Comparison of Global Optimization Algorithms for Inverse Design of Substrate Metal Density for Low Warpage Design in Ultra-Thin Packages

An inverse design framework incorporating a physics-based surrogate model and global optimization is proposed to assist in the design of low warpage ultra-thin packages by adjusting the metal densities over substrate subsections and layers. The surrogate model is derived from two finite element analysis (FEA) models. The first one describes the relationship between the metal density in the substrate layer to the coefficient of thermal expansion (CTE) while the second one describes the relationship between in-plane CTE variation of the substrate to the warpage profile. Results from these two FEA models are used to train separate artificial neural networks (ANN). When these ANNs are run sequentially, the surrogate model can accurately determine the warpage profile for any set of metal densities. Three global optimization algorithms, Particle Swarm Optimization (PSO), Genetic Algorithm (GA) and Cross Entropy (CE) were then evaluated using this surrogate model. Three case studies consisting of different warpage profiles (original and 20% reduced warpage) and constraints to the optimization search space (±20% or ±50% change to metal density) were then evaluated using these algorithms. For all three cases, the three algorithms converged to similar solutions, indicating that indeed the global minimum has been attained and determined. However, GA took a significantly longer time to converge than PSO and CE. Based on these results, PSO and CE are recommended to be suitable algorithms to carry out inverse design for this type of problem.