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

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 [1–15]. 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 [16–25]. 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 [23–25]. 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 [16–18, 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, 𝐿²(𝑅). 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 [18–22].

Daubechies’ functions are easy to construct [16, 17]. For an even integer 𝐿, we have the Daubechies’ scaling function 𝜙(𝑥) and wavelet 𝜓(𝑥) satisfying𝜙(𝑥)=^𝐿−1_𝑖=0𝑝_𝑖𝜙(2𝑥−𝑖)𝜓(𝑥)=¹_{𝑖=2−𝐿}(−1)^𝑖𝑝_1−𝑖𝜙(2𝑥−𝑖). (2.1) The fundamental support of the scaling function 𝜙(𝑥) is in the interval [0,𝐿−1] while that of the corresponding wavelet 𝜓(𝑥) is in the interval [1−𝐿/2,𝐿/2]. The parameter 𝐿 will be referred to as the degree of the scaling function 𝜙(𝑥). The coefficients 𝑝_𝑖 are called the wavelet filter coefficients. Daubenchies [16, 17] established these wavelet filter coefficients to satisfy the following conditions:^𝐿−1_𝑖=0𝑝_𝑖=2,_{𝐿−1𝑖=0}𝑝_𝑖𝑝_𝑖−𝑚=𝛿_0,𝑚,_{1𝑖=2−𝐿}(−1)^𝑖𝑝_1−𝑖𝑝_𝑖−2𝑚=0forinteger,_{𝐿−1𝑖=0}(−1)^𝑖𝑖^𝑚𝑝_𝑖𝐿=0,𝑚=0,1,…,2−1, (2.2) where 𝛿_0,𝑚 is the Kronecker delta function. Correspondingly, the constructed scanling function 𝜙(𝑥) and wavelet 𝜓(𝑥) have the following properties:^∞_−∞𝜙(𝑥)d𝑥=1,_∞−∞𝜙(𝑥−𝑖)𝜙(𝑥−𝑚)d𝑥=𝛿_𝑖,𝑚forintegers𝑖,𝑚_∞−∞,𝜙(𝑥)𝜓(𝑥−𝑚)d𝑥=0forinteger𝑚,_∞−∞𝑥^𝑚𝐿𝜓(𝑥)d𝑥=0,𝑚=0,1,…,2−1. (2.3)

Denote by 𝐿²(𝑅) the space of square-integrable functions on the real line. Let 𝑉_𝐽 and 𝑊_𝐽 be the subspace generated, respectively, as the 𝐿²-closure of the linear spans of 𝜙_𝐽,𝑖(𝑥)=2^𝐽/2𝜙(2^𝐽𝑥−𝑖) and 𝜓_𝐽,𝑖(𝑥)=2^𝐽/2𝜓(2^𝐽𝑥−𝑖), 𝐽,𝑖∈𝑍. ℤ denotes the set of integers. Then (2.3) implies that 𝑉_𝐽+1=𝑉_𝐽⊕𝑊_𝐽,𝑉₀⊂𝑉₁⊂⋯𝑉_𝐽⊂𝑉_𝐽+1,𝑉_𝐽+1=𝑉₀⊕𝑊₀⊕𝑊₁⊕⋯𝑊_𝐽, (2.4) Equation (2.4) presents the multiresolution properties of wavlets. Any function 𝑓∈𝐿²(𝑅), may be approximated by the multiresolution apparatus described above, by its projection 𝑃^𝑉_𝐽𝑓 onto the subspace𝑉_𝐽𝑃^𝑉_𝐽𝑓=_𝑖∈𝑍𝑓_𝐽,𝑖𝜙_𝐽,𝑖(𝑥). (2.5)

