (10)-(12) with the discretized one. Due to plasticity as well as contact conditions, this mechanical subsystem consists of two coupled inequalities rather than equations. In order to solve it numerically, this system is either regularized and penalized [16, 28] or replaced with the equivalent complementarity system [32, 37, 38]. In this second approach, the inequality-type conditions (4) and (7) describing the plastic yield and friction are reformulated and transformed into equivalent multiplicative equality-type conditions. In finite element formulation, these conditions are represented in the form of matrices \(N_{p}\) and \(N_{c}\) reflecting the plasticity and friction conditions.