Induced seismicity (earthquakes caused by injection or extraction of fluids in Earth's subsurface) is a major, new hazard in the United States, the Netherlands, and other countries, with vast economic consequences if not properly managed. Addressing this problem requires development of predictive simulations of how fluid-saturated solids containing frictional faults respond to fluid injection/extraction. Here we present a numerical method for linear poroelasticity with rate-and-state friction faults. A numerical method for approximating the fully coupled linear poroelastic equations is derived using the summation-by-parts-simultaneous-approximation-term (SBP-SAT) framework. Well-posedness is shown for a set of physical boundary conditions in 1D and in 2D. The SBP-SAT technique is used to discretize the governing equations and show semi-discrete stability and the correctness of the implementation is verified by rigorous convergence tests using the method of manufactured solutions, which shows that the expected convergence rates are obtained for a problem with spatially variable material parameters. Mandel's problem and a line source problem are studied, where simulation results and convergence studies show satisfactory numerical properties. Furthermore, two problem setups involving fault dynamics and slip on faults triggered by fluid injection are studied, where the simulation results show that fluid injection can trigger earthquakes, having implications for induced seismicity. In addition, the results show that the scheme used for solving the fully coupled problem, captures dynamics that would not be seen in an uncoupled model. Future improvements involve imposing Dirichlet boundary conditions using a different technique, extending the scheme to handle curvilinear coordinates and three spatial dimensions, as well as improving the high-performance code and extending the study of the fault dynamics.

This doctoral thesis is concerned with some theoretical and practical questions related to backward stochastic differential equations (BSDEs) and more specifically their connection with some parabolic partial differential equations (PDEs). The thesis is made of three parts. In the first part, we study the probabilistic representation for a class of multidimensional PDEs with quadratic nonlinearities of a special form. We obtain a representation formula for the PDE solution in terms of the solutions to a Lipschitz BSDE. We then use this representation to obtain an estimate on the gradient of the PDE solutions by probabilistic means. In the course of our analysis, we are led to prove some results for the associated multidimensional quadratic BSDEs, namely an existence result and a partial uniqueness result. In the second part, we study the well-posedness of a very general quadratic reflected BSDE driven by a continuous martingale. We obtain the comparison theorem, the special comparison theorem for reflected BSDEs (which allows to compare the increasing processes of two solutions), the uniqueness and existence of solutions, as well as a stability result. The comparison theorem (from which uniqueness follows) and the special comparison theorem are obtained through natural techniques and minimal assumptions. The existence is based on a perturbative procedure, and holds for a driver whis is Lipschitz, or slightly-superlinear, or monotone with arbitrary growth in y. Finally, we obtain a stability result, which gives in particular a local Lipschitz estimate in BMO for the martingale part of the solution. In the third and last part, we study the time-discretization of BSDEs having nonlinearities that are monotone but with polynomial growth in the primary variable. We show that in that case, the explicit Euler scheme is likely to diverge, while the implicit scheme converges. In fact, by studying the family of θ-schemes, which are mixed explicit-implicit, θ characterizing the degree of implicitness, we find that the scheme converges when the implicit component is dominant (θ ≥ 1/2 ). We then propose a tamed explicit scheme, which converges. We show that the implicit-dominant schemes with θ > 1/2 and our tamed explicit scheme converge with order 1/2 , while the trapezoidal scheme (θ = 1/2) converges with order 7/4.

