Radio-frequency microelectromechanical systems (RF MEMS) are widely used for contact actuators and capacitive switches. These devices typically consist of a metallic membrane which is activated by a time-periodic electrostatic force and makes periodic contact with a contact pad. The increase in switch capacitance at contact causes the RF signal to be deflected and the switch thus closes. Membrane motion is damped by the surrounding gas, typically air or nitrogen. As the switch opens and closes, the flow transitions between the continuum and rarefied regimes. Furthermore, creep is a critical physical mechanism responsible for the failure in these devices, especially those operating at high RF power. Simultaneous and accurate modeling of all these different physics is required to understand the dynamical membrane response in these devices and to estimate device lifetime and to improve MEMS reliability. It is advantageous to model fluid and structural mechanics and electrostatics within a single comprehensive numerical framework to facilitate coupling between them. In this work, we develop a single unified finite volume method based numerical framework to study this multi-physics problem in RF MEMS. Our objective required us to develop structural solvers, fluid flow solvers, and electrostatic solvers using the finite volume method, and efficient mechanisms to couple these different solvers. A particular focus is the development of flow solvers which work efficiently across continuum and rarefied regimes. A number of novel contributions have been made in this process. Structural solvers based on a fully implicit finite volume method have been developed for the first time. Furthermore, strongly implicit fluid flow solvers have also been developed that are valid for both continuum and rarefied flow regimes and which show an order of magnitude speed-up over conventional algorithms on serial platforms. On parallel platforms, the solution techniques developed in this thesis are shown to be significantly more scalable than existing algorithms. The numerical methods developed are used to compute the static and dynamic response of MEMS. Our results indicate that our numerical framework can become a computationally efficient tool to model the dynamics of RF MEMS switches under electrostatic actuation and gas damping. / text

It is well established that non-equilibrium flows cannot properly be described by traditional hydrodynamics, namely, the Navier-Stokes-Fourier (NSF) equations. Such flows occur, for example, in micro-electro-mechanical systems (MEMS), and ultra vacuum systems, where the dimensions of the devices are comparable to the mean free path of a gas molecule. Therefore, the study of non-equilibrium effects in gas flows is extremely important. The general interest of the present study is to explore boundary value problems for moderately rarefied gas flows, with an emphasis on numerical solutions of the regularized 13--moment equations (R13). Boundary conditions for the moment equations are derived based on either phenomenological principles or on microscopic gas-surface scattering models, e.g., Maxwell's accommodation model and the isotropic scattering model. Using asymptotic analysis, several non-linear terms in the R13 equations are transformed into algebraic terms. The reduced equations allow us to obtain numerical solutions for multidimensional boundary value problems, with the same set of boundary conditions for the linearized and fully non-linear equations. Some basic flow configurations are employed to investigate steady and unsteady rarefaction effects in rarefied gas flows, namely, planar and cylindrical Couette flow, stationary heat transfer between two plates, unsteady and oscillatory Couette flow. A comparison with the corresponding results obtained previously by the DSMC method is performed. The influence of rarefaction effects in the lid driven cavity problem is investigated. Solutions obtained from several macroscopic models, in particular the classical NSF equations with jump and slip boundary conditions, and the R13--moment equations are compared. The R13 results compare well with those obtained from more costly solvers for rarefied gas dynamics, such as the Direct Simulation Monte Carlo (DSMC) method. Flow and heat transfer in a bottom heated square cavity in a moderately rarefied gas are investigated using the R13 and NSF equations. The results obtained are compared with those from the DSMC method with emphasis on understanding thermal flow characteristics from the slip flow to the early transition regime. The R13 theory gives satisfying results including flow patterns in fair agreement with DSMC in the transition regime, which the conventional Navier-Stokes-Fourier equations are not able to capture. / Graduate / 0548 / anirudh@uvic.ca

