In this dissertation the heat flux prediction capabilities of Residual Distribution (RD) schemes for hypersonic flow fields are investigated. Two canonical configurations are considered: the flat plate and the blunt body (cylinder) problems, with a preference for the last one. Both simple perfect gas and more complex thermo-chemical non-equilibrium (TCNEQ) thermodynamic models have been considered.<p><p>The unexpected results identified early in the investigation lead to a thorough analysis to identify the causes of the unphysical hypersonic heating.<p><p>The first step taken is the assessment of the quality of flow field and heat transfer predictions obtained with RD methods for subsonic configurations. The result is positive, both for flat plate and cylinder configurations, as RD schemes produce accurate flow solutions and heat flux predictions whenever no shock waves are present, irrespective of the gas model employed.<p><p>Subsonic results prove that hypersonic heating anomalies are a consequence of the presence of a shock wave in the domain and/or the way it is handled numerically.<p><p>Regarding hypersonic flows, the carbuncle instability is discarded first as the cause of the erroneous stagnation heating. The anomalies are shown next to be insensitive to the kind and level of dissipation introduced via the (quasi-)positive contribution P to blended B schemes. Additionally, insufficient mesh resolution locally over the region where the shock wave is captured numerically is found to be irrelevant.<p><p>Capturing the bow shock in a manner that total enthalpy is preserved immediately before and after the numerical shock wave is, on the contrary, important for correct heating prediction.<p><p>However, a carefully conceived shock capturing term is, by itself, not sufficient to guarantee correct heating predictions, since the LP scheme employed (be it stand-alone in a shock fitting context or combined into a blended scheme for a shock capturing computation) needs to be immune to spurious recirculations in the stagnation point. <p><p>Once the causes inducing the heating anomalies identified, hypersonic shocked flows in TCNEQ conditions are studied.<p><p>In order to alleviate the computational effort necessary to handle many species non-equilibrium (NEQ) models, the extension of an entropic (or symmetrizing) variables formulation RD to the nS species, two temperature TCNEQ model is accomplished, and the savings in computational time it allows are demonstrated.<p><p>The multi-dimensional generalization of Roe-like linearizations for the TCNEQ model is addressed next: a study on the existence conditions of the linearized state guaranteeing discrete conservation is conducted.<p><p>Finally, the new dissipative terms derived for perfect gas are adapted to work under TCNEQ conditions; the resulting numerical schemes are free of the temperature undershoot and Mach number overshoot problem afflicting standard CRD schemes. / Doctorat en Sciences de l'ingénieur / info:eu-repo/semantics/nonPublished

This thesis is concerned with the further development and analysis of a class of Computational Fluid Dynamics (CFD) methods for the numerical simulation of compressible flows on unstructured grids, known as Residual Distribution (RD).<p>The RD method constitutes a class of discretization schemes for hyperbolic systems <p>of conservation laws, which forms an attractive alternative to the more classical Finite Volume methods, particularly since it allows better representation of the flow physics by genuinely multi-dimensional upwinding and offers second-order accuracy on a compact stencil.<p><p>Despite clear advantages of RD schemes, they also have some unexpected anomalies in common with Finite Volume methods and an attempt to resolve them is presented. The most notable anomaly is the violation of the entropy condition, which as a consequence allows unphysical expansion shocks to exist in the numerical solution. In the thesis the genuinely multi-dimensional character of this anomaly is analyzed and a multi-dimensional entropy fix is presented and shown to avoid expansion shocks. Another infamous anomaly is the carbuncle phenomenon, an instability observed in many numerical solutions with strong shocks, such as the bow shock on a blunt body in hypersonic flow. The occurence of the carbuncle phenomenon with RD methods is analyzed and a novel formulation for a shock fix, based on an anisotropic diffusion term added in the shock layer, is presented and shown to cure the anomaly in 2D and 3D hypersonic flow problems.<p><p>In the present work an effort has been made also to an objective and quantitative assessment of the merits of the RD method for typical aerodynamical engineering applications, such as the transonic flow over airfoils and wings.<p>Validation examples including inviscid, laminar as well as high Reynolds number turbulent flows <p>and comparisons against results from state-of-the-art Finite Volume methods are presented.<p>It is shown that the second-order multi-dimensional upwind RD schemes have an accuracy which is at least as good as second-order FV methods using dimension-by-dimension upwinding and that their main advantage lies in providing excellent monotone shock capturing. / Doctorat en Sciences de l'ingénieur / info:eu-repo/semantics/nonPublished

