The perfectly matched layer (PML) is a novel technique to simulate the absorption of waves in unbounded domains. The underlying equations are often a system of second order hyperbolic partial differential equations. In the numerical treatment, second order systems are often rewritten and solved as first order systems. There are several benefits with solving the equations in second order formulation, though. However, while the theory and numerical methods for first order hyperbolic systems are well developed, numerical techniques to solve second order hyperbolic systems are less complete. We construct a strongly well-posed PML for second order systems in two space dimensions, focusing on the equations of linear elasto-dynamics. In the continuous setting, the stability of both first order and second order formulations are linearly equivalent. We have found that if the so-called geometric stability condition is violated, approximating the first order PML with standard central differences leads to a high frequency instability at most resolutions. In the second order setting growth occurs only if growing modes are well resolved. We determine the number of grid points that can be used in the PML to ensure a discretely stable PML, for several anisotropic elastic materials. We study the stability of the PML for problems where physical boundaries are important. First, we consider the PML in a waveguide governed by the scalar wave equation. To ensure the accuracy and the stability of the discrete PML, we derived a set of equivalent boundary conditions. Second, we consider the PML for second order symmetric hyperbolic systems on a half-plane. For a class of stable boundary conditions, we derive transformed boundary conditions and prove the stability of the corresponding half-plane problem. Third, we extend the stability analysis to rectangular elastic waveguides, and demonstrate the stability of the discrete PML. Building on high order summation-by-parts operators, we derive high order accurate and strictly stable finite difference approximations for second order time-dependent hyperbolic systems on bounded domains. Natural and mixed boundary conditions are imposed weakly using the simultaneous approximation term method. Dirichlet boundary conditions are imposed strongly by injection. By constructing continuous strict energy estimates and analogous discrete strict energy estimates, we prove strict stability.

<p>A numerical high order difference method is developed for solution of the incompressible Navier-Stokes equations. The solution is determined on a staggered curvilinear grid in two dimensions and by a Fourier expansion in the third dimension. The description in curvilinear body-fitted coordinates is obtained by an orthogonal mapping of the equations to a rectangular grid where space derivatives are determined by compact fourth order approximations. The time derivative is discretized with a second order backward difference method in a semi-implicit scheme, where the nonlinear terms are linearly extrapolated with second order accuracy.</p><p>An approximate block factorization technique is used in an iterative scheme to solve the large linear system resulting from the discretization in each time step. The solver algorithm consists of a combination of outer and inner iterations. An outer iteration step involves the solution of two sub-systems, one for prediction of the velocities and one for solution of the pressure. No boundary conditions for the intermediate variables in the splitting are needed and second order time accurate pressure solutions can be obtained.</p><p>The method has experimentally been validated in earlier studies. Here it is validated for flow past a circular cylinder as an example of a physical test case and the fourth order method is shown to be efficient in terms of grid resolution. The method is applied to external flow past a parabolic body and internal flow in an asymmetric diffuser in order to investigate the performance in two different curvilinear geometries and to give directions for future development of the method. It is concluded that the novel formulation of boundary conditions need further investigation.</p><p>A new iterative solution method for prediction of velocities allows for larger time steps due to less restrictive convergence constraints.</p>

A numerical high order difference method is developed for solution of the incompressible Navier-Stokes equations. The solution is determined on a staggered curvilinear grid in two dimensions and by a Fourier expansion in the third dimension. The description in curvilinear body-fitted coordinates is obtained by an orthogonal mapping of the equations to a rectangular grid where space derivatives are determined by compact fourth order approximations. The time derivative is discretized with a second order backward difference method in a semi-implicit scheme, where the nonlinear terms are linearly extrapolated with second order accuracy. An approximate block factorization technique is used in an iterative scheme to solve the large linear system resulting from the discretization in each time step. The solver algorithm consists of a combination of outer and inner iterations. An outer iteration step involves the solution of two sub-systems, one for prediction of the velocities and one for solution of the pressure. No boundary conditions for the intermediate variables in the splitting are needed and second order time accurate pressure solutions can be obtained. The method has experimentally been validated in earlier studies. Here it is validated for flow past a circular cylinder as an example of a physical test case and the fourth order method is shown to be efficient in terms of grid resolution. The method is applied to external flow past a parabolic body and internal flow in an asymmetric diffuser in order to investigate the performance in two different curvilinear geometries and to give directions for future development of the method. It is concluded that the novel formulation of boundary conditions need further investigation. A new iterative solution method for prediction of velocities allows for larger time steps due to less restrictive convergence constraints. / QC 20101221

