Robust topology optimization of cellular structure for mitigating dynamic instability in elastic structure

Abstract

This study presents an optimization method using cellular structures to mitigate dynamic instability. Dynamic instability, characterized by positive real parts of complex eigenvalues, arises from asymmetric system matrices. It often leads to problematic squeal noise and severe vibrations in engineering systems. The conventional optimization approach of cellular structures consists of three main steps: the homogenization to capture the material properties, the topology optimization considering the dynamic instability, and the reconstruction step to check the accuracy of the optimization result. First, homogenized constitutive matrices are computed and approximated using polynomial curves. Second, topology optimization is carried out based on these fitted curves. Finally, the reconstruction step is conducted to validate finite element analysis (FEA) results. The difference between the optimized and the reconstructed microstructure model in the conventional approach exists because the reconstruction does not satisfy the assumption of homogenization, and the shape of the unit cell differs between idealized and reconstructed models. This discrepancy between the optimal homogenized FE model and the reconstructed microstructure can cause serious dynamic instability after the reconstruction step. To mitigate this dynamic instability caused by the discrepancy, the present method employs an increased number of repeated structures by employing the concepts of unit cell groups and element groups. Additionally, a pseudo-robust optimization approach is presented to improve the robustness of the optimization result. Through this approach, a stable optimal objective value matching the reconstructed objective value is achieved, thereby successfully eliminating dynamic instability.