This thesis presents the author's contribution <p>to the design and implementation of COOLFluiD,<p>an object oriented software platform for <p>the high performance simulation of multi-physics phenomena on unstructured grids. In this context, the final goal has been to provide a reliable tool for handling high speed aerothermodynamic <p>applications. To this end, we introduce a number of design techniques that have been developed in order to provide the framework with flexibility<p>and reusability, allowing developers to easily integrate new functionalities such as arbitrary mesh-based data structures, numerical algorithms (space discretizations, time stepping schemes, linear system solvers, ),and physical models. <p>Furthermore, we describe the parallel algorithms <p>that we have implemented in order to efficiently <p>read/write generic computational meshes involving <p>millions of degrees of freedom and partition them <p>in a scalable way: benchmarks on HPC clusters with <p>up to 512 processors show their effective suitability for large scale computing. <p>Several systems of partial differential equations, <p>characterizing flows in conditions of thermal and <p>chemical equilibrium (with fixed and variable elemental fractions)and, particularly, nonequilibrium (multi-temperature models) <p>have been integrated in the framework. <p>In order to simulate such flows, we have developed <p>two state-of-the-art flow solvers: <p>1- a parallel implicit 2D/3D steady and unsteady cell-centered Finite Volume (FV) solver for arbitrary systems of PDE's on hybrid unstructured meshes; <p>2- a parallel implicit 2D/3D steady vertex-centered Residual Distribution (RD) solver for arbitrary systems of PDE's on meshes with simplex elements (triangles and tetrahedra). <p>The FV~code has been extended to handle all <p>the available physical models, in regimes ranging from incompressible to hypersonic. <p>As far as the RD code is concerned, the strictly conservative variant of the RD method, denominated CRD, has been applied for the first time in literature to solve high speed viscous flows in thermochemical nonequilibrium, yielding some preliminary outstanding results on a challenging double cone flow simulation. <p>All the developments have been validated on real-life testcases of current interest in the aerospace community. A quantitative comparison with experimental measurements and/or literature has been performed whenever possible. / Doctorat en Sciences de l'ingénieur / info:eu-repo/semantics/nonPublished

La simulation numérique est aujourd'hui un outils majeur dans la conception des objets aérodynamiques, que ce soit dans l'aéronautique, l'automobile, l'industrie navale, etc... Un des défis majeurs pour repousser les limites des codes de simulation est d'améliorer leur précision, tout en utilisant une quantité fixe de ressources (puissance et/ou temps de calcul). Cet objectif peut être atteint par deux approches différentes, soit en construisant une discrétisation fournissant sur un maillage donné une solution d'ordre très élevé, soit en construisant un schéma compact et massivement parallèlisable, de manière à minimiser le temps de calcul en distribuant le problème sur un grand nombre de processeurs. Dans cette thèse, nous tentons de rassembler ces deux approches par le développement et l'implémentation de Schéma Distribuant le Résidu (RDS) d'ordre très élevé et de compacité maximale. Ce manuscrit commence par un rappel des principaux résultats mathématiques concernant les Lois de Conservation hyperboliques (CLs). Le but de cette première partie est de mettre en évidence les propriétés des solutions analytiques que nous cherchons à approcher, de manière à injecter ces propriétés dans celles de la solution discrète recherchée. Nous décrivons ensuite les trois étapes principales de la construction d'un schéma RD d'ordre très élevé : - la représentation polynomiale d'ordre très élevé de la solution sur des polygones et des polyèdres; - la description de méthodes distribuant le résidu de faible ordre, compactes et conservatives, consistantes avec une représentation polynomiale des données de très haut degré. Parmi elles, une attention particulière est donnée à la plus simple, issue d'une généralisation du schéma de Lax-Friedrichs (\LxF); - la mise en place d'une procédure préservant la positivité qui transforme tout schéma stable et linéaire, en un schéma non linéaire d'ordre très élevé, capturant les chocs de manière non oscillante. Dans le manuscrit, nous montrons que les schémas obtenus par cette procédure sont consistants avec la CL considérée, qu'ils sont stables en norme $\L^{\infty}$ et qu'ils ont la bonne erreur de troncature. Même si tous ces développements théoriques ne sont démontrés que dans le cas de CLs scalaires, des remarques au sujet des problèmes vectoriels sont faites dès que cela est possible. Malheureusement, lorsqu'on