Mechanical response identification of local interconnections in board-level packaging structures under projectile penetration using Bayesian regularization

Modern warfare demands weapons capable of penetrating substantial structures, which presents significant challenges to the reliability of the electronic devices that are crucial to the weapon's performance. Due to miniaturization of electronic components, it is challenging to directly measure or numerically predict the mechanical response of small-sized critical interconnections in board-level packaging structures to ensure the mechanical reliability of electronic devices in projectiles under harsh working conditions. To address this issue, an indirect measurement method using the Bayesian regularization-based load identification was proposed in this study based on finite element (FE) predictions to estimate the load applied on critical interconnections of board-level packaging structures during the process of projectile penetration. For predicting the high-strain-rate penetration process, an FE model was established with elasto-plastic constitutive models of the representative packaging materials (that is, solder material and epoxy molding compound) in which material constitutive parameters were calibrated against the experimental results by using the split-Hopkinson pressure bar. As the impact-induced dynamic bending of the printed circuit board resulted in an alternating tensile-compressive loading on the solder joints during penetration, the corner solder joints in the edge regions experience the highest S₁₁ and strain, making them more prone to failure. Based on FE predictions at different structural scales, an improved Bayesian method based on augmented Tikhonov regularization was theoretically proposed to address the issues of ill-posed matrix inversion and noise sensitivity in the load identification at the critical solder joints. By incorporating a wavelet thresholding technique, the method resolves the problem of poor load identification accuracy at high noise levels. The proposed method achieves satisfactorily small relative errors and high correlation coefficients in identifying the mechanical response of local interconnections in board-level packaging structures, while significantly balancing the smoothness of response curves with the accuracy of peak identification. At medium and low noise levels, the relative error is less than 6%, while it is less than 10% at high noise levels. The proposed method provides an effective indirect approach for the boundary conditions of localized solder joints during the projectile penetration process, and its philosophy can be readily extended to other scenarios of multiscale analysis for highly nonlinear materials and structures under extreme loading conditions.