Ce travail de thèse est consacré à deux thèmes de recherche en Calcul Scientifique liés par l'approximation numérique de problèmes en mécanique des fluides. Le premier thème concerne l'approximation numérique des équations de Stokes, modélisant les écoulements de fluides incompressibles à vitesse faible. Ce thème est présent dans plusieurs travaux en Calcum Scientifique. La discrétisation en temps est réalisée à l'aide de la méthode de projection. La discrétisation en espace utilise la méthode des éléments finis hybrides qui permet d'imposer de façon exacte la contrainte d'incompressibilité. Cette approche est originale : la méthode des éléments mixtes hybrides est couplée avec une méthode d'éléments finis standards. L'ordre de convergence des deux méthodes est préservé. Le second thème concerne la mise au point de méthodes numériques de type volumes finis pour la résolution de l'équation Level Set. Ces équations interviennent de manière essentielle dans la résolution des problèmes de propagation d'interfaces. Dans cette partie, nous avons développé une nouvelle méthode d'ordre 2 de type MUSCL pour résoudre le système hyperbolique résultant de l'équation Level Set. Nous illustrons ces propriétés par des applications numériques. En particulier nous avons regardé le cas du problème des deux demi-plans pour lequel notre schéma donne une approximation pour le gradient de la fonction Level Set. Par ailleurs, l'ordre de précision attendu est obtenu avec les normes L1 et Linfini pour des fonctions régulières. Pour finir, il est à noter que notre méthode peut être facilement étendue aux problèmes d'Hamilton-Jacobi du premier et du second ordre

Oscillatory integral operators have been of interest to both mathematicians and physicists ever since the emergence of the work Theorie Analytique de la Chaleur of Joseph Fourier in 1822, in which his chief concern was to give a mathematical account of the diffusion of heat. For example, oscillatory integrals naturally arise when one studies the behaviour at infinity of the Fourier transform of a Borel measure that is supported on a certain hypersurface. One reduces the study of such a problem to that of having to obtain estimates on oscillatory integrals. However, sub-level set operators have only come to the fore at the end of the 20th Century, where it has been discovered that the decay rates of the oscillatory integral I(lambda) above may be obtainable once the measure of the associated sub-level sets are known. This discovery has been fully developed in a paper of A. Carbery, M. Christ and J.Wright. A principal goal of this thesis is to explore certain uniformity issues arising in the study of sub-level set estimates.

