An Interior Penalty Method with C0C^0 Finite Elements for the Approximation of the Maxwell Equations in Heterogeneous Media: Convergence Analysis with Minimal Regularity

The present paper proposes and analyzes an interior penalty technique using $C^0$-finite elements to solve the Maxwell equations in domains with heterogeneous properties. The convergence analysis for the boundary value problem and the eigenvalue problem is done assuming only minimal regularity in Lipschitz domains. The method is shown to converge for any polynomial degrees and to be spectrally correct.Comment: 36 page

A sharp cartesian method for the simulation of air-water interface

Abstract: We firstly present a sharp cartesian method for the simulation of incompressible flows with high density and viscosity ratios, like air-water interfaces. This method is inspired from the second-order cartesian method for elliptic problems with immersed interfaces developed i

A sharp cartesian method for the simulation of air-water interface

International audienceWe firstly present a sharp cartesian method for the simulation of incompressible flows with high density and viscosity ratios, like air-water interfaces. This method is inspired from the second-order cartesian method for elliptic problems with immersed interfaces developed in [1]. Then, because a high-order interface description is necessary in this context, we present a level-set technique allowing to maintain in the course of time a third-order accuracy of the level-set itself, and thus a first order accuracy of the curvature

A new approach to modelling the two way coupling for momentum transfer in a hollow-cone spray

[EN] A new approach to modelling the interaction between droplets and the carrier phase is suggested. The new model is applied to the analysis of a spray injected into a chamber of quiescent air, using an Eulerian-Lagrangian approach. The conservative formulation of the equations for mass, momentum and energy transport is used for the analysis of the carrier phase. The dispersed phase is modelled using the Lagrangian approach with droplets represented by individual parcels. The implementation of the Discontinuous Galerkin method (ForestDG), based on a topological representation of the computational mesh by a hierarchical structure consisting of oct- quad- and binary trees, is used in our analysis. Adaptive mesh refinement (h-refinement) enables us to increase the spatial resolution for the computational mesh in the vicinity of the points of interest such as interfaces, geometrical features, or flow discontinuities. The local increase in the expansion order (p-refinement) at areas of high strain rates or vorticity magnitude results in an increase of the order of the accuracy of discretisation of shear layers and vortices. The initial domain consists of a graph of unitarian-trees representing hexahedral, prismatic and tetrahedral elements. The ancestral elements of the mesh can be split into self-similar elements allowing each tree to grow branches to an arbitrary level of refinement. The connectivity of the elements, their genealogy and their partitioning are described by linked lists of pointers. These are attached to the tree data structure which facilitates the on-the-fly splitting, merging and repartitioning of the computational mesh by rearranging the links of each node of the tree. This enables us to refine the computational mesh in the vicinity of the droplet parcels aiming to accurately resolve the coupling between the two phases.The authors are grateful to EPSRC (grants EP/K005758/1 and EP/M002608/1) for financial supportPapoutsakis, A.; Sazhin, S.; Begg, S.; Danaila, I.; Luddens, F. (2017). A new approach to modelling the two way coupling for momentum transfer in a hollow-cone spray. En Ilass Europe. 28th european conference on Liquid Atomization and Spray Systems. Editorial Universitat Politècnica de València. 448-455. https://doi.org/10.4995/ILASS2017.2017.4671OCS44845

Theoretical and numerical analysis of the magnetohydrodynamics equations (application to dynamo action)

On s'intéresse dans ce mémoire aux équations de la magnétohydrodynamique (MHD) dans des milieux hétérogènes, i.e. dans des milieux pouvant présenter des variations (éventuellement brutales) de propriétés physiques. En particulier, on met ici l'accent sur la résolution des équations de Maxwell dans des milieux avec des propriétés magnétiques inhomogènes. On présentera une méthode non standard pour résoudre ce problème à l'aide d'éléments finis de Lagrange. On évoquera ensuite l'implémentation dans le code SFEMaNS, développé depuis 2002 par J.-L. Guermond, C. Nore, J. Léorat, R. Laguerre et A. Ribeiro, ainsi que les premiers résultats obtenus dans les simulations de dynamo. Nous nous intéresserons par exemple au cas de la dynamo dite de Von Kármán, afin de comprendre l'expérience VKS2. En outre, nous aborderons des cas de dynamo en précession, ou encore le problème de la dynamo au sein d'un écoulement de Taylor-Couette.We focus on the magnetohydrodynamics (MHD) equations in hetereogeneous media, i.e. media with (possibly brutal) variations on the physical properties. In particular, we are interested in solving the Maxwell equations with discontinuous magnetic properties. We introduce a method that is, to the best of our knowledge, new to solve this problem using only Lagrange Finite Elements. We then discuss its implementation in SFEMaNS, a numerical code developped since 2002 by J.-L. Guermond, C. Nore, J. Léorat, R. Laguerre and A. Ribeiro. We show the results of the first dynamo simulations we have been able to make with this solver. For instance, we present a kinematic dynamo in a VKS setup, as well as some results about dynamo action induced either by a Taylor-Couette flow, or by a precessionnally driven flow.PARIS11-SCD-Bib. électronique (914719901) / SudocSudocFranceF

Quantum turbulence simulations using the Gross-Pitaevskii equation: high-performance computing and new numerical benchmarks

