## 3 Finite Element Simulation Method
### Stationary Study  
In the TPS method, the sensor, which consists of a double helix nickel strip embedded in Kapton, serves as the core component. Consequently, a stationary study of the sensor must be conducted first. This study aims to obtain the initial resistance value of the sensor at room temperature and to observe the variation in resistance values at different meshes. In this simulation, the geometry is composed solely of a double helical nickel strip, as illustrated in Figure 2b. Correspondingly, the resistance values calculated from the simulation, along with the parameters modeled using Equation (7), are presented in Table 2.  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Category** & \begin{tabular}{c} **Width** \\ **(\(\upmu\)m)** \\ \end{tabular} & \begin{tabular}{c} **Kapton Radius** \\ **(\(\upmu\)m)** \\ \end{tabular} & \
\begin{tabular}{c} **Thickness** \\ **(\(\upmu\)m)** \\ \end{tabular} &
\begin{tabular}{c} **Total Thickness** \\ **(\(\upmu\)m)** \\ \end{tabular} \\ \hline Sensor I & 200 & 6.4 & 10 & 50 \\ Sensor II & 300 & 14 & 10 & 50 \\ \hline \hline \end{tabular}
\end{table}
Table 1: The properties of the sensors used in the simulation.  
Figure 2: (a) Complete geometric model in finite element simulation. (b) Geometric structure of double helix nickel strip embedded in Kapton layer.  
Then, the AC/DC physical field interface is chosen within the physical field settings of the finite element simulation software (COMSOL 6.1.0.252). The terminals and grounding connections are applied to the two ends of the nickel strip, while the remaining boundaries are configured to be electrically insulated. The ambient temperature for the simulation setup is consistently maintained at room temperature.  
The dimensions of the sensor were optimized to make them as close as possible to those of the actual sensor, both in terms of size and resistance values. At this juncture, the resistance value derived from the simulation will be employed to ascertain the power dissipated in the transient simulation, whereby \(P=I^{2}R\) and \(I\) represents the applied current.  
In finite element simulation methods, the size of the mesh affects the accuracy of the computational results [51; 52; 53; 54]. Figure 3 shows the resistance variation curve of the sensor at different meshes, where number 1 to number 9 denote extremely coarse to extremely fine meshes, respectively. The resistance value of the reference is calculated using \(R=\rho L/s\), where \(\rho\), \(L\), and \(s\) represent the resistivity, total length of the heating wire, and cross-sectional area of the heating wire, respectively. At this juncture, the resistance values of Sensor I and Sensor II were calculated to be 11.946 \(\Omega\) and 21.521 \(\Omega\), respectively. Among them, the maximum discrepancy between the calculated values of the nine meshes and the theoretical values is less than 1%. Similarly, the maximum error in the calculated values is less than 1%. But the case with the extremely fine mesh is closest to the actual situation. Consequently, the subsequent simulation process of the two types of sensors in this paper is conducted using an extremely fine grid to ensure the accuracy of the calculated results.