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International audience; Fused Filament Fabrication (FFF) is a promising technology that is largely developed in small series, as this technology optimizes supply chains by reducing production time and costs. However, its shortcomings have slowed its adoption as a dominant production technology. Among its weaknesses, this work focuses on geometric and dimensional accuracy within tolerance range. There is a need for understanding the sources of geometrical inaccuracies and for methods of characterizing them, in order to modify the input parameters to eventually obtain the desired geometry. This work first focuses on the geometric and dimensional accuracy of parts printed by the FFF process by studying the influence of the inner radius of a cylindrical part, the type of material and the type of filling pattern. The levels with the greatest dimensional dispersion are the largest radius, the nylon material, and the hexagonal filling pattern. Secondly, a defect characterization method associated with a parametric mathematical model is developed. The 3D scanner enables the retrieval of the coordinates of the printed geometry; this allows to characterize the errors with respect to the theoretical 3D model and to modelize the printed part by a series of ellipses of which we obtain the analytical equations, as a first step of a correction process.

International audience; In this article, we consider the time dependent convection-diffusion-reaction equation coupled with the Darcy equation. We propose a numerical scheme based on finite element methods for the discretization in space and the implicit Euler method for the discretization in time. We establish optimal a posteriori error estimates with two types of computable error indicators, the first one linked to the time discretization and the second one to the space discretization. Finally, numerical investigations are performed and presented.

International audience; In this paper, we study in two and three space dimensions, the a posteriori error estimates for the Large Eddy Simulation applied to the Navier-Stokes system. We begin by introducing the Navier-Stokes and the corresponding Large Eddy Simulation (LES) equations. Then we introduce the corresponding discrete problem based on the finite element method. We establish an a posteriori error estimation with three types of error indicators related to the filter of the LES method, to the discretization and to the linearization. Finally, numerical investigations are shown and discussed.

International audience; Classical finite volume schemes for the Euler system are not accurate at low Mach number and some fixes have to be used and were developed in a vast literature over the last two decades. The question we are interested in in this article is: What about if the porosity is no longer uniform? We first show that this problem may be understood on the linear wave equation taking into account porosity. We explain the influence of the cell geometry on the accuracy property at low Mach number. In the triangular case, the stationary space of the Godunov scheme approaches well enough the continuous space of constant pressure and divergence-free velocity, while this is not the case in the Cartesian case. On Cartesian meshes, a fix is proposed and accuracy at low Mach number is proved to be recovered. Based on the linear study, a numerical scheme and a low Mach fix for the non-linear system, with a non-conservative source term due to the porosity variations, is proposed and tested.

International audience; We propose and analyse a parallel method, both in the time and space directions, that couples the Parareal algorithm with the Optimized Schwarz waveform relaxation (OSWR) method, with only few OSWR iterations in the fine propagator and with a simple coarse propagator deduced from the Backward Euler method. The analysis of this coupled method is presented for a one-dimensional advection-reaction-diffusion equation. We prove a general convergence result for this method via energy estimates. Numerical results for two-dimensional advection-diffusion problems and for a diffusion equation with strong heterogeneities are presented, to illustrate the performance of the coupled Parareal-OSWR algorithm.

International audience; In this article, we study the time dependent convection-diffusion-reaction equation coupled with the Darcy equation. We propose and analyze two numerical schemes based on finite element methods for the discretization in space and the implicit Euler method for the discretization in time. An optimal a priori error estimate is then derived for each numerical scheme. Finally, we present some numerical experiments that confirm the theoretical accuracy of the discretization.