2.2. Wavelet Interpolation Scaling Function

For a given Daubechies’ scaling function, its autocorrelation function 𝜃(𝑥) can be defined as follows [21]: 𝜃(𝑥)=^∞_−∞𝜙(𝜏)𝜙(𝜏−𝑥)d𝜏. (2.6) The function satisfies the following interpolating property𝜃(𝑘)=𝛿_0,𝑘,𝑘∈𝑍, (2.7) and has a symmetric support [−(𝐿−1),(𝐿−1)]. The derivative of the function 𝜃(𝑘) may be computed by differentiating the convolution product𝜃^(𝑠)(𝑘)=(−1)^(𝑠)∞_−∞𝜙(𝜏)𝜙^(𝑠)(𝜏−𝑘)d𝜏. (2.8) Let 𝜃(𝑥) act as the scaling function, we have𝜃_𝐽,𝑘(𝑥)=𝜃(2^𝐽𝑥−𝑘),𝑘∈𝑍. (2.9) For a set of dyadic grids of the type 𝑥^𝐽_𝑘∈𝑅: 𝑥^𝐽_𝑘=2^−𝐽𝑘, where 𝑘,𝐽∈𝑍, the 𝜃_𝐽,𝑘(𝑥) verifies the interpolation property at the dyadic points: 𝜃_𝐽,𝑘(𝑥^𝐽_𝑛)=𝛿_𝑛,𝑘. Let 𝑉^𝑥_𝐽 be the linear span of the set {𝜃(2^𝐽𝑥−𝑘),𝑘∈𝑍}. It can be proved that {𝑉^𝑥_𝐽} forms a multiresolution analysis, where 𝜃_𝐽,𝑘(𝑥) acts as the role of scaling function (the so-called interpolation scaling function), and the set {𝜃(2^𝐽𝑥−𝑘),𝑘∈𝑍} is a Riesz’s basis for 𝑉^𝑥_𝐽. For a function 𝑓∈𝐻¹(𝑅), an interpolation operator 𝐼_𝐽∶𝐻¹(𝑅)→𝑉^𝑥_𝐽 can be defined [21]: 𝐼_𝐽(𝑓)=_𝑘𝑓^𝐽_𝑘𝜃(2^𝐽𝑥−𝑘),𝑘∈𝑍, (2.10) where 𝑓^𝐽_𝑘=𝑓(𝑥^𝐽_𝑘)=𝑓(2^−𝐽𝑘). Thus, for a function 𝑓(𝑥) defined on 𝑥∈[0,1], 𝑓(𝑥) has the following approximation𝑓(𝑥)=^{2𝐽+(𝐿−1)}_{𝑘=−(𝐿−1)}𝑓_𝐽,𝑘𝜃(2^𝐽𝑥−𝑘)=¹_{𝑘=−(𝐿−1)}𝑓_𝐽,𝑘𝜃(2^𝐽𝑥−𝑘)+^2𝐽_𝑘=0𝑓_𝐽,𝑘𝜃(2^𝐽𝑥−𝑘)+^{2𝐽+𝐿−1}_𝑘=2𝐽+1𝑓_𝐽,𝑘𝜃(2^𝐽𝑥−𝑘). (2.11)

