## 3 Finite Element Simulation Method
### Transient Simulation  
The complete model for transient simulation is shown in Figure 2a. As the sensor is capable of functioning as a resistive heating source, the following equation is satisfied [55]:  
\[\
abla\cdot\overrightarrow{J}=q \tag{8}\]  
where \(\overrightarrow{J}\) and \(q\) are used to denote current density and charge density, respectively. In the finite element simulation software, the current is generated by utilizing the terminal and ground at the boundaries of both ends of the nickel strip. All domains in the model, \begin{table}
\begin{tabular}{c c c c c c} \hline
**Category** & \begin{tabular}{c} \(\mathbf{a_{i}}\) \\ **(\(\mathbf{\mu}\)m)** \\ \end{tabular} & \begin{tabular}{c} \(\mathbf{a_{f}}\) \\ **(\(\mathbf{\mu}\)m)** \\ \end{tabular} & \begin{tabular}{c} **Number of** \\ **Laps** \\ \end{tabular} & \
\begin{tabular}{c} **Spacing** \\ **(\(\mathbf{\mu}\)m)** \\ \end{tabular} &
\begin{tabular}{c} **Resistance** \\ **(\(\mathbf{\Omega}\))** \\ \end{tabular} \\ \hline Sensor I & 200 & 6400 & 16 & 200 & 11.885 \\ Sensor II & 300 & 14,000 & 20 & 300 & 21.340 \\ \hline \end{tabular}
\end{table} 
Table 2: The modeling parameters and calculated resistance values of the two types of sensors. Figure 3: The variation curve of sensor resistance with the mesh at room temperature. with the exception of the nickel strips, are electrically insulated (\(\overrightarrow{n}\cdot\overrightarrow{l}=0\)). In addition to the boundary of the model, there is an adiabatic boundary, i.e., \(\overrightarrow{n}\cdot\overrightarrow{q}=0\), where \(\overrightarrow{m}\) and \(\overrightarrow{q}\) represent the normal vector and the heat flow density vector, respectively. In the simulation, 293 K is set as the initial ambient temperature. Through temperature coupling, Joule heating is used as the heat source for the heat transfer module. In order to transfer heat to the Kapton layer and subsequently to the sample, a pair of thermal contact boundaries is defined and employed. The joint conductance (\(h_{\text{i}}\)) can be defined as the sum of the constriction conductance (\(h_{\text{c}}\)) and the thermal conductivity through the interstitial medium, (\(h_{g}\)), that is to say, \(h_{\text{i}}=h_{\text{c}}+h_{g}\). The conventional Copper-Mikic-Yovanovich (CMY) [56] model establishes a relationship between \(h_{\text{c}}\) and several other variables, including the microscopic surface average slope \(m_{asp}\), the mean height of the cusp angle \(\sigma_{asp}\), the contact pressure \(P_{r}\), and the surface microhardness \(H_{\text{c}}\). This relationship is expressed in the following equation [56]:  
\[\mathbf{h_{i}}=\mathbf{h_{c}}+\mathbf{h_{g}}=\mathbf{1.25\kappa_{c}}\frac{\mathbf{m_{asp}}}{\sigma _{asp}}(\frac{P_{r}}{\mathbf{H_{c}}})^{\mathbf{0.95}}+\frac{\mathbf{k_{g}}}{Y+\mathbf{M_{g}}} \tag{9}\]  
where \(\kappa_{g}\), \(Y\), and \(M_{\text{g}}\) denote the interstitial gas thermal conductivity, mean separation thickness, and gas parameters, respectively. In Figure 4, we present the temperature distribution along the centerline of the sensor and the two-dimensional temperature distribution within the PMMA sample, which provides a deep understanding of the heat transfer process. As shown in Figure 4a, it is evident that the temperature at the center of the sensor significantly exceeds that of the surrounding area as time progresses. This observed temperature gradient reflects the design characteristic of the heat source being concentrated at the center of the sensor, which leads to the rapid concentration and effective transfer of heat. This phenomenon not only reveals the efficiency of the sensor but also highlights its importance in thermal conductivity measurements. We particularly note that, in the initial stages, the center temperature rises at a significantly faster rate, after which the heat gradually diffuses into the surrounding area, resulting in a typical non-uniform temperature distribution.