A two-stage topology optimization framework for designing multi-scale elastic metamaterials with cross-scale dual bandgaps

Elastic metamaterials exhibit exceptional capabilities in controlling vibration and wave propagation. However, designing a single metamaterial architecture that possesses complete bandgaps at frequency scales separated by orders of magnitude remains a formidable challenge. To address this issue, this paper proposes a systematic two-stage topology optimization framework. In the first stage, a bi-material microstructural unit cell is optimized to generate a complete Bragg-scattering bandgap at a targeted high frequency range. The optimized microstructure is then characterized by numerical homogenization and utilized as an equivalent homogeneous material. In the second stage, the macroscale distribution of this equivalent material within a base matrix is optimized to create a distinct complete bandgap at a targeted low frequency range. A key contribution of this work is the introduction of a novel Quadratic Exclusion Aggregation (QEA) constraint, which is formulated from a series of individual quadratic exclusion (QE) constraints. Compared with conventional ipsilateral frequency (IF) constraint formulations, the proposed QEA constraint is designed to improve robustness and computational stability by reducing scale-dependent parameter tuning across different optimization stages and by employing a Kreisselmeier–Steinhauser (KS) function for effective aggregation. Numerical examples demonstrate the effectiveness of the proposed framework, successfully yielding a hierarchical metamaterial exhibiting two distinct complete bandgaps at frequencies separated by several orders of magnitude. The proposed framework, together with the improved constraint formulation, provides a systematic methodology for the on-demand design of advanced metamaterials with multiscale wave-manipulation functionalities.