Gradient-based topology optimization of soft dielectrics as tunable phononic crystals

Dielectric elastomers are active materials that undergo large deformations and change their instantaneous moduli when they are actuated by electric fields. By virtue of these features, composites made of soft dielectrics can filter waves across frequency bands that are electrostatically tunable. To date, to improve the performance of these adaptive phononic crystals, such as the width of these bands at the actuated state, metaheuristics-based topology optimization was used. However, the design freedom offered by this approach is limited because the number of function evaluations increases exponentially with the number of design variables. Here, we go beyond the limitations of this approach, by developing an efficient gradient-based topology optimization method for maximizing the width of the band gaps in an exemplary case study. We employ a finite element formulation of the governing equations, and use the properties of each element as the design variables. In order to iteratively update the design variables, we employ gradient-based optimization, namely the Method of Moving Asymptotes. We carry out and implement fully analytical sensitivity analysis for computing the gradient of the objective function with respect to each one of the design variables. The numerical results of the method developed here demonstrate prohibited frequency bands that are indeed wider that those that were generated using metaheuristics-based topology optimization, while the computational cost to identify them is reduced by orders of magnitude.