In the aim to find the simplest and most efficient shape of a noise absorbing wall to dissipate the acoustical energy of a sound wave, we consider a frequency model described by the Helmholtz equation with a damping on the boundary. The well-posedness of the model is shown in a class of domains with d-set boundaries (N -- 1 $\le$ d < N). We introduce a class of admissible Lipschitz boundaries, in which an optimal shape of the wall exists in the following sense: We prove the existence of a Radon measure on this shape, greater than or equal to the usual Lebesgue measure, for which the corresponding solution of the Helmholtz problem realizes the infimum of the acoustic energy defined with the Lebesgue measure on the boundary. If this Radon measure coincides with the Lebesgue measure, the corresponding solution realizes the minimum of the energy. For a fixed porous material, considered as an acoustic absorbent, we derive the damping parameters of its boundary from the corresponding time-dependent problem described by the damped wave equation (damping in volume)., SIAM Journal on Control and Optimization, Society for Industrial and Applied Mathematics, In press

We propose a method to explain the behaviour of the Godunov finite volume scheme applied to the linear wave equation with Coriolis source term at low Froude number. In particular, we use the Hodge decomposition and we study the properties of the modified equation associated to the Godunov scheme. Based on the structure of the discrete kernel of the linear operator discretized by using the Godunov scheme, we clearly explain the inaccuracy of the classical Godunov scheme at low Froude number and we introduce a way to modify it to recover a correct accuracy.

International audience; We present an application of the discrete duality finite volume method to the numerical approximation of the 2D Stokes or (unsteady) Navier–Stokes equations associated to Dirichlet boundary conditions. The finite volume method is based on the use of discrete differential operators obeying some discrete duality principles. The scheme may be seen as an extension of the classical MAC scheme to almost arbitrary meshes, thanks to an appropriate choice of degrees of freedom. Different numerical examples over triangular, cartesian, quadrangular and locally refined meshes are led in order to illustrate the possibilities and weaknesses of the method.

The study deals with collocated Godunov type finite volume schemes applied to the two-dimensional linear wave equation with Coriolis source term. The purpose is to explain the wrong behaviour of the classic scheme and to modify it in order to avoid accuracy issues around the geostrophic equilibrium and in geostrophic adjustment processes. To do so, a Hodge-like decomposition is introduced. Then three different well-balanced strategies are introduced. Some properties of the associated modified equations are proven and then extended to the semi-discrete case. Stability of fully discrete schemes under a suitable CFL condition is established thanks to a Von Neumann analysis. Some numerical results reinforce the purpose and exhibit the concrete improvements achieved by the application of these new techniques in both linear and nonlinear cases.

International audience; The shallow water equations can be used to model many phenomena in geophysical fluid mechanics. For large scales, the Coriolis force plays an important role and the geostrophic equilibrium which corresponds to the balance between the pressure gradient and the Coriolis force is an important feature. In this communication , we investigate the stability condition and the behavior of the so-called Apparent Topography scheme which is capable of capturing a discrete version of the geostrophic equilibrium.

FVCA 2017 : International Conference on Finite Volumes for Complex Applications; This benchmark proposes test-cases to assess innovative finite volume type methods developped to solve the equations of incompressible fluid mechanics. Emphasis is set on the ability to handle very general meshes, on accuracy, robustness and computational complexity. Two-dimensional as well as three-dimensional tests with known analytical solutions are proposed for the steady Stokes and both steady and unsteady Navier-Stokes equations, as well as classical lid-driven cavity tests.

International audience

We present an application of the discrete duality finite volume method to the numerical approximation of the vorticity-velocity-pressure formulation of the two-dimensional Stokes equations, associated to various nonstandard boundary conditions. The finite volume method is based on the use of discrete differential operators obeying some discrete duality principles. The scheme may be seen as an extension of the classical Marker and Cell scheme to almost arbitrary meshes, thanks to an appropriate choice of degrees of freedom. The efficiency of the scheme is illustrated by numerical examples over unstructured triangular and locally refined nonconforming meshes, which confirm the theoretical convergence analysis led in the article. © 2014 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 31: 1–30, 2015

