We consider topology optimization of lumped and continuous nonlinear metamaterial systems. Structures that consist of alternating layers of material with high impedance contrast are a simple example of a continuous phononic crystal that may exhibit nonlinearity. Analysis of this system, subject to prescribed constraints, shows that optimal design for a bilayered system consists of a thin nonlinear layer. Optimal, in this context, refers to a design which maximizes the frequency shift (and thus bandgap shift) at the edge of the first Brillouin Zone for the acoustic branch. Computer simulations of this system validate the predicted dispersion behavior. Optimization of two-dimensional arrays is presented using lumped-parameter models with nonlinear spring elements. Pattern-search algorithms identify topology (discrete mass distributions) that produce large increases in complete bandgap width. The analytical expression used in calculating nonlinear frequency shifts reveals that the largest contributions to the frequency shift are primarily produced from the resonant components of the system. Optimizing continuous multidimensional unit cells using a commercial finite-element code is briefly addressed.

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