25 research outputs found

    Capturing nonclassical shocks in nonlinear elastodynamic with a conservative finite volume scheme

    Full text link
    For a model of nonlinear elastodynamics, we construct a finite volume scheme which is able to capture nonclassical shocks (also called undercompressive shocks). Those shocks verify an entropy inequality but are not admissible in the sense of Liu. They verify a kinetic relation which describes the jump, and keeps an information on the equilibrium between a vanishing dispersion and a vanishing diffusion. The scheme pre-sented here is by construction exact when the initial data is an isolated nonclassical shock. In general, it does not introduce any diffusion near shocks, and hence nonclas-sical solutions are correctly approximated. The method is fully conservative and does not use any shock-tracking mesh. This approach is tested and validated on several test cases. In particular, as the nonclassical shocks are not diffused at all, it is possible to obtain large time asymptotics

    A Non-dissipative Reconstruction Scheme for the Compressible Euler Equations

    Full text link
    We present a finite volume scheme, first on the Burgers equations, then on the Euler equations, based on a conservative reconstruction of shocks inside each cells of the mesh. Its main features are the following points. First, the scheme is exact whenever the initial datum is a pure shock, in the sense that the approximate solution is the exact solution averaged over the cells of the mesh. Second, the scheme has in general a very low numerical diffusion and the shocks have a width of one or two cells. Third, no spurious oscillations in the momentum appear behind slowly moving shocks, which is not the case in most of the scheme developed so far. We also present prospective result on the full Euler equations with energy. The wall heating phenomenon, which is an artificial elevation of the temperature when a shock reflects on a wall, is also drastically diminished

    Convergence of finite volumes schemes for the coupling between the inviscid Burgers equation and a particle

    Get PDF
    International audienceIn this paper, we prove the convergence of a class of finite volume schemes for the model of coupling between a Burgers fluid and a pointwise particle introduced in [LST08]. In this model, the particle is seen as a moving interface through which an interface condition is imposed, which links the velocity of the fluid on the left and on the right of the particle and the velocity of the particle (the three quantities are all not equal in general). The total impulsion of the system is conserved through time.The proposed schemes are consistent with a “large enough” part of the interface conditions. The proof of convergence is an extension of the one of [AS12] to the case where the particle moves under the influence of the fluid. It yields two main difficulties: first, we have to deal with time-dependent flux and interface condition, and second with the coupling between and ODE and a PDE

    Semaine d'Etude Mathématiques et Entreprises 5 : Détection de marques de cylindres sur une ligne sidérurgique, ou comment séparer des sources périodiques dans une image bruitée

    Get PDF
    Lors de la fabrication de tôe par laminage, des défauts périodiques peuvent être marqués par les cylindres qui compressent le métal. Un contrôle automatique est incorporé dans le processus afin de détecter les défauts par traitement numérique des images prises par des capteurs. Ces images sont de grandes tailles et fortement bruitées de sorte que les défauts sont souvent à peine perceptibles. Arcelor Mittal propose ici de réfléchir sur des méthodes robustes et performantes qui permettraient l'extraction des composantes périodiques des images. Ce document est la synthèse des méthodes et pistes que nous avons explorées lors de cette cinquième édition de la SEME à l'Ecole des Mines de Nancy

    Problèmes d’interfaces et couplages singuliers dans les systèmes hyperboliques : analyse et analyse numérique

    No full text
    In this work, we study two problems concerning hyperbolic systems involving interfaces. The first one concerns the study of models of coupling between a compressible fluid and a pointwise particle. The second one deals with the sharp numerical approximation of shocks, which are discontinuities that appear in the solutions of hyperbolic systems.In the first two parts of the manuscript, we introduce different models of fluid-particle couplings. The fluid and the particle interact on each other through a drag force, which brings their velocities closer to one another. The coupling is singular because it can be written as the product of a discontinuous function by a Dirac measure. However, the system can be precisely defined as follows. The particle is seen as an interface through which interface conditions linking the properties of the fluid with those of the particle are imposed. When the fluid follows the compressible Burgers equations, we prove the convergence of a family of finite volume schemes and obtain the existence of a solution when the initial data has total bounded variation. In the more difficult case where the fluid is described by the isothermal Euler equations, we prove the existence and uniqueness of a selfsimilar solution to the Riemann problem, when the particle is motionless. Numerical experiments are also presented.In the last part of this work, we build non diffusive numerical schemes for different hyperbolic systems. These finite volume schemes are built to be exact when the initial data is an isolated shock. They are based on a discontinuous reconstruction of the solution at the beginning of each time step, in order to reconstruct shocks inside some specific cells of the mesh. The schemes we present have a very low numerical diffusion and, when the reconstruction operator is well chosen, they are able to correctly approximate the solution on various problematic test cases. These cases include slowly moving shocks, strong shocks and shock reflections for gas dynamics, as well as the apparition of nonclassical shocks for systems that are not truely nonlinear.Dans ce travail, nous nous intéressons à deux problèmes de la théorie des systèmes hyperboliques faisant intervenir des interfaces. Le premier concerne des modèles de couplages entre un fluide compressible et une particule ponctuelle et le second concerne la capture numérique précise des chocs, ces discontinuités qui apparaissent dans les solutions des systèmes hyperboliques.Sur la première thématique, nous commençons par introduire les différents modèles, dans lesquels la particule et le fluide interagissent à travers une force de frottement qui tend à rapprocher leurs vitesses. Le couplage est singulier car il fait intervenir le produit d’une fonction discontinue par une mesure de Dirac. On peut toutefois définir précisément le système en voyant la particule comme une interface à travers laquelle des relations liant les propriétés du fluide et celle de la particule sont imposées. Lorsque le fluide suit une équation de Burgers, nous démontrons la convergence d’une classe de schéma numérique, et nous obtenons l’existence d’une solution au problème de Cauchy pour une donnée initiale à variation totale bornée. Dans le cas plus complexe où le fluide est décrit par les équa- tions d’Euler isothermes, on prouve l’existence et l’unicité d’une solution autosemblable au problème de Riemann lorsque la particule est immobile. Des simulations numériques sont également présentées.La dernière partie de la thèse est consacrée à la construction de schémas non diffusifs pour les systèmes hyperboliques. Ces schémas, de type volumes finis, sont construits pour être exact lorsque la donnée initiale est un choc isolé. Ils sont basé sur une reconstruction discontinue de la solution au début de chaque itération en temps, dans le but de reconstituer des chocs à l’intérieur de certaines cellules du maillage. Cette stratégie mène à des schémas très peu diffusifs qui, lorsque l’opérateur de reconstruction est bien choisi, approchent correctement les solutions de cas tests problématiques (chocs lents, chocs forts, réflexions pour la dynamique des gaz, chocs non classiques pour les systèmes qui ne sont pas vraiment non linéaires)

    Error estimate for the upwind scheme for the linear transport equation with boundary data

    No full text
    International audienceWe study the upwind finite volume scheme on a general mesh for the initial and boundary-value problem associated with a linear transport equation. For any BV initial and boundary data we prove the (optimal) convergence rate 1/2 in the L 1-norm. Compared to previous works, our main contribution is to take into account the boundary data and to relax some regularity assumptions on the velocity field. As an intermediate result, we also provide a complete proof of the BV regularity of weak solutions of such a general transport problem
    corecore