Nous présentons une nouvelle méthode de type frontière immergée (immersed boundary method, ou méthode IB) pour le calcul d'écoulements visqueux incompressibles en géométries irrégulières. Dans les méthodes IB , la frontière irrégulière de la géométrie n'est pas alignée avec la grille de calcul, et le point crucial de leur développement demeure le traitement numérique des cellules fluides qui sont coupées par la frontière irrégulière, appelées cut-cells. La partie dédiée à la résolution des équations de Navier-Stokes de notre méthode IB, appelée méthode LS-STAG , repose sur la méthode MAC pour grilles cartésiennes décalées, et sur l'utilisation d'une fonction de distance signée (la fonction level-set ) pour représenter précisément les frontières irrégulières du domaine. L'examen discret des lois globales de conservation de l'écoulement (masse, quantité de mouvement et énergie cinétique) a permis de bâtir une discrétisation unifiée des équations de Navier-Stokes dans les cellules cartésiennes et les cut-cells . Cette discrétisation a notamment la propriété de préserver la structure à 5 points du stencil original et conduit à une méthode extrêmement efficace sur le plan du temps de calcul en comparaison à un solveur non-structuré. La précision de la méthode est évaluée pour l'écoulement de Taylor-Couette et sa robustesse éprouvée par l'étude de divers écoulements instationnaires, notamment autour d'objets profilés. Le champ d'application de notre solveur Newtonien s'étend au cas d'écoulements en présence de géométries mobiles, et la méthode LS-STAG s'avère être un outil prometteur puisqu'affranchie des étapes systématiques (et coûteuses) de remaillage du domaine. Finalement, la première application d'une méthode IB au calcul d'écoulements de fluides viscoélastiques est présentée. La discrétisation de la loi constitutive est basée sur la méthode LS-STAG et sur l'utilisation d'un arrangement totalement décalé des variables dans tout le domaine assurant le couplage fort requis entre les variables hydrodynamiques et les composantes du tenseur des contraintes élastiques. La méthode est appliquée au fluide d'Oldroyd-B en écoulement dans une contraction plane 4:1 à coins arrondis. / This thesis concerns the development of a new Cartesian grid / immersed boundary (IB) method for the computation of incompressible viscous flows in two-dimensional irregular geometries. In IB methods, the computational grid is not aligned with the irregular boundary, and of upmost importance for accuracy and stability is the discretization in cells which are cut by the boundary, the so-called ``cut-cells''. In this thesis, we present a new IB method, called the LS-STAG method, which is based on the MAC method for staggered Cartesian grids and where the irregular boundary is sharply represented by its level-set function. This implicit representation of the immersed boundary enables us to calculate efficiently the geometry parameters of the cut-cells. We have achieved a novel discretization of the fluxes in the cut-cells by enforcing the strict conservation of total mass, momentum and kinetic energy at the discrete level. Our discretization in the cut-cells is consistent with the MAC discretization used in Cartesian fluid cells, and has the ability to preserve the 5-point Cartesian structure of the stencil, resulting in a highly computationally efficient method. The accuracy and robustness of our method is assessed on canonical flows at low to moderate Reynolds number~: Taylor Couette flow, flows past a circular cylinder, including the case where the cylinder has forced oscillatory rotations. We extend the \em LS-STAG \em method to the handling of moving immersed boundaries and present some results for the transversely oscillating cylinder flow in a free-stream. Finally, we present the first IB method that handles flows of viscoelastic fluids. The discretization of the constitutive law equation is based on the \em LS-STAG \em method and on the use of a fully staggered arrangement of unknowns, which ensures a strong coupling between all flow variables in the whole domain. The resulting method is applied to the flow of an Oldroyd-B fluid in a 4:1 planar contraction with rounded corner.

Numerical simulation of cell motility is one of the difficult problems in computational science. It belongs to a class of problems which involve moving interfaces between flowing or deforming media. Different numerical techniques are being developed for different application areas and in this work an attempt is made to apply two popular numerical techniques used in the field of computational multiphase flows to a cell motility problem. An unsteady cell motility problem is considered to simulate numerically based on a two-dimensional mathematical model. Two important numerical methods, the Level set method and the Front tracking methods are applied to the cell motility problem to study several cases and to verify the convergence of the solution. With the assumption of no mechanical or physical obstructions to the cell, the results of the numerical simulations show that the domain shapes converge to a circular shape as they reach the steady state condition. The final steady state velocities with which the domains move and the final steady state area to which they converge are observed to be independent of domain shapes. Moreover all shapes converge to exactly same radius of circle and move with same velocity after reaching steady state condition.

