## 4 Results and Discussions  
In this paper, we analyze the physical modular functionality of the TPS method (Figure 1) and utilize the finite element simulation method to model the critical elements for measuring the thermophysical properties of materials using the TPS method. This includes both transient and stationary studies. Additionally, the Archimedes spiral equation is employed to ensure that the geometry of the design closely resembles that of the actual sensor. Correspondingly, the parameters used in the simulation are detailed in Tables 1 and 3. As shown in Figure 6, panels (a) and (b) illustrate the relative resistance and mesh density variation curves for Type I and Type II TPS sensors, respectively. The horizontal axis, labeled with numbers 1 to 9, represents grid sizes ranging from very coarse to very fine meshes. The red curve indicates the variation in mesh density, while the black curve shows the corresponding relative resistance variation in the sensor. The relative change in resistance refers to the absolute value of the difference between the numerical results of the current mesh computation and those of the previous mesh, divided by the results of the previous mesh. This aims to investigate the extent to which the numerical results of the current mesh increase relative to those of the previous mesh as the number of mesh elements increases [59]. It is apparent from the graph that as the mesh density increases,  
Figure 4: (**a**) Temperature profiles of the sensor along its centerline at 120 mW. The center of the double helix nickel strip is located at the origin of the coordinate system, and the duration of the simulation is 20 s with a step size of 10 ms. (**b**) Two-dimensional temperature distribution of the ZX-axis cross-section of the PMMA and the sensor. Figure 5: (**a**) The plot of current density distribution in the X-Y plane of the detector. (**b**) The line current density distribution curve along the Y-axis through the center of the detector. the curve of relative resistance variation approaches zero. However, increasing the mesh density further consumes more computational resources and extends the simulation time. Additionally, for finer mesh resolutions, increasing the mesh density beyond a certain point contributes little to improving the accuracy of the resistance calculation [60; 61; 62]. In this case, selecting a mesh type with a lower count and a relative change rate approaching zero offers a cost-effective balance between improving computational efficiency and ensuring accuracy. This is also reflected in Figure 3, where the sensor's resistance converges to a constant value. Consequently, a finer mesh will be employed in subsequent simulations to balance computational accuracy and efficiency. Furthermore, the simulation results exhibit an error of less than 1% when compared to theoretical values. In the stationary study, the current density distribution in the double helix nickel strip is also investigated in order to gain insight into the causes of heat source distribution, as shown in Figure 5a. From this distribution, it can be seen that the current density increases with increasing geometric curvature, except for a small region linked inside the helix (where the geometric curvature tends to zero). In the central region, due to the significant curvature changes at the edges of the spiral, the current density on both sides of the nickel strip connecting the two spiral lines is higher than at the origin, resulting in a V-shaped pattern, as shown in Figure 5b. The current density increases along the inner curve, with this gradient rise progressively intensifying as it approaches the central ring. The direct manifestation of this is in the form of high temperatures generated at the location of the central area of the sensor, as shown in each of the curves in Figure 4a.