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Mathematical Problems in Engineering
Volume 2010 (2010), Article ID 586718, 25 pages
http://dx.doi.org/10.1155/2010/586718
Research Article

A Wavelet Interpolation Galerkin Method for the Simulation of MEMS Devices under the Effect of Squeeze Film Damping

1School of Mechanical Engineering, Southeast University, Jiangning, Nanjing 211189, China
2College of Electronic Science and Engineering, Nanjing University of Posts and Telecommunications, Nanjing 210003, China

Received 29 March 2009; Revised 17 September 2009; Accepted 27 October 2009

Academic Editor: Stefano Lenci

Copyright © 2010 Pu Li and Yuming Fang. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

This paper presents a new wavelet interpolation Galerkin method for the numerical simulation of MEMS devices under the effect of squeeze film damping. Both trial and weight functions are a class of interpolating functions generated by autocorrelation of the usual compactly supported Daubechies scaling functions. To the best of our knowledge, this is the first time that wavelets have been used as basis functions for solving the PDEs of MEMS devices. As opposed to the previous wavelet-based methods that are all limited in one energy domain, the MEMS devices in the paper involve two coupled energy domains. Two typical electrically actuated micro devices with squeeze film damping effect are examined respectively to illustrate the new wavelet interpolation Galerkin method. Simulation results show that the results of the wavelet interpolation Galerkin method match the experimental data better than that of the finite difference method by about 10%.

1. Introduction

Modeling and simulation of MEMS devices play an important role in the design phase for system optimization and for the reduction of design cycles. The performances of MEMS devices are represented by partial-differential equations (PDEs) and associated boundary conditions. In the past two decades, there have been extensive, and successful, works focused on solving the partial-differential equations of MEMS [115]. A detailed review of the works is available in [1]. In the previous works, Galerkin method was widely used to reduce the partial-differential equations to ordinary-differential equations (ODEs) in time and then solve the reduced equations either numerically or analytically. The previous works differ from each other in the choice of the basis functions.

The basis set can be chosen arbitrarily, as long as its elements satisfy all of the boundary conditions and are sufficiently differentiable. To enhance convergence, the basis set has to be chosen to resemble the behavior of the device. For example, two ways have been used to generate the basis set for the reduced-order models of MEMS devices [1]. The first way [4, 9] uses the undamped linear model shapes of the undeflected microstructure as basis functions. For simple structures with simple boundary conditions, the mode shapes are found analytically. For complex structures or complex boundary conditions, the linear mode shapes are obtained numerically using the finite element method. The second way [2] conducts experiments or solves the PDEs using FEM or FDM to generate snapshots under a training signal, then applies a modal analysis method (one of the variation of the proper orthogonal decomposition method [6]) to the time series to extract the mode shapes of the device structural elements.

In the past two decades also, a new numerical concept was introduced and is gaining increasing popularity [1625]. The method is based on the expansion of functions in terms of a set of basis functions called wavelets. Indeed wavelets have many excellent properties such as orthogonality, compact support, exact representation of polynomials to a certain degree, and flexibility to represent functions at different levels of resolution. Indeed a complete basis can be generated easily by a signal function through dilatation and translation. The wavelet-based methods may be classified as wavelet-Galerkin method [19, 20], wavelet-collocation method [21, 22], and wavelet interpolation Galerkin method [2325]. Among the three methods, the wavelet-Galerkin method is the most common one because of its implementation simplicity. The method is a Galerkin scheme using scaling or wavelet functions as the trial and weight functions. However, both scaling and wavelet functions do not satisfy the boundary conditions. Thus the treatment of general boundary conditions is a major difficulty for the application of the wavelet-Galerkin method, especially for the bounded region problems, even though different efforts [19, 20] have been made. For the wavelet-collocation method, boundary conditions can be treated in a satisfactory way [21]. In the method, trial functions are a class of interpolating functions generated by autocorrelation of the usual compactly supported Daubechies scaling functions. However, the method requires the calculation of higher-order derivatives (up to the second derivatives for second-order parabolic problems) of the wavelets. Due to the derivatives of compactly supported wavelets being highly oscillatory, it is difficult to compute the connection coefficients by the numerical evaluation of integral [18]. The wavelet interpolation Galerkin method is a Galerkin scheme that both trial and weight functions are a class of interpolating functions generated by autocorrelation of the usual compactly supported Daubechies scaling functions. For the method, the boundary conditions [24] can be treated easily and the formulations are derived from the weak form; thus only the first derivatives of wavelets (for second-order parabolic problems) are required.

