The Direct Parametrization of Invariant Manifolds Applied to Model Order Reduction of Microstructures

This chapter addresses the generation of reduced-order models of nonlinear microstructures thorough the technique known as Direct Parametrization of Invariant Manifolds. Two important families of devices are considered, resorting to piezo and electrostatic actuation, respectively. The multiphysics nature of these problems, together with the presence of important geometrical nonlinearities, makes the derivation of reduced order models particularly challenging. While piezo actuation is treated in a simplified manner assuming that the polarization hysteresis cycles are known from experiments, the fully coupled electrostatic case requires specific care. Indeed, the parametrisation method relies upon automated asymptotic expansions which can be applied only to polynomial nonlinearities. As a consequence, the continuous problem is first rewritten in the form of a mixed formulation that the reduction technique can directly handle. Practical examples are presented addressing both micromirrors and arch resonators. Numerical results show a remarkable performance in terms of accuracy and wealth of nonlinear effects that can be accounted for, including the transition from hardening to softening of the primary resonance and internal resonances between different modes of the resonators.