During the CMP process, a wafer is rotated and pressed face down against a rotating polishing pad. Polishing slurry is delivered on the top of pad continuously and forms a thin lubricating film between the wafer and the pad. In this study, a three-dimensional slurry flow model based on a generalized Reynolds equation is developed, which can apply to a rough pad with the compressibility of the pad, and the multi-grid method is used to reduce computational time. According to the force and moment balance equations, the tilted angles and the slurry film thickness can be evaluated. When the pad surface is rough, the squeeze term differentiated by time should be considered in this model due to the rotation of the pad. The influences of applied load, pad speed, wafer speed, pad compressibility, and surface roughness pattern on the tilted angles and the slurry film thickness are investigated. Results show that the variation of the tilted angles becomes more significant for the anisotropic than that for the isotropic during the rotation of the pad. And the slurry film thickness at the center of the wafer increases as applied load decreases or pad speed increases or wafer speed decreases or the compressibility of the pad increases.

The "Ahrens method" is a very simple method for sampling from univariate distributions. It is based on rejection from piecewise constant hat functions. It can be applied analogously to the multivariate case where hat functions are used that are constant on rectangular domains. In this paper we investigate the case of distributions with so called orthounimodal densities. Technical implementation details as well as their practical limitations are discussed. The application to more general distributions is considered. (author's abstract) / Series: Research Report Series / Department of Statistics and Mathematics

We present an immersed interface method for the incompressible Navier Stokes equations capable of handling rigid immersed boundaries. The immersed boundary is represented by a set of Lagrangian control points. In order to guarantee that the no-slip condition on the boundary is satisfied, singular forces are applied on the fluid at the immersed boundary. The forces are related to the jumps in pressure and the jumps in the derivatives of both pressure and velocity, and are interpolated using cubic splines. The strength of singular forces is determined by solving a small system of equations at each time step. The Navier-Stokes equations are discretized on a staggered Cartesian grid by a second order accurate projection method for pressure and velocity. / Singapore-MIT Alliance (SMA)

We present a multi-grid method for a class of structured generalized Lyapunov matrix equations. Such equations need to be solved in each step of the Newton method for algebraic Riccati equations, which arise from linear-quadratic optimal control problems governed by partial differential equations. We prove the rate of convergence of the two-grid method to be bounded independent of the dimension of the problem under certain assumptions. The multi-grid method is based on matrix-matrix multiplications and thus it offers a great potential for a parallelization. The efficiency of the method is demonstrated by numerical experiments.

Viscoelastic fluid flow with immersed boundaries of complex geometry is widely found both in nature and engineering processes. Examples include haemocytes moving in human blood flow, self-propulsion of microscopic organisms in complex liquids, hydraulic fracturing with sand in oil flow, and suspension flow with viscoelastic medium. Computational modelling of such systems is important for understanding complex biological processes and assisting engineering designs. Conventional simulation methods use conformed meshes to resolve the boundaries of complex geometry. Dynamically updating the conformed mesh is computationally expensive and makes parallelization difficult. In comparison, Cartesian grid methods are more promising for large scale parallel simulation. Using Cartesian grid methods to simulate viscoelastic fluid flow with complex boundaries is a relatively unexplored area. In this thesis, a sharp interface Cartesian grid method (SICG) and a smoothed interface immersed boundary method (SIIB) are developed in order to simulate viscoelastic fluids in complex geometries. The SICG method shows a better prediction of the stress on stationary boundaries while the SIIB method shows reduced non-physical oscillations in the computation of drag and lift forces on moving boundaries. Parallel implementations of both solvers are developed. Convergence of the numerical schemes is shown and the implementations are validated with a few benchmark problems with both stationary and moving boundaries. This study also focuses on the simulation of flows past 2D cylindrical or 3D spherical particles. Lateral migration of particles induced by inertial and viscoelastic effects are investigated with different flow types. Equilibrium positions of inertia-induced migration are reported as a function of the particle Reynolds number and the blockage ratio. The migration in the viscoelastic fluid is simulated from zero elastic number to a finite elastic number. The inclusion of both inertial and viscoelastic effects on the lateral migration of a particle is the first of its kind. New findings are reported for the equilibrium positions of a spherical particle in square duct flow, which suggest the need for future experimental and computational work.