Wavelets have proven to be an efficient tool of analysis in many fields including the solution of PDEs. However, few papers in MEMS area give attention to the wavelet-based methods. This paper presents a new wavelet interpolation Galerkin method for the numerical simulation of MEMS devices under the effect of squeeze film damping. To the best of our knowledge, this is the first time that wavelets have been used as basis functions for solving the PDEs of MEMS devices. As opposed to the previous wavelet-based methods that are all limited in one energy domain, the MEMS devices in the paper involve two coupled energy domains. The squeeze film damping effect on the dynamics of microstructures has already been extensively studied. We stress that our intention here is not to discover new physics to the squeeze film damping.

The outline of this paper is as follows. Section 2 presents a brief introduction to some major concepts and properties of wavelets. In Sections 3 and 4, two typical electrically actuated micro devices with squeeze film damping effect are examined respectively to illustrate the wavelet interpolation Galerkin method. Section 5 calculates the frequency responses and the quality factors using the present method, and compares the calculated results with those generated by experiment [26, 27], by the finite difference method, and by other published analytical models [15, 26]. Finally, a conclusion is given in Section 6.

2. Basic Concepts of Daubechies’ Wavelets and Wavelet Interpolation

In this section, we shall give a brief introduction to the concepts and properties of Daubechies’ wavelets. More detailed discussions can be found in [1618, 21].

2.1. Daubechies’ Orthonormal Wavelets

Daubechies [16, 17] constructed a family of orthonomal bases of compactly supported wavelets for the space of square-integrable funcntions, . Due to the fact that they possess several useful properties, such as orthogonality, compact support, exact representation of polynomials to a certain degree, and ability to represent functions at different levels of resolution, Daubechies’ wavelets have gained great interest in the numerical solutions of PDEs [1822].

Daubechies’ functions are easy to construct [16, 17]. For an even integer , we have the Daubechies’ scaling function and wavelet satisfying The fundamental support of the scaling function is in the interval while that of the corresponding wavelet is in the interval . The parameter will be referred to as the degree of the scaling function . The coefficients are called the wavelet filter coefficients. Daubenchies [16, 17] established these wavelet filter coefficients to satisfy the following conditions: where is the Kronecker delta function. Correspondingly, the constructed scanling function and wavelet have the following properties:

Denote by the space of square-integrable functions on the real line. Let and be the subspace generated, respectively, as the -closure of the linear spans of and , . denotes the set of integers. Then (2.3) implies that Equation (2.4) presents the multiresolution properties of wavlets. Any function , may be approximated by the multiresolution apparatus described above, by its projection onto the subspace

2.2. Wavelet Interpolation Scaling Function

For a given Daubechies’ scaling function, its autocorrelation function can be defined as follows [21]: The function satisfies the following interpolating property and has a symmetric support . The derivative of the function may be computed by differentiating the convolution product Let act as the scaling function, we have For a set of dyadic grids of the type : , where , the verifies the interpolation property at the dyadic points: . Let be the linear span of the set . It can be proved that forms a multiresolution analysis, where acts as the role of scaling function (the so-called interpolation scaling function), and the set is a Riesz’s basis for . For a function , an interpolation operator can be defined [21]: where . Thus, for a function defined on , has the following approximation

In this paper, wavelet collocation scheme is applied on , where and . Therefore, instead of the values of at , and , we may use some values which are extrapolated from the values in those dyadic points internal to the interval . As described in [21, 22], we define where where the coefficients and are defined by where and represent Lagrange interpolation polynomials, defined by

An analogous manner can be given for two-dimensional problem. By using tensor products, it is then possible to define a multiresolution on the square . The two-dimensional scaling function is defined by . Let be the linear span of the set ; thus the set forms a multiresolution analysis and the set is a Riesz basis for . Therefore, for a function defined on , it has the following approximation:

