## 3 Finite Element Simulation Method
### Numerical Simulation Settings  
The numerical simulation of the TPS test system, utilizing finite element simulation method, necessitates the undertaking of several preparatory steps [43; 44; 45]. These essential steps include the creation of a geometric model, the selection of appropriate materials, the designation of the desired physical field type, the configuration of the boundary conditions, the definition of the mesh, and the choice of solver type within the standard simulation program [46; 47; 48]. In the TPS method, a current is passed through a double helix sensor element to generate heat. As shown in Equation (6), both the temperature and resistance of the detector element rise over time. The form of the resulting transient temperature response curve is influenced by the thermal conductivity of the material surrounding the sensor element. For instance, the temperature of a sample with a high thermal conductivity rises at a slower rate.  
This paper presents the initial resistance of the sensor element at room temperature, as calculated based on steady-state studies. Additionally, temperature response curves for different samples are obtained based on transient studies. The double helix geometry of the sensor is modeled based on the Archimedes spiral equation, as shown below [49; 50]:  
\[\left\{\begin{array}{l}x=(a_{i}+r_{b}\theta)cos\theta\\ y=(a_{i}+r_{b}\theta)sin\theta\\ r_{b}=(a_{f}-a_{i})/2n\pi\end{array}\right. \tag{7}\]  
where \(a_{i}\) and \(a_{f}\) represent the minimum and maximum radii of the spiral, respectively. Meanwhile, \(n\) is employed to indicate the number of turns of the spiral ring. Additionally, \(r_{b}\) represents the spiral growth rate, while \(\theta\) denotes the angle of rotation of the spiral. The dimensions of the simulation are essentially identical to those of the hot disc method  
Figure 1: (**a**) Diagram of actual experimental setup. (**b**) The main body of the testing device.  
sensor employed in the experiment. The width of the double helix nickel wire and the spacing between neighboring nickel wires can be controlled with greater ease through the utilization of this equation.  
In transient simulations, it is necessary to embed a double helix nickel strip within the Kapton layer, which can enhance the insulation and mechanical strength of the actual detector. Meanwhile, two samples of the same material must also be positioned at the two ends of the detector to form a similar sandwich structure, as illustrated in Figure 2a, which depicts the geometry of the detailed simulation model. Figure 2b illustrates the geometry of the double helix nickel strip. The materials used include PMMA (polymethyl methacrylate), stainless steel, ceramic, nickel, and Kapton [35; 36]. The thermal conductivities in the simulation are 0.19 W\(\cdot\)m\({}^{-1}\cdot\)K\({}^{-1}\), 13.56 W\(\cdot\)m\({}^{-1}\cdot\)K\({}^{-1}\), 1.45 W\(\cdot\)m\({}^{-1}\cdot\)K\({}^{-1}\), 90.7 W\(\cdot\)m\({}^{-1}\cdot\)K\({}^{-1}\), and 0.15 W\(\cdot\)m\({}^{-1}\cdot\)K\({}^{-1}\), respectively. The specific heats are 1420 J\(\cdot\)kg\({}^{-1}\cdot\)K\({}^{-1}\), 8000 J\(\cdot\)kg\({}^{-1}\cdot\)K\({}^{-1}\), 2300 J\(\cdot\)kg\({}^{-1}\cdot\)K\({}^{-1}\), 445 J\(\cdot\)kg\({}^{-1}\cdot\)K\({}^{-1}\), and 1420 J\(\cdot\)kg\({}^{-1}\cdot\)K\({}^{-1}\), respectively. The densities are 1190 kg\(\cdot\)m\({}^{-3}\), 460 kg\(\cdot\)m\({}^{-3}\), 657 kg\(\cdot\)m\({}^{-3}\), 8900 kg\(\cdot\)m\({}^{-3}\), and 1090 kg\(\cdot\)m\({}^{-3}\), respectively. The dimensions of the corresponding sensors are shown in Table 1.  
This paper aims to use the finite element simulation method to analyze the heat transfer process in TPS method, and to use the model to test other samples. It is mainly completed via two kinds of research simulation: stationary and transient.