## 3 Results and Discussion  
### Thermal Simulation Analysis of Packaging Defects  
The impact of different packaging defects on the thermal behavior of BGA structures was systematically analyzed through steady-state and transient simulations. Four scenarios were considered: normal packaging, missing solder balls, missing solder ball and pad, and bridging. The heat dissipation pathways under these conditions are schematically illustrated in Figure 5. Under normal conditions, the heat generated by the package substrate can be evenly transferred through the conduction path of substrate-solder pad-solder ball-PCB. However, in the case of a missing solder ball, the heat cannot be conducted through the above path but only conducts to the pad or conducts through the normal solder balls around the missing ball and finally conducts heat laterally on the PCB. When both the solder ball and its corresponding pad are missing, the heat conduction mode is similar to that of missing solder ball. The heat on the package substrate is conducted around the normal solder balls, and it is also conducted laterally on the PCB. In the case of bridging, where two or more solder balls are unintentionally connected, an additional lateral heat conduction path is introduced. This alters the local temperature distribution by enabling heat to flow not only through the normal vertical path but also across the bridged solder balls, potentially impacting the system's overall thermal performance. Under steady-state thermal conditions, with a constant temperature of 200 \({}^{\circ}\)C applied to the top surface of the FR-4 package substrate, the resulting temperature distribution on the top surface of the PCB is illustrated in Figure 6. Under normal conditions, as shown in Figure 6a, the temperature field exhibits a symmetrical diffusion pattern, with heat transferred from the highest-temperature region at the center of the chip to the PCB through the solder ball array. The temperatures of the four central solder balls are significantly higher than those of the twelve peripheral solder balls, indicating that the peripheral solder balls dissipate heat more efficiently due to enhanced thermal convection. In contrast, the slower heat dissipation of the central solder balls leads to heat accumulation. Meanwhile, the four corners of the square PCB represent the lowest temperature zones, forming cold regions as both adjacent sides facilitate heat dissipation via convection. In the absence of solder balls, as illustrated in Figure 6b, the original symmetrical temperature distribution is disrupted. A cold spot with a temperature below 170 \({}^{\circ}\)C emerges in the upper-right corner of the solder ball array, and the temperature gradient of the adjacent solder balls increases. Figure 5: Schematic illustration of heat dissipation pathways under different packaging defect scenarios: normal structure with uniform heat conduction; missing solder ball causing heat diversion to adjacent paths; missing pad and ball showing similar diversion effects; bridging enabling additional lateral heat conduction. This suggests that heat is redirected around the missing region toward neighboring solder balls, which is consistent with the thermal resistance network analysis. At this point, the four central solder balls still maintain relatively high temperatures, and the cold zones at the PCB corners remain prominent. When the pad is further removed (Figure 6c), the resulting temperature distribution remains highly similar to the case involving only missing solder balls. This implies that a pad not in direct contact with the PCB has a limited effect on the overall temperature field. Under solder ball bridging conditions (Figure 6d), the temperature field largely reverts to a nearly symmetrical diffusion pattern, with no distinct cold spots observed. However, in the lower-left bridging region, the temperatures of the two connected solder balls tend to equalize due to lateral heat conduction between them.