3. Wavelet Interpolation Galerkin Method for a Parallel Plate Microresonator under the Effect of Squeeze Film Damping

3.1. Governing Equations

In this section, we examine the example of a rectangular parallel plate under the effect of squeeze film damping. As shown in Figure 1, the rectangular parallel plate is excited by a conventional voltage. The voltage is composed of a dc component and a small ac component , . The plate is rigid. The displacement of the plate under the electric force is composed of a static component to the dc voltage, denoted by , and a small dynamic component due to the ac voltage, denoted by , , that is, The equation of motion that governs the displacement of the plate is written as where is the mass of the plate, is the are of the plate, is the stiffness of the spring, is the zero-voltage air gap spacing, is the dielectric constant of the gap medium, is the force acting on the plate owing to the pressure of the squeeze gas film between the plate and the substrate.

fig1
Figure 1: A schematic drawing of an electrically actuated microplate under the effect of squeeze film damping.

We expand (3.2) in a Taylor series around and up to first order and rewrite (3.2) as where , . The force acting on the plate owing to the pressure of the squeeze gas film is given by where and are the length and width of plate, is the absolute pressure in the gap and is the ambient pressure. The pressure is governed by the nonlinear Reynolds equation [3] where and is the effective viscosity of the fluid in the gap. In this section, all edges of the rectangular plate are ideally vented; thus the pressure boundary conditions for the case in Figure 1 are

For convenience, we introduce the nondimensional variables where is a timescale, , is the nature frequency of the plate. Substituting (3.7) into (3.3)–(3.6), we obtain where , , , and . The nondimensional boundary conditions are

As mentioned above, the microplate is under small oscillation around and therefore the pressure variation from ambient in the squeeze film is also small, is given by where . Substituting (3.11) into (3.9), and linearizing the outcome around and , we obtain The boundary conditions for the case are

For a harmonic excitation, the ac component voltage is given by Usually, the excitation frequency is approximate to the natural frequency . The steady-state solution of (3.8) and (3.12) may be expressed by where ,, , and . The nondimensional boundary conditions are

As mentioned above, the microplate is under small torsion oscillation around and therefore the pressure variation from ambient in the squeeze film is also small, is given by where . Substituting (4.7) into (4.5), and linearizing the outcome around and , we obtain The boundary conditions for the case are

For a harmonic excitation, the ac component voltage is given by Correspondingly, the steady-state solution of (4.4) and (4.8) may be expressed by where is the complex amplitude to be determined. Substituting (4.11) into (4.8), we obtain The boundary conditions are

4.2. Wavelet Interpolation Method for Squeeze Film Damping Equations

In this section, the approximate solution of can be approximated by (3.21). The weak form functional of (4.12) is From the necessary conditions for the determination of the minimum , we obtain Substituting (3.21) into (4.15), leads to This is a linear system where is an unknown coefficients’matrix, is a matrix, and is a matrix. The entries in are of the form

4.2.1. Squeeze Film Damping of the Torsion Plate

The numerical solution of (4.17) can be written as The elements of can be expressed as where and are the real and imaginary parts of , respectively. Using (4.20) and (4.4), the force acting on the plate owing to the pressure of the squeeze gas film can be rewritten as where and are the spring and damping components of the force Substituting (4.21) into (4.4), leads to The quality factor and the damped natural frequency are given in(3.32)

5. Comparisons with Experiments

Veijola et al. [26] conducted experiments to measure the frequency response of an accelerometer under the effect of squeeze film damping. Minikes et al. [27] measured the quality factors of two torsion rectangular mirrors at low pressure. In this section, the experimental results presented by Veijola et al. [26] and Minikes et al. [27] were used to verify the wavelet interpolation Galerkin method.

