Finite element formulation of general boundary conditions for incompressible flows

We study the finite element formulation of general boundary conditions for incompressible flow problems. Distinguishing between the contributions from the inviscid and viscid parts of the equations, we use Nitsche's method to develop a discrete weighted weak formulation valid for all values of the viscosity parameter, including the limit case of the Euler equations. In order to control the discrete kinetic energy, additional consistent terms are introduced. We treat the limit case as a (degenerate) system of hyperbolic equations, using a balanced spectral decomposition of the flux Jacobian matrix, in analogy with compressible flows. Then, following the theory of Friedrich's systems, the natural characteristic boundary condition is generalized to the considered physical boundary conditions. Several numerical experiments, including standard benchmarks for viscous flows as well as inviscid flows are presented

Mixed finite element approximation for a coupled petroleum reservoir model

In this paper, we are interested in the modelling and the finite element approximation of a petroleum reservoir, in axisymmetric form. The flow in the porous medium is governed by the Darcy-Forchheimer equation coupled with a rather exhaustive energy equation. The semi-discretized problem is put under a mixed variational formulation, whose approximation is achieved by means of conservative Raviart-Thomas elements for the fluxes and of piecewise constant elements for the pressure and the temperature. The discrete problem thus obtained is well-posed and a posteriori error estimates are also established. Numerical tests are presented validating the developed code

NUMERICAL COUPLING OF 2.5D RESERVOIR AND 1.5D WELLBORE MODELS IN ORDER TO INTERPRET THERMOMETRICS

International audienceThe paper deals with the numerical coupling of an axisymmetric reser- voir model, governed by Darcy-Forchheimer's equation together with a nonstandard energy balance, and a quasi 1D wellbore model described by the compressible Navier- Stokes equation

A dG method for the Stokes equations related to nonconforming approximations

We study a discontinuous Galerkin method for the Stokes equations with a new stabilization of the viscous term. On the one hand, it allows us to recover, as the stabilization parameter tends towards infinity, some stable and well-known nonconforming approximations of the Stokes problem. On the other hand, we can easily define an a posteriori error indicator, based on the reconstruction of a locally conservative H(div)-tensor. An a priori error analysis is also carried out, yielding optimal convergence rates. Numerical tests illustrating the accuracy and the robustness of the scheme are presented

Variational approach for the multiscale modeling of an estuarian river. Part 1 : Derivation and numerical approximation of a 2D horizontal model

The paper is devoted to the 2D hydrodynamical modeling and numerical approximation of an estuarian river flow. A new 2D horizontal model is derived and analyzed as a conforming approximation of the 3D time-discretized problem. It provides the three-dimensional velocity field as well as the pressure, which remains an unknown of the problem. Thanks to the framework of weak formulations, we avoid any closure problem usually encountered in the shallow water system and we can next employ finite elements for the approximation of the 2D model. The discrete problem is shown to be well-posed and finally, numerical tests are presented. The developed code is validated by means of comparisons with the classical shallow water model as well as with measured data

Numerical analysis of a Riccati type matrix transport equation

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Nitsche's Extended Finite Element Method for a Fracture Model in Porous Media

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Discretization of Giesekus model for polymer flows preserving positivity

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Analyse de méthodes mixtes d'éléments finis en mécanique

The research work of this habilitation is in the field of Numerical Analysis of Partial Differential Equations and concerns the modeling, the discretization, the a priori and a posteriori analysis of numerical schemes and the numerical simulation of different problems from mechanics. A leading thread in my research is the use and study of finite element methods (conforming, nonconforming, mixed, discontinuous Galerkin, stabilized) and of mixed formulations. The considered fields of application are: solid mechanics, petroleum engineering and fluid mechanics, both Newtonian and non-Newtonian. Thus, some problems in linear elasticity were studied, such as the discretization of two bending thin plate models endowed with physical boundary conditions. Single-phase but also multi-phase flows with heat transfer in porous media, described by the Darcy-Forchheimer equations together with an exhaustive energy law, as well as the thermomechanical coupling between a petroleum wellbore (fluid media) and a reservoir (porous media), were also considered, within the framework of a collaboration with the company Total. Finally, I have addressed several topics in fluid mechanics such as: the robust discretization of Stokes equation by a discontinuous Galerkin method related to nonconforming finite elements, the treatment of non-standard boundary conditions for the Navier-Stokes equations, the multi-dimensional hierarchical modeling of fluvial hydrodynamics, the realistic simulation of polymer flows and the stability of numerical schemes with respect to physical parameters, in particular for the Giesekus model.Les travaux de recherche de cette habilitation se situent dans le domaine de l'Analyse Numérique des Equations aux Dérivées Partielles et portent sur la modélisation, la discrétisation, l'analyse a priori et a posteriori de schémas et la simulation numérique de différents problèmes issus de la mécanique. Un fil conducteur de ces travaux est l'utilisation et l'étude des méthodes d'éléments finis (conformes, non-conformes, mixtes, de Galerkin discontinus, stabilisés) et des formulations mixtes. Les domaines d'application abordés sont la mécanique des solides élastiques, l'ingénierie pétrolière et la mécanique des fluides, newtoniens et non-newtoniens. Ainsi, des problèmes d'élasticité linéaire, comme la discrétisation de deux modèles de plaque mince en flexion munie de conditions aux limites physiques, ont été considérés. Des écoulements anisothermes dans les milieux poreux, décrits par les équations de Darcy-Forchheimer avec un bilan d'énergie exhaustif dans les cas mono et multi-phasique, ainsi qu'un couplage thermo-mécanique puits - réservoir pétrolier ont aussi été étudiés, dans le cadre d'une collaboration industrielle avec Total. Enfin, plusieurs questions en mécanique des fluides ont été abordées, comme la discrétisation robuste des équations de Stokes par une méthode de Galerkin discontinue en lien avec les éléments finis non-conformes, le traitement des conditions aux limites non-standard pour les équations de Navier-Stokes, la modélisation hiérarchique multi-dimensionnelle des écoulements fluviaux à surface libre, la simulation réaliste des écoulements de liquides polymères et la stabilité des schémas numériques par rapport aux paramètres physiques, en particulier pour le modèle de Giesekus

A positivity preserving discontinuous Galerkin method with applications in polymer flows

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