On s'interesse aux problèmes elliptiques avec coéficients variables à travers des interfaces intérieures. La solution et ses dérivées normales peuvent subir des variations significatives à travers les frontières intérieures. On présente une méthode compacte aux différences finies sur des maillages adaptés de type octree conçues pour une résolution en parallèle. L'idée principale est de minimiser l'erreur de troncature sur la discretisation locale, en fonction de la configuration du maillage, en rapprochant une convergence à l'ordre deux. On montrera des cas 2D et 3D des résultat liés à des applications concrètes. / We consider problems governed by a linear elliptic equation with varying coéficients across internal interfaces. The solution and its normal derivative can undergo significant variations through these internal boundaries. We present a compact finite-difference scheme on a tree-based adaptive grid that can be efficiently solved using a natively parallel data structure. The main idea is to optimize the truncation error of the discretization scheme as a function of the local grid configuration to achieve second order accuracy. Numerical illustrations relevant for actual applications are presented in two and three-dimensional configurations.

Many physical problems lead to boundary value problems for partial differential equations, which can be solved with the finite element method. In order to construct adaptive solution algorithms or to measure the error one aims at reliable a posteriori error estimators. Many such estimators are known, as well as their theoretical foundation. Some boundary value problems yield so-called anisotropic solutions (e.g. with boundary layers). Then anisotropic finite element meshes can be advantageous. However, the common error estimators for isotropic meshes fail when applied to anisotropic meshes, or they were not investigated yet. For rectangular or cuboidal anisotropic meshes a modified error estimator had already been derived. In this paper error estimators for anisotropic tetrahedral or triangular meshes are considered. Such meshes offer a greater geometrical flexibility. For the Poisson equation we introduce a residual error estimator, an estimator based on a local problem, several Zienkiewicz-Zhu estimators, and an L_2 error estimator, respectively. A corresponding mathematical theory is given.For a singularly perturbed reaction-diffusion equation a residual error estimator is derived as well. The numerical examples demonstrate that reliable and efficient error estimation is possible on anisotropic meshes. The analysis basically relies on two important tools, namely anisotropic interpolation error estimates and the so-called bubble functions. Moreover, the correspondence of an anisotropic mesh with an anisotropic solution plays a vital role. AMS(MOS): 65N30, 65N15, 35B25

1 Abstract: Time-dependent and three-dimensional flow of Newtonian fluid is studied in context of two biomechanical applications, flow in cerebral aneurysms and flow in stenotic valves. In the first part of the thesis, the computational meshes obtained from the medical imaging techniques are used for the computation of hemodynamic parameters associated with the rupture potency of the cerebral aneurysms. The main result is the computation within twenty geometries of aneurysms. It is shown that the aneurysm size has more important role in wall shear stress distribution than the fact whether the aneurysm is ruptured or unruptured. The second part of the thesis is addressed to the flow in stenotic valves. It is shown that the method cur- rently used in medical practice is based on assumptions which are too restrictive to be apply to blood flow in the real case. The full continuum mechanics model is presented with physiologically relevant boundary conditions and it is shown that results are consistent with measured data obtained from literature. Then we focus on the obtaining the pressure field from the velocity field. The presented method provides more accurate pressure approximation than commonly used Pressure Poisson Equation. The last chapter of the thesis is dedicated to Nitsche's method for treating slip boundary...