International audience; This article is composed of three self-consistent chapters that can be read independently of each other. In Chapter 1, we define and we analyze the low Mach number problem through a linear analysis of a perturbed linear wave equation. Then, we show how to modify Godunov type schemes applied to the linear wave equation to make this scheme accurate at any Mach number. This allows to define an all Mach correction and to propose a linear all Mach Godunov scheme for the linear wave equation. In Chapter 2, we apply the all Mach correction proposed in Chapter 1 to the case of the non-linear barotropic Euler system when the Godunov type scheme is a Roe scheme. A linear stability result is proposed and a formal asymptotic analysis justifies the construction in this non-linear case by showing how this construction is related with the linear analysis of Chapter 1. At last, we apply in Chapter 3 the all Mach correction proposed in Chapter 1 in the case of the full Euler compressible system. Numerous numerical results proposed in Chapters 1, 2 and 3 justify the theoretical results and show that the obtained all Mach Godunov type schemes are both accurate and stable for all Mach numbers. We also underline that the proposed approach can be applied to other schemes and allows to justify other existing all Mach schemes.

We introduce continuous tools to study the low Mach number behavior of the Godunov scheme applied to the linear wave equation with porosity on cartesian meshes. More precisely, we extend the Hodge decomposition to a weighted L2 space in the continuous case and we study the properties of the modified equation associated to this Godunov scheme. This allows to partly explain the inaccuracy of the Godunov scheme at low Mach number on cartesian meshes and to propose two corrections: a first one named low Mach and a second one named all Mach. These results are preliminary since it remains to prove them in the discrete case.

By studying the structure of the discrete kernel of the linear acoustic operator discretized with a Godunov scheme, we clearly explain why the behaviour of the Godunov scheme applied to the linear wave equation deeply depends on the space dimension and, especially, on the type of mesh. This approach allows us to explain why, in the periodic case, the Godunov scheme applied to the resolution of the compressible Euler or Navier-Stokes system is accurate at low Mach number when the mesh is triangular or tetrahedral and is not accurate when the mesh is a 2D (or 3D) cartesian mesh. This approach confirms also the fact that a Godunov scheme remains accurate when it is modified by simply centering the discretization of the pressure gradient.

International audience; An efficient and fully computable a posteriori error bound is derived for the discrete duality finite volume discretization of the Laplace equation on very general twodimensional meshes. The main ingredients are the equivalence of this method with a finite element like scheme and tools from the finite element framework. Numerical tests are performed with a stiff solution on highly nonconforming locally refined meshes and with a singular solution on triangular meshes.

A new finite volume method is presented for discretizing the two-dimensional Maxwell equations. This method may be seen as an extension of the covolume type methods to arbitrary, possibly non-conforming or even non-convex, n-sided polygonal meshes, thanks to an appropriate choice of degrees of freedom. An equivalent formulation of the scheme is given in terms of discrete differential operators obeying discrete duality principles. The main properties of the scheme are its energy conservation, its stability under a CFL-like condition, and the fact that it preserves Gauss' law and divergence free magnetic fields. Second-order convergence is demonstrated numerically on non-conforming and distorted meshes.

We compare two models used to compute the internal hydrodynamics of a gas centrifuge. The scoop action is taken into account through boundary conditions on the flow entering the bowl of the centrifuge in the first model, and through sinks and drag forces in the chambers of the centrifuge in the second. The numerical approximations of the models are based on a finite volume scheme on staggered rectangular grids and on a fixed-point iterative method. Convergence of the approximations is studied numerically on a family of refined grids and comparisons of the two models are discussed for the Iguacu centrifuge. It appears that linear computations on rough grids are sufficient in the first model to correctly predict the separative power of the centrifuge, while other parameters like the return flow or the drag forces require finer meshes and non-linear computations in the second model.

International audience; We derive an a posteriori error estimation for the discrete duality finite volume (DDFV) discretization of the stationary Stokes equations on very general twodimensional meshes, when a penalty term is added in the incompressibility equation to stabilize the variational formulation. Two different estimators are provided: one for the error on the velocity and one for the error on the pressure. They both include a contribution related to the error due to the stabilization of the scheme, and a contribution due to the discretization itself. The estimators are globally upper as well as locally lower bounds for the errors of the DDFV discretization. They are fully computable as soon as a lower bound for the inf-sup constant is available. Numerical experiments illustrate the theoretical results and we especially consider the influence of the penalty parameter on the error for a fixed mesh and also of the mesh size for a fixed value of the penalty parameter. A global error reducing strategy that mixes the decrease of the penalty parameter and adaptive mesh refinement is described.