A finite element (FE) based level set method is proposed for structural topology optimization problems in this thesis. The level set method has become a popular tool for structural topology optimization in recent years because of its ability to describe smooth structure boundaries and handle topological changes. There are commonly two stages in the optimization process: the stress analysis stage and the boundary evolution stage. The first stage is usually performed with the finite element method (FEM) while the second is often realized by solving the level set equation with the finite difference method (FDM). The first motivation for developing the proposed method is the desire to unify the techniques of both stages within a uniform framework. In addition, there are many problems involving irregular design domains in practice, the FEM is more powerful than the FDM in dealing with these problems. This is the second motivation for this study. / Numerical examples are involved in this thesis to illustrate the reliability of the proposed method. Problems on both regular and irregular design domains are considered and different meshes are tested and compared. / Solving the level set equation with the standard Galerkin FEM might produce unstable results because of the hyperbolic characteristic of this equation. Therefore, the streamline diffusion finite element method (SDFEM), a stabilized method, is employed to solve the level set equation. In addition to the advantage of simplicity, this method generates a system of equations with a constant, symmetric, and positive defined coefficient matrix. Furthermore, this matrix can be diagonalized by virtue of the lumping technique in structural dynamics. This makes the cost in solving and storing quite low. It is more important that the lumped coefficient matrix may help to improve the stability under some circumstance. / The accuracy of the finite element based level set method (FELSM) is compared with that of the finite difference based level set method (FDLSM). The FELSM is a first-order accurate algorithm but we prove that its accuracy is enough for structural optimization problems considered in this study. Even higher-order accurate FDLSM schemes are used, the numerical results are still the same as those obtained by FELSM. It is also shown that if the Courant-Friedreichs-Lewy (CFL) number is large, the FELSM is more robust and accurate than FDLSM. / The reinitialization equation is also solved with the SDFEM and an extra diffusion term is added to improve the stability near the boundary. We propose a criterion to select the factor of the diffusion term. Due to numerical errors and the diffusion term, boundary will drift during the process of reinitialization. To constrain the boundary from moving, a Dirichlet boundary condition is enforced. Within the framework of FEM, this enforcement can be conveniently preformed with the Lagrangian multiplier method or the penalty method. / Velocity extension is discussed in this thesis. A natural extension method and a partial differential equation (PDE)-based extension method are introduced. Some related topics, such as the "ersatz" material approach and the recovery of stresses, are discussed as well. / Xing, Xianghua. / Adviser: Michael Yu Wang. / Source: Dissertation Abstracts International, Volume: 71-01, Section: B, page: 0628. / Thesis (Ph.D.)--Chinese University of Hong Kong, 2009. / Includes bibliographical references (leaves 102-113). / Electronic reproduction. Hong Kong : Chinese University of Hong Kong, [2012] System requirements: Adobe Acrobat Reader. Available via World Wide Web. / Electronic reproduction. Ann Arbor, MI : ProQuest Information and Learning Company, [200-] System requirements: Adobe Acrobat Reader. Available via World Wide Web. / Abstracts in English and Chinese.

Lui, Fung Yee. / Thesis (M.Phil.)--Chinese University of Hong Kong, 2008. / Includes bibliographical references (leaves 83-92). / Abstracts in English and Chinese. / Acknowledgement --- p.iii / Abbreviation --- p.xii / Chapter 1 --- Introduction --- p.1 / Chapter 1.1 --- Background --- p.1 / Chapter 1.2 --- Related Work --- p.6 / Chapter 1.2.1 --- Parametric Optimization Method and Radial Basis Functions --- p.6 / Chapter 1.3 --- Contribution and Organization of the Dissertation --- p.7 / Chapter 2 --- Level Set Method for Structure Shape and Topology Optimization --- p.8 / Chapter 2.1 --- Primary Ideas of Shape and Topology Optimization --- p.8 / Chapter 2.2 --- Level Set models of implicit moving boundaries --- p.11 / Chapter 2.2.1 --- Representation of the Boundary via Level Set Method --- p.11 / Chapter 2.2.2 --- Hamilton-Jacobin Equations --- p.13 / Chapter 2.3 --- Numerical Techniques --- p.13 / Chapter 2.3.1 --- Sign-distance function --- p.14 / Chapter 2.3.2 --- Discrete Computational Scheme --- p.14 / Chapter 2.3.3 --- Level Set Surface Re-initialization --- p.16 / Chapter 2.3.4 --- Velocity Extension --- p.16 / Chapter 3 --- Structure Topology Optimization with Discrete Level Sets --- p.18 / Chapter 3.1 --- A Level Set Method for Structural Shape and Topology Optimization --- p.18 / Chapter 3.1.1 --- Problem Definition --- p.18 / Chapter 3.2 --- Shape Derivative: an Engineering-oriented Deduction --- p.21 / Chapter 3.2.1 --- Sensitivity Analysis --- p.23 / Chapter 3.2.2 --- Optimization Algorithm --- p.28 / Chapter 3.3 --- Limitations of Discrete Level Set Method --- p.30 / Chapter 4 --- RBF based Parametric Level Set Method --- p.32 / Chapter 4.1 --- Introduction --- p.32 / Chapter 4.2 --- Radial Basis Functions Modeling --- p.33 / Chapter 4.2.1 --- Inverse Multiquadric (IMQ) Radial Basis Functions --- p.38 / Chapter 4.3 --- Parameterized Level Set Method in Structure Topology Optimization --- p.39 / Chapter 4.4 --- Parametric Shape and Topology Structure Optimization Method with Radial Basis Functions --- p.42 / Chapter 4.4.1 --- Changing Coefficient Method --- p.43 / Chapter 4.4.2 --- Moving Knot Method --- p.45 / Chapter 4.4.3 --- Combination of Changing Coefficient and Moving Knot method --- p.46 / Chapter 4.5 --- Numerical Implementation --- p.48 / Chapter 4.5.1 --- Sensitivity Calculation --- p.48 / Chapter 4.5.2 --- Optimization Algorithms --- p.49 / Chapter 4.5.3 --- Numerical Examples --- p.52 / Chapter 4.6 --- Summary --- p.65 / Chapter 5 --- Conclusion and Future Work --- p.80 / Chapter 5.1 --- Conclusion --- p.80 / Chapter 5.2 --- Future Work --- p.81 / Bibliography --- p.83

