# Interdisciplinary Applied Mathematics

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FIGURE 18.25. Mixed-level approach for modeling squeezed film damping in MEMS (see (Schrag and Wachutka, 2002), for details).

Compact, lumped Elements &

FN models of other parts and physical domains

 r> System model of microsystem

V

System Simulation

plate are rectified using error-compensating compact models. These compact models are in the form of lumped circuit elements such as resistances or constants, which can be determined from a few FEM simulations. The FN model in the sense of the Kirchhoffian network theory describes the squeezed film damping by two conjugate variables, namely, the pressure differencepik between two adjacent nodes (“across variables”) and the corresponding mass flow rate Qik (“through variable”) (Schrag and Wachutka, 2002). The mass balance equation is satisfied automatically as a result of the Kirchhoffian laws. However, correct formulation of the mass flowrate at each node must be done separately. The FN model can be implemented into a general-purpose system simulator and applied to arbitrary device geometries. The flowchart for the method is shown in Figure 18.25.

###### 18.2.4 Black Box Models

The Arnoldi method has been used in (Chen and Kang, 2001a), to solve a MEMS micromirror device for both small and large deflections in the presence of fluid damping. The fluid damping equation (nonlinear isothermal Reynolds equation) has been linearized using Taylors series, and the Arnoldi method has been used to construct a reduced-order linear model for small angular deflections. For large angular deflections, both the linear and the second-order nonlinear terms from the fluid equation were retained in the Taylors expansion, and the Arnoldi method has been applied to construct a weakly nonlinear model. The accuracy can be increased by considering higher-order terms in the Taylors expansion, but the computational cost goes up, restricting the use of the process. The trajectory piecewise-linear approach overcomes some of these difficulties, as described in Chapter 17.

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