considère le schéma \LxF, le problème algébrique non linéaire associé à la recherche de la solution stationnaire est en général mal posé. En particulier, on observe l'apparition de modes parasites de haute fréquence dans les régions de faible gradient. Ceux-ci sont éliminés grâce à un terme supplémentaire de stabilisation dont les effets et l'évaluation numérique sont précisément détaillés. Enfin, nous nous intéressons à une discrétisation correcte des conditions limites pour le schéma d'ordre élevé proposé. Cette théorie est ensuite illustrée sur des cas test scalaires bidimensionnels simples. Afin de montrer la généralité de notre approche, des maillages composés uniquement de triangles et des maillages hybrides, composés de triangles et de quandrangles, sont utilisés. Les résultats obtenus par ces tests confirment ce qui est attendu par la théorie et mettent en avant certains avantages des maillages hybrides. Nous considérons ensuite des solutions bidimensionnelles des équations d'Euler de la dynamique des gaz. Les résultats sont assez bons, mais on perd les pentes de convergence attendues dès que des conditions limite de paroi sont utilisées. Ce problème nécessite encore d'être étudié. Nous présentons alors l'implémentation parallèle du schéma. Celle-ci est analysée et illustrée à travers des cas test tridimensionnel de grande taille. / Numerical simulations are nowadays a major tool in aerodynamic design in aeronautic, automotive, naval industry etc... One of the main challenges to push further the limits of the simulation codes is to increase their accuracy within a fixed set of resources (computational power and/or time). Two possible approaches to deal with this issue are either to contruct discretizations yielding, on a given mesh, very high order accurate solutions, or to construct compact, massively parallelizable schemes to minimize the computational time by means of a high performance parallel implementation. In this thesis, we try to combine both approaches by investigating the contruction and implementation of very high order Residual Distribution Schemes (RDS) with the most possible compact stencil. The manuscript starts with a review of the mathematical theory of hyperbolic Conservation Laws (CLs). The aim of this initial part is to highlight the properties of the analytical solutions we are trying to approximate, in order to be able to link these properties with the ones of the sought discrete solutions. Next, we describe the three main steps toward the construction of a very high order RDS: - The definition of higher order polynomial representations of the solution over polygons and polyhedra; - The design of low order compact conservative RD schemes consistent with a given (high degree) polynomial representation. Among these, particular accest is put on the simplest, given by a generalization of the Lax-Friedrich's (\LxF) scheme; - The design of a positivity preserving nonlinear transformation, mapping first-order linear schemes onto nonlinear very high order schemes. In the manuscript, we show formally that the schemes obtained following this procedure are consistent with the initial CL, that they are stable in $L^{\infty}$ norm, and that they have the proper truncation error. Even though all the theoretical developments are carried out for scalar CLs, remarks on the extension to systems are given whenever possible. Unortunately, when employing the first order \LxF scheme as a basis for the construction of the nonlinear discretization, the final nonlinear algebraic equation is not well-posed in general. In particular, for smoothly varying solutions one observes the appearance of high frequency spurious modes. In order to kill these modes, a streamline dissipation term is added to the scheme. The analytical implications of this modifications, as well as its practical computation, are thouroughly studied. Lastly, we focus on a correct discretization of the boundary conditions for the very high order RDS proposed. The theory is then extensively verified on a variety of scalar two dimensional test cases. Both triangular, and hybrid triangular-quadrilateral meshes are used to show the generality of the approach. The results obtained in these tests confirm all the theoretical expectations in terms of accuracy and stability and underline some advantages of the hybrid grids. Next, we consider solutions of the two dimensional Euler equations of gas dynamics. The results obtained are quite satisfactory and yet, we are not able to obtain the desired convergence rates on problems involving solid wall boundaries. Further investigation of this problem is under way. We then discuss the parallel implementation of the schemes, and analyze and illustrate the performance of this implementation on large three dimensional problems. Due to the preliminary character and the complexity of these three dimensional problems, a rather qualitative discussion is made for these tests cases: the overall behavior seems to be the correct one, but more work is necessary to assess the properties of the schemes in three dimensions.

