This research is mainly focused on the development of the adaptive Cartesian grid methods for compressibl  e flow. At first, the ghost cell method and its applications for inviscid compressible flow on adaptive tree Cartesian grid are developed. The proposed method is successfully used to evaluate various inviscid compressible flows around complex bodies. The mass conservation of the method is also studied by numerical analysis. The extension to three-dimensional flow is presented. Then, an h-adaptive Runge–Kutta discontinuous Galerkin (RKDG) method is presented in detail for the development of high accuracy numerical method under Cartesian grid. This method combined with the ghost cell immersed boundary method is also validated by well documented test problems involving both steady and unsteady compressible flows over complex bodies in a wide range of Mach numbers. In addition, in order to suppress the failure of preserving positivity of density or pressure, which may cause blow-ups of the high order numerical algorithms, a positivity-preserving limiter technique coupled with h-adaptive RKDG method is developed. Such a method has been successfully implemented to study flows with the large Mach number, strong shock/obstacle interactions and shock diffraction. The extension of the method to viscous flow under the adaptive Cartesian grid with hybrid overlapping bodyfitted grid is developed. The method is validated by benchmark problems and has been successfully implemented to study airfoil with ice accretion. Finally, based on an open source code, the detached eddy simulation (DES) is developed for massive separation flow, and it is used to perform the research on aerodynamic performance analysis over the wing with ice accretion.

The world population is expected to increase to approximately 11 billion by 2100. The ageing population (aged 60 and over) is projected to exceed the number of children in 2047. This will be a situation without precedent. The number of citizens with disorders of old age like Dementia will rise to 115 million worldwide by 2050. The estimated cost of Dementia will also increase, from $604 billion in 2010, to $1,117 billion by 2030. At the same time, medical expertise, evidence-driven policymaking and commissioning of services are increasingly evolving the definitive architecture of comprehensive long-term care to account for these changes. Technological advances, such as those provided by computational science and biomedical engineering, will allow for an expansion in our ability to model and simulate an almost limitless variety of complex problems that have long defied traditional methods of medical practice. Numerical methods and simulation offer the prospect of improved clinically relevant predictive information, and of course optimisation, enabling more efficient use of resources for designing treatment protocols, risk assessment and urgently needed management of a long term care system for a wide spectrum of brain disorders. Within this paradigm, the importance of the relationship of senescence of cerebrospinal fluid transport to dementia in the elderly make the cerebral environment notably worthy of investigation through numerical and computational modelling. Hydrocephalus can be succinctly described as the abnormal accumulation (imbalance between production and circulation) of cerebrospinal fluid (CSF) within the brain. Using hydrocephalus as a test bed, one is able to account for the necessary mechanisms involved in the interaction between cerebral fluid production, transport and drainage. The current state of knowledge about hydrocephalus, and more broadly integrative cerebral dynamics and its associated constitutive requirements, advocates that poroelastic theory provides a suitable framework to better understand the disease. In this work, Multiple-network poroelastic Theory (MPET) is used to develop a novel spatio-temporal model of fluid regulation and tissue