This thesis develops the methodology for solving initial boundary value problems with the use of summation-by-parts discretizations. The combination of high orders of accuracy and a systematic approach to construct provably stable boundary and interface procedures makes this methodology especially suitable for scientific computations with high demands on efficiency and robustness. Most classes of high order methods can be applied in a way that satisfies a summation-by-parts rule. These include, but are not limited to, finite difference, spectral and nodal discontinuous Galerkin methods. In the first part of this thesis, the summation-by-parts methodology is extended to the time domain, enabling fully discrete formulations with superior stability properties. The resulting time discretization technique is closely related to fully implicit Runge-Kutta methods, and may alternatively be formulated as either a global method or as a family of multi-stage methods. Both first and second order derivatives in time are considered. In the latter case also including mixed initial and boundary conditions (i.e. conditions involving derivatives in both space and time). The second part of the thesis deals with summation-by-parts discretizations on multi-block and hybrid meshes. A new formulation of general multi-block couplings in several dimensions is presented and analyzed. It collects all multi-block, multi-element and  hybrid summation-by-parts schemes into a single compact framework. The new framework includes a generalized description of non-conforming interfaces based on so called summation-by-parts preserving interpolation operators, for which a new theoretical accuracy result is presented.

This thesis presents the construction, the analysis and the verication of compact residual discretizations for the solution of conservation laws on unstructured meshes. The schemes considered belong to the class of residual distribution (RD) or fluctuation splitting (FS) schemes. 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. 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. 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.

La thématique étudiée dans ce mémoire est axée sur la préservation de la stabilité dynamique de véhicules évoluant en environnement naturel. En effet, la mobilité en milieu tout-terrain est une activité particulièrement pénible et dangereuse en raison de la nature difficile de l'environnement de conduite et de la reconfigurabilité des machines. Le caractère changeant et incertain des interactions rencontrées entre des véhicules à dynamique complexe et variable et leur environnement entraîne régulièrement des risques accrus de renversement et/ou de perte de contrôle (dévalement, dérapage déclenché par une perte soudaine d'adhérence) pour le conducteur. Une forte accidentalité mortelle est, en effet, recensée dans ce secteur, en particulier, dans le milieu agricole ou le renversement de véhicule est classé comme étant la première cause de mortalité au travail. A l'heure actuelle, les approches existantes sur la stabilité d'engins agricoles sont qualifiées à juste titre de passives car elles ne permettent pas d'éviter que les accidents ne se produisent. Par ailleurs, la transposition directe des solutions de sécurité active du secteur de l'automobile (ABS, ESP) s'est révélée inadaptée aux véhicules tout-terrain a cause des hypothèses simplificatrices (routes plates et homogènes, conditions d'adhérence constantes, etc.) dont souffre la conception de ces dispositifs. Ainsi, le développement de systèmes actifs de sécurité prenant en compte les spécificités de la conduite en milieu tout-terrain se révèle être la meilleure voie d'amélioration à suivre. Eu égard à ces circonstances, ce projet se propose d'adresser cette problématique en étudiant des métriques de stabilité pertinentes permettant d'estimer et d'anticiper en temps réel les risques afin de permettre des actions correctives pour la préservation de l'intégrité des machines tout-terrain. Afin de faciliter l'industrialisation du dispositif actif de sécurité conçu, l'une des contraintes sociétales et commerciales de ce projet a été l'utilisation de capteurs compatibles avec le coût des machines visées. L'objectif ambitieux de cette étude a été atteint par différentes voies. En premier lieu, une approche de modélisation multi-échelle a permis de caractériser l'évolution dynamique de véhicules en milieu tout-terrain. Cette approche à dynamique partielle a offert l'avantage de développer des modèles suffisamment précis pour être représentatifs du comportement réel de l'engin mais tout en présentant une structure relativement simple permettant la synthèse d'asservissements performants. Puis, une étude comparative des avantages et des inconvénients des trois grandes familles de métriques répertoriées dans la littérature a permis de mettre en exergue l'intérêt des métriques analytiques à modèle dynamique par rapport aux catégories de critères de stabilité dits statiques et empiriques. Enfin, l'analyse approfondie des métriques dynamiques a facilité le choix de trois indicateurs (Lateral and Longitudinal Load Transfer (LLT), Force Angle Stability Measurement (FASM) et Dynamic Energy Stability Measurement (DESM)) qui sont représentatifs d'un risque imminent de renversement du véhicule. La suite du mémoire s'appuie sur la théorie d'observation pour l'estimation en ligne des variables non directement mesurables en milieu tout-terrain telles que les rigidités de glissement et dérive du pneumatique. Jumelée aux différents modèles dynamiques du véhicule, la synthèse d'observateurs a permis donc d'estimer en temps réel les efforts d'interaction pneumatiques-sol nécessaires à l'évaluation des indicateurs d'instabilité. Le couplage de ces modèles multi-échelles à la théorie d'observation a ainsi constitué un positionnement original à même de briser la complexité de la caractérisation de la stabilité de véhicules à dynamiques complexes et incertaines. (...) / This work is focused on the thematic of the maintenance of the dynamic stability of off-road vehicles. Indeed, driving vehicles in off-road environment remains a dangerous and harsh activity because of the variable and bad grip conditions associated to a large diversity of terrains. Driving difficulties may be also encountered when considering huge machines with possible reconfiguration of their mechanical properties (changes in mass and centre of gravity height for instance). As a consequence, for the sole agriculture sector, several fatal injuries are reported per year in particular due to rollover situations. Passive protections (ROllover Protective Structure - ROPS) are installed on tractors to reduce accident consequences. However, protection capabilities of these structures are very limited and the latter cannot be embedded on bigger machines due to mechanical design limitations. Furthermore, driving assistance systems (such as ESP or ABS) have been deeply studied for on-road vehicles and successfully improve safety. These systems usually assume that the vehicle Center of Gravity (CG) height is low and that the vehicles are operating on smooth and level terrain. Since these assumptions are not satisfied when considering off-road vehicles with a high CG, such devices cannot be applied directly. Consequently, this work proposes to address this research problem by studying relevant stability metrics able to evaluate in real time the rollover risk in order to develop active safety devices dedicated to off-road vehicles. In order to keep a feasible industrialization of the conceived active safety device, the use of compatible sensors with the cost of the machines was one of the major commercial and societal requirements of the project. The ambitious goal of this study was achieved by different routes. First, a multi-scale modeling approach allowed to characterize the dynamic evolution of off-road vehicles. This partial dynamic approach has offered the advantage of developing sufficiently accurate models to be representative of the actual behavior of the machine but having a relatively simple structure for high-performance control systems. Then, a comparative study of the advantages and drawbacks of the three main families of metrics found in the literature has helped to highlight the interest of dynamic stability metrics at the expense to categories of so-called static and empirical stability criteria. Finally, a thorough analysis of dynamic metrics has facilitated the choice of three indicators (Longitudinal and Lateral Load Transfer (LLT), Force Angle Stability Measurement (FASM) and Dynamic Energy Stability Measurement (DESM)) that are representative of an imminent rollover risk. The following of the document is based on the observation theory for estimating online of variables which are not directly measurable in off-road environment such as slip and cornering stiffnesses. Coupled to the dynamic models of the vehicle, the theory of observers has helped therefore to estimate in real time the tire-soil interaction forces which are necessaries for evaluating indicators of instability. The coupling of these multiscale models to the observation theory has formed an original positioning capable to break the complexity of the characterization of the stability of vehicles having complex and uncertain dynamics. (...)

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