<p>Pharmacometric modelling is an increasingly used method for analysing the outcome from clinical trials in drug development. The model building process is complex and involves testing, evaluating and diagnosing a range of plausible models aiming to make an adequate inference from the observed data and predictions for future studies and therapy. </p><p>The aim of this thesis was to advance the approaches used in pharmacometrics by introducing improved models and methods for application in essential parts of model building procedure: (i) structural model development, (ii) stochastic model development and (iii) model diagnostics. </p><p>As a contribution to the structural model development, a novel flexible structural model for drug absorption, a transit compartment model, was introduced and evaluated. This model is capable of describing various drug absorption profiles and yet simple enough to be estimable from data available from a typical trial. As a contribution to the stochastic model development, three novel methods for parameter distribution estimation were developed and evaluated; a default NONMEM nonparametric method, an extended grid method and a semiparametric method with estimated shape parameters. All these methods are useful in circumstances when standard assumptions of parameter distributions in the population do not hold. The new methods provide less biased parameter estimates, better description of variability and better simulation properties of the model. As a contribution to model diagnostics, the most commonly used diagnostics were evaluated for their usefulness. In particular, diagnostics based on individual parameter estimates were systematically investigated and circumstances which are likely to misguide modelers towards making erroneous decisions in model development, relating to choice of structural, covariate and stochastic model components were identified. </p><p>In conclusion, novel approaches, insights and models have been provided to the pharmacometrics community. </p><p>Implementation of these advances to make model building more efficient and robust has been facilitated by development of diagnostic tools and automated routines.</p>

L’objectif du présent travail est d’étudier expérimentalement et numériquement la fissuration des bois tropicaux issus de la forêt gabonaise, à savoir : le Milicia excelsa (iroko),l’ Aucoumea klaineana pierre (okoumé) et Pterocarpus soyauxii (padouk). Cette étude est axée sur le processus de croissance de la fissure en mode d’ouverture et en mode mixte des trois essences à l’aide d’éprouvettes Compact Tension Shear (CTS) et Mixed Mode CrackGrowth (MMCG) modifiées. Les éprouvettes sont montées dans un système Arcan et placées dans une machine d’essai électromécanique. La méthode de la grille est utilisée pour mesurer les champs de déplacements et de déformations au voisinage de la fissure. Ces cartes ainsi obtenues permettent d’obtenir l’ouverture et la longueur de la fissure tout au long de l’essai. Les courbes force en fonction de l’ouverture de la fissure sont déduites des mesures. La méthode de la complaisance à déplacement imposé est utilisée pour calculer le taux de restitution d’énergie G en mode d’ouverture de fissure et en mode mixte. Les résultats ont montré un effet de l’épaisseur sur la fissuration. Ces résultats ont aussi permis de mettre en évidence une proportionnalité de la densité par rapport aux paramètres de rupture de ces essences. En mode mixte les valeurs de l’évolution de G sont présentées en fonction de la longueur de fissure après découplage des modes (séparation du mode 1 et du mode 2). On observe pour l’okoumé par exemple, que les rapports des taux de restitution d’énergie des deux modes sont constants, ce qui justifie le caractère intrinsèque des paramètres de fissuration obtenus. Des comparaisons faites avec les résultats issus de de la littérature sur les essences tempérées de même densité ont montré des similitudes. / The objective of the present work is to study experimentally and numerically the crackingbehavior of tropical woods from the Gabonese forest, namely : Milicia excelsa (iroko), Aucoumeaklaineana pierre (okume) and Pterocarpus soyauxii (padouk). This study focuseson the crack growth process in crack opening mode and mixed mode of these three speciesusing Modified Tension Shear (CTS) and Mixed Mode Crack Growth (MMCG) specimens.The specimens are mounted in an Arcan system and placed in an electromechanical testingmachine. The grid method is used to measure the displacement and strains fields near the tipcrack. These maps make it possible to obtain the opening and the length of the crack duringthe tests. The curves force showing the face as a function of the crack opening are deducedfrom the measurements. The experimental critical energy release rate G is evaluated by thecompliance method in imposed displacement. The results show an effect of the thickness oncracking. These results also made it possible to demonstrate a proportionality of the densitywith the parameters of rupture of these species. In mixed mode, after decoupling the modes(separation of mode 1 and mode 2), the values of G are presented as a function of to the cracklength. For Okume, for example, it was observed that the ratios of the energy release ratesof the two modes are constant, which justifies the intrinsic character of the cracking parameterswere obtained. Comparisons made with results from the literature review on temperatespecies of the same density showed similarities.

