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Due to the inevitable presence of random defects, unpredictable grain boundaries in macroscopic samples, stress concentration at clamping points, and unknown load distribution in the investigation of graphene sheets, uncertainties are crucial and challenging issues that require more exploration. The application of the Kriging surrogate model in vibration analysis of graphene sheets is proposed in this study. The Latin hypercube sampling method effectively propagates the uncertainties in geometrical and material properties of the finite element model. The accuracy and convergence of the Kriging surrogate model are confirmed by a comparison with the reported references. The uncertainty analysis for both Zigzag and Armchair graphene sheets are compared and discussed.

Electro-migration (EM) induced by electric current under the conditions of multi-physical fields in integrated circuits and electronic packaging is a crucial factor that affects reliability and safety of the entire system. The uncertainty quantification and propagation in the EM process are the challenging issues that deserve more attention. In this article, the uncertainties in the related material parameters of the Sn–Ag–Cu (SAC) solder and copper conductors are taken into consideration based on the stochastic finite-element model. The corners and edges of the contact surface in the SAC solder are the most dynamic and active places with the maximum concentration gradient. This reaches a satisfied agreement with the experimental results and parallel numerical investigations in the literatures. The extreme values of the concentration and its gradient in each time step are computed and recorded. The accuracy and convergence of results are confirmed by the comparison of different period durations and time step scales with discrete time points. Furthermore, the probability density distribution, mean, and variance of the extreme values in different time steps are recorded and compared. Based on the huge database provided by the Monte Carlo-based stochastic finite-element model (MC-SFEM), the correlations between the material parameters and the concentration as well as the concentration gradients are analyzed. The proposed MC-SFEM is a feasible and effective model for the comprehensive analysis of EM with the potential in uncertainty and reliability analysis. [ABSTRACT FROM AUTHOR]

The ball grid array (BGA) is a type of popular and competitive package in electronic flip-chip packaging due to its feasibility in high density integrated circuits and convenience in the product design process. However, the effects of geometrical parameters on the product reliability and safety under complicated operating situations are not clear. In this article, an independent solder ball and four typical BGA cases are compared and analyzed based on the finite element (FE) method. The coupled random shear stress and thermal temperature are simulated in the FE models by the Latin hypercube sampling (LHS) method. According to the sensitivity analysis, the edges of the solder ball are the most dangerous places, which has the qualitative agreement with the experimental results. The complete grid array in the first BGA case with homogeneous stress and strain distribution is the most reliable and competitive design. Furthermore, the normal and Weibull distributions are not suitable to present the stochastic response of solder balls in flip-chip packaging under random coupled mechanical and thermal stress. In order to effectively improve packaging performance and reliability, the radius of the solder ball acts as the key factor, while the upper and lower height of the solder ball, as well as the pitch along the X- and Y-directions, are all feasible and potential for the geometrical optimization. However, the small scale of the solder ball causing microstress concentration points and discontinuous volumes is the essential challenge for industrial manufacturing. The work in this article provides helpful references to the industrial electronic package geometrical optimal design. [ABSTRACT FROM AUTHOR]