This paper is concerned with the numerical investigation of Quantum Turbulence (QT) described by the Gross-Pitaevskii (GP) equation. Numerical simulations are performed using a parallel (MPI-OpenMP) code based on a pseudo-spectral spatial discretization and second order splitting for the time integration. We start by revisiting (in the framework of high-performance/high-accuracy computations) well-known GP-QT settings, based on the analogy with classical vortical flows: Taylor-Green (TG) vortices and Arnold-Beltrami-Childress (ABC) flow. Two new settings are suggested to build the initial condition for the QT simulation. They are based on the direct manipulation of the wave function by generating a smoothed random phase (SRP) field, or seeding random vortex rings (RVR) pairs. The new initial conditions have the advantage to be simpler to implement than the TG and ABC approaches, while generating statistically equivalent QT fields. Each of these four GP-QT settings is described in detail by defining corresponding benchmarks that could be used to validate/calibrate new GP codes. We offer a comprehensive description of the numerical and physical parameters of each benchmark. We analyze the results in detail and present values, spectra and structure functions of main quantities of interest (energy, helicity, etc.) that are useful to describe the turbulent flow. Some general features of QT are identified, despite the variety of initial states.Comment: 61 pages, 21 figure

Equations de la MHD en milieu hétérogène

Dynamo effect is one of the most commonly accepted explanation for the existence of a magnetic field on Earth. Aiming to the numerical simulation of the VKS2 experiment (one of the successful experiments highlighting dynamo effect), a numerical code (SFEMaNS) has been developed. It combines the use of Fourier decomposition in an azimuthal direction, and a Lagrange Finite Element Solver in meridian planes. This choice of FE is a challenging task and requires a non-standard approach. Results have been successfully confronted to experimental results and to other numerical simulations

Numerical dynamo action in cylindrical containers

The purpose of this paper is to present results from numerical simulations of dynamo action in relation with two magnetohydrodynamics (MHD) experiments using liquid sodium in cylindrical containers. The first one is the von Kármán sodium (VKS) experiment from Cadarache (France), the second one is a precession-driven dynamo experiment from the DREsden sodium facility for DYNamo and thermohydraulic studies (DRESDYN)

An efficient Adaptive Mesh Refinement (AMR) algorithm for the Discontinuous Galerkin method: Applications for the computation of compressible two-phase flows

We present an Adaptive Mesh Refinement (AMR) method suitable for hybrid unstructured meshes that allows for local refinement and de-refinement of the computational grid during the evolution of the flow. The adaptive implementation of the Discontinuous Galerkin (DG) method introduced in this work (ForestDG) is based on a topological representation of the computational mesh by a hierarchical structure consisting of oct- quad- and binary trees. Adaptive mesh refinement (h-refinement) enables us to increase the spatial resolution of the computational mesh in the vicinity of the points of interest such as interfaces, geometrical features, or flow discontinuities. The local increase in the expansion order (p-refinement) at areas of high strain rates or vorticity magnitude results in an increase of the order of accuracy in the region of shear layers and vortices. A graph of unitarian-trees, representing hexahedral, prismatic and tetrahedral elements is used for the representation of the initial domain. The ancestral elements of the mesh can be split into self-similar elements allowing each tree to grow branches to an arbitrary level of refinement. The connectivity of the elements, their genealogy and their partitioning are described by linked lists of pointers. An explicit calculation of these relations, presented in this paper, facilitates the on-the-fly splitting, merging and repartitioning of the computational mesh by rearranging the links of each node of the tree with a minimal computational overhead. The modal basis used in the DG implementation facilitates the mapping of the fluxes across the non conformal faces. The AMR methodology is presented and assessed using a series of inviscid and viscous test cases. Also, the AMR methodology is used for the modelling of the interaction between droplets and the carrier phase in a two-phase flow. This approach is applied to the analysis of a spray injected into a chamber of quiescent air, using the Eulerian–Lagrangian approach. This enables us to refine the computational mesh in the vicinity of the droplet parcels and accurately resolve the coupling between the two phases

Analyse théorique et numérique des équations de la magnétohydrodynamique : application à l'effet dynamo

We focus on the magnetohydrodynamics (MHD) equations in hetereogeneous media, i.e. media with (possibly brutal) variations on the physical properties. In particular, we are interested in solving the Maxwell equations with discontinuous magnetic properties. We introduce a method that is, to the best of our knowledge, new to solve this problem using only Lagrange Finite Elements. We then discuss its implementation in SFEMaNS, a numerical code developped since 2002 by J.-L. Guermond, C. Nore, J. Léorat, R. Laguerre and A. Ribeiro. We show the results of the first dynamo simulations we have been able to make with this solver. For instance, we present a kinematic dynamo in a VKS setup, as well as some results about dynamo action induced either by a Taylor-Couette flow, or by a precessionnally driven flow.On s'intéresse dans ce mémoire aux équations de la magnétohydrodynamique (MHD) dans des milieux hétérogènes, i.e. dans des milieux pouvant présenter des variations (éventuellement brutales) de propriétés physiques. En particulier, on met ici l'accent sur la résolution des équations de Maxwell dans des milieux avec des propriétés magnétiques inhomogènes. On présentera une méthode non standard pour résoudre ce problème à l'aide d'éléments finis de Lagrange. On évoquera ensuite l'implémentation dans le code SFEMaNS, développé depuis 2002 par J.-L. Guermond, C. Nore, J. Léorat, R. Laguerre et A. Ribeiro, ainsi que les premiers résultats obtenus dans les simulations de dynamo. Nous nous intéresserons par exemple au cas de la dynamo dite de Von Kármán, afin de comprendre l'expérience VKS2. En outre, nous aborderons des cas de dynamo en précession, ou encore le problème de la dynamo au sein d'un écoulement de Taylor-Couette