In this paper, wavelet collocation scheme is applied on 𝑥∈[0,1], where 𝑥^𝐽_𝑘=2^−𝐽𝑘 and 𝑘=0,1,…,2^𝐽. Therefore, instead of the values of 𝑓(𝑥) at 𝑥^𝐽_𝑘, 𝑘=−(𝐿−1),…,−1 and 𝑘=2^𝐽+1,…,(𝐿−1), we may use some values which are extrapolated from the values in those dyadic points internal to the interval 𝑥∈[0,1]. As described in [21, 22], we define𝑓(𝑥)=^2𝐽_𝑘=0𝑓_𝐽,𝑘𝜃(2^𝐽𝑥−𝑘), (2.12) where𝜃(2^𝐽𝑥−𝑘)=𝜃(2^𝐽𝑥−𝑘)+⁻¹_{𝑛=−(𝐿−1)}𝑎_𝑛𝑘𝜃(2^𝐽𝑥−𝑛),𝑘=0,1,…,2𝑀−1𝜃(2^𝐽𝑥−𝑘),𝑘=2𝑀,…,2^𝐽−2𝑀𝜃(2^𝐽𝑥−𝑘)+^{2𝐽+𝐿−1}_𝑛=2𝐽+1𝑏_𝑛𝑘𝜃(2^𝐽𝑥−𝑛),𝑘=2^𝐽−2𝑀+1,…,2^𝐽, (2.13) where the coefficients 𝑎_𝑛𝑘 and 𝑏_𝑛𝑘 are defined by𝑎_𝑛𝑘=𝑙¹_𝑘(𝑥^𝐽_𝑛),𝑏_𝑛𝑘=𝑙²_𝑘(𝑥^𝐽_𝑛), (2.14) where 𝑙¹_𝑘(𝑥) and 𝑙²_𝑘(𝑥) represent Lagrange interpolation polynomials, defined by𝑙¹_𝑘(𝑥)=^2𝑀−1_{𝑖=0𝑖≠𝑘}𝑥−𝑥^𝐽_𝑖𝑥^𝐽_𝑘−𝑥^𝐽_𝑖,𝑙²_𝑘(𝑥)=^2𝐽_{𝑖=2𝐽−2𝑀+1𝑖≠𝑘}𝑥−𝑥^𝐽_𝑖𝑥^𝐽_𝑘−𝑥^𝐽_𝑖. (2.15)

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 𝑥,𝑦∈[0,1]. The two-dimensional scaling function is defined by Θ^𝐽_𝑘,𝑘′(𝑥,𝑦)=∑^2𝐽_𝑘=0∑^2𝐽_𝑘′=0𝑓^𝐽_𝑘,𝑘′𝜃(2^𝐽𝑥−𝑘)𝜃(2^𝐽𝑦−𝑘^′). Let 𝑉_𝐽^𝑥𝑦=𝑉^𝑥_𝐽⊗𝑉^𝑦_𝐽 be the linear span of the set {𝜃(2^𝐽𝑥−𝑘)𝜃(2^𝐽𝑦−𝑘^′),𝐽,𝑘,𝑘^′∈𝑍}; thus the set {𝑉_𝐽^𝑥𝑦} forms a multiresolution analysis and the set {𝜃(2^𝐽𝑥−𝑘)𝜃(2^𝐽𝑦−𝑘^′),𝑘,𝑘^′∈𝑍} is a Riesz basis for {𝑉_𝐽^𝑥𝑦}. Therefore, for a function 𝑓(𝑥,𝑦) defined on 𝑥,𝑦∈[0,1], it has the following approximation:𝑓(𝑥,𝑦)=^2𝐽_𝑘=0^2𝐽_𝑘′=0𝑓^𝐽_𝑘,𝑘′𝜃(2^𝐽𝑥−𝑘)𝜃(2^𝐽𝑦−𝑘^′). (2.16)

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 𝑉₀ and a small ac component 𝑣(𝑡), 𝑉₀≫𝑣(𝑡). 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 𝑧₀, and a small dynamic component due to the ac voltage, denoted by 𝑧(𝑡), 𝑧₀≫𝑧(𝑡), that is, 𝑧_𝐸(𝑡)=𝑧₀+𝑧(𝑡). (3.1) The equation of motion that governs the displacement of the plate is written as𝑚_plate𝑧̈_𝐸+𝑘_spring𝑧_𝐸=𝜀𝐴_plate(𝑉₀+𝑣)²2(𝑔₀−𝑧_𝐸)²−𝑓(𝑡), (3.2) where 𝑚_plate is the mass of the plate, 𝐴_plate is the are of the plate, 𝑘_spring is the stiffness of the spring, 𝑔₀ is the zero-voltage air gap spacing, 𝜀 is the dielectric constant of the gap medium, 𝑓(𝑡) is the force acting on the plate owing to the pressure of the squeeze gas film between the plate and the substrate.

