Final Report of AMoSS : Advanced Modeling on Shear Shallow Flows for Curved Topography : water and granular flows

Our aim was to derive new models and numerical strategies for shear shallow flows on curved topography. This modeling was supported by physical and numerical experiments, investigations and validations. The initial scientific plan was organized in seven tasks. We will reconsider the initial tasks and gives some detail of the achievements of the last tree year’s of collaboration

Shock capturing computations with stabilized Powell-Sabin elements

International audienceIn the last recent years, thanks to the increasing power of the computational machines , the interest in more and more accurate numerical schemes is growing. Methods based on high-order approximations are nowadays the common trend in the computational research community, in particular for CFD applications. This work is focused on Powell-Sabin (PS) finite elements, a finite element method (FEM) based on PS splines. PS splines are piecewise quadratic polynomials with a global C1 continuity , defined on conforming triangulations. Despite its attractive characteristics, so far this scheme hasn't had the attention it deserves. PS splines are adapted to unstructured meshes and, contrary to classical tensor product B-splines, they are particularly suited for local refinement , a crucial aspect in the analysis of highly convective and anisotropic equations. The additional global smoothness of the C1 space has a beneficial stabilization effect in the treatment of advection-dominated equations and leads to a better capturing of thin layers. Finally, unlike most of other typology of high-order finite elements, the numerical unknowns in PS elements are located in the vertices of the triangulation, leading to an easy treatment of the parallel aspects. Some geometrical issues related to PS elements are discussed here, in particular, the generation of the control triangles and the imposition of the boundary conditions. The PS FEM method is used to solve the compressible Euler equation in supersonic regime. A classical shock-capturing technique is used to reduce the oscillation around the discontinuity, while a variational multiscale formulation is used to introduce numerical diffusion in the streamwise direction. Some typical numerical examples are used to evaluate the performance of the PS discretization

Liquid and liquid-gas flows at all speeds : Reference solutions and numerical schemes

This paper present some reference solutions to be used in order to validate and improve numerical schemes for multiphase flows. We address here exact one-dimensional liquid and liquid-gas compressible flows solutions in nozzles. The exact solution is first derived for the compressible single liquid phase Euler equations and extends the well known ideal gas dynamic nozzle flow solutions.The all Mach number scheme is then derived. A preconditioned Riemann solver is built and embedded into the Godunov explicit scheme. It is shown that this method converges to exact solutions but needs too small time steps to be efficient. An implicit version is then derived, in one dimension first and second in the frame of 3D unstructured meshes

Propositions de méthodes et modèles eulériens pour les problèmes à interfaces entre fluides compressibles en présence de transfert de chaleur

On évalue différentes formulations euleriennes aptes au traitement de problèmes à interfaces entre fluides compressibles. La difficulté dans ce type de problème réside dans le calcul des variables thermodynamiques dans les zones de diffusion numériques produites aux interfaces. En effet, tout schéma eulérien diffuse artificiellement les discontinuités de contact (ou interfaces) et produit donc un mélange artificiel pour lequel la détermina- tion de l'état thermodynamique est difficile. De plus, lorsque l'état thermodynamique est mal déterminé, les méthodes échouent très rapidement en raison de pressions négatives ou d'arguments négatifs dans le calcul de la vitesse du son. Les modèles et les méthodes de résolution qui sont évaluées n'ont jamais été examinées pour le calcul de la température aux interfaces. L'examen des défauts et avantages de ces formulations nous conduit à en rejeter certaines et à en proposer une nouvelle, très efficace. Ce nouveau modèle est accompagné de son schéma numérique. On présente ensuite le traitement des transferts diffusifs aux interfaces, puis un exemple de résolution en deux dimensions d'espace. L'évaluation est effectuée sur une série de problèmes possédant des solutions exactes

Multidimensional Riemann Problem with Self-Similar Internal Structure – Part III– A Multidimensional Analogue of the HLLI Riemann Solver for Conservative Hyperbolic Systems

Just as the quality of a one-dimensional approximate Riemann solver is improved by the inclusion of internal sub-structure, the quality of a multidimensional Riemann solver is also similarly improved. Such multidimensional Riemann problems arise when multiple states come together at the vertex of a mesh. The interaction of the resulting one-dimensional Riemann problems gives rise to a strongly-interacting state. We wish to endow this strongly-interacting state with physically-motivated sub-structure. The fastest way of endowing such sub-structure consists of making a multidimensional extension of the HLLI Riemann solver for hyperbolic conservation laws. Presenting such a multidimensional analogue of the HLLI Riemann solver with linear sub-structure for use on structured meshes is the goal of this work. The multidimensional MuSIC Riemann solver documented here is universal in the sense that it can be applied to any hyperbolic conservation law. The multidimensional Riemann solver is made to be consistent with constraints that emerge naturally from the Galerkin projection of the self-similar states within the wave model. When the full eigenstructure in both directions is used in the present Riemann solver, it becomes a complete Riemann solver in a multidimensional sense. I.e., all the intermediate waves are represented in the multidimensional wave model. The work also presents, for the very first time, an important analysis of the dissipation characteristics of multidimensional Riemann solvers. The present Riemann solver results in the most efficient implementation of a multidimensional Riemann solver with sub-structure. Because it preserves stationary linearly degenerate waves, it might also help with well-balancing. Implementation-related details are presented in pointwise fashion for the one-dimensional HLLI Riemann solver as well as the multidimensional MuSIC Riemann solver. Several stringent test problems drawn from hydrodynamics, MHD and relativistic MHD are presented to show that the method works very well on structured meshes. Our results demonstrate the versatility of our method. The reader is also invited to watch a vide