displacement in various scales within the cerebral environment. The model is discretised in a variety of formats, through the established finite difference method, finite difference – finite volume coupling and also the finite element method. Both chronic and acute hydrocephalus was investigated in a variety of settings, and accompanied by emerging surgical techniques where appropriate. In the coupled finite difference – finite volume model, a key novelty was the amalgamation of anatomically accurate choroid plexuses with their feeding arteries and a simple relationship relaxing the constraint of a unique permeability for the CSF compartment. This was done in order to account for Aquaporin-4 sensitisation. This model is used to demonstrate the impact of aqueductal stenosis and fourth ventricle outlet obstruction. The implications of treating such a clinical condition with the aid of endoscopic third (ETV) and endoscopic fourth ventriculostomy (EFV) are considered. It was observed that CSF velocity in the aqueduct, along with ventricular displacement, CSF pressure, wall shear stress and pressure difference between lateral and fourth ventricles increased with applied stenosis. The application of ETV reduced the aqueductal velocity, ventricular displacement, CSF pressure, wall shear stress and pressure difference within nominal levels. The greatest reversal of the effects of atresia come by opting for ETV rather than the more complicated procedure of EFV. For the finite difference model incorporating nonlinear permeability, qualitatively similar results were obtained in comparison to the pertinent literature, however, there was an overall amplification of ventriculomegaly and transparenchymal pressure difference using this model. A quantitative and qualitative assessment is made of hydrocephalus cases involving aqueductal stenosis, along with the effects to CSF reabsorption in the parenchyma and subarachnoid space. The finite element discretisation template produced for the n^th- dimensional transient MPET system allowed for novel insight into hydrocephalus. In the 1D formulation, imposing the breakdown of the blood-CSF barrier responsible for clearance resulted in an increase in ventricular displacement, transparenchymal venous pressure gradient and transparenchymal CSF pressure gradient, whilst altering the compliance proved to markedly alter the rate of change of displacement and CSF pressure gradient. The influence of Poisson's ratio was investigated through the use of the dual-grid solver in order to distinguish between possible over or under prediction of the ventricular displacement. In the 2D model based on linear triangles, the importance of the MPET boundary conditions is acknowledged, along with the quality of the underlying mesh. Interesting results include that the fluid content is highest in the periventricular region and the skull, whilst after longer time scales, the peak CSF content becomes limited to the periventricular region. Venous fluid content is heavily influenced by the Biot-Willis constant, whilst both the venous and CSF/ISF compartments show to be strongly influenced by breakdown in the blood-CSF barrier. Increasing the venous compliance effects the arterial, capillary and venous compartments. Decreasing the venous compliance shows an accumulation of fluid, possibly helping to explain why the ventricles can be induced to compress rather than expand under decreased compliance. Finally, a successful application of the 3D-MPET template is shown for simple geometries. It is envisaged that future observations into the biology of cerebral fluid flow (such as perivascular CSF-ISF fluid exchange) and its interaction with the surrounding parenchyma, will demand the evolution of the MPET model to reach a level of complexity that could allow for an experimentally guided exploration of areas that would otherwise prove too intricate and intertwined under conventional settings.

