**3. ROM (Reduced Order Model)**

156 Microelectromechanical Systems and Devices

MEMS typically involve multiple energy domains such as kinetic energy, elastic deformation, electrostatic or magneto static stored energy and fluidic interactions. The difficulty in the modeling of MEMS devices is mainly caused by the tight coupling between the multiple energy domains. Individual physical effects are governed by partial differential equations (PDE), typically nonlinear. When these equations become coupled, the computational challenges of highly meshed numerical simulation become

FEM relies on highly localized interpolation functions (or *mesh element functions*) for approximation of the solution of PDE. These mesh element functions are generated by meshing the domain of interest and parameterize the desired solution locally on each mesh element. This parameterized solution converts a continuous (PDE) problem to a coupled system of ordinary differential equations (ODE) that can be integrated in time. The resulting ODE system usually has many degrees of freedom (perhaps **several variables per mesh element**). If a fine mesh is required, the problem size grows rapidly, with a corresponding rapid growth in computational cost for explicit dynamic simulation. Consequently, it is very expensive to use FEM model in system-level simulations during MEMS iterative design*.* As a result, FEM models are mostly used to analyze the performance of MEMS components and

By reading ANSYS binary FULL file it is possible to assemble a MEMS component statespace model in the form of first order systems or second order ordinary differential

where *Ar, Er ,Cr, Br, M ,D, K, B, C-* are the system matrices, *Br, B* are the input and the *Cr, C* output matrices, *f* is input force. In mechanics matrices *M, D* and *K* are known as the *mass, damping* and *stiffness* matrices correspondingly. Usually damping is included in the model as Rayleigh damping. The damping matrix *D* is computed as a linear combination of the

D=α M +β K,

In (1) the state space vector *z* is defined through the unknowns deflections *u*(*x,t)* and pressures *p(x,y,t)* into the node points being automatically generated in MEMS structure*:*

> *u u z uu p p t t*

1 <sup>1</sup> <sup>11</sup> [ ... ... ... ] *<sup>N</sup> <sup>T</sup> N MN*

0

*<sup>D</sup> M K B Qx E A BC z M M Rx* 

00 0 *r r r r*

 *+ Arz* = *Brf , Y=Crz* (1)

Mx'' + Dx' + Kx =Bf , Y=QTx +RTx' , (2)

(3)

(4)

**2. FEM/FDM model** 

to couple their multiphysics effects.

 *Erz'*

stiffness *K* and the mass *M* matrices:

where *α, β* are constant coefficients*.*

second equations (2) can be transfer to the first (1).

formidable.

equations (ODE)

By defining

It would be easier and more intuitive for the designer to explore the design space if the MEMS model had only a few variables with a clear relationship between them and the overall device performance. *Reduced-order models (ROM)*, also called *macromodels,* lend themselves very well to these purposes. The main idea behind the reduced order model is that the number of ordinary differential equations (ODE) needed to simulate the system has been reduced from perhaps many thousands in the case of the full FEM simulation, to just a few basis function coordinates (fig.1).

Fig. 1. Reduced order model illustration

Such the macromodel simulation can be very efficient computationally compared to the FEM model. A designer can use the FEM model for different component geometry and materials trying and the ROM model for investigation of different input forces effect (fig.2).

Fig. 2. Compact reduced order model in MEMS design [33]

Macromodels of Micro-Electro-Mechanical Systems (MEMS) 159

Let's see what happens to the deformation pattern on the structure at each one of these natural frequencies. Modes are further characterized as either *rigid body* or *flexible body*  modes. All structures can have up to *six rigid body* modes, three translational modes and three rotational modes. Many deformation problems are caused, or at least amplified by the excitation of one or more flexible body modes. Fig.5 shows some of the common fundamental (low frequency) modes of a plate. The fundamental modes are given names like those shown in Fig.5. The higher frequency mode shapes are usually more complex in

Modal decomposition method uses a weighted sum of *n* mode shapes (modal amplitudes or eigenvalues) *qi*, basis function (an eigenvector) *φi (x, y, z))* of the mechanical structure to

> 1 ( , , ,) () ( , , ) *n eq i i i uxyzt u q t xyz*

where *ueq* is the initial displacement produced by the initial load. For MEMS it is usually sufficient that few modes accurately describe dynamical response of the system. This approach is equivalent to the projection of the original PDE, describing the MEMS behavior,

By inserting (5) into the equation of motion (2), taking **x** = φ *eλt* and multiplying by φ*Ti* from the left, and using the orthogonality of eigenvectors, the equation (2) can be reduced to

Note that the orthogonality condition does not apply in the case of multiple eigenvalues. In

The decoupling of modes yields that transfer functions can be written as a sum of *modal transfer functions*. Using Fourier transforming expansion and equations (6), the transfer

> () () ( )( ) *n n T*

 

This relation is the basis of modal analysis. It relates the *measurable* transfer functions to the modal properties ωi, σi, and *φi.* Each *i*- th mode contributes with a modal transfer matrix *Hi* to the complete transfer matrix. Fig.6 shows an example of a theoretical transfer function (7) with three *modal peaks* corresponding to modes at 1, 2, and 4 Hz which are all damped with a logarithmic decrement of 1% (*−σi/fi* = 0*.*01). Also the individual modal transfer functions (7)

By the way the eigenvalues *qk* and the corresponding eigenvectors *φ<sup>k</sup>* for *k* = 1*,* 2*, . . ., n* can

(Mλ2 + Dλ + K)φ =0 . (8)

*i i ii ii*

 

k+ σ<sup>2</sup>

*i i*

(7)

*j jj j* 

2 2 2( ) *<sup>T</sup> i ii i i i qq f*

The eigenvectors satisfy the orthogonality conditions *φTk Mφi= φTk Dφi = φTk Kφi = 0*  for *λk \_*≠ *λi*. With the normalization of the eigenvectors *φTk Mφ<sup>k</sup> ≡ 1*, it can be shown that

such case, special modal analysis techniques are needed for decoupling of modes [7].

 

φTk Dφk = −2σk and φTk Kφk = ω<sup>2</sup>

1 1

 

*k*

are plotted from which the complete transfer function is computed.

be found as an solution of the modified eigenvalue problem

*H H*

, (5)

, *i=1,2,…n* (6)

k if λk = σk +jωk.

 

appearance, and therefore don't have common names.

on the subspace defined by the basis functions.

represent its deflection *u*:

matrix can be derived as
