Kevin L. Manktelow

Topology Optimization of Nonlinear Metamaterials

Optimal topology for periodic structures exhibiting nonlinearity is considered for two-dimensional plane stress systems. 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. Parametric analyses and a genetic algorithm implementation identify topologies that produce large increases in complete bandgap width or group velocity variation. Analysis of Bloch wavemodes which produce large nonlinear frequency shifts reveals that the largest contributions to the frequency shift are primarily produced from localized strains in thin layers of compliant, nonlinear matrix material.Copyright

Toplogy Design and Optimization of Nonlinear Metamaterials

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.Copyright

Intensity-Dependent Dispersion in Nonlinear Phononic Layered Systems

Phononic crystals are typically considered to operate in regimes where a linear constitutive relationship provides an adequate representation. For high intensity wave propagation, however, weak nonlinearities can affect performance. For example, a cubic nonlinearity gives rise to frequency shifting and thus a shift in band gap location. In the study of nonlinear optics, a cubic term has been treated using a quasi-linear constitutive relationship with intensity dependent properties. This technique is explored herein for generating nonlinear dispersion relationships for the elastic case. In addition, a perturbation method developed previously for discrete systems, used in conjunction with a finite element discretization, is proposed as an alternative dispersion analysis tool. Simulations of the fully nonlinear governing equations are provided as validation of the predicted dispersion curves.Copyright