This study establishes a power-law gradient model and investigates the influence of power-law exponents on bandgaps by considering unit cell symmetry and the biased filling fraction. The transfer matrix of the graded material is constructed using the exponential of the Magnus series, and the dispersion relation is derived via the transfer matrix method (TMM). Single- and multi-objective optimization of the first bandgap is conducted by integrating TMM with genetic algorithm (GA) and non-dominated sorting genetic algorithm (NSGA-II). Key findings include: (1) A clear correlation exists between the power-law exponent and the biased filling fraction, where increasing the biased filling fraction widens the first bandgap, with symmetric unit cells yielding larger bandwidths under identical biased filling fraction; (2) bandwidth and central frequency exhibit a strong trade-off, where broadening the bandwidth inevitably increases the central frequency; (3) Pareto-optimal power-law exponents are not restricted to boundary values. These results offer practical guidance for selecting power-law exponents and balancing bandwidth and central frequency in graded phononic metamaterial design.
Li et al. (Sun,) studied this question.