Cyclic steady states of nonlinear electro-mechanical devices excited at resonance

Abstract

SUMMARYWe present an efficient numerical method to solve for cyclic steady states of nonlinear electro-mechanical devices excited at resonance. Many electro-mechanical systems are designed to operate at resonance, where the ramp-up simulation to steady state is computationally very expensive -especially when low damping is present. The proposed method relies on a Newton-Krylov shooting scheme for the direct calculation of the cyclic steady state, as opposed to a naïve transient time-stepping from zero initial conditions. We use a recently developed high-order Eulerian-Lagrangian finite element method in combination with an energy-preserving dynamic contact algorithm in order to solve the coupled electro-mechanical boundary value problem. The nonlinear coupled equations are evolved by means of an operator split of the mechanical and electrical problem with an explicit as well as implicit approach. The presented benchmark examples include the first three fundamental modes of a vibrating nanotube, as well as a micro-electro-mechanical disk resonator in dynamic steady contact. For the examples discussed, we observe power law computational speed-ups of the form S D 0:6 0:8 , where is the linear damping ratio of the corresponding resonance frequency.

Newton‐Krylov method for computing the cyclic steady states of evolution problems in nonlinear mechanics

“…[19][20][21][22][23][24][25] In application to periodic problems, Newton-Krylov algorithm can be considered as a shooting method for calculating the cyclic state. 1,18 The main idea here consists in considering the periodicity condition as an equation for the unknown initial state. Indeed, if we have the correct initial state, the resulting state, obtained by solving the associated initial value problem on one time period with this initial data, will be identical to this initial state, within the imposed shift in space.…”

“…We define the Jacobian action using the tangent matrices of the evolution problem, calculated and stored during each time step of the forward time integration of the problem. As proposed in the works of Govindjee et al, 1 and Brandstetter and Govindjee, 18 we use the matrix-free generalized minimal residual (GMRES 28 ) method to solve the linear system, which appears at each global Newton step.…”

“…Examples of application of shooting technique to periodic problems include the adjoint-based optimal control methods (as in the works of Bristeau et al 16 and Ambrose and Wilkening 17 ) and the Newton-Krylov shooting algorithm. 1,18 The Newton-Krylov method is the subject of this work, following the pioneering work of Govindjee et al, 1 and will be discussed in more details.…”

A Two-Step Hybrid Approach for Modeling the Nonlinear Dynamic Response of Piezoelectric Energy Harvesters

“…where and are the Lame constants, C 1 and C 2 are further material constants to be calibrated, and I 1 , I 4 , and I 5 are computed for a transversely isotropic material according to [45,46] and they are equal to…”