Robust Optimization: Design in MEMS - University of California ...

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7Chapter 1IntroductionMicro-electro-mechanical systems (MEMS) is a new field that integrates mechanicalsystems and electronics through a silicon substrate using a microfabrication technique[1]. This new field largely emerged after polycrystalline silicon was discoveredto have excellent mechanical properties [2]. This realization prompted experimentationof many simple mechanical devices such as the double-folded flexure resonator[3]. The MEMS community was able to borrow substantially from the relativelymature IC industry and as a result grew quickly. The cutting edge of MEMS technologycurrently incorporates over 100 different materials and numerous fabricationprocesses. Some examples of current MEMS devices includes 3-axis accelerometers,microfluidics, folding mirrors, biomedical needles, micromotors, optical switches andgyroscopes. A good overview on the field of MEMS can be found in [4].While uncertainty is inherent in all systems, it is particularly important in MEMSdesign for several reasons. First, there is the issue of scale. While most MEMS featuresare on the order of micrometers, typical geometric tolerances for most processes areon the order of tenths of microns [5]. With minimum feature sizes of 2 microns, therelative uncertainty can reach 5% or more which is embarrassing when compared withmacroscopic manufacturing techniques.Second, the materials used in MEMS devices are poorly characterized. This islargely a result of the infancy of the field and the number of new materials andprocesses being used. We are well aware that metals, plastics, and other material
8properties are largely dependent on how they are manufactured, similarly the propertiesof MEMS materials are dependent on how they are fabricated. With a number ofdevelopments occurring in the past 15 years, the characterization of mechanical, electrical,and thermal properties has barely scratched the surface for the overwhelmingcombinations of materials and processes. Further complicating the characterizationprocess is the difficulty of making measurements at the micrometer scale. For thesereasons, most microfabrication materials have large uncertainties associated with theirproperties. For example, Sharpe and Jackson [6] characterized polycrystalline siliconin the MUMPs R process and measured values of 162 ± 14 GPa for Young’s modulusand 1.56 ± 0.25 GPa for tensile strength, which are relative uncertainties of 9% and16%, respectively.Finally, there are several other sources of uncertainty such as surface roughness,side-wall etch angle, and grain size. These factors, however, are typically much morecomplicated and difficult to characterize than the aforementioned two.Uncertainty is clearly significant at the micro-scale, however it is often neglectedin the MEMS design process. In order to design highly reliable systems that can befabricated with high yields, it will be necessary to account for the effects of uncertaintyespecially with the current trends to make more complex and integrated systems. Inthis report we will address a strategy for robust design through optimization. Ourfocus will be on dimensional and material uncertainties.
Page 1 and 2: Robust Optimization:Design in MEMSb
Page 4 and 5: 36 Conclusion 46Bibliography 48A Al
Page 6 and 7: 5List of Tables3.1 Probabilities wi
Page 11 and 12: 10performance, f(x, 0), is equal to
Page 13 and 14: 12function f(x), there are several
Page 15 and 16: 14Chapter 3Optimization AlgorithmIn
Page 17 and 18: 16paribus. This can be very effecti
Page 19 and 20: 18to the active linear constraints
Page 21 and 22: 20can all the constraints be satisf
Page 23 and 24: 22Chapter 4Optimization ExamplesWe
Page 25 and 26: 2475007300220200t 1t2t 37100180Cost
Page 27 and 28: 26design variables, other than the
Page 29 and 30: 28Chapter 5Robust Design Examples5.
Page 31 and 32: 30where ¯f is the w 2 design , Σ
Page 33 and 34: 32design along the line c 2 (x) = 0
Page 35 and 36: 34550500450(5)global solutionPath 1
Page 37 and 38: 36Figure 5.6: Diagram of Six Variab
Page 39 and 40: 38problem is well-defined. Before a
Page 41 and 42: 405.2.3 Robust DesignWe will use th
Page 43 and 44: 42evident by noting that the c 2 (x
Page 45 and 46: 44Robust Uncorrelated DesignAverage
Page 47 and 48: 46Chapter 6ConclusionIn this report
Page 49 and 50: 48Bibliography[1] What is MEMS Tech
Page 51: 50Appendix AAlkylation Constantsi c

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