## 2 Methods  
A transient 3D thermal analysis was conducted using the Heat Transfer in Solids interface in COMSOL Multiphysics (version 5.2). The governing equation for heat conduction is expressed as:  
\[\rho C_{\text{P}}\frac{\partial T}{\partial t}=\
abla\cdot(k\
abla T)+Q \tag{1}\]  
where \(\rho\) is the density (kg/m\({}^{3}\)), \(C_{\text{P}}\) is the specific heat capacity (J/kg\(\cdot\)K), \(k\) is the thermal conductivity (W/m\(\cdot\)K), \(T\) is the temperature (K), and \(Q\) is the volumetric heat generation rate (W/m\({}^{3}\)). The equation for radiation is expressed as:  
\[-n\cdot q=\varepsilon\sigma\Big{(}T_{\text{amb}}^{4}-T^{4}\Big{)} \tag{2}\]  
where \(T_{\text{amb}}\) is the ambient temperature (K), \(q\) is the heat flux vector, \(n\) is the unit outward normal vector, \(\varepsilon\) is the surface emissivity, and \(\sigma\) is the Stefan-Boltzmann constant (W/m\({}^{2}\cdot\)K\({}^{4}\)). In this study, convection was neglected based on the assumption that heat conduction dominates over convection due to the small size of the LED die, as supported by previous studies [21, 36, 37, 38, 39, 40]. Figure 1a illustrates the LED module model used in this study. The simulation domains include an LED chip, solder layer, substrate, copper layers, thermal interface material (TIM), and heat sink (Figure 1b). The LED chip with a footprint of 1 \(\times\) 1 mm\({}^{2}\), representative of high-power LEDs [32, 41], is mounted at the center of the 2.8 \(\times\) 2.8 mm\({}^{2}\) substrate with the solder interlayer. Solid copper layers are laminated on the top and bottom surfaces of the substrate (25 \(\upmu\)m thick for the top layer and 20 \(\upmu\)m thick for the bottom layer). The substrate is attached to the heat sink via TIM. Thermal vias are vertical holes with a diameter of 0.1 mm, fully filled with copper, a design commonly used in high-density printed circuit boards (PCBs) [32]. The detailed material properties and geometric parameters of each component are summarized in Table 1.  
The initial temperature for all domains was set to 25 \({}^{\circ}\)C. A uniform heat flux of 1.5 \(\times\) 105 W/m\({}^{2}\) was applied to the top surface of the LED chip to simulate heat generation. Adiabatic boundary conditions (\(-\)n\(\cdot\)q = 0) were imposed on the lateral surfaces of all domains except for the heat sink to prevent heat loss. The outer surfaces were subject to surface-to-ambient radiation boundary conditions with an emissivity of 0.85. Two-dimensional temperature distributions were extracted from the top surface of the LED chip (Monitor 1) and the substrate (Monitor 2), as shown in Figure 1b. To balance computational efficiency and accuracy, a physics-controlled mesh (normal setting) was applied, with locally refined elements in regions of high thermal gradients, particularly around the LED chip and the substrate (Figure 1c). Insufficient mesh refinement in these critical areas can lead to errors in temperature estimation and diminished local resolution. The mesh size in the reference model, including the heat sink, consisted of over 30,000 domain elements and 1000 edge elements. The minimum element quality, based on skewness, was 0.0078, and the average quality was 0.5235, both of which are suitable for stable numerical calculations. Mesh refinement had no noticeable effect on the resulting temperature distribution. In this study, \(T_{\text{LED}}\) is defined as the maximum temperature within the LED chip, and \(\Delta T_{\text{LED}}\) represents the temperature difference across the chip. Similarly, \(T_{\text{sub}}\) and \(\Delta T_{\text{sub}}\) denote the maximum temperature and the temperature difference within the substrate, respectively. Temperature data were collected after the system reached a steady thermal state. As shown in Figure 2a, \(T_{\text{LED}}\) rapidly increased during the initial stage and began to plateau around 600 s, indicating that the thermal equilibrium had been reached. A time point of 1500 s was chosen for analysis to ensure that the system had fully stabilized.