(a) Side view of the microplate

(b) Top view of the microplate

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 𝑉₀ and 𝑧₀ up to first order and rewrite (3.2) as𝑚_plate𝑧̈+𝑘_𝐸𝑧=𝜀𝐴_plate𝑉₀𝑔²₀𝑣−𝑓(𝑡), (3.3) where 𝑘_E=𝑘_spring−(𝜀𝐴_plate𝑉²₀/(𝑔₀−𝑧₀)³), 𝑔₀=𝑔₀−𝑧₀. The force 𝑓(𝑡) acting on the plate owing to the pressure of the squeeze gas film is given by 𝑓(𝑡)=^𝑙𝑦₀^𝑙𝑥₀(𝑝(𝑥,𝑦,𝑡)−𝑝₀)d𝑥d𝑦, (3.4) where 𝑙_𝑥 and 𝑙_𝑦 are the length and width of plate, 𝑝(𝑥,𝑦,𝑡) is the absolute pressure in the gap and 𝑝₀ is the ambient pressure. The pressure 𝑝(𝑥,𝑦,𝑡) is governed by the nonlinear Reynolds equation [3]𝜕𝜕𝑥(ℎ³𝑝𝜕𝑝𝜕𝜕𝑥)+𝜕𝑦(ℎ³𝑝𝜕𝑝𝜕𝑦)=12𝜂_eff(ℎ𝜕𝑝𝜕𝑡+𝑝𝜕ℎ𝜕𝑡), (3.5) where ℎ(𝑥,𝑡)=𝑔₀−𝑧₀−𝑧(𝑡)=𝑔₀−𝑧(𝑡) and 𝜂_eff 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 are𝑝(𝑥,0,𝑡)=𝑝(𝑥,𝑙_𝑦,𝑡)=𝑝(0,𝑦,𝑡)=𝑝(𝑙_𝑥,𝑦,𝑡)=𝑝₀. (3.6)

For convenience, we introduce the nondimensional variables𝑥𝑋=𝑙_𝑥𝑦,𝑌=𝑙_𝑦𝑧,𝑍=𝑔₀,𝑝𝑃=𝑝₀𝑡,𝑇=𝑆ℎ,𝐻=𝑔₀=1−𝑍, (3.7) where 𝑇 is a timescale, 𝑆=𝑚_plate/𝑘_E=1/𝜔_𝑛, 𝜔_𝑛 is the nature frequency of the plate. Substituting (3.7) into (3.3)–(3.6), we obtaind²𝑍d𝑇²+𝑍=𝛼𝑉₀𝑣−𝑃_non¹₀(𝑃−1)d𝑋d𝑌, (3.8)𝜕𝜕𝑋(𝐻³𝑃𝜕𝑃𝜕𝑋)+𝛽²𝜕𝜕𝑌(𝐻³𝑃𝜕𝑃𝜎𝜕𝑌)=𝑆𝜕𝑃(𝐻+𝑃𝜕𝑇𝜕𝐻𝜕𝑇), (3.9) where 𝛼=𝜀𝐴_plate/𝑘_𝐸𝑔³₀, 𝑃_non=𝑝₀𝑙_𝑥𝑙_𝑦/𝑘_𝐸𝑔₀, 𝜎=12𝜂_eff𝑙²_𝑥/𝑔²₀𝑝₀, and 𝛽=𝑙_𝑥/𝑙_𝑦. The nondimensional boundary conditions are𝑃(𝑋,0,𝑇)=𝑃(𝑋,1,𝑇)=𝑃(1,𝑌,𝑇)=𝑃(0,𝑌,𝑇)=1. (3.10)