We present a finite volume method based on the integration of the Laplace equation on both the cells of a primal almost arbitrary two-dimensional mesh and those of a dual mesh obtained by joining the centers of the cells of the primal mesh. The key ingredient is the definition of discrete gradient and divergence operators verifying a discrete Green formula. This method generalizes an existing finite volume method that requires "Voronoi-type" meshes. We show the equivalence of this finite volume method with a non-conforming finite element method with basis functions being P 1 on the cells, generally called "diamond-cells", of a third mesh. Under geometrical conditions on these diamond- cells, we prove a first-order convergence both in the H 1 norm and in the L 2 norm. Superconvergence results are obtained on certain types of homothetically refined grids. Finally, numerical experiments confirm these results and also show second-order convergence in the L 2 norm on general grids. They also indicate that this method performs particularly well for the approximation of the gradient of the solution, and may be used on degenerating triangular grids. An example of application on non- conforming locally refined grids is given.

This paper deals with the linear response of a plasma in a one-dimensional bounded geometry under the action of a time-periodic electric field. The nonlinear Vlasov equation is solved by following the characteristic curves until they reach the boundary of the domain, where the distribution function of the incoming particles is supposed to be known and independent of time. Then, a first-order Taylor expansion in the velocity variable is performed, thanks to an approximation of the exact characteristics by the unperturbed ones. The resulting first-order correction to the distribution function is finally integrated over velocities to yield the dielectric function. The special case of a plane wave for the electric field is examined and the results are compared with the more usual unbounded case: the integral does not present any singularity in the vicinity of resonant particles and the dielectric function depends on the distance to the boundary and tends to the usual infinite-geometry value when this distance tends to infinity, with a rate of convergence proportional to its inverse square root. Numerical examples are provided for illustration.

We present an extension of the Optimized Schwarz Waveform Relaxation method with Robin transmission conditions to finite volume schemes of DDFV type (Discrete Duality Finite Volumes) for solving heterogeneous time-dependent advection-diffusion problems. We propose a new strategy which is well adapted to domain decomposition for coupling upwind discretization of the convection with diffusion in the context of a finite volume method. The method is proven to be well posed and we prove the convergence of the iterative algorithm. Then we present numerical results to illustrate the method.

A self-consistent nonlinear model of an isotope separation process based on selective ion cyclotron resonance heating in a magnetized plasma is presented, and its numerical resolution is described. The response of the electrons to the electromagnetic field is modeled by a cold and linear conductivity tensor, while a particle method is used to solve nonlinear Vlasov equations for the ions. The resolution of the time-harmonic Maxwell equations is achieved by a finite-element method. Both steps are coupled by an iterative procedure, which shows fast convergence. Results are presented for the case of a solenoidal launching antenna.

Usually, non-stationary numerical calculations in electromagnetics are based on the hyperbolic evolution equations for the electric and magnetic fields and leave Gauss' law out of consideration because the latter is a consequence of the former and of the charge conservation equation in the continuous case. However, in the simulation of the self-consistent movement of charged particles in electromagnetic fields, it is a well-known fact that the approximation of the particle motion introduces numerical errors and that, consequently, the charge conservation equation is not satisfied on the dicrete level. Then, in order to avoid the increase of errors in Gauss' law, a divergence cleaning step which solves a Poisson equation for a correction potential is often added. In the present paper, a new method for incorporating Gauss' law into non-stationary electromagnetic simulation codes is developed, starting from a constrained formulation of the Maxwell equations. The resulting system is hyperbolic, and the divergence errors propagate with the speed of light to the boundary of the computational domain. Furthermore, the basic ideas of the numerical approximation are introduced and the extended hyperbolic system is treated numerically within the framework of high-resolution finite-volume schemes. Simulation results obtained with this new technique for pure electromagnetic wave propagation and for an electromagnetic particle-in-cell computation are presented and compared with other methods.