In the present work we implemented parallel version of a computational fluid dynamics code. This code is based on Discontinuous Galerkin Method (DGM), which is due to its favourable properties suitable for parallelization. In the work we describe the Navier-Stokes equations and their discretization using DGM. We explain the advantages of usage of the DGM and formulate the serial algorithm. Next we focus on the parallel implementation of the algorithm and several particular issues connected to the parallelization. We present the numerical experiments showing the efficiency of the parallel code in the last chapter.

This thesis is concerned on the simulation of wet steam flow using discontinuous Galerkin method. Wet steam flow equations consist of Naviere-Stokes equations for compressible flow and Hill's equations for condensation of water vapor. The first part of this thesis describes the mathematical formulation of wet steam model and the derivation of Hill's equations. The model equations are discretized with the aid of discontinuous Galerkin method and backward difference formula which leads to implicit scheme represented by nonlinear algebraic system. This system is solved using Newton-like method. The derived scheme was implemented in program ADGFEM which is used for solving non-stationary convective-diffusive problems. The numerical results are presented in the last part of this thesis. 1

O estudo da dinâmica estrutural de risers oceânicos apresenta instigantes desafios aos pesquisadores da área da engenharia de estruturas, uma vez que os meios tradicionais de análises dinâmicas lineares nem sempre se ajustam às suas complexas particularidades. No atual estágio do desenvolvimento científico da área de engenharia de estruturas, a aplicação de técnicas de análise dinâmica não linear, dentro de determinadas hipóteses, mostra-se como uma das alternativas possíveis e viáveis à tradicional análise dinâmica linear. Com vistas a uma nova abordagem do problema, o presente trabalho adota uma metodologia de análise não linear dinâmica de risers oceânicos em configuração de lançamento de catenária, conjugada a uma técnica de processamento de Modelos de Ordem Reduzida para o estudo dos fenômenos dinâmicos manifestados por risers. Trata-se de um método de modelagem local, restrito à região de contato unilateral do riser com o solo, considerado este último um meio viscoelástico. Os resultados da aplicação desta metodologia são demonstrados nos estudos de caso apresentados com comparações com modelos numéricos (Método dos Elementos Finitos) e modelos físicos. / The dynamic study of offshore risers still demands large efforts from structural engineering researchers, since these systems may behave in a way that is not well modeled and understood using simply linear dynamic theories. Nevertheless, the current development stage of non linear dynamic theories gives hope that their use for the analyses of such systems can be of great value, even though, this must be carefully done specially by the analyst. The present work refers to a non linear dynamic methodology application to offshore risers, particularly steel catenary risers, by a technique known as reduced-order modeling, in the study of dynamic phenomena that these structures may present. The model is local, which means that it represents the touch-down zone of the riser-soil system. The soil modeling was presumed to be viscoelastic. The results obtained in case studies are compared with those from numerical (Finite Element Method) and small scale physical models.

Cette thèse est consacrée à l'étude de problèmes de stabilisation exponentielle par retour d'état ou "feedback" des équations de Navier-Stokes dans un domaine borné Ω ⊂ Rd, d = 2 ou 3. Le cas d'un contrôle localisé sur la frontière du domaine est considéré. Le contrôle s'exprime en fonction du champ de vitesse à l'aide d'une loi de feedback non-linéaire. Celle-ci est fournie grâce aux techniques d'estimation a priori via la procédure de Faedo-Galerkin laquelle consiste à construire une suite de solutions approchées en utilisant une base de Galerkin adéquate. Cette loi de feedback assure la décroissance exponentielle de l'énergie du problème discret correspondant et grâce au résultat de compacité, nous passons à la limite dans le système satisfait par les solutions approchées. Le chapitre 1 étudie le problème de stabilisation des équations de Navier- Stokes autour d'un état stationnaire donné, tandis que le chapitre 2 examine le problème de stabilisation autour d'un état non-stationnaire prescrit. Le chapitre 3 est consacré à l'étude de la stabilisation du problème de Navier-Stokes avec des conditions aux bords mixtes (Dirichlet- Neumann) autour d'un état d'équilibre donné. Enfin, nous présentons dans le chapitre 4, des résultats numériques dans le cas d'un écoulement autour d'un obstacle circulaire / In this thesis we study the exponential stabilization of the two and three-dimensional Navier- Stokes equations in a bounded domain Ω, by means of a boundary control. The Control is expressed in terms of the velocity field by using a non-linear feedback law. In order to determine a feedback law, we consider an extended system coupling the Navier-Stokes equations with an equation satisfied by the control on the domain boundary. While most traditional approaches apply a feedback controller via an algebraic Riccati equation, the Stokes-Oseen operator or extension operators, a Galerkin method is proposed instead in this study. The Galerkin method permits to construct a stabilizing boundary control and by using energy a priori estimation technics, the exponential decay is obtained. A compactness result then allows us to pass to the limit in the nonlinear system satisfied by the approximated solutions. Chapter 1 deals with the stabilization problem of the Navier-Stokes equations around a given steady state, while Chapter 2 examines the stabilization problem around a prescribed non-stationary state. Chapter 3 is devoted to the stabilization of the Navier-Stokes problem with mixed-boundary conditions (Dirichlet-Neumann), around to a given steady-state. Finally, we present in Chapter 4, numerical results in the case of a flow around a circular obstacle