Pharmacometric modelling is an increasingly used method for analysing the outcome from clinical trials in drug development. The model building process is complex and involves testing, evaluating and diagnosing a range of plausible models aiming to make an adequate inference from the observed data and predictions for future studies and therapy. The aim of this thesis was to advance the approaches used in pharmacometrics by introducing improved models and methods for application in essential parts of model building procedure: (i) structural model development, (ii) stochastic model development and (iii) model diagnostics. As a contribution to the structural model development, a novel flexible structural model for drug absorption, a transit compartment model, was introduced and evaluated. This model is capable of describing various drug absorption profiles and yet simple enough to be estimable from data available from a typical trial. As a contribution to the stochastic model development, three novel methods for parameter distribution estimation were developed and evaluated; a default NONMEM nonparametric method, an extended grid method and a semiparametric method with estimated shape parameters. All these methods are useful in circumstances when standard assumptions of parameter distributions in the population do not hold. The new methods provide less biased parameter estimates, better description of variability and better simulation properties of the model. As a contribution to model diagnostics, the most commonly used diagnostics were evaluated for their usefulness. In particular, diagnostics based on individual parameter estimates were systematically investigated and circumstances which are likely to misguide modelers towards making erroneous decisions in model development, relating to choice of structural, covariate and stochastic model components were identified. In conclusion, novel approaches, insights and models have been provided to the pharmacometrics community. Implementation of these advances to make model building more efficient and robust has been facilitated by development of diagnostic tools and automated routines.

We present a multi-grid method for a class of structured generalized Lyapunov matrix equations. Such equations need to be solved in each step of the Newton method for algebraic Riccati equations, which arise from linear-quadratic optimal control problems governed by partial differential equations. We prove the rate of convergence of the two-grid method to be bounded independent of the dimension of the problem under certain assumptions. The multi-grid method is based on matrix-matrix multiplications and thus it offers a great potential for a parallelization. The efficiency of the method is demonstrated by numerical experiments.

Le bois est en général étudié comme un matériau orthotrope et homogène, mais en réalité il existe des gradients de propriétés mécaniques et physiques dans chaque direction. En effet, le bois est constitué d’une alternance de cernes annuels de croissance constitués de bois d’été, de printemps et de transition. Ces cernes sont plus réguliers dans la plupart des résineux que dans les feuillus. Dans le présent document, on s’intéresse à l’étude et à l’identification des propriétés de diffusion hydrique d’un résineux à l’échelle du cerne, le sapin blanc du Massif Central. Une telle étude nécessite l’utilisation d’une technique de mesure de champ sans contact, ici la méthode de la grille. Cette dernière offre la possibilité d’observer l’hétérogénéité du bois à l’échelle du cerne avec des champs de déformations. Elle permet donc d’obtenir la réponse sous sollicitations hydriques des bois d’été et de printemps. Cette technique de mesure est couplée à une méthode d’identification inverse des paramètres de comportement hydrique que nous avons mise en place. Elle permet d’obtenir les coefficients de diffusion des bois d’été et de printemps dans la direction tangentielle de diffusion, ainsi que leurs coefficients d’expansion hydrique dans la direction radiale à partir des cartes de déformations obtenues par la méthode de la grille. La méthode de la grille est également utilisée pour l’étude du comportement du bois sous environnement constant ou variable, et sous contrainte mécanique de compression radiale. Des méthodes d’identification sont également proposées afin de déterminer les propriétés mécaniques selon la direction radiale du bois à l’échelle du cerne, ceci à partir des cartes de déformation. / Wood is generally studied as an orthotropic and homogeneous material, but in reality there are gradients of mechanical and physical properties in each direction. Indeed, wood consists of summer and spring woods which manifest annual growth rings. These latter are more regular in most of softwoods than in hardwoods. In this paper, we are interested in studying and identifying the properties of a resinous under water diffusion, namely the white fir of the Massif Central. This study requires the use of a contacless full-field measurement technique. The grid method is used here. This latter offers the opportunity to observe the heterogeneity of wood at the ring scale through the strain fields. It thus provides the response of latewood and earlywood under water stress. This measurement technique is coupled with an inverse method to identify the water behavior parameters. This inverse method provides latewood and earlywood diffusion coefficients along the tangential diffusion direction, as well as their coefficients of expansion in the radial direction from the strain maps. The grid method is also used to study the behavior of wood under constant or variable environment when it is subjected to radial compression. Some identification methods are also proposed to determine from the deformation maps the mechanical properties along the radial direction of wood at the ring scale.