This thesis presents a framework for simulating the fluid dynamical behavior of complex moving boundary problems in a high-performance computing environment. The framework is implemented in the pELAFINT3D software package. Moving boundaries are evolved in a seamless fashion through the use of distributed narrow band level set methods and the effect of moving boundaries is incorporated into the flow solution by a novel Cartesian grid method. The proposed Cartesian grid approach builds on the concept of a ghost fluid method where boundary conditions are applied through least-squares polynomial extrapolations. The method is hybridized such that computational cells adjacent to moving boundaries change discretization schemes smoothly in time to avoid the introduction of strong oscillations in the pressure field. The hybridization is shown to have minimal effect on accuracy while significantly suppressing pressure oscillations. The computational capability of the Cartesian grid approach is enhanced with a massively parallel adaptive meshing algorithm. Local mesh enrichment is effected through the use of octree refinement, and a scalable mesh pruning algorithm is used to reduce the memory footprint of the Cartesian grid for geometries which are not well bounded by a rectangular cuboid. The computational work is kept in a well-balanced state through the use of an adaptive repartitioning strategy. The numerical scheme is validated against many benchmark problems and the composite approach is demonstrated to work well on tens of thousands of computational cores. A simulation of the closure phase of a mechanical heart valve was carried out to demonstrate the ability of the pELAFINT3D package to compute high Reynolds number flows with complex moving boundaries and a wide disparity in length scales. Finally, a novel image-to-computation algorithm was implemented to demonstrate the flexibility the current method allows in designing new applications.

The subject of this thesis is the development of an accurate, general and robust numerical method capable of predicting the flow behavior of two-phase immiscible fluids, separated by a well defined interface. In the quest of an accurate and robust numerical method for the modeling of two-phase flows, one has to keep in mind the intrinsic properties and difficulties associated with the problem: (i) those flows are mostly three-dimensional, (ii) some flows are steady, others unsteady, (iii) the interface might encounter a lot of topology changes (like merger or break-up), (iv) large jumps of density and viscosity might exist across the interface (e.g. ratio of density of 1/1000 for water and air), (v) surface tension forces may play a very important role in the interface dynamics. Hence, the influence of this force should be accurately evaluated and incorporated into the model, (vi) mass conservation is of primary importance. All these issues are addressed in this thesis, and special techniques are proposed for their treatment, which enables to construct the desired computational method. The chosen computational method combines a pressure stabilized finite element method for the Navier Stokes equations with a discontinuous Galerkin (DG) method for the level set equation. Such a combination of those two numerical methods results in a simple and effective algorithm that allows to simulate diverse flow regimes presenting large density and viscosity ratios (ratio up to 1/1000).

Simultaneous Localization and Mapping (SLAM) algorithms are used by autonomous robots to build or update maps of an environment while maintaining their position simultaneously. A fundamental open problem in SLAM is the e ective representation of the map in unknown, ambiguous, complex, dynamic environments. Representing such environments in a suitable manner is a complex task. Existing approaches to SLAM use map representations that store individual features (range measurements, image patches, or higher level semantic features) and their locations in the environment. The choice of how the map is represented produces limitations which in many ways are unfavourable for application in real-world scenarios. In this thesis, a new approach to SLAM is explored that rede nes sensing and robot motion as acts of deformation of a di erentiable surface. Distance elds and level set methods are utilized to de ne a parallel to the components of the SLAM estimation process and an algorithm is developed and demonstrated. The variational framework developed is capable of representing complex dynamic scenes and spatially varying uncertainty for sensor and robot models. / UOIT

Zugl.: Aachen, Techn. Hochsch., Diss., 2008