Gradient-Based Topology Optimization of Localized Defect Modes with Bandgap Preservation in Phononic Crystals
Phononic crystals can confine elastic waves through localized defect states within bandgaps, offering promising opportunities for vibration control and energy localization. However, designing defect states at prescribed frequencies while maintaining adequate separation from other in-gap modes remains a significant challenge. Existing optimization approaches generally treat the target mode indirectly and provide limited control over competing localized modes. This study presents a gradient-based two-stage topology optimization framework for the frequency placement of localized defect modes in periodic elastic media. First, a host unit cell is optimized to create a bandgap around a prescribed frequency. Subsequently, only the defect cell is modified to attract a selected localized mode toward the target frequency while repelling non-target modes away from the central region of the bandgap. The formulation incorporates a smooth mode-selection function that combines mode attraction and repulsion within a unified objective, enabling automatic tracking of the relevant modes throughout the optimization process. Because the localized defect branches of interest are nearly flat, the optimization is performed using only the $\Gamma$-point eigenspectrum, while the corresponding dispersion relations over a reduced irreducible Brillouin zone are evaluated afterwards for verification. Numerical examples involving two material systems and two supercell sizes demonstrate accurate placement of localized resonances, clear separation from competing in-gap modes, and substantial preservation of the host bandgap. The resulting structures exhibit strong elastic-wave localization, highlighting the potential of the proposed approach for the design of phononic devices for vibration confinement and energy trapping.