Phononic crystals are periodic elastic structures that exhibit frequency bandgaps within which elastic waves cannot propagate. The formation and width of these bandgaps depend on the spatial distribution of material properties within periodic phononic unit cells. In this work, we propose novel topology optimization formulations to achieve multi-material design of phononic crystals for bandgap optimization at targeted frequencies, where we discover the influence of spatially varying properties of base materials on the band structures that block and permit elastic wave propagation. Specifically, we model wave propagation by Floquet-Bloch theory and solve the resulting eigenvalue problem by the finite element method along the boundary of the irreducible Brillouin zone. We first propose a multivariate weight-based method via a novel interpolation scheme that maps the design variables of an arbitrary number of candidate materials into effective density and stiffness for elastic wave-propagation calculations, where bandgaps arise from the contrast in the stiffness and inertia among material constituents. We then develop two formulations for bandgap optimization. The first formulation maximizes the gap between two adjacent frequency bands using a bound-based approach, where we introduce two auxiliary scalar variables as additional design variables and incorporate them directly into the design objective, yielding a linear, differentiable design objective, with the eigenfrequencies entering the formulation through associated constraints. The second formulation places a bandgap over a user-defined target frequency range through a dynamic band-pairing strategy, where we adaptively identify the relevant pair of adjacent bands amid optimization iterations. Towards the end, we validate the bandgap performance of our designs against the transmission loss spectra computed from COMSOL Multiphysics.

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