The macroscopic chemistry method for modelling non-equilibrium reacting gas flows with the direct simulation Monte Carlo (DSMC) method is developed and tested. In the macroscopic method, the calculation of chemical reactions is decoupled from the DSMC collision routine. The number of reaction events that must be performed in a cell is calculated with macroscopic rate expressions. These expressions use local macroscopic information such as kinetic temperatures and density. The macroscopic method is applied to a symmetrical diatomic gas. For each dissociation event, a single diatom is selected with a probability based on internal energy and is dissociated into two atoms. For each recombination event, two atoms are selected at random and replaced by a single diatom. To account for the dissociation energy, the thermal energies of all particles in the cell are adjusted. The macroscopic method differs from conventional collision-based DSMC chemistry procedures, where reactions are performed as an integral part of the collision routine. The most important advantage offered by the macroscopic method is that it can utilise reaction rates that are any function of the macroscopic flow conditions. It therefore allows DSMC chemistry calculations to be performed using rate expressions for which no conventional chemistry model may exist. Given the accuracy and flexibility of the macroscopic method, it has significant potential for modelling reacting non-equilibrium gas flows. The macroscopic method is tested by performing DSMC calculations and comparing the results to those obtained using conventional DSMC chemistry models and experimental data. The macroscopic method gives density profiles in good agreement with experimental data in the chemical relaxation region downstream of a strong shock. Within the shock where strongly non-equilibrium conditions prevail, the macroscopic method provides good agreement with a conventional chemistry model. For the flow over a blunt axisymmetric cylinder, which also exhibits strongly non-equilibrium conditions, the macroscopic method also gives reasonable agreement with conventional chemistry models. The ability of the macroscopic method to utilise any rate expression is demonstrated by using a two-temperature rate model that accounts for dissociation-vibration coupling effects that are important in non-equilibrium reacting flows. Relative to the case without dissociation-vibration coupling, the macroscopic method with the two-temperature model gives reduced dissociation rates in vibrationally cold flows, as expected. Also, for the blunt cylinder flow, the two-temperature model gives reduced surface heat fluxes, as expected. The macroscopic method is also tested with a number density dependent form of the equilibrium constant. For zero-dimensional chemical relaxation, the resulting relaxation histories are in good agreement with those provided by an exact Runge-Kutta solution of the relaxation behaviour. Reviews of basic DSMC procedures and conventional DSMC chemistry models are also given. A method for obtaining the variable hard sphere parameters for collisions between particles of different species is given. Borgnakke-Larsen schemes for modelling internal energy exchange are examined in detail. Both continuous rotational and quantised vibrational energy modes are considered. Detailed derivations of viscosity and collision rate expressions for the generalised hard sphere model of Hassan and Hash [Phys. Fluids 5, 738 (1993)] and the modified version of Macrossan and Lilley [J. Thermophys. Heat Transfer 17, 289 (2003)] are also given.

This thesis deals with the direct simulation Monte Carlo (DSMC) method of analysing gas flows. The DSMC method was initially proposed as a method for predicting rarefied flows where the Navier-Stokes equations are inaccurate. It has now been extended to near continuum flows. The method models gas flows using simulation molecules which represent a large number of real molecules in a probabilistic simulation to solve the Boltzmann equation. Molecules are moved through a simulation of physical space in a realistic manner that is directly coupled to physical time such that unsteady flow characteristics are modelled. Intermolecular collisions and moleculesurface collisions are calculated using probabilistic, phenomenological models. The fundamental assumption of the DSMC method is that the molecular movement and collision phases can be decoupled over time periods that are smaller than the mean collision time. Two obstacles to the wide spread use of the DSMC method as an engineering tool are in the areas of simulation configuration, which is the configuration of the simulation parameters to provide a valid solution, and the time required to obtain a solution. For complex problems, the simulation will need to be run multiple times, with the simulation configuration being modified between runs to provide an accurate solution for the previous run�s results, until the solution converges. This task is time consuming and requires the user to have a good understanding of the DSMC method. Furthermore, the computational resources required by a DSMC simulation increase rapidly as the simulation approaches the continuum regime. Similarly, the computational requirements of three-dimensional problems are generally two orders of magnitude more than two-dimensional problems. These large computational requirements significantly limit the range of problems that can be practically solved on an engineering workstation or desktop computer. The first major contribution of this thesis is in the development of a DSMC implementation that automatically adapts the simulation. Rather than modifying the simulation configuration between solution runs, this thesis presents the formulation of algorithms that allow the simulation configuration to be automatically adapted during a single run. These adaption algorithms adjust the three main parameters that effect the accuracy of a DSMC simulation, namely the solution grid, the time step and the simulation molecule number density. The second major contribution extends the parallelisation of the DSMC method. The implementation developed in this thesis combines the capability to use a cluster of computers to increase the maximum size of problem that can be solved while simultaneously allowing excess computational resources to decrease the total solution time. Results are presented to verify the accuracy of the underlying DSMC implementation, the utility of the solution adaption algorithms and the efficiency of the parallelisation implementation.