Ce travail de thèse, propose un schéma numérique d'ordre élevé, distribuantle résidu (RD) pour l'approximation d'un problème hydro-sédimentairehyperbolique non conservatif, couplant les modèles de Grass et de Saint-Venant. Il fait appel à des méthodes de Runge-Kutta à variation totale diminuanteet de stabilisation (méthode de décentrement amont, dit Upwind),avec ou sans adjonction de limiteurs et présente de bonnes propriétés.L'une des facettes importantes de ce qui a été réalisée, repose sur la conceptionet le développement d'un programme Python 2D-espace, sous la formed'un logiciel faisant appel à un ensemble de modules créés pour l'occasion.Le développement du code de calcul, qui se propose d'approcher la solutiondu problème hydro-sédimentaire, a été e_ectué avec une orientation Objetet pour être e_cace sur calculateur parallèle (utilisant le parallélisme multithreadsOpenMP). L'une des particularités du schéma numérique dans cecadre, est liée à son application à des quadrangles.Un programme 1D-espace, qui se présente également sous forme de logiciel,a aussi été mis en place. Pour des raisons de portabilité et d'e_catité, il aété écrit multilangages (Python-Fortran : via numpy.ctypes pour Python etvia l'interface standard de Fortran pour C). Le schéma RD avec ou sansadjonction de limiteurs de _ux, a été implémenté à la manière d'un schémaprédicteur-correcteur. Des comparaisons avec d'autres schémas ont été e_ectuées a_n de montrer son e_cacité, son ordre de précision élevé a été mis enévidence, et la C-propriété a été testée. Les tests ont révélé que, pour le casd'un transport d'un pro_l sédimentaire parabolique, c'est le limiteur de _uxMUSCL MinMod, qui est le plus adapté parmi ceux testés.Dans le cas scalaire, des tests numériques ont été réalisés a_n de validerle second ordre de précision. / The present work, proposes a high order Residual Distribution (RD) numericalscheme to solve the non conservative hyperbolic problem, coupling Shallow Water and Grass equations. It uses Total Value Diminishing Runge Kutta and stabilisation Upwind methods, with or without limiters. It also has some good properties.A part of the work realised in this thesis, is about the conception and the developpement of a 2D-space Python program, under the form of a software,using a set of moduls created for the occasion. the code developpement, whichis said to approach the _uid-sediment model, coupling Shallow-Water and sedimentequations, has been made with an Object orientation and in orderto be e_cient on parallel architecture (using multithreads OpenMP parallelism). One of the features of the scheme in this case, is due to its application on quadrangles.A 1D-space program, also writen as a software, has been estabished. In order to be portable and e_cient, It has been developped multilinguals (Python- Fortran : by numpy.ctypes for Python and by standart interface FORTRAN for C). The RD scheme with or without Flux Limiters, has been implemented like predictor-corrector one. Comparisons with other schemes results have been realised, in order to show its e_ciency, moreover its high order accuracy has been focus on, and the C-proprerty has been tested. The tests show that MUSCL MinMod _ux limiters, is the most adaptated for a dune test case, between all tested.In the scalar case, numerical tests have been realised, for validating the secondorder of accuracy.