High order stabilized finite element method for MHD plasma modeling

International audienceThe understanding of Magnetohydrodynamic (MHD) instabilities is quite essential for the optimization of magnetically confined plasma. For example, the ITER (International Thermonuclear Experimental Reactor) scenario is expected to generate oscillations in the plasma core, modes around the outward limit of the plasma confinement zone, or local reconfiguration of the magnetic field topology. Numerical simulations play an important role in the investigation of the non-linear behavior of these instabilities and the interpretation of experimental observations. The study of plasma instabilities requires the use of high order methods which can be difficult in the context of finite volume (FV). Hence, the use of finite element methods seems to be reasonable for numerical precision, adaptability and flexibility to complex geometries. Moreover, High order smooth (C1) finite element can be combined with the potential formulation to achieve the solenoidal condition of the magnetic field. As MHD instabilities are usually dominated byconvection, numerical schemes must take into account the effects of unresolved scales in order to insure stability of the numerical approach. In this context, stabilized finite element method (FEM) can provide a useful framework for the numerical approximation. Galerkin finite element gives rise to a centered approximation of differential operators. This is suitable for elliptic like operators but can lead to nonphysical behaviors when flows are dominated by the effect of hyperbolic operators (convection). The variational multi-scale (VMS) formulation provides attractive guidelines for the development of stabilized schemes that take into account the hyperbolic nature of the considered systems. In this frame of work, stabilizationis achieved by an additional contribution to the weak formulation which mimics the effects of the unresolved scales over the resolved ones. The critical point of this strategy is the design of a scaling matrix used to adjust the numerical dissipation such as to preserve the order of accuracy of the Galerkin method. This is very good for smooth solutions however it might generate spurious Gibbs oscillations associated to spectral truncation in the wave-number space. Hence, discontinuity capturing is also often used to enforce the total variation stability where the solution develops sharp gradients. We will present applications to MHD and Reduced-MHD in tokamak (toroidal) geometry using the non-linear finite element code JOREK where the VMS stabilization will be discussed in terms of the Taylor-Galerkin formulatio

Shock capturing computations with stabilized Powell-Sabin elements

International audienceIn the last recent years, thanks to the increasing power of the computational machines , the interest in more and more accurate numerical schemes is growing. Methods based on high-order approximations are nowadays the common trend in the computational research community, in particular for CFD applications. This work is focused on Powell-Sabin (PS) finite elements, a finite element method (FEM) based on PS splines. PS splines are piecewise quadratic polynomials with a global C1 continuity , defined on conforming triangulations. Despite its attractive characteristics, so far this scheme hasn't had the attention it deserves. PS splines are adapted to unstructured meshes and, contrary to classical tensor product B-splines, they are particularly suited for local refinement , a crucial aspect in the analysis of highly convective and anisotropic equations. The additional global smoothness of the C1 space has a beneficial stabilization effect in the treatment of advection-dominated equations and leads to a better capturing of thin layers. Finally, unlike most of other typology of high-order finite elements, the numerical unknowns in PS elements are located in the vertices of the triangulation, leading to an easy treatment of the parallel aspects. Some geometrical issues related to PS elements are discussed here, in particular, the generation of the control triangles and the imposition of the boundary conditions. The PS FEM method is used to solve the compressible Euler equation in supersonic regime. A classical shock-capturing technique is used to reduce the oscillation around the discontinuity, while a variational multiscale formulation is used to introduce numerical diffusion in the streamwise direction. Some typical numerical examples are used to evaluate the performance of the PS discretization

A Godunov-Type Solver for the Numerical Approximation of Gravitational Flows

International audienceWe present a new numerical method to approximate the solutions of an Euler-Poisson model, which is inherent to astrophysical flows where gravity plays an important role. We propose a discretization of gravity which ensures adequate coupling of the Poisson and Euler equations, paying particular attention to the gravity source term involved in the latter equations. In order to approximate this source term, its discretization is introduced into the approximate Riemann solver used for the Euler equations. A relaxation scheme is involved and its robustness is established. The method has been implemented in the software HERACLES and several numerical experiments involving gravitational flows for astrophysics highlight the scheme

Self-organized populations interacting under pursuit-evasion dynamics

International audienceWe discuss the modelling of interacting populations through pursuit-evasion ---- or attraction-repulsion ---- principles~: preys try to escape chasers, chasers are attracted by the presence of preys. We construct a hierarchy of models, ranging from ODEs systems with finite numbers of individuals of each population, to hydrodynamic systems. First order macroscopic models look like generalized ''two-species Keller-Segel equations''. But, due to cross--interactions, we can show that the system does not exhibit any blow up phenomena in finite time. We also obtain second order models, that have the form of systems of balance laws, derived from kinetic models. We bring out a few remarkable features of the models based either on mathematical analysis or numerical simulations

H1-parametrizations of complex planar physical domains in Isogeometric analysis

International audienceIsogeometric analysis (IGA) is a method for solving geometric partial differential equations(PDEs). Generating parameterizations of a PDE's physical domain is the basic and important issues within IGA framework. In this paper , we present a global H 1-parameterization method for a planar physical domain with complex topology