Cette thèse porte sur la simulation tridimensionnelle (3D) des décharges couronnes à l'aide du calcul haute performance. Lorsqu'on applique une impulsion de haute tension entre une pointe et un plan, les lignes de champ électrique fortement resserrées autour de la pointe induisent la propagation simultanée de plusieurs streamers et la formation d'une décharge couronne de structure arborescente. Dans ces conditions, seule une simulation électro-hydrodynamique 3D est apte à reproduire cette structure et fournir les ordres de grandeur de l'énergie déposée et de la concentration des espèces créées durant la phase de décharge. Cependant, cette simulation 3D est très consommatrice en temps et mémoire de calcul et n'est désormais accessible que grâce à l'accroissement permanent de la puissance des ordinateurs dédié au calcul haute performance. Dans le cadre d'une simulation électro-hydrodynamique 3D, une attention particulière doit être prise concernant l'efficacité des solveurs à résoudre les équations elliptiques 3D car leur contribution en termes de temps de calcul peut dépasser 80% du temps global de la simulation. Ainsi, une partie de manuscrit est consacrée aux tests de performances de méthodes de résolution d'équations elliptiques directes ou itératives telle que SOR R&B, BiCGSTAB et MUMPS, en utilisant le calcul massivement parallèle et les librairies MPI. Les calculs sont réalisés sur le supercalculateur EOS du réseau CALMIP, avec un nombre de cœurs de calcul allant jusqu'à 1800, et un nombre de mailles atteignant 8003 (soit plus 1/2 Milliard de mailles). Les tests de performances sont réalisés en statique sur le calcul du potentiel géométrique et en dynamique en propageant une densité de charge d'espace analytique caractéristique des streamers. Pour réaliser une simulation complète 3D de la décharge il faut également intégrer au programme un algorithme capable de résoudre les équations de transport de particule chargée à fort gradients de densité caractéristiques aux streamers. Dans ce manuscrit, l'algorithme MUSCL est testé dans différentes conditions de propagation d'un cube de densité (à vitesse homogène ou non homogène spatialement) afin d'optimiser le transport des densités d'espèces chargées impliquées. Le code 3D, conçu pour résoudre le modèle électro- hydrodynamique complet de la décharge (couplant les équations de transport, de Poisson et de cinétique réactionnelle) est ensuite validé par la confrontation des résultats 3D et 2D dans une condition de simulation présentant une symétrie de révolution autour de l'axe de propagation d'un streamer. Enfin, les premiers résultats des simulations 3D de la phase décharge avec la propagation d'un ou plusieurs streamers asymétriques sont présentés et analysés. Ces simulations permettent de suivre la structure arborescente de la décharge lorsqu'on applique une tension pulsée entre une pointe et un plan. L'initiation de la structure arborescente est étudiée en fonction de la position de spots plasmas et de leur influence sur l'amorçage des streamers. / This work is devoted to the three dimensional (3D) simulation of streamer corona discharges in air at atmospheric pressure using high-performance parallel computing. When a pulsed high-voltage is applied between a tip and a plane in air, the strong electric field lines constricted around the tip induce the simultaneous propagation of several streamers leading to a corona discharge with a tree structure. Only a true 3D electro-hydrodynamics simulation is able to reproduce this branching and to provide the orders of magnitude of the local deposited energy and the concentration of the species created during the discharge phase. However, such a 3D simulation which requires large computational memory and huge time calculation is nowadays accessible only when performed with massively parallel computation. In the field of 3D electro-hydrodynamics simulations, a special attention must be paid to the efficiency of solvers in solving 3D elliptic equations because their contribution can exceed 80% of the global computation time. Therefore, a specific chapter is devoted to test the performance of iterative and direct methods (such as SOR R&B, BiCGSTAB and MUMPS) in solving elliptic equations, using the massively parallel computation and the MPI library. The calculations are performed on the supercomputer EOS of the CALMIP network, with a number of computing cores and meshes increasing up to respectively 1800 and 8003 (i.e. more than 1/2 Billion meshes). The performances are compared for the calculation of the geometric potential and in a dynamic simulation conditions consisting in the propagation of an analytical space charge density characteristic of the streamers. To perform a complete 3D simulation of the streamer discharge, must also involve a robust algorithm able to solve the coupled conservation equations of the charged particle density with very sharp gradients characteristic of the streamers. In this manuscript, the MUSCL algorithm is tested under different propagation conditions of a cubic density (with uniform or non-uniform velocity field). The 3D code, designed to solve the complete electro-hydrodynamics model of the discharge (coupling the conservation equations, the Poisson equation and the chemical kinetics) is validated by comparing the 3D and 2D results in a simulation conditions presenting a rotational symmetry around the propagation axis of a mono-filamentary streamer. Finally, the first results of the 3D simulations of the discharge phase with the propagation of one or several asymmetric streamers are presented and analyzed. These simulations allow to follow the tree structure of a corona discharge when a pulsed voltage is applied between a tip and a plane. The ignition of the tree structure is studied as a function of the initial position of the plasma spots.

[pt] A equação diferencial parcial de Poisson é de fundamental importância em várias áreas de pesquisa, dentre elas: matemática, física e engenharia. Para resolvê-la numericamente utilizam-se vários métodos, tais como os já tradicionais métodos das diferenças finitas e dos elementos finitos. Este trabalho propõe um método para resolver a equação de Poisson, utilizando uma abordagem de sistema de partículas conhecido como SPH, do inglês Smoothed Particles Hydrodynamics. O método proposto para a solução da equação de Poisson e os operadores diferenciais discretos definidos no método SPH, chamados de operadores SPH, são utilizados neste trabalho em duas aplicações: na decomposição de campos vetoriais; e na simulação numérica de escoamentos de fluidos monofásicos e bifásicos utilizando a equação de Navier-Stokes. / [en] Poisson`s equation is of fundamental importance in many research areas in engineering and the mathematical and physical sciences. Its numerical solution uses several approaches among them finite differences and finite elements. In this work we propose a method to solve Poisson`s equation using the particle method known as SPH (Smoothed Particle Hydrodynamics). The proposed method together with an accurate analysis of the discrete differential operators defined by SPH are applied in two related situations: the Hodge-Helmholtz vector field decomposition and the numerical simulation of the Navier-Stokes equations.

