The mismatch in thermal expansion coefficients among different materials leads to nonuniform deformation in a package, known as warpage. The warpage has become a significant reliability issue in advanced packaging that requires careful management. To address the issue, we have developed a high-fidelity warpage modeling approach that contains preprocessing followed by matrix calculations. This method transforms a three-dimensional packaging structure into a two-dimensional thermo-mechanical coupling problem, simplifying the packaging analysis. We apply finite element methods to the model and obtain the balance between accuracy and efficiency. Experimental results show that our model offers superior accuracy compared to the widely used Suhir’s model, achieving average speedups of 11.0X in preprocessing and 12130X in matrix calculations, with an overall speedup of 233.9X and only 1.0% error compared to the Ansys commercial analysis tool. This high efficiency and accuracy make our model suitable for large-scale optimization, package simulation, and modeling applications.

Warpage caused by the manufacturing thermal process can significantly reduce product yield in advanced packaging. As a result, numerical simulations such as finite element methods (FEMs) are often used to analyze warpage effects. However, constrained by the mesh generation and large matrix-solving requirements in finite element methods, optimizing for warpage can be time-consuming. This paper presents a fundamental physical model, training framework, and methodology for a warpage surrogate model based on DeepONets, a physics-informed operator learning framework. Experimental results show that our warpage model achieves an average speedup of 435X compared to traditional solvers while maintaining a minimal average warpage error of just 1.9%.

The mismatch of thermal expansion coefficients between different materials incurs nonuniform deformation in a package called warpage. The warpage effect has evolved into a crucial reliability concern that demands careful management in advanced packaging. Considering the severe impact of the effect, we develop an efficient, high-fidelity warpage modeling of preprocessing followed by matrix calculation to address the issue by transforming a three-dimensional packaging structure to a two-dimensional thermo-solid coupling problem. The transformation provides an efficient measure to reduce the complexity of the packaging analysis. We then apply formal numerical methods to this model for a better trade-off between accuracy and efficiency. Experimental results show that our model averagely achieves respective speedups of 11.0X and 12130X in preprocessing and matrix calculation and an overall speedup of 233.9X with only 1.0% error over the Ansys commercial analysis tool. In particular, such high efficiency and accuracy enable our model for large-scale optimization, simulation, and modeling applications.