Abstract Micro-electromechanical systems (MEMS) are subject to inevitable and inherent uncertainties in their dimensional and material parameters. Those lead to variability in their performance and reliability. Manufacturing processes leave substantial variability in the shape and geometry of the device due to its small dimensions and high feature complexity, while the material properties of a component are inherently subject to scattering. The effects of these variations have to be considered and a modeling methodology is needed in order to ensure required MEMS performance under uncertainties. Furthermore, in the design of high-Q micro-resonators, dissipation mechanisms may have detrimental effects on the quality factor (Q). One of the major dissipation phenomena to consider is thermoelastic damping, so that performances are directly related to the thermoelastic quality factor, which has to be predicted accurately. The purpose of this research is to develop a numerical method to analyze the effects of geometric and material property random variations on the thermoelastic quality factor of micro-resonators. The extension of the Perturbation Stochastic Finite Element Method (PSFEM) to the analysis of strongly coupled multiphysic phenomena allows the quantification of the influence of uncertainties, making available a new efficient numerical tool to MEMS designers. Résumé Dans le domaine des microsystèmes électromécaniques (MEMS), les micro-résonateurs jouent un rôle important pour le développement de micro-capteurs de plus en plus précis (ex : micro-accéléromètres). Dans cette optique daugmentation de la précision, les pertes dénergie qui limitent les performances des micro-résonateurs doivent être identifiées et quantifiées. Le facteur limitant des micro-résonateurs actuels est leur facteur de qualité thermo-élastique, qui doit donc être prédit de manière précise. De plus, suite à la tendance actuelle de miniaturisation et complexification accrues des MEMS, les sources de dispersions sont très nombreuses, à la fois sur les constantes physiques des matériaux utilisés et sur les paramètres géométriques. La mise au point doutils numériques permettant de prendre en compte les incertitudes de manière efficace est donc primordiale afin daméliorer les prestations densemble du microsystème et dassurer un certain niveau de robustesse et de fiabilité. Le but de cette recherche est de développer une méthode numérique pour analyser les effets des variations aléatoires des propriétés matérielles et géométriques sur le facteur de qualité thermo-élastique de micro-résonateurs. Pour ce faire, lapproche dite perturbative de la méthode des éléments finis stochastiques (PSFEM) est étendue à lanalyse de phénomènes multiphysiques fortement couplés, fournissant ainsi aux acteurs de lindustrie des MEMS un nouvel outil de conception efficace.

The first objective of this dissertation is to present hybrid numerical-analytical approaches and reduced-order models to simulate microelectromechanical systems (MEMS) in multi-physics fields. These include electric actuation (AC and DC), squeeze-film damping, thermoelastic damping, and structural forces. The second objective is to investigate MEMS phenomena, such as squeeze-film damping and dynamic pull-in, and use the latter to design a novel RF-MEMS switch. In the first part of the dissertation, we introduce a new approach to the modeling and simulation of flexible microstructures under the coupled effects of squeeze-film damping, electrostatic actuation, and mechanical forces. The new approach utilizes the compressible Reynolds equation coupled with the equation governing the plate deflection. The model accounts for the slip condition of the flow at very low pressures. Perturbation methods are used to derive an analytical expression for the pressure distribution in terms of the structural mode shapes. This expression is substituted into the plate equation, which is solved in turn using a finite-element method for the structural mode shapes, the pressure distributions, the natural frequencies, and the quality factors. We apply the new approach to a variety of rectangular and circular plates and present the final expressions for the pressure distributions and quality factors. We extend the approach to microplates actuated by large electrostatic forces. For this case, we present a low-order model, which reduces significantly the cost of simulation. The model utilizes the nonlinear Euler-Bernoulli beam equation, the von K´arm´an plate equations, and the compressible Reynolds equation. The second topic of the dissertation is thermoelastic damping. We present a model and analytical expressions for thermoelastic damping in microplates. We solve the heat equation for the thermal flux across the microplate, in terms of the structural mode shapes, and hence decouple the thermal equation from the plate equation. We utilize a perturbation method to derive an analytical expression for the quality factor of a microplate with general boundary conditions under electrostatic loading and residual stresses in terms of its structural mode shapes. We present results for microplates with various boundary conditions. In the final part of the dissertation, we present a dynamic analysis and simulation of MEMS resonators and novel RF MEMS switches employing resonant microbeams. We first study microbeams excited near their fundamental natural frequencies (primary-resonance excitation). We investigate the dynamic pull-in instability and formulate safety criteria for the design of MEMS sensors and RF filters. We also utilize this phenomenon to design a low-voltage RF MEMS switch actuated with a combined DC and AC loading. Then, we simulate the dynamics of microbeams excited near half their fundamental natural frequencies (superharmonic excitation) and twice their fundamental natural frequencies (subharmonic excitation). For the superharmonic case, we present results showing the effect of varying the DC bias, the damping, and the AC excitation amplitude on the frequency-response curves. For the subharmonic case, we show that if the magnitude of the AC forcing exceeds the threshold activating the subharmonic resonance, all frequency-response curves will reach pull-in. / Ph. D.

Im Fokus dieser Arbeit stehen Dämpfungseffekte schwingfähiger Mikroelektromechanischer Systeme (MEMS), die nach dem kapazitiven Wirkprinzip arbeiten. Die verschiedenen Dissipationsprozesse und die zugehörigen analytischen Modelle sowie numerischen Berechnungsmöglichkeiten auf physikalischer Ebene werden vorgestellt und mit eigenen experimentellen Ergebnissen verglichen. Der Schwerpunkt liegt dabei auf der fluidischen Dämpfung im Kontinuum und bei leichter Verdünnung, was bei den meisten kapazitiven MEMS den dominierenden Verlusteffekt darstellt.:1 Überblick 2 Grundlagen zur Beschreibung von Mikrosystemen 3 Herstellung und Charakterisierung 4 Fluidische Dämpfung 5 Weitere dissipative Effekte mikromechanischer Systeme 6 Zusammenfassung und Ausblick / This thesis focuses on damping effects of vibrational micro-electromechanical systems (MEMS) with capacitive working principle. The different dissipation processes and the associated analytical models as well as numerical calculation possibilities on a physical level are presented and compared to own experimental results. The main emphasis is on fluidic damping in the continuum regime and with slight rarefaction, which is the dominant loss effect in most capacitive MEMS.:1 Überblick 2 Grundlagen zur Beschreibung von Mikrosystemen 3 Herstellung und Charakterisierung 4 Fluidische Dämpfung 5 Weitere dissipative Effekte mikromechanischer Systeme 6 Zusammenfassung und Ausblick