# Interdisciplinary Applied Mathematics

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1.    Find    the    linear    modes    for    the    elastic    problem assuming    the    no-load

condition.

2. Perform quasi-static simulation over a design space that includes the deformations described by a superposition of p modes and develop a suitable potential energy function for other conservative forces (e.g., electrostatic) and large-amplitude elastic effects (e.g., for stress stiffening). Create analytical expressions for the variation of potential energy as a function of the selected mode set (this function is nonlinear and must include products of modal amplitudes, etc.).

3. Replace the right-hand side of the modal dynamic formulation with suitable derivatives of the potential energy functions with respect to modal amplitudes. The net result is a small coupled set of 2p (2 state variables per mode) ODEs that can be easily integrated forward in time, without requiring any conversions to and from the original meshed space.

The advantage with modal methods is that they break open the coupled-domain problem. The original modal basis functions are obtained from a

(16.13)

Vreal

single energy domain, e.g., elasticity, together with the associated mass distribution, and the nonlinear potential energy functions can be computed one energy domain at a time. Therefore, it is not necessary to perform complex self-consistent coupled-domain simulations. This approach requires many (single-energy domain) simulations combined with fitting parameters to obtain analytical functions. So it is difficult to do it manually. It is possible to automate the procedure for nonlinear conservative problems. There are limits to the basis-function approach. Thus far, it has been difficult to calculate accurately the stress stiffening of an elastic body undergoing large-amplitude deformation using superposition of modal coordinates. Additionally, when the device undergoes nonlinear motion, such as contact, modal approaches fail. However, the class of microsystems that can now be handled with the automated basis-function approach is large enough to be interesting. More examples using modal basis functions for MEMS simulations are given in (Gabbay and Senturia, 1998; Varghese et al., 1999).

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