As mentioned above, the microplate is under small oscillation around 𝑔₀ and therefore the pressure variation from ambient in the squeeze film is also small, 𝑃(𝑋,𝑌,𝑇) is given by𝑝𝑃(𝑋,𝑌,𝑇)=𝑝₀=1+𝑃(𝑋,𝑌,𝑇), (3.11) where |𝑃(𝑋,𝑌,𝑇)|≪1. Substituting (3.11) into (3.9), and linearizing the outcome around 𝑝₀ and 𝑔₀, we obtain 𝜕²𝑃𝜕𝑋²+𝛽²𝜕²𝑃𝜕𝑌²−𝜎𝑆𝜕𝑃𝜎𝜕𝑇=−𝑆𝜕𝑍𝜕𝑇. (3.12) The boundary conditions for the case are𝑃(𝑋,0,𝑇)=𝑃(𝑋,1,𝑇)=𝑃(0,𝑌,𝑇)=𝑃(1,𝑌,𝑇)=0. (3.13)

For a harmonic excitation, the ac component voltage v(t) is given by𝑣(𝑇)=𝑣₀𝑒^{𝑗𝜔𝑇𝑆}. (3.14) Usually, the excitation frequency 𝜔 is approximate to the natural frequency 𝜔_𝑛. The steady-state solution of (3.8) and (3.12) may be expressed by𝑍(𝑇)=𝐴𝑒^{𝑗𝜔𝑇𝑆},𝑃(𝑋,𝑌,𝑇)=𝐴⋅𝑃_𝐴(𝑋,𝑌)𝑒^{𝑗𝜔𝑇𝑆}, (3.15)(3.16) where 𝛼=(𝑥²₂−𝑥²₁)𝜀𝑙_𝑦/𝑘_𝑇−𝐸𝑔²₀𝛾_max,𝑃_non=𝑝₀𝑙²_𝑥𝑙_𝑦/𝑘_𝑇−𝑒𝛾_max, 𝜎=12𝜂𝑙²_𝑥/𝑔²₀𝑝₀, and 𝛽=𝑙_𝑥/𝑙_𝑦. The nondimensional boundary conditions are𝑃(𝑋,0,𝑇)=𝑃(𝑋,1,𝑇)=𝑃(0,𝑌,𝑇)=𝑃(1,𝑌,𝑇)=1. (4.6)

As mentioned above, the microplate is under small torsion oscillation around 𝛾₀=0 and therefore the pressure variation from ambient in the squeeze film is also small, 𝑃(𝑋,𝑌,𝑇) is given by 𝑝𝑃(𝑋,𝑌,𝑇)=𝑝₀=1+𝑃(𝑋,𝑌,𝑇), (4.7) where |𝑃(𝑋,𝑌,𝑇)|≪1. Substituting (4.7) into (4.5), and linearizing the outcome around 𝑝₀ and 𝛾₀, we obtain 𝜕²𝑃𝜕𝑋²+𝛽²𝜕²𝑃𝜕𝑌²−𝜎𝑆𝜕𝑃=𝜕𝑇2𝜎𝑆1(𝑋−2)𝜕𝜗𝜕𝑇. (4.8) The boundary conditions for the case are𝑃(𝑋,0,𝑇)=𝑃(𝑋,1,𝑇)=𝑃(0,𝑌,𝑇)=𝑃(1,𝑌,𝑇)=0. (4.9)

For a harmonic excitation, the ac component voltage 𝑣(𝑇) is given by𝑣(𝑇)=𝑣₀𝑒^{𝑗𝜔𝑇𝑆}. (4.10) Correspondingly, the steady-state solution of (4.4) and (4.8) may be expressed by𝜗(𝑇)=𝐴𝑒^{𝑗𝜔𝑇𝑆},𝑃(𝑋,𝑌,𝑇)=𝐴⋅𝑃_𝐴(𝑋,𝑌)𝑒^{𝑗𝜔𝑇𝑆}, (4.11) where 𝐴 is the complex amplitude to be determined. Substituting (4.11) into (4.8), we obtain 𝜕²𝑃_𝐴(𝑋,𝑌)𝜕𝑋²+𝛽²𝜕²𝑃_𝐴(𝑋,𝑌)𝜕𝑌²−𝑗𝜎𝜔𝑃_𝐴1(𝑋,𝑌)=𝑗2𝜎𝜔(𝑋−2). (4.12) The boundary conditions are𝑃_𝐴(𝑋,0)=𝑃_𝐴(𝑋,1)=𝑃_𝐴(0,𝑌)=𝑃_𝐴(1,𝑌)=0. (4.13)