Lors de simulations numériques d’ordre élevé, la discrétisation sous-paramétrique du domaine de calcul peut générer des erreurs dominant l’erreur liée à la discrétisation des variables. De nombreux travaux proposent d’utiliser l’analyse isogéométrique afin de mieux représenter les géométries et de résoudre ce problème.Nous présenterons dans ce travail le couplage du schéma aux résidus distribués limité et stabilisé de Lax-Frieirichs avec l’analyse isogéométrique. En particulier, nous construirons une famille de fonctions de base permettant de représenter exactement les coniques et définies tant sur les éléments triangulaires que quadrangulaires : les fonctions de base de Bernstein rationnelles. Nous nous intéresserons ensuite à la génération de maillages précis pour l’analyse isogéométrique. Notre méthode consiste à créer un maillage courbe à partir d’un maillage linéaire par morceaux de la géométrie. Le maillage obtenu en sortie de notre procédure est non-structuré, conforme et assure la continuité de nos fonctions de base sur tout le domaine. Pour finir, nous décrirons les différentes méthodes d’adaptation de maillages développées : l’élévation d’ordre et le raffinement isotrope. Bien évidemment, la géométrie exacte du maillage courbe d’entrée est préservée au cours des processus d’adaptation. / During high order simulations, the approximation error may be dominated by the errors linked to the sub-parametric discretization used for the geometry representation. Many works propose to use an isogeometric analysis approach to better represent the geometry and hence solve this problem. In this work, we will present the coupling between the limited stabilized Lax-Friedrichs residual distributed scheme and the isogeometric analysis. Especially, we will build a family of basis functions defined on both triangular and quadrangular elements and allowing the exact representation of conics : the rational Bernstein basis functions. We will then focus in how to generate accurate meshes for isogeometric analysis. Our idea is to create a curved mesh from a classical piecewise-linear mesh of the geometry. We obtain a conforming unstructured mesh which ensures the continuity of the basis functions over the entire mesh. Last, we will detail the curved mesh adaptation methods developed : the order elevation and the isotropic mesh refinement. Of course, the adaptation processes preserve the exact geometry of the initial curved mesh.

Cette thèse présente la construction de schémas distribuant le résidu (RD) d'ordre très élevés, pour la discrétisation d'équations d'advection-diffusion multidimensionnelles et stationnaires sur maillages non structurés. Des schémas linéaires ainsi que des schémas non linéaires sont considérés. Une approximation de la solution polynomiale par morceaux et continue sur chaque élément est adoptée, de plus une procédure de reconstruction du gradient que celle de la solution numérique est utilisée afin d'avoir une représentation continue de la solution numérique et de son gradient. Il est montré que le gradient doit être reconstruit avec la même précision de la solution, sans quoi la précision formel du schéma numérique est perdue dans les cas où les effets de diffusion prévalent sur les effets d'advection, et aussi quand l'advection et la diffusion sont également importants. Ensuite, la méthode est étendue à des systèmes d'équations, en particulier aux équations de Navier-Stokes et aux équations RANS. La précision, l'efficacité et la robustesse du solveur RD implicite sont démontrées sur plusieurs cas tests. / The construction of compact high-order Residual Distribution schemes for the discretizationof steady multidimensional advection-diffusion problems on unstructuredgrids is presented. Linear and non-linear scheme are considered. A piecewise continuouspolynomial approximation of the solution is adopted and a gradient reconstructionprocedure is used in order to have a continuous representation of both thenumerical solution and its gradient. It is shown that the gradient must be reconstructedwith the same accuracy of the solution, otherwise the formal accuracy ofthe numerical scheme is lost in applications in which diffusive effects prevail overthe advective ones, and when advection and diffusion are equally important. Thenthe method is extended to systems of equations, with particular emphasis on theNavier-Stokes and RANS equations. The accuracy, efficiency, and robustness of theimplicit RD solver is demonstrated using a variety of challenging aerodynamic testproblems.