In this thesis, we construct a new hydrodynamical algorithm able of handling general compressible reactive flow problems, based on a finite-volume method inspired by the original MUSCL scheme of van Leer (1979). The algorithm is of second-order in the smooth part of the flow and avoids dimensional splitting. It uses MPI to achieve parallelism, and includes an astrophysical equation of state and a nuclear reaction network. It proves to be robust to tests cases. In particular it reproduces quite well the reactive and non-reactive results obtained with two different numerical methods (Fryxell & al. 1989, Busegnies & al. 2007). Moreover the time-dependent results are in agreement with the corresponding steady state solution. This gives us confidence in applying it to an astrophysical situation which has never been studied, the propagation of a detonation in conditions relevant to superbursts. The algorithm is described in (Noel & al. 2007). In a firt step we obtain the detonation profiles in pure carbon and in a mixture of carbon and iron. In both cases we underline the large difference between the total reaction length and the length on which some species burn. This difference leads to enormous numerical difficulties because all the length scales cannot be resolved at the same time in a single simulation. We show that the carbon detonation might be studied in a partial resolution approach like the one of Gamezo & al. (1999). In a second step we construct a new reduced nuclear reaction network able to reproduce the energy production due to the photo-disintegrations of heavy elements, like ruthenium, which are thought to occur during superbursts in mixed H/He accreting systems. Using this new nuclear network we simulate detonations in mixture of carbon and ruthenium. An interesting feature is that, in this case, all the reaction lengths can be resolved in the same simulation. This makes the C/Ru detonations easier to study in future multi-dimensional simulations than the pure carbon ones (Noel & al. 2007b). Finally we perform some numerical experiments which show that our algorithm is able to deal with initially inhomogeneous medium, and that the multi-dimensional simulations are attainable even if they are quite computational time consuming. - B. Van Leer, J. Comp. Phys., 21, 101, 1979 - Fryxell, B.A., Muller, E., and Arnett, W.D., Technical report MPA 449, 1989 - Busegnies, Y., Francois, J. and Paulus, G., Shock Waves, 11, 2007 - Gamezo, V.N., Wheeler, J.C., Khokhlov, A.M., and Oran, E.S., ApJ, 512, 827, 1999 - Noël, C., Busegnies, Y., Papalexandris, M.V. & al., A&A, 470, 653, 2007 - Noël, C., Goriely, S., Busegnies, Y. & Papalexandris, M.V., submitted to A&A, 2007b / Un algorithme parallèle basé sur une méthode aux volumes finis inspirée du schéma MUSCL de Van Leer (1979) a été construit. Il a été développé sur base de la méthode de Lappas & al. (1999) qui permet de résoudre simultanément toutes les dimensions spatiales. Cette méthode se base sur la construction de surfaces appropriées dans l'espace-temps, le long desquelles les équations de bilan se découplent en équations plus simples à intégrer. Cet algorithme est actuellement le seul à éviter le "splitting" des dimensions spatiales. Dans les modèles conventionnels (PPM, FCT, etc.), l'intégration spatiale des équations est réalisée de manière unidimensionnelle pour chaque direction. Un réseau de réactions nucléaires ainsi qu'une équation d'état astrophysique ont été inclus dans l'algorithme et celui-ci a ensuite été soumis à une grande variété de cas tests réactifs et non réactifs. Il a été comparé à d'autres codes généralement utilisés en astrophysique (Fryxell & al. 1989, Fryxell & al. 2000, Busegnies & al. 2007) et il reproduit correctement leurs résultats. L'algorithme est décrit dans Noël & al. (2007). Sur base de cet algorithme, les premières simulations de détonation dans des conditions thermodynamiques représentatives des Superbursts ont été réalisées. Différentes compositions du milieu ont été envisagées (carbone pur, mélange de carbone et de fer, mélange de carbone et de cendres du processus rp). Dans la plupart des systèmes où des Superbursts ont été observés, la matière accrétée est un mélange d'hydrogène et d'hélium. Dans ce cas, des phases de combustion précédant le Superburst produisent des nucléides plus lourd que le fer (Schatz & al. 2003). Ces nucléides peuvent être photodésintégrés durant le Superburst. Pour prendre en compte ces réactions endothermiques de photodésintégration, nous avons construit un nouveau réseau réduit de réactions