4.2. Wavelet Interpolation Method for Squeeze Film Damping Equations

In this section, the approximate solution of 𝑃_𝐴(𝑋,𝑌) can be approximated by (3.21). The weak form functional of (4.12) is𝑊(𝑃_𝐴)=_Ω{12[(𝜕𝑃_𝐴)𝜕𝑋²+𝛽²(𝜕𝑃_𝐴)𝜕𝑌²+𝑗𝜎𝜔𝑃²_𝐴1]+𝑗2𝜎𝜔(𝑋−2)𝑃_𝐴}d𝑋d𝑌. (4.14) From the necessary conditions for the determination of the minimum 𝑊, we obtain 𝛿𝑊(𝑃_𝐴)=_Ω[𝜕𝛿𝑃_𝐴𝜕𝑋𝜕𝑃_𝐴𝜕𝑋+𝛽²𝜕𝛿𝑃_𝐴𝜕𝑌𝜕𝑃_𝐴𝜕𝑌+𝑗𝜎𝜔𝛿𝑃_𝐴𝑃_𝐴1+𝑗2𝜎𝜔(𝑋−2)𝛿𝑃_𝐴]d𝑋d𝑌=0. (4.15) Substituting (3.21) into (4.15), leads to^2𝐽−1_𝑘=1^2𝐽_𝑘⁻¹_′=1{_Ω[𝜕Θ^𝐽_𝑚,𝑛𝜕𝑋𝜕Θ^𝐽_𝑘,𝑘′𝜕𝑋+𝛽²𝜕Θ^𝐽_𝑚,𝑛𝜕𝑌𝜕Θ^𝐽_𝑘,𝑘′𝜕𝑌+𝑗𝜎𝜔Θ^𝐽_𝑚,𝑛Θ^𝐽_𝑘,𝑘′]d𝑋d𝑌}𝑝^𝐽_𝑘,𝑘′=−𝑗2𝜎𝜔_Ω1(𝑋−2)Θ^𝐽_𝑚,𝑛d𝑋d𝑌,for𝑚,𝑛=1,2,…,(2^𝐽−1). (4.16) This is a (2^𝐽−1)²×(2^𝐽−1)² linear systemΘ𝐩=−𝑗𝜎𝜔𝐄, (4.17) where 𝐩=[𝑝^𝐽_1,1𝑝^𝐽_1,2⋯𝑝^𝐽_1,2𝐽−1𝑝^𝐽_2,1𝑝^𝐽_2,2⋯𝑝^𝐽_{2𝐽−1,2𝐽−1}]^𝑇 is an (2^𝐽−1)²×1unknown coefficients’matrix, 𝐄=2[∬_Ω(𝑋−1/2)Θ^𝐽_1,1d𝑋d𝑌∬_Ω(𝑋−1/2)Θ^𝐽_1,2d𝑋d𝑌⋯∬_Ω(𝑋−(1/2))Θ^𝐽_{2𝐽−1,2𝐽−1}d𝑋d𝑌]^𝑇is a (2^𝐽−1)²×1 matrix, and Θ is a (2^𝐽−1)²×(2^𝐽−1)² matrix. The entries in Θ are of the formΘ(𝑖,𝑗)=Θ([(𝑚−1)(2^𝐽−1)+𝑛],[(𝑘−1)(2^𝐽−1)+𝑘^′])=_Ω[𝜕𝜃(2^𝐽𝑋−𝑚)𝜕𝑋𝜃(2^𝐽𝜕𝑌−𝑛)𝜃(2^𝐽𝑋−𝑘)𝜕𝑋𝜃(2^𝐽𝑌−𝑘^′)+𝛽²𝜃(2^𝐽𝜕𝑋−𝑚)𝜃(2^𝐽𝑌−𝑛)𝜕𝑌𝜃(2^𝐽𝜕𝑋−𝑘)𝜃(2^𝐽𝑌−𝑘^′)𝜕𝑌+𝑗𝜎𝜔𝜃(2^𝐽𝑋−𝑚)𝜃(2^𝐽𝑌−𝑛)𝜃(2^𝐽𝑋−𝑘)𝜃(2^𝐽𝑌−𝑘^′)]d𝑋d𝑌. (4.18)

