Optimization under Uncertainty of a Chain of Nonlinear Resonators using a Field Representation

Chains of resonators are known to exhibit interesting dynamic phenomena. For instance, band gaps are frequency ranges within which waves do not propagate or are strongly attenuated. Through the introduction of nonlinearities, other interesting behaviors can be achieved to manipulate waves and optimize the dynamic behavior of a chain. This study proposes the design optimization under uncertainty of such a chain of resonators for the purpose of vibration mitigation. The stochastic optimization accounts for the presence of design uncertainties (e.g., nonlinear stiffness properties) as well as aleatory uncertainties (e.g., loading conditions). The algorithm is tailored to account for discontinuities in the chain's response due to nonlinearities. In addition, a field formulation is used to define the properties of the resonators along the chain and reduce the dimensionality of the optimization problem. It is shown that the combination of the stochastic optimization algorithm and the field representation leads to robust designs that could not be achieved with optimal properties constant over the chain.