5.1. Comparisons with the Experimental Results of Veijola et al. [26]

In [26], Veijola et al. simulated the frequency response of an accelerometer with a spring-mass- damper model with a parallel-plate electrostatic force. The damping coefficient was estimated by the Blech model [28]. The spring constants and the gas pressures were estimated by curve fitting the experimental measurements. They compared their simulations with experimental data and found good agreement. The parameters for the accelerometer are listed in Table 1.

tab1
Table 1: The parameters of the accelerometer presented by Veijola et al. [26].

In this subsection, we use the wavelet interpolation Galerkin method to predict the frequency response of the accelerometer. Various numerical tests have been conducted by changing the degree of the Daubechies wavelet and the number of the scale. Better accuracy can be achieved by increasing and . The higher is, the smoother the scaling function becomes. The price for the high smoothness is that its supporting domain gets larger. The higher is, the more accurate the solution becomes. The number of differential equations and the CPU time increase significantly as increases. In this work, only the solutions for and are presented, as the results for higher resolutions are indistinguishable from the exact solution.

For comparison purpose, we give the frequency responses of the accelerometer calculated by Blech’s model [28] and the finite difference method, respectively. Blech [28] expanded the air film pressure into an assumed double sine series and derived an analytical expression for the spring and damping forces. In this subsection, the number of terms for the double sine series is taken as (), which shows good convergence. For the finite difference method, we use the following approximate formulae for a node () on the microplate: In this subsection, we assume that ===; thus the element size of the finite difference method is equal to the wavelet interpolation Galerkin method. Substituting (5.1) into (3.17), we obtain

Figure 3 shows the comparisons of the frequency response obtained by different methods. As expected, the wavelet interpolation Galerkin method, Blech’s model and the finite difference method give almost same results. The three results agree well with the experimental results [26] except for one data at resonance peak of the amplitude frequency response. The reason for this discrepancy is that the damping coefficient is slightly underestimated by the three methods, respectively. Table 2 shows the Comparison of the damping obtained by different methods. In the experiment [26], the squeeze film damping is dominant. Obviously, the accuracy of the finite difference method is less than the wavelet interpolation Galerkin method and Blech’s model. The wavelet interpolation Galerkin method and the Blech model give almost identical results. Figure 4 shows the real part and the imaginary part of the air film pressure calculated by the wavelet interpolation Galerkin method.

tab2
Table 2: A Comparison of the damping ratios obtained by different methods with the experimental data [26].
fig3
Figure 2: Comparisons of the frequency response obtained by the wavelet Galerkin method, the Blech model and the experimental data of Veijola et al. [26].
fig4
Figure 3: The air film pressure distribution calculated by the wavelet interpolation Galerkin method.
5.2. Comparisons with the Experimental Results of Minikes et al. [27]

Minikes et al. [27] measured the quality factors of two rectangular torsion mirrors at low pressure and plotted the curves of the quality factors as a function of air pressure in the range from 10−2 torr to 102 torr. The structure of the two torsion mirrors is identical with the structure shown in Figure 2. The two torsion mirrors have similar dimensions in terms of surface area and inertial moment, but have different gaps between the mirror and the actuation electrodes. The parameters for the two torsion mirrors are listed in Table 3.

tab3
Table 3: The parameters of two torsion mirrors [27].

In Table 3, the values of the two torsional natural frequencies are measured under the dc bias voltage. Based on the two torsional natural frequencies and two moments of inertia, we determined the torsional stiffness . The extracted torsional stiffness for mirror 1 and mirror 2 are 2.46110−6 and 2.36210−6 Nm/rad, respectively. Now we use the wavelet interpolation Galerkin method to predict the quality factors of the two torsion mirrors. Various numerical tests have been conducted by changing the degree of the Daubechies wavelet and the number of the scale. Only the solutions for and are presented, as the results for higher resolutions are indistinguishable from the exact solution.

For comparison purpose, we give the quality factors calculated by Pan’s model [15] and the finite difference method, respectively. Pan et al. [15] expanded the air film pressure into an assumed double sine series and derived an analytical expression for the spring and damping torques. In this work, the number of terms for the double sine series is taken as (), which shows good convergence. For the finite difference method, we use (5.1) and (4.12) for a node () on the mirror; thus obtain In this subsection, we assume that == =; thus the finite difference method yields the same grids as the wavelet interpolation Galerkin method.