4.2.1. Squeeze Film Damping of the Torsion Plate

The numerical solution of (4.17) can be written as𝐩=−𝑗𝜎𝜔Θ⁻¹𝐄. (4.19) The elements of p can be expressed as𝑝^𝐽_𝑘,𝑘′=−𝑝^𝐽,𝑅_𝑘,𝑘′−𝑗𝑝^𝐽,𝐼_𝑘,𝑘′for𝑚,𝑛=1,2,…,(2^𝐽−1), (4.20) where 𝑝^𝐽,𝑅_𝑘,𝑘′ and 𝑝^𝐽,𝐼_𝑘,𝑘′ are the real and imaginary parts of 𝑝^𝐽_𝑘,𝑘′, 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 as𝑃_non¹₀(1𝑃−1)(𝑋−2)d𝑋d𝑌=−𝐾_𝑎⋅𝜗(𝑇)−𝐶_𝑎⋅𝑑𝜗(𝑇)𝑑𝑇, (4.21) where 𝐾_𝑎⋅𝜗(𝑇) and C_a⋅(d𝜗(𝑇)/d𝑇) are the spring and damping components of the force𝐾_𝑎=𝑃_non^2𝐽−1_𝑘=1^2𝐽_𝑘⁻¹_′=1𝑝^𝐽,𝑅_𝑘,𝑘′¹₀𝜃(2^𝐽𝑋−𝑘)𝜃(2^𝐽𝑌−𝑘^′1)(𝑋−2𝐶)d𝑋d𝑌_𝑎=𝑃_non𝜔𝑆^{2𝐽−1𝑘=12𝐽𝑘−1′=1}𝑝^𝐽,𝐼_𝑘,𝑘′¹₀𝜃(2^𝐽𝑋−𝑘)𝜃(2^𝐽𝑌−𝑘^′1)(𝑋−2)d𝑋d𝑌 (4.22) Substituting (4.21) into (4.4), leads to𝑑²𝜗𝑑𝑇²+𝐶_𝑎⋅𝑑𝜗𝑑𝑇+(𝐾_𝑎+1)𝜗=𝛼𝑉₀𝑣(𝑇),𝜗(𝑇)=𝐴𝑒^{𝑗𝜔𝑇𝑆}=𝑎𝑉₀𝑣₀𝐾_𝑎⋅1+11−(𝜔²(𝑆²/(𝐾_𝑎+1)))+(𝑗𝜔(𝐶_𝑎𝑆/(𝐾_𝑎𝑒+1)))^{𝑗𝜔𝑇𝑆}. (4.23) 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.