Le travail présenté dans ce manuscrit est une contribution à l’étude des vibrations non-linéaires des poutres isotropes et en composite, en mouvement de rotation. Le modèle mathématique utilisé est basé sur la formulation intrinsèque et géométriquement exacte de Hodges, dédiée au traitement des poutres ayant des grands déplacements et de petites déformations. La résolution est faite dans le domaine fréquentiel suite à une discrétisation spatio-temporelle, en utilisant l’approximation de Galerkin et la méthode de l’équilibrage harmonique, avec des conditions aux limites correspondantes aux poutres encastrées-libres. Le systéme dynamique final est traité par des méthodes de continuation : la méthode asymptotique numérique et la méthode pseudo-longueur d’arc. Des algorithmes basés sur ces méthodes de continuation ont été développés et une étude comparative de convergence a été menée. Cette étude a cerné les aspects : statique, analyse modale linéaire, vibrations libres non-linéaires et les vibrations forcées non-linéaires des poutres rotatives. Ces algorithmes de continuations ont été testés pour le calculs des courbes de réponse sur des cas traités dans la littérature. La résonance interne et la stabilité des solutions obtenues sont étudiées / The work presented in this manuscript is a contribution to the non-linear vibrations of the isotropic beams and composite rotating beams study. The mathematical model used is based on the intrinsic formulation and geometrically exact of Hodges, developped for beams subjected to large displacements and small deformations. The resolution is done in the frequency domain after a spatial-temporal dicretisation, by using the Galerkin approximation and the the harmonic balance method, with boundary conditions corresponding to the clamped-free. The final dynamic system is treated by continuation methods : asymptotic numerical method and the pseudo-arc length method, whose algorithms based on these continuation methods were developed and a convergence study was carried out. This study surround the aspects : statics, linear modal analysis, non-linear free vibrations and the non-linear forced vibrations of the rotating beams. These continuation algorithms were tested for the response curves calculations on cases elaborated in the literature. Internal resonance and the stability of the solutions obtained are studied

Distributed optimal control problems with transient convection dominated diffusion convection reaction equations are considered. The problem is discretized in space by using three types of discontinuous Galerkin (DG) method: symmetric interior penalty Galerkin (SIPG), nonsymmetric interior penalty Galerkin (NIPG), incomplete interior penalty Galerkin (IIPG). For time discretization, Crank-Nicolson and backward Euler methods are used. The discretize-then-optimize approach is used to obtain the finite dimensional problem. For one-dimensional unconstrained problem, Newton-Conjugate Gradient method with Armijo line-search. For two-dimensional control constrained problem, active-set method is applied. A priori error estimates are derived for full discretized optimal control problem. Numerical results for one and two-dimensional distributed optimal control problems for diffusion convection equations with boundary layers confirm the predicted orders derived by a priori error estimates.

Numerical solutions of mathematical modelizations of heat and mass transfer in cubical and cylindrical reactors of solar adsorption refrigeration systems are studied. For the resolution of the equations describing the coupling between heat and mass transfer, Bubnov-Galerkin method is used. An exact solution for time dependent heat transfer in cylindrical multilayered annulus is presented. Separation of variables method has been used to investigate the temperature behavior. An analytical double series relation is proposed as a solution for the temperature distribution, and Fourier coefficients in each layer are obtained by solving some set of equations related to thermal boundary conditions at inside and outside of the cylinder.

In this thesis, we analyze an adaptive discontinuous finite element method for symmetric second order linear elliptic operators. Moreover, we obtain a fully computable convergence analysis on the broken energy seminorm in first order symmetric interior penalty discontin- uous Galerkin finite element approximations of this problem. The method is formulated on nonconforming meshes made of triangular elements with first order polynomial in two di- mension. We use an estimator which is completely free of unknown constants and provide a guaranteed numerical bound on the broken energy norm of the error. This estimator is also shown to provide a lower bound for the broken energy seminorm of the error up to a constant and higher order data oscillation terms. Consequently, the estimator yields fully reliable, quantitative error control along with efficiency. As a second problem, explicit expression for constants of the inverse inequality are given in 1D, 2D and 3D. Increasing mathematical analysis of finite element methods is motivating the inclusion of mesh dependent terms in new classes of methods for a variety of applications. Several inequalities of functional analysis are often employed in convergence proofs. Inverse estimates have been used extensively in the analysis of finite element methods. It is char- acterized as tools for the error analysis and practical design of finite element methods with terms that depend on the mesh parameter. Sharp estimates of the constants of this inequality is provided in this thesis.