Topology Optimization of Void Distributions in Phononic Crystals for Broadening Elastic-Wave Bandgaps in Vibration-Sensitive Structures

Phononic crystals are periodic media whose spatially varying mass density andelastic stiffness can produce frequency intervals in which propagating Bloch waves areabsent. This paper presents a theoretical and computational framework for designingvoid distributions in periodic elastic unit cells to broaden bandgaps relevant tovibration-sensitive structures. The framework combines linear elastodynamics, Bloch–Floquet periodicity, generalized eigenvalue analysis, and density-based topologyoptimization. The design objective is formulated in terms of the global extrema ofadjacent dispersion branches, with an optional penalty that centers the gap arounda prescribed target frequency. The formulation also discusses volume, connectivity,minimum-feature-size, strength, and manufacturing constraints. A robust-designextension accounts for uncertainty in material properties, geometry, and prestress.Since an infinite-crystal bandgap does not by itself guarantee a specified attenuationin a finite structure, the proposed workflow includes finite-slab harmonic-responseand transmission analyses. The equations and methodology constitute a researchframework rather than new numerical or experimental findings. Application-specificperformance must be established through converged computation and, where possible,experimental validation.