Les résultats présentés dans cette thèse portent sur la construction et la simulation numérique de modèles théoriques de plasmas en présence d'une paroi absorbante. Ces modèles se basent sur des systèmes de Vlasov-Poisson ou Vlasov-Ampère à deux espèces en présence de conditions limites. Les solutions stationnaires recherchées vérifient l'équilibre des flux de charges dans la direction perpendiculaire à la paroi. Cette propriété s'appelle l'ambipolarité. A travers l'étude d'une équation de Poisson non linéaire, on montre le caractère bien posé d'un système de Vlasov-Poisson stationnaire 1d-1v pour lequel on détermine des distributions de particules entrantes et un potential au mur qui induisent l'ambipolarité et une densité de charge positive. On donne également une estimation de la taille de la couche limite au mur. Ces résultats sont illustrés numériquement. On prouve ensuite la stabilité linéaire des solutions stationnaires électroniques pour un modèle de Vlasov-Ampère instationnaire. Enfin, on étudie un modèle de Vlasov-Poisson stationnaire 1d-3v en présence d'un champ magnétique constant et parallèle à la paroi. On détermine les distributions de particules entrantes et un potentiel au mur qui induisent l'ambipolarité. On étudie une équation de Poisson non linéaire associée au modèle à l'aide d'une fonctionnelle non linéaire d'énergie qui admet des minimiseurs. On établit des bornes de paramètres à l'intérieur desquelles notre modèle s'applique et on propose une interprétation des résultats. / This thesis focuses on the construction and the numerical simulation theoretical models of plasmas in interaction with an absorbing wall. These models are based on two species Vlasov-Poisson or Vlasov-Ampère systems in the presence of boundary conditions. The expected stationary solutions must verify the balance of the flux of charges in the orthogonal direction to the wall. This feature is called the ambipolarity.Through the study of a non linear Poisson equation, we prove the well-posedness of 1d-1v stationary Vlasov-Poisson system, for which we determine incoming particles distributions and a wall potential that induces the ambipolarity as well as a non negative charge density hold. We also give a quantitative estimates of the thickness of the boundary layer that develops at the wall. These results are illustrated numerically. We prove the linear stability of the electronic stationary solution for a non-stationary Vlasov-Ampère system. Finally, we study a 1d-3v stationary Vlasov-Poisson system in the presence of a constant and parallel to the wall magnetic field . We determine incoming particles distributions and a wall potential so that the ambipolarity holds. We study a non linear Poisson equation through a non linear functional energy that admits minimizers. We established some bounds on the numerical parameters inside which, our model is relevant and we propose an interpretation of the results.

This thesis extends the utility of the Monte Carlo approach to PDE-based methods presented in the paper Monte Carlo Geometry Processing. In particular, we implement this method on the GPU using CUDA, and investigate more viable methods of estimating the source integral when solving Poisson’s equation with intricate source terms. This is the case for a large group of gradient-domain methods in computer graphics, where source terms are represented by discrete volumetric data on regular grids. We develop unbiased source integral estimators like image-based importance sampling (IBIS) and biased estimators like source integral caching (SIC) and evaluate these against existing GPU-accelerated finite difference solvers for gradient-domain applications. By decoupling the source integration step from the WoS-algorithm, we find that the SIC method can improve performance by several orders of magnitude, making it competitive with existing finite difference solvers in many cases. We further investigate the viability of distance fields for accelerated distance queries and find that these can provide significant performance improvements compared to BVHs without meaningfully affecting bias. / <p>Examensarbetet är utfört vid Institutionen för teknik och naturvetenskap (ITN) vid Tekniska fakulteten, Linköpings universitet</p>

