# A Parameter-Driven Methodology of Wheel Flat Modeling for Wheel-Rail Impact Dynamics
## 2 Three-Dimensional Wheel Flat Model
### Geometric Constraints on Wheel Flats  
During the edge worn or completely worn stage, the profile of the flat is geometrically determined, resulting in the following constraints for any section in the wheel flat region during the wear process.  
For any \(y\in[-L_{w}/2,L_{w}/2]\), \(x\in[-\beta^{\prime}L_{0}/2,-\beta^{\prime}L_{0}/2+\gamma^{\prime}L_{0}]\cup[ \beta^{\prime}L_{0}/2-\gamma^{\prime}L_{0},\beta^{\prime}L_{0}/2]\), the existence of constant constraints is shown in Equation (13)  
The wheel flat region is discretized into 50 parallel sections along the lateral direction of the wheel tread. Within each section, the wheel flat profile is further discretized into 50 points. \(\beta\in(1,5]\) is defined for each specific value of \(\beta\), where \(\gamma\in(\frac{\beta-1}{2},\frac{\beta}{2}]\). Therefore, the feasible domain search space corresponding to each section is subdivided into a 20 \(\times\) 20 grid based on the control parameters. Each grid cell corresponds to a specific combination of the control parameters, denoted as \(\beta_{0}\) and \(\gamma_{0}\). Subsequently, each grid cell is systematically evaluated to determine if all discretized points within the current section satisfy Equation (13) within the numerical computation tolerance. This evaluation process is translated into solving a convex optimization problem. If all points in a grid cell meet the criteria of Equation (13), the cell is marked as valid; otherwise, it is marked as invalid. Finally, the intersection of all valid grid cells across sections defines the distribution of labeled cells, and the boundary of the feasible domain is refined using nearest neighbor interpolation.  
The feasible domain of control parameters is shown in Figure 5.  
In Figure 5, the entire search space is partitioned into three regions by the upper and lower boundaries. The yellow-marked region (Stage II) corresponds to the feasible domain of the control parameters, depicted by Equation (14), where \(\gamma_{L}\) and \(\gamma_{U}\) denote the values of the upper and lower boundaries, respectively.  
\[\left\{\begin{array}{l}\gamma\in[\gamma_{L},\gamma_{U}]\left(\beta<\beta_{C }\right)\\ \gamma\in[\gamma_{L},\gamma_{U})\left(\beta\geq\beta_{C}\right)\end{array}\right. \tag{14}\]  
In the feasible domain, point A corresponds to the control parameter \(\beta=1\),\(\gamma=0\), indicating that the length of the worn region of the wheel flat is 0, representing the new flat stage. As the wheel flat undergoes edge wear, the length of the flat increases, at which time the value of \(\beta\) increases, and the range of \(\gamma\) expands accordingly. When \(\beta=\beta_{C}(\beta_{C}=1.8)\)  
Figure 4: The process of feasible domain search.  
it reaches \(\gamma_{U}=\beta/2\) for the first time and maintains this relationship thereafter. Therefore, point C marks the critical transition between the edge-worn stage and the completely worn stage. Subsequently, at Point D, \(\beta>\beta_{C}\) and \(\gamma\in[\gamma_{L},\gamma_{U})\) correspond to Stage II, and at Point F, \(\beta>\beta_{C}\) and \(\gamma=\beta/2\) correspond to Stage III.  
Moreover, the critical value of the worn and fresh wheel flat of the haversine method is \(\pi/2\), which closely matches the critical value of the model presented in this paper, indicating consistency in depicting the completely worn stage.  
The search space contains two types of error regions in addition to the feasible domain. Point B represents the first type of error, violating the \(g(x,y)\geq F_{1}(x,y)\) constraint in Equation (13), while Point E corresponds to the second type of error, violating \(g(x,y)\geq r(x,y)\) in Equation (13).