Tables 4 and 5 show the comparisons of quality factors obtained by the wavelet interpolation Galerkin method, the finite difference method and Pan’s model for the mirror 1 and 2, respectively. As shown in Tables 4 and 5, the qualify factors obtained by the wavelet interpolation Galerkin method are almost 1%~4% higher than Pan’s model. However the qualify factors obtained by the finite difference method are almost 10%~15% higher than Pan’s model. The result of the wavelet interpolation Galerkin method matches the result of Pan’s model better than that of the finite difference method.

tab4
Table 4: A comparison of quality factors obtained by Pan’s model, the wavelet interpolation Galerkin method and the finite difference method for mirror 1.
tab5
Table 5: A comparison of quality factors obtained by Pan’s model, the wavelet interpolation Galerkin method and the finite difference method for mirror 2.

Figure 5 shows the comparisons of quality factors obtained by different methods. As expected, the wavelet interpolation Galerkin method, the finite difference method and Pan’s model give almost same results. Above = 10 torr, the viscous damping is dominant, the three methods give results in good agreement with the experimental results [27]. Below = 10 torr, the accuracy of the three methods decreases as the pressure decreases. The main reason for this trend are as follows. The three methods are based on Reynolds equation, which is derived from the Navier-Stokes equations and the continuity equation. The main assumption is that the gas in the gap can be treated as a continuum. Below = 10 torr, the Knudsen number , the gas in the gap cannot be treated as a continuum. Thus the three methods fail to give a good prediction. Below = 0.1 torr, the influence of squeeze film damping begins to vanish and the qualify factors reaches a plateau that is dominated by the intrinsic damping.

fig5
Figure 4: Comparison of the quality factors obtained by the wavelet Galerkin method, the Pan model and the experimental data of Minikes et al. [27].

Tables 6 and 7 list the comparisons of quality factors between p = 10 and 760 torr for mirrors 1 and 2, respectively. Obviously, the accuracy of the finite difference method is less than the wavelet interpolation Galerkin method and Pan’s model. Figure 6 shows the real part and the imaginary part of the air film pressure calculated by the wavelet interpolation Galerkin method at = 1 torr. The air film pressure looks similar to the results calculated by Pan’s model.

tab6
Table 6: A comparison of quality factors between = 10 and 760 torr for mirror 1.
tab7
Table 7: A comparison of quality factors between = 10 and 760 torr for mirror 2.
fig6
Figure 5: The air film pressure distribution calculated by the wavelet interpolation Galerkin method at = 1 torr.

6. Summary and Conclusions

A new wavelet interpolation Galerkin method has been developed for the numerical simulation of MEMS devices under the effect of squeeze film damping. The air film pressure are expressed as a linear combination of a class of interpolating functions generated by autocorrelation of the usual compactly supported Daubechies scaling functions. To the best of our knowledge, this is the first time that wavelets have been used as basis functions for solving the PDEs of MEMS devices. As opposed to the previous wavelet-based methods that are all limited in one energy domain, the MEMS devices in the paper involve two coupled energy domains. Two typical electrically actuated micro devices with squeeze film damping effect are examined respectively to illustrate the wavelet interpolation Galerkin method. The method is validated by comparing its results with available theoretical and experimental results. The accuracy of the method is higher than the finite difference method.

In this paper, the wavelet interpolation Galerkin method is not suitable to solve problems defined on nonrectangular domains, since higher-dimensional wavelets are constructed by employing the tensor product of the one-dimensional wavelets and so their application is restricted to rectangular domains. In this paper, both trial and weight functions are a class of interpolating functions generated by autocorrelation of the first-generation wavelets. Future area of research is based on the second-generation wavelets [29]. The main advantage of the second-generation wavelets is that the wavelets are constructed in the spatial domain and can be custom designed for complex domains. This work is currently under way.

Acknowledgments

This work was supported by the National Natural Science Foundation of China (Grants No. 60806036 and 50675034), the Natural Science Foundation of Jiangsu Province (Grant No. BK2009286), and Natural Science Research Plan of the universities in Jiangsu Province (Grant No 08KJB510014).

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