nucléaires qui a été incorporé dans l'algorithme hydrodynamique (Noël & al. 2007b). Ce réseau réduit reproduit globalement l'énergétique d'un réseau complet et a permis de faire la première simulation numérique de détonation dans des conditions caractéristiques de systèmes accréteurs d'un mélange hydrogène-hélium. Finallement quelques simulations multidimensionelles préliminaires ont éte réalisées. - Busegnies, Y., Francois, J. and Paulus, G., Shock Waves, 11, 2007 - Fryxell, B.A., Muller, E., and Arnett, W.D., Technical report MPA 449, 1989 - Fryxell, B.A., Olson, K., Ricker, P. & al., ApJS, 131, 273, 2000 - Lappas, T., Leonard, A. and Dimotakis, P.E., SIAM J. Sci. Comput., 20, 1064, 1999 - Noël, C., Busegnies, Y., Papalexandris, M.V. & al., A&A, 470, 653, 2007 - Noël, C., Goriely, S., Busegnies, Y. & Papalexandris, M.V., submitted to A&A, 2007b - Röpke, F. K. PhD thesis, Technischen Universitat Munchen, 2003 - Schatz, H., Bildsten, L., Cumming, A. and Ouellette, M., Nuclear Physics A, 718, 247, 2003 - Van Leer, B. Comp. Phys., 21, 101, 1979 - Weinberg, N.N. and Bildsten, L., ArXiv e-prints, 0706.3062, 2007

Mathematical models describing the Q-switched laser generation, which is a widely used laser technique for producing short intense pulses of light, belong to the class of semi-nonlinear models where only source terms nonlinearly depend on the solution. Numerical methods for solution of systems of semi-nonlinear partial differential equations have been extensively studied in many papers. Schrödinger-type equations, parabolic-type equations or general diffusion-reaction models arise in nonlinear optics. Such differential problems are solved mainly by finite-difference and Galerkin methods. The convergence analysis is based on the stability analysis of the linearized problems. The construction and theoretical analysis of discrete schemes for one-dimensional problem give a basis for a numerical solution of more general two-dimensional and three-dimensional problems where a diffraction process is taken into account. The two-dimensional problem simulates the dynamics of high-power semiconductor lasers. To solve the problems simulating propagation of photon fluxes in the nonlinear disperse medium, the finite-difference time-domain method is used. However, the major drawback of this method is that the computational domain must be sufficiently large. In order to restrict the computational domain and to solve the problem only in the region of interest, special artificial boundary conditions are investigated. The three-dimensional problem simulates an interaction of counter propagating... [to full text] / Disertacijoje nagrinėjami kai kurių lazerių fizikos ir netiesinės optikos uždavinių skaitinės analizės metodai. Tiriami trys pagrindiniai atvejai: bėgančias plokščias bangas aprašantis vienmatis, bėgančias difraguojančias bangas nagrinėjantis dvimatis ir lazerio pluoštų sąveiką netiesinėje Kero terpėje modeliuojantis trimatis modeliai. Šiuos uždavinius sieja pernešimo diferencialinės lygtys dalinėmis išvestinėmis, aprašančios į priešingas puses sklindančias lazerio bangas. Dvimačiame ir trimačiame uždaviniuose sprendžiamos dalinių išvestinių Šrėdingerio (ang. Schrödinger) tipo diferencialinės lygtys. Šiems matematiniams modeliams sudarytos baigtinių skirtumų schemos, atlikta jų analizė ir pagrindimas. Skaitinių eksperimentų realizacijai sukurti lygiagretieji algoritmai, jie yra būtini atliekant didelių resursų reikalaujančius skaičiavimus. Disertaciją sudaro įvadas, keturi skyriai, rezultatų apibendrinimas, naudotos literatūros ir autoriaus publikacijų disertacijos tema sarašai. Įvadiniame skyriuje aptariama tiriamoji problema, darbo aktualumas, aprašomas tyrimų objektas, formuluojamas darbo tikslas bei uždaviniai, aprašoma tyrimų metodika, darbo mokslinis naujumas, darbo rezultatų praktinė reikšmė, ginamieji teiginiai. Įvado pabaigoje pristatomos disertacijos tema autoriaus paskelbtos publikacijos ir pranešimai konferencijose bei disertacijos struktūra. Pirmasis skyrius skirtas mokslinės literatūros apžvalgai ir supažindinimui su netiesinės optikos sąvokomis bei... [toliau žr. visą tekstą]