The macroscopic chemistry method for modelling non-equilibrium reacting gas flows with the direct simulation Monte Carlo (DSMC) method is developed and tested. In the macroscopic method, the calculation of chemical reactions is decoupled from the DSMC collision routine. The number of reaction events that must be performed in a cell is calculated with macroscopic rate expressions. These expressions use local macroscopic information such as kinetic temperatures and density. The macroscopic method is applied to a symmetrical diatomic gas. For each dissociation event, a single diatom is selected with a probability based on internal energy and is dissociated into two atoms. For each recombination event, two atoms are selected at random and replaced by a single diatom. To account for the dissociation energy, the thermal energies of all particles in the cell are adjusted. The macroscopic method differs from conventional collision-based DSMC chemistry procedures, where reactions are performed as an integral part of the collision routine. The most important advantage offered by the macroscopic method is that it can utilise reaction rates that are any function of the macroscopic flow conditions. It therefore allows DSMC chemistry calculations to be performed using rate expressions for which no conventional chemistry model may exist. Given the accuracy and flexibility of the macroscopic method, it has significant potential for modelling reacting non-equilibrium gas flows. The macroscopic method is tested by performing DSMC calculations and comparing the results to those obtained using conventional DSMC chemistry models and experimental data. The macroscopic method gives density profiles in good agreement with experimental data in the chemical relaxation region downstream of a strong shock. Within the shock where strongly non-equilibrium conditions prevail, the macroscopic method provides good agreement with a conventional chemistry model. For the flow over a blunt axisymmetric cylinder, which also exhibits strongly non-equilibrium conditions, the macroscopic method also gives reasonable agreement with conventional chemistry models. The ability of the macroscopic method to utilise any rate expression is demonstrated by using a two-temperature rate model that accounts for dissociation-vibration coupling effects that are important in non-equilibrium reacting flows. Relative to the case without dissociation-vibration coupling, the macroscopic method with the two-temperature model gives reduced dissociation rates in vibrationally cold flows, as expected. Also, for the blunt cylinder flow, the two-temperature model gives reduced surface heat fluxes, as expected. The macroscopic method is also tested with a number density dependent form of the equilibrium constant. For zero-dimensional chemical relaxation, the resulting relaxation histories are in good agreement with those provided by an exact Runge-Kutta solution of the relaxation behaviour. Reviews of basic DSMC procedures and conventional DSMC chemistry models are also given. A method for obtaining the variable hard sphere parameters for collisions between particles of different species is given. Borgnakke-Larsen schemes for modelling internal energy exchange are examined in detail. Both continuous rotational and quantised vibrational energy modes are considered. Detailed derivations of viscosity and collision rate expressions for the generalised hard sphere model of Hassan and Hash [Phys. Fluids 5, 738 (1993)] and the modified version of Macrossan and Lilley [J. Thermophys. Heat Transfer 17, 289 (2003)] are also given.

Kyoto University (京都大学) / 0048 / 新制・課程博士 / 博士(工学) / 甲第17512号 / 工博第3671号 / 新制||工||1558(附属図書館) / 30278 / 京都大学大学院工学研究科機械理工学専攻 / (主査)教授 青木 一生, 教授 稲室 隆二, 教授 斧 髙一 / 学位規則第4条第1項該当

The content of Chapter 1 is an author produced version of a paper published in Physics of Fluids. The final publication is available at AIP via http://dx.doi.org/10.1063/1.3691262. The content of Chapters 2 and 4 is an author produced version of papers published in Journal of Statistical Physics. The final publications are available at Springer via http://dx.doi.org/10.1007/s10955-012-0512-z and http://dx.doi.org/10.1007/s10955-015-1364-0, respectively. / 京都大学 / 0048 / 新制・課程博士 / 博士(工学) / 甲第19682号 / 工博第4137号 / 新制||工||1638(附属図書館) / 32718 / 京都大学大学院工学研究科機械理工学専攻 / (主査)教授 青木 一生, 教授 髙田 滋, 教授 稲室 隆二 / 学位規則第4条第1項該当 / Doctor of Philosophy (Engineering) / Kyoto University / DFAM

京都大学 / 0048 / 新制・課程博士 / 博士(工学) / 甲第19686号 / 工博第4141号 / 新制||工||1639(附属図書館) / 32722 / 京都大学大学院工学研究科航空宇宙工学専攻 / (主査)教授 稲室 隆二, 教授 青木 一生講師杉元 宏 / 学位規則第4条第1項該当 / Doctor of Philosophy (Engineering) / Kyoto University / DFAM

京都大学 / 0048 / 新制・課程博士 / 博士(工学) / 甲第21757号 / 工博第4574号 / 新制||工||1713(附属図書館) / 京都大学大学院工学研究科機械理工学専攻 / (主査)教授 西脇 眞二, 教授 髙田 滋, 教授 鈴木 基史 / 学位規則第4条第1項該当 / Doctor of Philosophy (Engineering) / Kyoto University / DFAM

The abstract kinetic equation Tψ’=-Aψ is studied with partial range boundary conditions in two geometries, in the half space x≥0 and on a finite interval [0, r]. T and A are abstract self-adjoint operators in a complex Hilbert space. In the case of the half space problem it is assumed that T is a (possibly) unbounded injection and A is a positive compact perturbation of the identity satisfying a regularity condition, while in the case of slab geometry T is a bounded injection and A is a bounded Fredholm operator with a finite dimensional negative part. Existence and uniqueness theory is developed for both models. Results are illustrated on relevant physical examples. / Ph. D.