## 5 Conclusions  
The wear and fatigue between wheels and rails greatly affect the maintenance and dynamic performance of railway vehicles. In the case of an urban railway with no rail lubrication system, it has been reported that severe damage to the wheel occurs in the curved sections of the rail, with the radius of curvature between R310 and R400. In the present study, we intended to design a wheel profile of urban railway which minimizes the wheel damage in a section with a radius of curvature of R350, that is, the median of the radii of curvature of this section. The wheel profile was created using PCHIP and a Pareto-optimal Solution that minimizes the wear and fatigue occurring on the wheel when a railway vehicle is driving in a curved section, which is obtained using NSGA-II, a multi-objective optimization algorithm. Through a Pareto-optimal analysis, the case where the wear number decreased by approximately 4.3% and the surface fatigue index decreased by 17.9% was generated the wheel profile with the smallest wear, the case where the wear number increased by 17.2% and the surface fatigue index decreased by 65.8% generated the wheel profile with the smallest fatigue, and the case where the wear number decreased by 2.4% and the surface fatigue index decreased by 52.8% was generated the wheel profile with the smallest and surface fatigue. The derailment, lateral pressure, and so on, of these optimized wheel profiles showed better performance than that of the initial wheel profile. We obtained as many Pareto-optimal Solutions as possible that minimized the wear and fatigue that occur on wheels by the method presented in the present study, and confirmed that these can be selected as needed to design a suitable wheel profile. In the future, we plan to conduct diverse experiments and optimization related to the analysis of rail and  
Figure 16: Lateral displacement of wheelset.  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Content** & **Initial** & **Wear-Optimized** & **Fatigue-Optimized** &
\begin{tabular}{c} **Both Wear and** \\ **Fatigue-Optimized** \\ \end{tabular} \\ \hline Wear index & 98.96 (100.0\%) & 94.61 (95.6\%) & 115.73 (116.9\%) & 96.52 (97.5\%) \\ Fatigue index & 0.3048 (100.0\%) & 0.2508 (82.3\%) & 0.1049 (34.4\%) & 0.1442 (47.3\%) \\ Derailment ratio & 0.4155 (100.0\%) & 0.3566 (85.8\%) & 0.3411 (82.1\%) & 0.3535 (85.1\%) \\ Lateral force & 29.26 (100.0\%) & 26.61 (90.9\%) & 23.86 (81.5\%) & 25.55 (87.3\%) \\ Vertical force & 85.04 (100.0\%) & 85.32 (100.3\%) & 84.91 (99.8\%) & 85.37 (100.4\%) \\ \hline \hline \end{tabular}
\end{table}
Table 3: Comparison of performance functions.  
wheel wear phenomena in the curved sections of the existing railways, and the correlation between cant and slack, the result of which is expected to be utilized to improve the service lives of wheels and rails and improve the curved section passing speed of vehicles.  
Conceptualization, H.-Y.C.; methodology, C.-Y.S.; software, H.-Y.C.; validation, C.-Y.S.; formal analysis, H.-Y.C.; investigation, H.-Y.C.; resources, C.-Y.S.; data curation, H.-Y.C.; writing--original draft preparation, C.-Y.S.; writing--review and editing, C.-Y.S.; visualization, H.-Y.C.; supervision, H.-Y.C.; project administration, H.-Y.C.; funding acquisition, None. All authors have read and agreed to the published version of the manuscript.  
This research received no external funding.  
Not applicable.  
The data presented in this study are available in this article.  
This work was supported by \"Regional Innovation Strategy (RIS)\" through the National Research Foundation of Korea (NRF) funded by the Ministry of Education (MOE), and also supported by the New & Renewable Energy Core Technology Program of the Korea Institute of Energy Technology Evaluation and Planning (KETEP) granted financial resource from the Ministry of Trade, Industry & Energy, Republic of Korea (No. 2021300000030).  
The authors declare no conflict of interest.