Disertacijoje nagrinėjami kai kurių lazerių fizikos ir netiesinės optikos uždavinių skaitinės analizės metodai. Tiriami trys pagrindiniai atvejai: bėgančias plokščias bangas aprašantis vienmatis, bėgančias difraguojančias bangas nagrinėjantis dvimatis ir lazerio pluoštų sąveiką netiesinėje Kero terpėje modeliuojantis trimatis modeliai. Šiuos uždavinius sieja pernešimo diferencialinės lygtys dalinėmis išvestinėmis, aprašančios į priešingas puses sklindančias lazerio bangas. Dvimačiame ir trimačiame uždaviniuose sprendžiamos dalinių išvestinių Šrėdingerio (ang. Schrödinger) tipo diferencialinės lygtys. Šiems matematiniams modeliams sudarytos baigtinių skirtumų schemos, atlikta jų analizė ir pagrindimas. Skaitinių eksperimentų realizacijai sukurti lygiagretieji algoritmai, jie yra būtini atliekant didelių resursų reikalaujančius skaičiavimus. Disertaciją sudaro įvadas, keturi skyriai, rezultatų apibendrinimas, naudotos literatūros ir autoriaus publikacijų disertacijos tema sarašai. Įvadiniame skyriuje aptariama tiriamoji problema, darbo aktualumas, aprašomas tyrimų objektas, formuluojamas darbo tikslas bei uždaviniai, aprašoma tyrimų metodika, darbo mokslinis naujumas, darbo rezultatų praktinė reikšmė, ginamieji teiginiai. Įvado pabaigoje pristatomos disertacijos tema autoriaus paskelbtos publikacijos ir pranešimai konferencijose bei disertacijos struktūra. Pirmasis skyrius skirtas mokslinės literatūros apžvalgai ir supažindinimui su netiesinės optikos sąvokomis bei... [toliau žr. visą tekstą] / Mathematical models describing the Q-switched laser generation, which is a widely used laser technique for producing short intense pulses of light, belong to the class of semi-nonlinear models where only source terms nonlinearly depend on the solution. Numerical methods for solution of systems of semi-nonlinear partial differential equations have been extensively studied in many papers. Schrödinger-type equations, parabolic-type equations or general diffusion-reaction models arise in nonlinear optics. Such differential problems are solved mainly by finite-difference and Galerkin methods. The convergence analysis is based on the stability analysis of the linearized problems. The construction and theoretical analysis of discrete schemes for one-dimensional problem give a basis for a numerical solution of more general two-dimensional and three-dimensional problems where a diffraction process is taken into account. The two-dimensional problem simulates the dynamics of high-power semiconductor lasers. To solve the problems simulating propagation of photon fluxes in the nonlinear disperse medium, the finite-difference time-domain method is used. However, the major drawback of this method is that the computational domain must be sufficiently large. In order to restrict the computational domain and to solve the problem only in the region of interest, special artificial boundary conditions are investigated. The three-dimensional problem simulates an interaction of counter propagating... [to full text]

This thesis presents an analysis on a class of asymmetric imperfectly discrim- inating multi-prize contests with the aim to investigate when more than one prize becomes optimal prize allocation if the average effort is to be maximized. We present n-person model with heterogeneous contestants who compete for two, possibly different, prizes. The contestants may differ in their relative abil- ities, i.e., parameters affecting their probabilities to win either of the prizes. Two different numerical methods for finding pure strategy Nash equilibria are employed. Depending on particular distributions of the abilities, we find two possible scenarios when the second prize becomes optimal. Furthermore, we ad- dress an issue of existence and uniqueness of a pure strategy Nash equilibrium with respect to the returns to scale in effort parameter. JEL Classification C63, D72 Keywords Imperfectly discriminating contests; Heteroge- neous abilities; Multiple prizes; Numerical meth- ods Author's e-mail lida.matyskova@centrum.cz Supervisor's e-mail gregor@fsv.cuni.cz

Le travail de cette thèse a porté sur le développement de techniques numériques pour l'homogénéisation de matériaux présentant à une petite échelle des hétérogénéités aléatoires. Sous certaines hypothèses, la théorie mathématique de l'homogénéisation stochastique permet d'expliciter les propriétés effectives de tels matériaux. Néanmoins, en pratique, la détermination de ces propriétés demeure difficile. En effet, celle-ci requiert la résolution d'équations aux dérivées partielles stochastiques posées sur l'espace tout entier. Dans cette thèse, cette difficulté est abordée de deux manières différentes. Les méthodes classiques d'approximation conduisent à approcher les propriétés effectives par des quantités aléatoires. Réduire la variance de ces quantités est l'objectif des travaux de la Partie I. On montre ainsi comment adapter au cadre de l'homogénéisation stochastique une technique de réduction de variance déjà