## 2. Wheel-Rail Contact  
To simulate wheel/rail contact, solid elements for wheels and shell elements for rails are used. The basic contact method of Ls-Dyna is the penalty method [19]. The concept is that when a slave node penetrates a master surface, the contact force is calculated from the amount of penetration and the contact stiffness. The contact force also increases in proportion to the increase of the amount of penetration. In Figure 2, t represents the position vector of the slave node \(n_{s}\), and the master segment surface \(s_{i}\) is related to \(n_{s}\). If \(n_{s}\) penetrates \(s_{i}\), the contact point coordinates (\(\xi_{c}\), \(\eta_{c}\)) must satisfy Equation (1). \[\frac{\partial r}{\partial\xi}(\xi_{c},\ \eta_{c})\cdot[t-r(\xi_{c},\ \eta_{c})]=0,\ \frac{ \partial r}{\partial\eta}(\xi_{c},\ \eta_{c})\cdot[t-r(\xi_{c},\ \eta_{c})]=0 \tag{1}\]  
The initial values are estimated through a least-squares projection iteration,  
\[\xi_{o}=0,\ \eta_{o}=0 \tag{2}\] \[\left[ \begin{array}{c}r_{\xi,\ \ \ }\\ r_{\eta,\ \ }\end{array} \right]\left[\begin{array}{c}\left[r_{\xi,\ \xi}\ r_{\eta} \right]\left[\frac{\Delta\xi}{\Delta\eta}\right]=\left[ \begin{array}{c}r_{ \xi,\ \ \ }\\ r_{\eta,\ \ }\end{array} \right]\left[r(\xi_{i},\ \eta_{i})-t\right],\] \[\xi_{i+1}=\xi_{i}+\Delta\xi,\ \eta_{i+1}=\eta_{i}+\Delta\eta\]  
From the Newton-Raphson iteration, the amount of penetration on the penetration coordinate can be calculated as (3). \[\left[\mathrm{H}\right]\!\left[\frac{\Delta\xi}{\Delta\eta}\right]=-\!\left[ \begin{array}{c}r_{\xi,\ \ \ }\\ r_{\eta,\ \ }\end{array} \right]\left[r(\xi_{i},\ \eta_{i})-t\right]\!, \tag{3}\] \[\left[\mathrm{H}\right]=\left\{\frac{r_{\xi}}{r_{\eta}}\right\} \!\left[r_{\xi,\ \ \ }r_{\eta}\right]+\left[ \begin{array}{cc}0&r_{\tau,\xi \eta}\\ r_{\tau,\xi\eta}&0\end{array} \right]\] \[\xi_{i+1}=\xi_{i}+\Delta\xi,\ \eta_{i+1}=\eta_{i}+\Delta\eta\]  
When the amount of the slave node coordinate penetrates into the master surface, Equation (4) is obtained. \[l=\eta_{i}\times\left[t-r(\xi_{c},\eta_{c})\right]\ <0 \tag{4}\]  
At this point, the normal vector of the master surface at the contact point is:  
\[n_{i}=n_{i}(\xi_{c},\eta_{c}) \tag{5}\]  
If the slave node penetrates the master surface, the contact force vector becomes Equation (6). \[f_{s}=-lk_{i}\eta_{i}\ if\ l<0 \tag{6}\]  
In addition, considering the degree of freedom of \(\eta_{s}\), the contact force is given by Equation (7). \[f_{m}^{i}=\phi_{i}(\xi_{c},\eta_{c})f_{s}\ if\ l<0 \tag{7}\]  
The master segment consists of four nodes (\(i=\)1, 2, 3, 4), and the contact force \(k_{i}\) consists of the bulk modulus \(K_{i}\) and the surface area \(A_{i}\) of the master segment. The contact force obtained in the master segment is calculated as follows:  
\[k_{i}=\frac{f_{si}K_{i}A_{i}}{max(\textit{shell\ daonal})} \tag{8}\]  
Generally, contact search methods of one-way and two-way are useful for handling contact problems. The two-way contact method calculates the contact force between a master surface and a slave node at each calculation cycle. The one-way contact method shows good results in the dynamic stabilization of the vertical contact force, but it does not accurately simulate the contact between a flange and a rail. Therefore, a two-way contact method which can simulate the contact between a flange and a rail was applied [20]. Depending on the relative distance between the wheel and rail, large oscillations could occur because of the penalty method. The dynamic relaxation time was saved by appropriately adjusting the distance between the wheel and the rail before contact analysis. Another important factor in rolling contact simulation is to reduce vibration generated by wheel rail contact between elements. To minimize vibration, wheel tread and flange are finely modeled as shown in Figure 3. For simulation of wheel rail contact, friction coefficient between wheel and rail is applied as 0.3 and CONTACT_AUTOMATIC_SURFACE_TO_SURFACE keyword which is two way contact method is used to simulate flange and rail contact.