A time-domain Discontinuous Galerkin method for mechanical resonator quality factor computations

Numerical simulations are becoming increasingly important in the design of micromechanical resonators, in particular for the prediction of complex frequency response in high quality devices where damping is controlled by anchor losses. Frequency based approaches have been shown to predict these accurately, however, they require the solution of eigenvalue problems or the inversion of Helmholtz-type operators which are known to be very difficult for large-scale iterative solvers. We propose using a time-domain approach instead, where a broadband input signal is propagated through the system with a local explicit time-stepper. This is achieved using a new high-order Discontinuous Galerkin (DG) discretization for the linear elasticity equations, in particular a second-order formulation with Compact DG fluxes and a Runge–Kutta time integrator, where the block-diagonal mass matrices allow for efficient, stable, and accurate time stepping. Our solver scales well on distributed parallel computers, even in three spatial dimension and for large problem sizes. The resulting output signal is analyzed using a well-known filter diagonalization method, which is capable of finding accurate frequencies and quality factors for as little as a hundred periods of data. We validate the properties of our scheme on model problems, and demonstrate the feasibility of our proposed analysis process on two high quality factor disk resonators, using an axisymmetric formulation as well as full three dimensional simulations which is shown to scale well.