éprouvée dans d'autres domaines. Les travaux de la Partie II s'intéressent à des cas pour lesquels le matériau d'intérêt est considéré comme une petite perturbation aléatoire d'un matériau de référence. On montre alors numériquement et théoriquement que cette simplification de la modélisation permet effectivement une réduction très importante du coût calcul / In this thesis we investigate numerical methods for the homogenization of materials the structures of which, at fine scales, are characterized by random heterogenities. Under appropriate hypotheses, the effective properties of such materials are given by closed formulas. However, in practice the computation of these properties is a difficult task because it involves solving partial differential equations with stochastic coefficients that are additionally posed on the whole space. In this work, we address this difficulty in two different ways. The standard discretization techniques lead to random approximate effective properties. In Part I, we aim at reducing their variance, using a well-known variance reduction technique that has already been used successfully in other domains. The works of Part II focus on the case when the material can be seen as a small random perturbation of a periodic material. We then show both numerically and theoretically that, in this case, computing the effective properties is much less costly than in the general case

This research utilizes CFD to analyze blood flow through pathways representative of central shunts, commonly used as part of the Fontan procedure to treat cyanotic heart disease. In the first part of this research, a parametric study of steady, Newtonian blood flow through parabolic pathways was performed to demonstrate the effect that flow pathway curvature has on wall shear stress distribution and flow energy losses. In the second part, blood flow through two shunts obtained via biplane angiograms is simulated. Pressure boundary conditions were obtained via catheterization. Results showed that wall shear stresses were of sufficient magnitude to initiate platelet activation, a precursor for thrombus formation. Steady results utilizing time-averaged boundary conditions showed excellent agreement with the time-averaged results obtained from pulsatile simulations. For the points of interest in this research, namely wall shear stress distribution and flow energy loss, the Newtonian viscosity model was found to yield acceptable results.

Online stability assessment is an important problem that has not been solved completely yet. The purpose of this research is to tackle online transient stability assessment. Currently, most utility companies use step-by-step integration in order to set protective equipment so that they effectively work for critical contingencies. However, there are times an unforeseen contingency may occur which may cause the system to transit and the protective equipment to misoperate and does not isolate the disturbed part of the system. This research introduces a method that automatically determines a group of generators that participate in system separation and hence transient instability. The method consists of four phases: modeling and simulation, critical machines identification, online transient stability assessment, and critical clearing time calculation. In the modeling and simulation phase, the power system is built and the generators’ rotor angles and speeds are captured. In the critical machines identification phase, the average instantaneous rotor accelerating powers, coherency measures, the during-fault rotor angles and speeds characteristics, and the pre- and post-fault rotor angles are used to identify the Severely Disturbed Group (SDG) of machines. The results of this phase are used to calculate the kinetic energy of the SDG and potential energy of another (or possibly the same) group of generators. Utilization and success of the proposed method will be documented using results from the IEEE 39-Bus test system. Each step of each phase will be demonstrated as needed. The proposed method is compared to step-by-step integration and two direct methods. The suitability of the proposed method for operation will be shown in cases where the Y-Bus matrix and rotor angles and speeds are given. The proof of concept of the proposed method was used in simulating the test system and encouraging results of the simulation were published in ‎[1] and ‎[2]. The proof of concept is the foundation of the method proposed in this dissertation to determine transient stability of large-scale power systems.

