Page 278 - The Mechatronics Handbook

be zero, as will the sum of all values entering or leaving a given node (the “through” quantities). Thus,
for example, the sum of all forces and moments on each node must be zero, as must the sum of all
currents flowing into or out of a given node. This type of modeling is sometimes referred to as “lumped
parameter,” since quantities such as resistance and capacitance, which are in fact distributed along a graph
edge, are modeled as discrete components. In the electrical domain Kirchhoff’s laws are examples of these
rules. This method, which is routinely applied to electrical circuits in elementary network analysis courses
(see, e.g., [37]), can easily be applied to other energy domains by using correct domain equivalents (see,
e.g., [38]). A comprehensive discussion of the theory of nodal analysis can be found in [39]. In Fig. 13.5(a),
the cantilever beam has been divided into four “devices,” subbeams between node i and i + 1, i = 1, 2,
3, 4, where the positions of nodes i and i + 1 are described by (x i , y i , θ i ) and (x i+1 , y i+1 , θ i+1 ) the coordinates
and slope at P i and P i+1 . The beam is assumed to have uniform width W and thickness T, and each
subbeam is treated as a two-dimensional structure free to move in three-space. In [40] a modified version
of nodal analysis is used to develop numerical routines to simulate several MEMS behaviors, including
static and transient behavior of a beam-capacitor actuator. This modified method also adds position
coordinates z i and z i+1 and replaces the slope θ i at each node with a vector of slopes, θ ix , θ iy , and θ iz , giving
each node six degrees of freedom.
Since nodal analysis is based on linear elements represented as the edges in the underlying graph, it
cannot be used to model many complex structures and phenomena such as fluid flow or piezoelectricity.
Even for the cantilever beam, if the beam is composed of layers of two different materials (e.g., polysilicon
and metal), it cannot be adequately modeled using nodal analysis. The technique of finite element analysis
(FEA) must be used instead. For example, in some follow-up work to that reported in [36], nodal analysis
and symbolic computation gave essentially the same results, but the FEA results were significantly different.
Finite element analysis for the beam begins with the identification of subelements, as in Fig. 13.5(a), but
each element is treated as a true three-dimensional object. Elements need not all have the same shape, for
example, tetrahedral and cubic “brick” elements could be mixed together, as appropriate. In FEA, one cubic
element now has eight nodes, rather than two (Fig. 13.6), so computational complexity is increased. Thus,
developing efficient computer software to carry out FEA for a given structure can be a difficult task in itself.
But this general method can take into account many features that cannot be adequately addressed using
nodal analysis, including, for example, unaligned beam sections, and surface texture (Fig. 13.7). FEA, which
can incorporate static, transient, and dynamic behavior, and which can treat heat and fluid flow, as well as
electrical, mechanical, and other forces, is explained in detail in [41]. The basic procedure is as follows:

• Discretize the structure or region of interest into finite elements. These need not be homogeneous,
either in size or in shape. Each element, however, should be chosen so that no sharp changes in
geometry or behavior occur at an interior point.
• For each element, determine the element characteristics using a “local” coordinate system. This
will represent the equilibrium state (or an approximation if that state cannot be computed exactly)
for the element.
• Transform the local coordinates to a global coordinate system and “assemble” the element equa-
tions into one (matrix) equation.
• Impose any constraints implied by restricted degrees of freedom (e.g., a fixed node in a mechanical
problem).
• Solve (usually numerically) for the nodal unknowns.
• From the global solution, calculate the element resultants.

A Catalog of Resources for MEMS Modeling and Simulation
To make our discussion of the state-of-the-art of MEMS simulation less confusing, we first list some of
the tools and products available. This list is by no means comprehensive, but it will provide us with a
range of approaches for comparison. It should be noted that this list is accurate as of July 2001, but the
MEMS development community is itself developing, with both commercial companies and university

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