Inhomogeneous Sparse Ball Grid Arrays in Electronic Packaging Under Random Shear Stress and Temperature Coupling

The geometrical optimization of BGAs and the design of solder ball distributions are critical for ensuring mechanical reliability and cost efficiency in electronic packaging, yet further efforts are required, particularly in developing predictive and reliable computational modeling approaches. To this end, this paper investigates inhomogeneous sparse BGA configurations under coupled random shear stress and thermal conditions. A stochastic finite element model (SFEM) is developed, incorporating temperature-dependent nonlinear elastic modulus of solder (SAC305) and underfill materials, to build a database for uncertainty quantification. Three representative sparse configurations are analyzed. A normalized stress variance index (β) is introduced to quantify the variations caused by the randomness in the boundary conditions. Statistical results reveal that Case 3 achieves the most favorable balance between mechanical robustness and manufacturing efficiency, maintaining underfill continuity while mitigating displacement and stress concentration. Random temperature fluctuations induce nonlinear effects, whereas shear stress shows a more linear yet dominant influence on stress variations. The findings provide engineering-oriented guidelines for optimizing solder ball arrangements, highlighting that inhomogeneous sparse layouts can reduce cost and complexity without sacrificing reliability, thereby supporting large-scale integration and deployment in harsh service environments.