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Electrodialysis performance strongly depends on the boundary layer near ion exchange membranes. The thickness of the boundary layer has not been clearly evaluated due to its substantial fluctuation around the spacer geometry. In this study, the boundary layer thickness was defined with three statistical parameters: the mean, standard deviation, and correlation coefficient between the two boundary layers facing across the spacer. The relationship between the current and potential under conditions of the competitive transport between mono- and di-valent cations was used to estimate the statistical parameters. An uncertainty model was developed for the steady-state ionic transport in a two-dimensional cell pair. Faster ionic separations were achieved with smaller means, greater standard deviations, and more positive correlation coefficients. With the increasing flow velocity from 1.06 to 4.24 cm/s in the bench-scale electrodialyzer, the best fit values for the mean thickness reduced from 40 to less than 10 μm, and the standard deviation was in the same order of magnitude as the mean. For the partitioning of mono- and di-valent cations, a CMV membrane was examined in various KCl and CaCl₂ mixtures. The equivalent fraction correlation and separation factor responded sensitively to the composition of the mixture; however, the selectivity coefficient was consistent over the range of aqueous-phase ionic contents between 5 and 100 mN and the range of equivalent fractions of each cation between 0.2 and 0.8. It was shown that small analytic errors in measuring the concentration of the mono-valent cation are amplified when estimating the selectivity coefficient. To minimize the effects of such error propagation, a novel method employing the least square fitting was proposed to determine the selectivity coefficient. Each of thermodynamic factors, such as the aqueous- and membrane-phase activity coefficients, water activity, and standard state, was found to affect the magnitude of the selectivity coefficient. The overlimiting current, occurring beyond the electroneutral limit, has not been clearly explained because of the difficulty in solving the singularly perturbed Nernst-Planck-Poisson equations. The steady-state Nernst-Planck-Poisson equations were converted into the Painlevé equation of the second kind (P[subscript II] equation). The converted model domain is explicitly divided into the space charge and electroneutral regions. Given this property, two mathematical formulae were proposed for the limiting current and the width of the space charge region. The Airy solution of the P[subscript II] equation described the ionic transport in the space charge region. By using a hybrid numerical scheme including the fixed point iteration and Newton Raphson methods, the P[subscript II] equation was successfully solved for the ionic transport in the space charge and electroneutral regions as well as their transition zone. Above the limiting current, the sum of the ionic charge in the aqueous-phase electric double layer and in the space charge region remains stationary. Thus, growth of the space charge region involves shrinkage of the aqueous-phase electric double layer. Based on this observation, a repetitive mechanism of expansion and shrinkage of the aqueous-phase electric double layer was suggested to explain additional current above the limiting current. / text

Die experimentellen Untersuchungen zur Strömungs-Struktur-Interaktion kommen zu folgenden Ergebnissen. Die als entrainment bezeichnete Verhaltensweise von Fischen kann durch den gezielten Ausgleich von Auftriebs-, Widerstands- und einer erstmals nachgewiesenen Saugkraft und der damit verbundenen Reduzierung der lokomotorischen Energie erklärt werden. Des Weiteren zeigen die Experimente an einer wellenförmig bewegten Oberfläche, dass die Strömung an einem Wellenberg zyklisch zwischen laminarem und turbulentem Regime wechselt und dass diese Oszillation zur Reduzierung des Form- und Gesamtwiderstandes der geschleppten Oberfläche führt. Dünne, flexible Häutchen hairy flaps an der Rückseite eines Zylinders führen zu einer Änderung der Struktur der Wirbelablösung an diesem. Dadurch wird sowohl die auf den Zylinder wirkende Auftriebskraft als auch die Widerstandskraft um bis zu 65% reduziert. Für die Interaktion zwischen der Strömung um zwei hintereinander positionierte elastische Zylinder und ihrer Kinematik konnte die Synchronisierung (lock-in) ihrer Bewegung mit einer verbundenen Zerstörung der Wirbelstraße hinter dem zweiten Zylinder gezeigt werden. / The experimental investigations presented herein explain the behavioural adaptation of fish called entrainment for the first time. The results confirm a balance of lift-, drag- and a suction-force, explaining the reduction of locomotive energy. Furthermore, flow measurements around an undulating membrane affirm an oscillation between laminar and turbulent flow over one period of the motion and that this oscillation decreases the pressure- and drag-force of the towed membrane. Experiments on thin and flexible flaps attached at the lee-side of a cylinder, show that the flaps alter the natural vortex separation cycle in such a way that the vortices do not shed in a staggered side-by-side arrangement but in line in a row with the cylinder wake axis. Thus, flow fluctuations are reduced by 42% in stream-wise - and 35% in transversal direction at best, compared to a reference case without hairy-flaps. Finally, investigations on the flow around and on the kinematics of two flexible cylinders in a tandem arrangement demonstrate a synchronisation of their motion (lock-in), resulting in the destruction of the vortex-street behind the second cylinder.