Cette thèse présente la construction d’un schéma aux Résidus Distribués p-adaptatif pour la discrétisation des équations d’Euler ainsi qu’un schéma aux Résidus Distribués hp-adaptatif pour les équations de Navier- Stokes pénalisées. On rappelle tout d’abord les équations d’Euler et de Navier-Stokes ainsi que leurs versions non dimensionnelles. Les définitions et propriétés de base des schémas aux Résidus Distribués sont ensuite présentées. On décrit alors la construction d’un schéma aux Résidus Distribués p-adaptatif pour les équations d’Euler. La construction du schéma p-adaptatif est basée sur la possibilité d’exprimer le résidu total d’un élément K de degré k (au sens où l’élément fini (K; P; Sigma ) est un élément fini de degré k) comme une somme pondérée des résidus totaux de ses sous-éléments de degré 1. La solution discrète ainsi obtenue est en général discontinue à l’interface entre un élément subdivisé et un élément non subdivisé. Ceci contredit l’hypothèse de continuité de la solution qui est utilisée pour démontrer le théorème de Lax-Wendroff discret pour les schémas aux Résidus Distribués. Cependant, on montre que cette hypothèse peut être assouplie. La conséquence pratique est que si l’on emploie des quadratures particulières dans l’implémentation numérique, on peut quand même démontrer le théorème de Lax-Wendroff discret, ce qui garantit la convergence du schéma numérique vers une solution faible des équations d’origine. Les formules qui permettent d’exprimer le résidu total comme une somme pondérée des résidus totaux des sous-éléments sont à la base de la méthode de p-adaptation présentée ici. Dans le cas quadratique, la formule est obtenue avec les classiques fonctions de base de Lagrange en dimension deux et avec des fonctions de base de Bézier en dimension trois. Ces deux formules sont ensuite généralisées à des degrés polynomiaux quelconques en dimension deux et trois avec des fonctions de base de Bézier. Dans la deuxième partie de la thèse, on présente l’application du schéma p-adaptatif aux équations pénalisées de Navier-Stokes avec adaptation de maillage anisotrope. . En pratique, on combine le schéma p-adaptatif avec la méthode IBM-LS-AUM (Immersed Boundary Method with Level Sets and Adapted Unstructured Meshes). La méthode IBM-LS-AUM permet d’imposer les conditions aux bords grâce à la méthode de pénalisation et l’adaptation anisotrope du maillage à la solution numérique et à la level-set augmente la précision de la solution et de la représentation de la surface. Une fois la méthode IBM-LS-AUM combinée avec le schéma p-adaptatif, il est alors possible d’utiliser des éléments d’ordre élevés en-dehors de la zone où la pénalisation est appliquée. La méthode est robuste comme le montrent les diverses expérimentations numériques à des vitesses faibles à élevées et à différents nombres de Reynolds. / This thesis presents the construction of a p-adaptive Residual Distribution scheme for the steady Euler equations and a hp-adaptive Residual Distribution scheme for the steady penalized Navier-Stokes equations in dimension two and three. The Euler and Navier-Stokes equations are recalled along with their non dimensional versions. The basis definitions and properties of the steady Residual Distribution schemes are presented. Then, the construction of a p-adaptive Residual Distribution scheme for the Euler equations is considered. The construction of the p-adaptive scheme is based upon the expression of the total residual of an element of a given degree k (in the Finite Element sense) into the total residuals of its linear sub-elements. The discrete solution obtained with the p-adaptive scheme is then a one degree polynomial in the divided elements and a k-th degree polynomial in the undivided ones. Therefore, the discrete solution is in general discontinuous at the interface between a divided element and an undivided one. This is in apparent contradiction with the continuity assumption used in general to demonstrate the discrete Lax-Wendroff theorem for Residual Distribution schemes. However, as we show in this work, this constrain can be relaxed. The consequence is that if special quadrature formulas are employed in the numerical implementation, the discrete Lax-Wendroff theorem can still be proved, which guaranties the convergence of the p-adaptive scheme to a weak solution of the governing equations. The formulas that express the total residual into the combination of the total residuals of the sub-elements are central to the method. In dimension two, the formula is obtained with the classical Lagrange basis in the quadratic case and with the Bézier basis in dimension three. These two formulas are then generalized to arbitrary polynomial degrees in dimension two and three with a Bézier basis. In the second part of the thesis the application of the p-adaptive scheme to the penalized Navier-Stokes equations with anisotropic mesh adaptation is presented. In practice, the p-adaptive scheme is used with the IBM-LS-AUM (Immersed Boundary Method with Level Sets and Adapted Unstructured Meshes) method. The IBM-LS-AUM allows to impose the boundary conditions with the penalization method and the mesh adaptation to the solution and to the level-set increases the accuracy of the representation of the surface and the solution around walls. When the IBM-LSAUM is combined with the p-adaptive scheme, it is possible to use high-order elements outside the zone where the penalization is applied. The method is robust as shown by the numerical applications at low to large Mach numbers and at different Reynolds in dimension two and three.