International audience; Cell-centered and vertex-centered finite volume schemes for the Laplace equation with homogeneous Dirichlet boundary conditions are considered on a triangular mesh and on the Voronoi diagram associated to its vertices. A broken $P^1$ function is constructed from the solutions of both schemes. When the domain is two-dimensional polygonal convex, it is shown that this reconstruction converges with second-order accuracy towards the exact solution in the~$L^2$ norm, under the sufficient condition that the right-hand side of the Laplace equation belongs to~$H^1(\Omega)$.

We investigate the accuracy of the Godunov scheme applied to the variable cross-section acoustic equations. Contrarily to the constant cross-section case, the accuracy issue of this scheme in the low Mach number regime appears even in the one-dimensional case; on the other hand, we show that it is possible to construct another Godunov type scheme which is accurate in the low Mach number regime.

This document is a synthesis of a set of works concerning the development and the analysis of finite volume methods used for the numerical approximation of partial differential equations (PDEs) stemming from physics. In the first part, the document deals with colocalized Godunov type schemes for the Maxwell and wave equations, with a study on the loss of accuracy of this scheme at low Mach number. In the second part, discrete differential operators are built on fairly general, in particular very distorted or nonconforming, bidimensional meshes. These operators are used to approach the solutions of PDEs modelling diffusion, electro and magnetostatics and electromagnetism by the discrete duality finite volume method (DDFV) on staggered meshes. The third part presents the numerical analysis and some a priori as well as a posteriori error estimations for the discretization of the Laplace equation by the DDFV scheme. The last part is devoted to the order of convergence in the L^2 norm of the finite volume approximation of the solution of the Laplace equation in one dimension and on meshes with orthogonality properties in two dimensions. Necessary and sufficient conditions, relatively to the mesh geometry and to the regularity of the data, are provided that ensure the second-order convergence of the method.; Ce document synthétise un ensemble de travaux portant sur le développement et l'analyse de méthodes de volumes finis utilisées pour l'approximation numérique d'équations aux dérivées partielles issues de la physique. Le mémoire aborde dans sa première partie des schémas colocalisés de type Godunov d'une part pour les équations de l'électromagnétisme, et d'autre part pour l'équation des ondes acoustiques, avec une étude portant sur la perte de précision de ce schéma à bas nombre de Mach. La deuxième partie est consacrée à la construction d'opérateurs différentiels discrets sur des maillages bidimensionnels relativement quelconques, en particulier très déformés ou encore non-conformes, et à leur utilisation pour la discrétisation d'équations aux dérivées partielles modélisant des phénomènes de diffusion, d'électrostatique et de magnétostatique et d'électromagnétisme par des schémas de type volumes finis en dualité discrète (DDFV) sur maillages décalés. La troisième partie aborde ensuite l'analyse numérique et les estimations d'erreur a priori et a posteriori associées à la discrétisation par le schéma DDFV de l'équation de Laplace. La quatrième et dernière partie est consacrée à la question de l'ordre de convergence en norme L^2 de la solution numérique du problème de Laplace, issue d'une discrétisation volumes finis en dimension un et en dimension deux sur des maillages présentant des propriétés d'orthogonalité. L'étude met en évidence des conditions nécessaires et suffisantes relatives à la géométrie des maillages et à la régularité des données du problème afin d'obtenir la convergence à l'ordre deux de la méthode.

International audience; The cell-centered finite volume approximation of the Laplace equation in dimension one is considered. An exact expression of the error between the exact and numerical solutions is derived through the use of continuous and discrete Green functions. This allows to discuss convergence of the method in the L infinity and L2 norms with respect to the choice of the control points in the cells and with respect to the regularity of the data. Well-known second-order convergence results are recovered if those control points are properly chosen and if the data belongs to H1. Counterexamples are constructed to show that second-order may be lost if these conditions are not met.