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%.
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 [
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 [
In the past two decades also, a new numerical concept was introduced and is gaining increasing popularity [
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
In this section, we shall give a brief introduction to the concepts and properties of Daubechies’ wavelets. More detailed discussions can be found in [
Daubechies [
Daubechies’ functions are easy to construct [
Denote by
For a given Daubechies’ scaling function, its autocorrelation function
In this paper, wavelet collocation scheme is applied on
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
In this section, we examine the example of a rectangular parallel plate under the effect of squeeze film damping. As shown in Figure
A schematic drawing of an electrically actuated microplate under the effect of squeeze film damping.
Side view of the microplate
Top view of the microplate
We expand (
For convenience, we introduce the nondimensional variables
As mentioned above, the microplate is under small oscillation around
For a harmonic excitation, the ac component voltage
In this subsection, the approximate solution of
For the application of Galerkin method, (
The weak form functional of (
The numerical solution of (
A similar analysis as the one given for the parallel plate microresonator can be given for a torsion microplate.
In this section, we examine the example of a rectangular torsion microplate under the effect of squeeze film damping. As shown in Figure
A schematic drawing of a torsion microplate under the effect of squeeze film damping.
3-D view of the microplate
Side view of the microplate
For convenience, we introduce the nondimensional variables
As mentioned above, the microplate is under small torsion oscillation around
For a harmonic excitation, the ac component voltage
In this section, the approximate solution of
The numerical solution of (
Veijola et al. [
In [
The parameters of the accelerometer presented by Veijola et al. [
Parameters | Values |
---|---|
Mass | 4.9 |
Spring constant | 212.1 N/m |
Gap spacing | 3.95 |
Length of the moving masss | 2 960 |
Width of the moving masss | 1 780 |
Ambient pressure | 11 Pa |
Bias voltage | 9 V |
Effective viscosity | 10.2 |
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
For comparison purpose, we give the frequency responses of the accelerometer calculated by Blech’s model [
Figure
A Comparison of the damping ratios obtained by different methods with the experimental data [
Methods | Damping ratio | Peak value (dB) |
---|---|---|
Experimental data [ | 0.0239 | 18.3 |
Blech’s model (error) | 0.0159 (33.5%) | 21.7 |
The wavelet interpolation Galerkin method (error) | 0.0155 (35.1%) | 21.9 |
The finite difference method (error) | 0.0136 (43.1%) | 23.2 |
Comparisons of the frequency response obtained by the wavelet Galerkin method, the Blech model and the experimental data of Veijola et al. [
Amplitude response
Phase response
The air film pressure distribution calculated by the wavelet interpolation Galerkin method.
The real part of the pressure distribution
The imaginary part of the pressure distribution
Minikes et al. [
The parameters of two torsion mirrors [
Mirror 1 | Mirror 2 | |
---|---|---|
Width of the mirror | 500 | 500 |
Length of the mirror | 500 | 500 |
Thickness of the mirror | 30 | 30 |
Density of the mirror | 2300 kg/m3 | 2300 kg/m3 |
Gap spacing | 28 | 13 |
Torsional natural frequency | 13092.56 Hz | 12824.87 Hz |
In Table
For comparison purpose, we give the quality factors calculated by Pan’s model [
Tables
A comparison of quality factors obtained by Pan’s model, the wavelet interpolation Galerkin method and the finite difference method for mirror 1.
Pan’s model | The wavelet interpolation Galerkin method (error) | The finite difference method (error) | |
---|---|---|---|
0.08 | 5.784 | 5.987 | 6.586 × 104 (13.9%) |
0.10 | 4.483 | 4.627 | 5.160 × 104 (15.1%) |
0.50 | 7.070 | 7.330 | 8.006 × 103 (13.2%) |
1 | 3.306 | 3.388 × 103 (2.5%) | 3.694 × 103 (11.7%) |
3 | 1.061 | 1.080 × 103 (1.8%) | 1.173 × 103 (10.6%) |
6 | 5.680 | 5.895 × 102 (3.8%) | 6.526 × 102 (14.9%) |
10 | 3.979 | 4.052 × 102 (1.8%) | 4.454 × 102 (11.9%) |
30 | 2.451 | 2.505 × 102 (2.2%) | 2.721 × 102 (11.0%) |
60 | 2.074 × 102 | 2.160 × 102 (4.1%) | 2.327 × 102 (12.2%) |
100 | 1.994 × 102 | 2.035 × 102 (2.1%) | 2.240 × 102 (12.3%) |
760 | 1.850 × 102 | 1.896 × 102 (2.5%) | 2.109 × 102 (14.0%) |
A comparison of quality factors obtained by Pan’s model, the wavelet interpolation Galerkin method and the finite difference method for mirror 2.
Pan’s model | The wavelet interpolation Galerkin method (error) | The finite difference method (error) | |
---|---|---|---|
0.08 | 1.645 | 1.679 | 1.847 × 104 (12.3%) |
0.10 | 1.268 × 104 | 1.297 × 104 (2.3%) | 1.458 × 104 (15.0%) |
0.50 | 2.135 × 103 | 2.208 × 103 (3.4%) | 2.352 × 103 (10.2%) |
1 | 8.919 × 102 | 9.195 × 102 (3.1%) | 1.010 × 103 (13.2%) |
3 | 2.681 × 102 | 2.712 × 102 (1.2%) | 3.012 × 102 (12.3%) |
6 | 1.282 × 102 | 1.318 × 102 (2.8%) | 1.463 × 102 (14.1%) |
10 | 8.004 × 101 | 8.118 × 101 (1.4%) | 9.092 × 101 (13.6%) |
30 | 3.461 × 101 | 3.599 × 101 (4.0%) | 3.959 × 101 (14.4%) |
60 | 2.574 × 101 | 2.628 × 101 (2.1%) | 2.970 × 101 (15.4%) |
100 | 2.255 × 101 | 2.278 × 101 (1.0%) | 2.593 × 101 (15.0%) |
760 | 1.820 × 101 | 1.900 × 101 (4.4%) | 2.055 × 101 (12.9%) |
Figure
Comparison of the quality factors obtained by the wavelet Galerkin method, the Pan model and the experimental data of Minikes et al. [
Mirror 1 (
Mirror 2 (
Tables
A comparison of quality factors between
Experiment data [ | Pan’s model (error) | The wavelet interpolation Galerkin method (error) | The finite difference method (error) | |
---|---|---|---|---|
10 | 345 | 398 (15.4%) | 405 (17.4%) | 445 (29.0%) |
20 | 275 | 289 (5.1%) | 301 (9.4%) | 311 (13.1%) |
40 | 229 | 236 (3.1%) | 241 (5.2%) | 252 (10.0%) |
760 | 179 | 185 (3.4%) | 190 (6.1%) | 211 (17.9%) |
A comparison of quality factors between
Expertiment data [ | Pan’s model (error) | The wavelet interpolation Galerkin method (error) | The finite difference method (error) | |
---|---|---|---|---|
10 | 66 | 80.0 (21.2%) | 81.2 (23.0%) | 90.9 (37.7%) |
20 | 40 | 48.8 (22.0%) | 50.1 (25.3%) | 54.0 (35.0%) |
40 | 27 | 30.1 (11.5%) | 31.0 (14.8%) | 33.5 (24.1%) |
760 | 18 | 18.2 (1.1%) | 19.0 (5.6%) | 20.6 (14.4%) |
The air film pressure distribution calculated by the wavelet interpolation Galerkin method at
The real part of mirror 1
The imaginary part of mirror 1
The real part of mirror 2
The imaginary part of mirror 2
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 [
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).