FastStokes : a fast 3-D fluid simulation program for micro-electro-mechanical systems

Abstract

We have developed boundary integral equation formulas and a corresponding fast 3-D Stokes flow simulation program named FastStokes to accurately simulate viscous drag forces on geometrically complicated MEMS (micro- electro- mechanical systems) devices. Unlike the 3-D finite element or finite difference solvers which often take days to run to completion or fail when geometry gets complicated, the FastStokes 3.0 simulation program is capable of simulating complicated devices such as resonators, accelerometers, and micro-mirrors on PC computers in minutes. The FastStokes 3.0 simulation program is a fast 3-D boundary-element simulation program that uses only surface discretizations. The implementation of the Precorrected-FFT algorithm in combination with the GMRES algorithm substantially improves the speed of this simulation program. An efficient two-step approach that successfully handles the null space of the singular incompressible Stokes BEM operators is developed to avoid numerical errors and solution discontinuities. An analytical flat-panel kernel integration algorithm is implemented in FastStokes and an accurate curved-panel integration algorithm is also developed. Both an incompressible FastStokes solver and a compressible FastStokes solver have been developed and tested. They are not only fast, but also accurate. The incompressible FastStokes solver solves the steady incompressible Stokes equation; the effectiveness of this fast solver has been repeatedly proved by the close matches between numerical simulation results and experiments, within engineering accuracy (5-10% error).

(cont.) The numerical simulation results of a comb drive resonator, the ADXL 76 accelerometer, and a micro-mirror are given. The compressible FastStokes solver solves a linearized compressible Stokes equation that is also capable of capturing the weak air compression effect in MEMS devices. Therefore, the compressible FastStokes solver is a more general simulation program, and it is especially useful when the strength of the fluid compression effect is uncertain. The solutions of the compressible FastStokes are compared with the analytical solutions of the linearized compressible Reynolds equation. Numerical simulations of some common structures that may exhibit compression effect when packaged in gases are also given.