La mutation technologique du transport en général et aéronautique en particulier, engagée au niveau européen, conduit à une évolution vers des avions plus économiques et moins consommateurs de carburant. Ceci impacte fortement les systèmes de conditionnement d’air par une électrification partielle ne nécessitant plus de prélèvement d’air sur les réacteurs. Il est alors nécessaire d’assurer une large plage de débit à travers la turbine, élément de la turbomachine constituant le cœur du « pack » de conditionnement d’air, tout en fournissant le maximum de puissance possible sur l’ensemble de la plage. L’étage turbine classique ne peut pas assurer la plage de débit spécifiée. Il est donc remplacé par un étage turbine à section d’injection variable. Cet étage turbine doit fonctionner depuis la phase de maintenance au sol (faible débit, fort taux de détente) jusqu'en phase de croisière (fort débit, faible taux de détente), tout en assurant également son rôle sur les autres phases de vol et multiples cas de panne. La problématique est alors de concevoir une turbine dont la géométrie varie en fonctionnement et qui présente de très bons rendements sur une large plage de débit. Il est ainsi primordial de comprendre au préalable la complexité des écoulements pour ce type de géométrie, et comment le dispositif assurant la variation de section va influencer la topologie de l’écoulement dans l’étage turbine. En particulier, la présence de jeux dans les parties statiques de l’étage introduit une perturbation tourbillonnaire en amont du rotor. L’impact de cette perturbation sur l’écoulement principal, son interaction avec les écoulements secondaires, doit être détaillé. L’influence de la localisation de cette perturbation, de son intensité, doit être analysée, dans un contexte rendu très complexe par la variabilité de la géométrie. La compréhension des phénomènes mis en jeu responsables de la variation des performances dans l’étage turbine, permettra de définir une stratégie de dimensionnement à adopter. L’amélioration des performances de la turbine permettra ainsi de limiter la puissance demandée sur le moteur électrique afin de limiter la masse embarquée et donc la consommation de carburant. La méthodologie retenue pour aborder cette problématique, se décline en quatre volets. Un premier volet bibliographique pour s’approprier les phénomènes physiques liés à l’écoulement dans une turbine à géométrie variable et faire un état de l’art des solutions techniques existantes de géométrie variable des distributeurs de turbines centripètes. Un volet numérique dont l’objectif sera double. D'une part, de proposer une méthodologie de calcul robuste de prévision des performances et, d’autre part, de discriminer différentes options de dimensionnement dont la pertinence doit être démontrée sur l’ensemble de sa plage d’opérabilité. Un volet expérimental représentant la part principale de la thèse, consistera à mettre en place un module spécifique pour réaliser et analyser les essais pour des points de spécification représentatifs du fonctionnement de la turbine sur avion. Cela permettra de fournir une base de données d’analyse et de validation, et de quantifier les effets d’intégration. Ces études numérique et expérimentale seront conduites conjointement, afin que l’analyse de l‘écoulement profite de la complémentarité des deux approches. La dernière étape de cette étude a pour but la restitution des résultats obtenus et le savoir-faire vers l’industrie tant du point de vue de la prédiction des performances que de la méthodologie de dimensionnement des turbines à géométrie variable. / The technological mutation of transport in general and aeronautics in particular, engaged to the European level, leads to an evolution of more economical and fuel-efficient aircrafts. It strongly impacts the environmental control systems by a partial electrification which does not need an air bleeding on the engine anymore. Then it is necessary to insure a large output range through the turbine, element of the turbomachine which forms the heart of the air conditioning « pack », while providing the maximum amount of possible power on the whole range. The classical turbine stage cannot insure the specified output range. Then it is replaced by a variable geometry radial inflow turbine. This turbine stage has to function from the maintenance phase on the ground (weak output, strong expansion ratio) to the en route phase (strong output, weak expansion ratio). It also has to guarantee its role during the others phases of flight and in case of multiple failures power. So the problematic is to design a turbine such that its geometry varies in operation and adapt itself to the changing operating with the best possible efficiency on the widest possible range. Thus it is primordial to understand beforehand the complexity of flows for this kind of geometry, and how the variable geometry device affects the flow topology in the turbine stage. In particular, the presence of clearances in the static parts of the stage creates a vortex perturbation upstream from the rotor. The impact of this perturbation on the main flow, its interaction with secondary flows, must be detailed. The influence of the perturbation localisation, its intensity, must be analysed, in the complex variable geometry context. The understanding of phenomenon involved and responsible for the downgrade of performance in the turbine stage, will allow defining a specific strategy of design. The improvement of performance for the turbine will enable to restrict the required power on the electrical engine for limiting the on board weight, and then the fuel consumption. The selected methodology to broach this problematic, is divided into four parts. Firstly, a bibliographic part in order to appropriate physics phenomenon related to the flow in a variable geometry turbine will be conducted, together with a state of art about the different existing technological solutions. Secondly, some numerical simulations will be set to propose a methodology of robust calculations for performance prediction and, to discriminate different design options. The third step consists in an experimental phase representing the main work of the thesis. It will consist in the definition of a specific module instrumented for tests representative of the turbine on aircraft functioning. It will provide a database for analysing the flow and validating the numerical simulations, and to quantify the effects of integration. These numerical and experimental studies will be led jointly, such that the general analysis takes advantage of complementarity of both approaches. The last step of this study aims at conditioning the results achieved and the know-how for industrial application.