## 2 Temperature Field Fitting and Simulation of CRTS III Slab Track
### Temperature Field Fitting
\begin{table}
\begin{tabular}{c c c c c c c} \hline
**The Distance from Slab** & & & & & & \\
**Track Surface(mm)** & **6.00** (°C) & **8.00** (°C) & **10.00** (°C) & **12.00** (°C) & **14.00** (°C) & **16.00** (°C) & **18.00** (°C) \\ \hline
0 & 32.424 & 35.28 & 40.696 & 46.601 & 49.475 & 46.725 & 43.606 \\
10 & 32.802 & 35.26 & 40.404 & 45.97 & 48.887 & 46.655 & 43.778 \\
20 & 33.141 & 35.242 & 40.122 & 45.366 & 48.314 & 46.545 & 43.91 \\
30 & 33.442 & 35.225 & 39.852 & 44.79 & 47.757 & 46.396 & 44 \\
40 & 33.709 & 35.209 & 39.592 & 44.242 & 47.216 & 46.213 & 44.052 \\
50 & 33.945 & 35.195 & 39.343 & 43.721 & 46.692 & 45.999 & 44.067 \\
60 & 34.153 & 35.182 & 39.105 & 43.226 & 46.183 & 45.76 & 44.05 \\
70 & 34.337 & 35.17 & 38.878 & 42.757 & 45.691 & 45.5 & 44.002 \\
80 & 34.498 & 35.159 & 38.661 & 42.312 & 45.216 & 45.222 & 43.927 \\
90 & 34.64 & 35.149 & 38.454 & 41.891 & 44.756 & 44.93 & 43.828 \\
100 & 34.765 & 35.141 & 38.258 & 41.494 & 44.313 & 44.627 & 43.706 \\ \hline \end{tabular}
\end{table} 
Table 2: The vertical temperature at different positions in the center of slab track. The fitting toolbox in MATLAB was used to fit the temperature data. Generally speaking, the higher the order of the function in the fitting equation, the better the fitting effect, but the corresponding solution cost is also higher. Through calculation and comparison, it was found that the fitting of the cubic function of the slab track depth and temperature at 6:00 can achieve high accuracy, as shown in Figure 6. The fitting function is as follows:  
\[ \begin{array}{l}y=0.00000033x^{3}-0.0001812x^{2}\\ +0.03818x+32.44,R^{2}=0.9999\end{array}  \tag{5}\]  
The relationship between the slab track depth and the temperature at 14:00 can be achieved with high accuracy by quadratic function fitting, as shown in Figure 7. The fitting function is  
\[ \begin{array}{l}y=0.00007895x^{2}-0.05944x+\\ 49.47,R^{2}=1\end{array}  \tag{6}\] \begin{table}
\begin{tabular}{c c c c c c c c} \hline
**The Distance from Slab** & **600 (°C)** & **8:00 (°C)** & **10:00 (°C)** & **12:00 (°C)** & **14:00 (°C)** & **16:00 (°C)** & **18:00 (°C)** \\
**Track Surface(mm)** & & & & & & & \\ \hline
110 & 34.876 & 35.134 & 38.071 & 41.118 & 43.886 & 44.317 & 43.565 \\
120 & 34.973 & 35.129 & 37.894 & 40.763 & 43.474 & 44.002 & 43.408 \\
130 & 35.06 & 35.124 & 37.726 & 40.429 & 43.079 & 43.685 & 43.236 \\
140 & 35.137 & 35.121 & 37.567 & 40.115 & 42.699 & 43.367 & 43.051 \\
150 & 35.206 & 35.118 & 37.418 & 39.819 & 42.335 & 43.051 & 42.857 \\
160 & 35.269 & 35.117 & 37.277 & 39.541 & 41.985 & 42.739 & 42.654 \\
170 & 35.325 & 35.117 & 37.144 & 39.281 & 41.651 & 42.432 & 42.445 \\
180 & 35.376 & 35.119 & 37.02 & 39.036 & 41.332 & 42.13 & 42.231 \\
190 & 35.422 & 35.121 & 36.903 & 38.807 & 41.027 & 41.836 & 42.015 \\
200 & 35.465 & 35.124 & 36.794 & 38.593 & 40.735 & 41.55 & 41.796 \\
210 & 35.505 & 35.129 & 36.692 & 38.392 & 40.458 & 41.273 & 41.578 \\ \hline \end{tabular}
\end{table} 
Table 2: _Cont._  
Figure 6: The cubic function fitting results of the relationship between the slab track depth and the temperature at 6:00. Figure 7: The quadratic function fitting results of the relationship between the slab track depth and the temperature at 14:00. Using the same method, the temperature data under various working conditions were fitted to provide a basis for the subsequent solution of the temperature load applied in the displacement of the slab track.