Interdisciplinary Applied Mathematics

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17.1 Classification


Several techniques have been developed for reduced-order modeling or macromodeling of microsystems. Each technique has its own advantages and disadvantages, and the selection of a technique for a particular problem depends on a number of parameters such as the desired accuracy and nonlinearity. Many of the macromodels are created directly from physical-level simulations and often require human input at some stage of the process; i.e.,


/»Generalized KirchoffianX Networks


1.    Non-linear Static Models


i. Table based


2.    Linear Dynamic Models


i.    Krylov Subspace (Lanczos)


ii.    Moment Matching


3.    Non-linear Dynamic Models


i.    Krylov Subspace (Amoldi)


ii.    Trajectory Kecewise-Linear Approach


1.    Standard Galerkm


i.    Linear Modes


ii.    KL Decomposition


2.    Non-linear Galerkm


i.    Inertial Manifold theory


ii.    Post-processed Galerkm


1.    Equivalent Circuit Models


i.    Lumped Parameter


ii.    Distributed Parameter


2.    Description Language Models


i.    Element Stamps


ii.    Nodal Analysis

FIGURE 17.1. Classification of macro-models used in microsystem design.


there exists no systematic procedure to extract them automatically from the physical simulations. To identify macromodel extraction steps that can be automated in these cases is an important research topic in the field of microsystem simulation. In this section, we introduce the different macromodels and classify them into several broad categories. Figure 17.1 shows the classification of the various types of reduced-order models.

17.1.1 Quasi-Static Reduced-Order Modeling


Quasi-static macromodels are particularly useful for conservative systems with no dissipative terms. The distinction between energy domains in which the energy is strictly conserved (such as ideal elasticity, electromagnetic fields in linear lossless media, and inviscid flows) and those that have intrinsic dissipation mechanisms (fluidic viscosity, friction, heat flow, viscoelasticity, and internal loss mechanisms such as domain-wall motion that can lead to hysteresis (Senturia, 1998a)) is important, since dynamical behavior in a conservative domain can be derived from quasi-static behavior. All forces can be expressed as gradients of suitable potential-energy functions. If only conservative mechanisms are involved, one can use quasi-static simulations together with the mass distribution to fully characterize the dynamical behavior. Quasi-static macromodels are appropriate for cases in which a steady-state behavior is a reasonable assumption. In many cases, such as the squeeze film damping in a moving MEMS structure (see Chapter 18), such an assumption may be incorrect, in which case a dynamical macromodel is needed.

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