This thesis presents the construction, the analysis and the verication of compact residual discretizations for the solution of conservation laws on unstructured meshes. <p>The schemes considered belong to the class of residual distribution (RD) or fluctuation splitting (FS) schemes. <p>The methodology presented relies on three main elements: design of compact linear first-order stable schemes for linear hyperbolic PDEs, a positivity preserving procedure mapping stable first-order linear schemes onto nonlinear second-order schemes with non-oscillatory shock capturing capabilities, and a conservative formulation enabling to extend the schemes to nonlinear CLs. These three design steps, and the underlying theoretical tools, are discussed in depth. The nonlinear RD schemes resulting from this construction are tested on a large set of problems involving the solution of scalar models, and systems of CLs. This extensive verification fills the gaps left open, where no theoretical analysis is possible. <p>Numerical results are presented on the Euler equations of a perfect gas, on a two-phase flow model with highly nonlinear thermodynamics, and on the shallow-water equations. <p>On irregular grids, the schemes proposed yield quite accurate and stable solutions even on very difficult computations. Direct comparisone show that these results are more accurate than the ones given by FV and WENO schemes. Moreover, our schemes have a compact nearest-neighbor stencil. This encourages to further develop our approach, toward the design of very high-order schemes, which would represent a very appealing alternative, both in terms of accuracy and efficiency, to now classical FV and ENO/WENO discretizations. These schemes might also be very competitive with respect to very high-order DG schemes. / Doctorat en sciences appliquées / info:eu-repo/semantics/nonPublished

<p align="justify">This work deals with the development of numerical methods for compressible flow simulation with application to the interaction of fluid flows and structural bodies.</p><p><p><p align="justify">First, we develop numerical methods based on multidimensional upwind residual distribution (RD) schemes. Theoretical results for the stability and accuracy of the methods are given. Then, the RD schemes for unsteady problems are extended for computations on moving meshes. As a second approach, cell centered and vertex centered finite volume (FV) schemes are considered. The RD schemes are compared to FV schemes by means of the 1D modified equation and by the comparison of the numerical results for scalar problems and system of Euler equations. We present a number of two and three dimensional steady and unsteady test cases, illustrating properties of the numerical methods. The results are compared with the theoretical solution and experimental data.</p><p><p><p align="justify">In the second part, a numerical method for fluid-structure interaction problems is developed. The problem is divided into three distinct sub-problems: Computational Fluid Dynamics, Computational Solid Mechanics and the problem of fluid mesh movement. The problem of Computational Solid Mechanics is formulated as a system of partial differential equations for an anisotropic elastic continuum and solved by the finite element method. The mesh movement is determined using the pseudo-elastic continuum approach and solved again by the finite element method. The coupling of the problems is achieved by a simple sub-iterative approach. Capabilities of the methods are demonstrated on computations of 2D supersonic panel flutter and 3D transonic flutter of the AGARD 445.6 wing. In the first case, the results are compared with the theoretical solution and the numerical computations given in the references. In the second case the comparison with experimental data is presented.</p> / Doctorat en Sciences de l'ingénieur / info:eu-repo/semantics/nonPublished