# Interdisciplinary Applied Mathematics

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6. Steps 1 to 5 are repeated for each eigenmode.

(a)    (b)

FIGURE 18.24. (a) Pressure distribution under an oscillating flat plate at low frequency. (b) Pressure distribution under an oscillating flat plate at high frequency.

A modal decomposition of damping effects is acceptable, since the Reynolds squeezed film equation is linear. More examples using modal basis functions for MEMS simulations are given in (Gabbay and Senturia, 1998; Varghese et al., 1999). Figure 18.24(a) shows the pressure distribution under a flat plate for a low frequency of oscillation, while Figure 18.24(b) shows the pressure distribution under a flat plate for a high frequency of oscillation. At high    frequency,    the    fluid    cannot    easily escape    from    the    sides,    giving

roughly uniform and high pressure under the plate.

The Karhunen-Loeve decomposition method for model order reduction has also been effectively used for squeezed film damping in (Hung and Senturia, 1999). The method used is basically the same as that in the absence of damping. From a few full-simulation runs, snapshots are taken for the fluid pressure distribution and a set of pressure basis functions formed using an SVD analysis. These basis functions are then used in the dynamic simulation of the device, thereby reducing the order of the system. The nonlinear Reynolds equation can be reduced efficiently using this method with no linearization involved as in the case of the equivalent circuit representation. The trajectory piecewise-linear approach has also been used for modeling squeezed film damping in MEM devices; see (Rewienski and White, 2001), for details.

###### 18.2.3 Mixed-Level Simulation

A mixed-level formulation uses a hardware description language for modeling squeezed film damping (Schrag and Wachutka, 2002). From an FEM model of the microstructure built by any standard FEM tool, a netlist for finite network (FN) simulation is constructed utilizing the grid and the geometric information from the FEM model. The governing equations (the Reynolds equation and the mass continuity equation) are discretized and coded in VHDL-AMS (Schrag et al., 2001; Sattler et al., 2003). The limitations of the Reynolds equation due to edge effects and perforations on the

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