Interdisciplinary Applied Mathematics

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17.1.2 Dynamical Reduced-Order Modeling

Explicit dynamical formulation of microsystems can be very time-consuming, and computationally expensive to insert in a system-level simulator. As a result, it is difficult for the designer to use it in an iterative design cycle or to probe sensitivities to variations in the geometry and material constants by repeated simulations. This demands the development of dynamical reduced-order models for projecting the results of the fully meshed analysis onto physically meaningful reduced variables, containing algebraic dependencies on structural dimensions and material constants. Dynamical reduced-order modeling is much more challenging than the quasi-static reduced-order modeling, since the design space involves large motions and nonlinear forces. The various reduced-order modeling methods that fall under dynamical methods are shown in Figure 17.1. These methods can be broadly classified into three categories: (1) generalized Kirchhoffian networks, (2) black box models, and (3) Galerkin methods. In the following sections, we will look into the different methods that fall under these three categories in detail.

17.2 Generalized Kirchhoffian Networks

In this method, a complex microsystem is decomposed into components (or lumped elements) that interact with each other as constituent parts of a Kirchhoffian network (Voigt and Wachutka, 1997). Compact models with very few degrees of freedom are formulated for each of the components. All the system components are given a mathematical description in terms of conjugate thermodynamic state variables and the pertinent currents (fluxes or through quantities) and the driving forces (affinities or across quantities) such as mass flow and pressure gradient, electrical current, and voltage drop. A system component is called a “block” and is characterized by the number and    nature    of its    terminals,    which    allow    for    the    exchange    of    flux

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