Tandis que les humains explorent la nature sans relâche, ils font également attention à être plus conscients d'eux-mêmes, à mieux connaître ce qui les entourent, et, par exemple, à être informés de leurs positions, vitesses, ou trajectoires où qu'ils se trouvent. Les systèmes de positionnement par satellites (GNSS) ont fourni une manière de le faire à l'extérieur, devenant un assistant indispensable pour nous. Après le succès de GPS et GLONASS, Galileo et BeiDou sont actuellement en cours de déploiement, offrant plus de choix pour le positionnement autonome ou assisté. Toutefois, les signaux GNSS sont vulnérables aux obstacles: il n'y a presque pas de service GNSS à l'intérieur des bâtiments, dans les tunnels, ou dans les parkings souterrains; la continuité des services de positionnement par satellites n'est toujours pas assurée dans les "canyons urbains". Afin de faire face à ce problème, cette thèse est consacrée au positionnement en environnements contraints (où le GNSS n'est pas disponible), en y incluant la conception, les algorithmes et les technologies correspondantes. Un état de l'art est élaboré sur les systèmes de positionnement à radio ou à l'inertie. Parmi les technologies possibles, la discussion ne se limite pas à l'approche basée sur GNSS, mais elle est centrée sur cette dernière à cause de ses avantages et aussi de l’implication profonde de notre groupe de recherche dans ce domaine. Nous avons, d'une part, étudié la possibilité ainsi que les limites du positionnement avec une précision centimétrique en déployant notre système Repealite (i.e. un système de positionnement à l'intérieur d'une base des émetteurs GNSS spéciaux); et proposé, d'autre part, une méthode de localisation en groupe d'objets dynamiques communicants. Cette méthode procède d'une problématique délicate que l'on rencontre dans les approches GNSS, qu'est la détermination de la position initiale du récepteur. On montre en quoi elle permet également d'aller au-delà de la portée des approches classiques GNSS / While the human beings explore the nature tirelessly, they also put significant concerns to be aware of themselves, to know better of the circumstances, and to be informed with their precise positions, velocities, trajectories, and so on, in local environments. The Global Navigation Satellite Systems (GNSS) have provided an efficient method to do so outdoors, and have already become an indispensable assistant of many people. After the success of GPS and GLONASS, Galileo and BeiDou are currently under deployment, offering more choices of the independent or collaborative positioning. However, the GNSS signal is vulnerable to obstructions: almost no GNSS services are available inside buildings, tunnels, or underground parkings; the services are not always coherent in urban canyons. In order to address this problem, this thesis is dedicated to the design frameworks for the positioning in GNSS-challenged environments, as well as the corresponding algorithms and technologies. A brief survey of the latest radio-based and inertial positioning/tracking systems is provided. Among the feasible technologies, the discussion is centered on but not limited to the GNSS-based approach, which is due to the inherited advantages of this approach and also the deep engagement of our research group in this domain. We have, on one hand, explored the possibility and limitations on the centimeter-accuracy positioning with our Repealite system (i.e. a GNSS-base indoor positioning system with specific features); on the other hand, a method of batch localization for the nodes in a network of dynamic communicating objects is proposed, which is originated from an issue of the GNSS-based approach - the resolution of the receiver initial point, but then it goes beyond the scope of the “classical” GNSS-based approach