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 𝐿 and the number 𝐽 of the scale. Better accuracy can be achieved by increasing 𝐿 and 𝐽. The higher 𝐿 is, the smoother the scaling function becomes. The price for the high smoothness is that its supporting domain gets larger. The higher 𝐽 is, the more accurate the solution becomes. The number of differential equations and the CPU time increase significantly as 𝐽 increases. In this work, only the solutions for 𝐿=6 and 𝐽=4 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 (5,5), which shows good convergence. For the finite difference method, we use the following approximate formulae for a node (𝑖,𝑗) on the microplate: 𝜕𝑃_𝐴|𝜕𝑋_𝑖,𝑗=𝑃_𝐴(𝑖+1,𝑗)−𝑃_𝐴(𝑖−1,𝑗)2Δ_𝑋,𝜕𝑃_𝐴|𝜕𝑌_𝑖,𝑗=𝑃_𝐴(𝑖,𝑗+1)−𝑃_𝐴(𝑖,𝑗−1)2Δ_𝑌,𝜕²𝑃_𝐴𝜕𝑋²|_𝑖,𝑗=𝑃_𝐴(𝑖+1,𝑗)−2𝑃_𝐴(𝑖,𝑗)+𝑃_𝐴(𝑖−1,𝑗)Δ²_𝑋,𝜕²𝑃_𝐴𝜕𝑌²|_𝑖,𝑗=𝑃_𝐴(𝑖,𝑗+1)−2𝑃_𝐴(𝑖,𝑗)+𝑃_𝐴(𝑖,𝑗−1)Δ²_𝑌. (5.1) In this subsection, we assume that Δ_𝑋=Δ_𝑌=1/2^𝐽=1/2⁶; thus the element size of the finite difference method is equal to the wavelet interpolation Galerkin method. Substituting (5.1) into (3.17), we obtain𝑃_𝐴(𝑖+1,𝑗)+𝑃_𝐴(𝑖−1,𝑗)+𝛽²𝑃_𝐴(𝑖,𝑗+1)+𝛽²𝑃_𝐴(𝑖,𝑗−1)Δ²_𝑋−(2+2𝛽²Δ²_𝑋+𝑗𝜎𝜔)𝑃_𝐴(𝑖,𝑗)=−𝑗𝜎𝜔. (5.2)

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.

Table 2 

A Comparison of the damping ratios obtained by different methods with the experimental data [26].

(a) Amplitude response

(b) Phase response

Figure 3 

Comparisons of the frequency response obtained by the wavelet Galerkin method, the Blech model and the experimental data of Veijola et al. [26].

(a) The real part of the pressure distribution

(b) The imaginary part of the pressure distribution

Figure 4 

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.

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 𝑘_𝑇−𝐸. The extracted torsional stiffness 𝑘_𝑇−𝐸 for mirror 1 and mirror 2 are 2.461×10⁻⁶ and 2.362×10⁻⁶ N⋅m/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 𝐿 and the number 𝐽 of the scale. Only the solutions for 𝐿=6 and 𝐽=4 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 (6,5), which shows good convergence. For the finite difference method, we use (5.1) and (4.12) for a node (𝑖,𝑗) on the mirror; thus obtain𝑃_𝐴(𝑖+1,𝑗)+𝑃_𝐴(𝑖−1,𝑗)+𝛽²𝑃_𝐴(𝑖,𝑗+1)+𝛽²𝑃_𝐴(𝑖,𝑗−1)Δ²_𝑋−(2+2𝛽²Δ²_𝑋+𝑗𝜎𝜔)𝑃_𝐴(𝑖,𝑗)=𝑗2𝜎𝜔(𝑋_𝑖−12). (5.3) In this subsection, we assume that Δ_𝑋=Δ_𝑌=1/2^𝐽 =1/2⁶; 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.

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.

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 𝑝₀ = 10 torr, the viscous damping is dominant, the three methods give results in good agreement with the experimental results [27]. Below 𝑝₀ = 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 𝑝₀ = 10 torr, the Knudsen number Kn>0.1, the gas in the gap cannot be treated as a continuum. Thus the three methods fail to give a good prediction. Below 𝑝₀ = 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.

(a) Mirror 1 (g0= 28 μm)

(b) Mirror 2 (g0= 13 μm)

Figure 5 

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 𝑝₀ = 1 torr. The air film pressure looks similar to the results calculated by Pan’s model.

Table 6 

A comparison of quality factors between 𝑝₀ = 10 and 760 torr for mirror 1.

Table 7 

A comparison of quality factors between 𝑝₀ = 10 and 760 torr for mirror 2.

(a) The real part of mirror 1

(b) The imaginary part of mirror 1

(c) The real part of mirror 2

(d) The imaginary part of mirror 2

Figure 6 

The air film pressure distribution calculated by the wavelet interpolation Galerkin method at $p_0$ = 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).