Interdisciplinary Applied Mathematics

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3. Nonlinear dynamic models: In these models, the modeling is done based on assumptions about the internal structure composed of basic functional blocks. A few control theory approaches are also available. Krylov subspace methods fall under this category.

17.3.1 Nonlinear Static Models

The method that falls in this category is table-based reduced-order modeling.

Table-Based Reduced-Order Modeling

Table-based reduced-order models use tables of numerical data to describe the relationship between variables (Wu and Carley, 2001). The table-based models are built directly from data obtained from measurements or device simulations without detailed knowledge of underlying physics. Hence, the difficulties and errors associated with extracting analytical models are eliminated and the process can be automated. For behavioral-level simulations, cubic spline interpolation is used to evaluate the functions. In addition to the nonlinear behavioral numerical model, a set of linear numerical models is constructed to assist in solving for operating points, to perform AC analysis, and to design closed-loop feedback systems. The procedure for table-based macromodeling can be described as:

1.    The    device is described    by    a    set    of    ODEs,    which are    solved    using

standard numerical methods.

2. The functions that describe the relationships between the variables are represented in numerical tables, obtained from device simulations.

3. During simulations, the models are evaluated by interpolating the data in the tables.

The microdevice shown in Figure 17.5 is modeled using the steps outlined above. The governing equation for the dynamics of the device is given by

d2x    dx

гаэё + са+/= /

where m is the effective mass, c is the damping factor, fs is the spring force, and fe is the electrostatic actuation force. First, a full simulation of the device is performed to generate the table of data. The mechanical part    (or    the    spring    force)    is computed    using    any    standard finite    element

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