# Interdisciplinary Applied Mathematics

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1. They are generally analytical, rather than numerical, permitting the designer to reason about the effects of design changes (dimensions and material properties) without having to resimulate at the physical level.

2. They have a minimal number of degrees of freedom.

3.    Both large- and small-amplitude excitations are represented.

4. They incorporate correct dependencies on device geometry and constitutive properties.

5. They should accurately capture both quasi-static and dynamic behavior.

6. They should be expressible in a simple-to-use form, i.e., an equation, a network analogy, or a small set of coupled ODEs.

7. They should be easy to connect to system-level simulators.

8. They should be sufficiently accurate when compared with experiments on suitable test devices and with fully meshed 3D simulations.

9. They should account for correct explicit energy conservation and dissipation behavior.

To clarify    the    concept    of    a macromodel    or    a reduced    model,    we    give

a specific    example,    taken    from    (Senturia    et    al.,    1997),    for    a    suspended

membrane of thickness b deflected at its center by an amplitude d under the action of a uniform pressure force p. Let us also denote by 2a the length of the membrane, by E the Young’s modulus, and by vp the Poisson ratio. The form of the pressure-deflection relation can be obtained analytically, for example, by employing power series assuming a circular thin membrane. This can be extended to more general shapes and nonlinear responses, for example,

C1b — + a2

p

С2E d3

a4 1 — i/p

where C1 and C2 are dimensionless constants that depend on the shape of the membrane, and f (vp) is a slowly varying function of the Poisson ratio. This function is determined from detailed finite-element simulations over a range    of    length    a,    thickness    b,    and material    properties    vp    and    E.

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