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1. Investigating the effect of process parameters for fused filament fabrication

Author

Asma Boumedine, Samir Lecheb, Khaled Benfriha, Pascal Omnes, Laboratoire Conception de Produits et Innovation (LCPI), Arts et Métiers Sciences et Technologies, HESAM Université - Communauté d'universités et d'établissements Hautes écoles Sorbonne Arts et métiers université (HESAM)-HESAM Université - Communauté d'universités et d'établissements Hautes écoles Sorbonne Arts et métiers université (HESAM), Université M'Hamed Bougara Boumerdes (UMBB), Commissariat à l'énergie atomique et aux énergies alternatives (CEA), Laboratoire Analyse, Géométrie et Applications (LAGA), and Université Paris 8 Vincennes-Saint-Denis (UP8)-Centre National de la Recherche Scientifique (CNRS)-Université Sorbonne Paris Nord

Subjects
Design of experiment (DOE), Mathematical modelling, Additive manufacturing, Génie des procédés [Sciences de l'ingénieur], Full factorial design, Dimensional accuracy, [SPI.GPROC]Engineering Sciences [physics]/Chemical and Process Engineering, Fused Filament Fabrication (FFF), Geometrical characterization, Industrial and Manufacturing Engineering, Composites
Abstract

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.

Published
2023
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2. Construction of a low Mach finite volume scheme for the isentropic Euler system with porosity

Author

Jonathan Jung, Stéphane Dellacherie, Pascal Omnes, Hydro-Québec - TransÉnergie et Équipement, DCMÉ, Prévisions de contrôle du réseau, Hydro-Québec TransÉnergie, Laboratoire de Mathématiques et de leurs Applications [Pau] (LMAP), Université de Pau et des Pays de l'Adour (UPPA)-Centre National de la Recherche Scientifique (CNRS), Computational AGility for internal flows sImulations and compaRisons with Experiments (CAGIRE), Inria Bordeaux - Sud-Ouest, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria)-Université de Pau et des Pays de l'Adour (UPPA), Commissariat à l'énergie atomique et aux énergies alternatives (CEA), Laboratoire Analyse, Géométrie et Applications (LAGA), Université Paris 8 Vincennes-Saint-Denis (UP8)-Centre National de la Recherche Scientifique (CNRS)-Université Sorbonne Paris Nord, and PLAFRIM

Subjects
Numerical Analysis, Finite volume method, Applied Mathematics, Mathematical analysis, Godunov's scheme, 010103 numerical & computational mathematics, Euler system, Numerical diffusion, Space (mathematics), 01 natural sciences, law.invention, Euler equations, 010101 applied mathematics, Computational Mathematics, symbols.namesake, Mach number, law, Modeling and Simulation, symbols, [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], Cartesian coordinate system, 0101 mathematics, Analysis, [MATH.MATH-NA]Mathematics [math]/Numerical Analysis [math.NA], Mathematics
Abstract

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.

Published
2021
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3. Optimal absorption of acoustical waves by a boundary

Author

Pascal Omnes, Frédéric Magoulès, Thi Phuong Kieu Nguyen, Anna Rozanova-Pierrat, Mathématiques et Informatique pour la Complexité et les Systèmes (MICS), CentraleSupélec, CEA-Direction des Energies (ex-Direction de l'Energie Nucléaire) (CEA-DES (ex-DEN)), Commissariat à l'énergie atomique et aux énergies alternatives (CEA), Laboratoire Analyse, Géométrie et Applications (LAGA), Université Paris 8 Vincennes-Saint-Denis (UP8)-Institut Galilée-Centre National de la Recherche Scientifique (CNRS)-Université Sorbonne Paris Nord, CentraleSupélec-Université Paris-Saclay, Université Paris 8 Vincennes-Saint-Denis (UP8)-Centre National de la Recherche Scientifique (CNRS)-Université Sorbonne Paris Nord, and Funding: This work was funded by the Pôle de Compétitivité Systematic (France) under thegrant OpenGPU, and the Pôle de Compétitivité CapDigital (France) under the grant Callisto-Sari.

Subjects
Absorption (acoustics), Control and Optimization, Helmholtz equation, Wave propagation, sound absorption, Boundary (topology), wave propagation, Robin boundary condition, [MATH.MATH-FA]Mathematics [math]/Functional Analysis [math.FA], 01 natural sciences, Mathematics - Analysis of PDEs, [MATH.MATH-MP]Mathematics [math]/Mathematical Physics [math-ph], FOS: Mathematics, [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], 35L405, 35J05, 35J25, 15A06, Shape optimization, fractals, 0101 mathematics, Mathematics, Applied Mathematics, 010102 general mathematics, Mathematical analysis, Dissipation, Lipschitz continuity, 010101 applied mathematics, Absorbing wall, shape optimization, [MATH.MATH-OC]Mathematics [math]/Optimization and Control [math.OC], [MATH.MATH-NA]Mathematics [math]/Numerical Analysis [math.NA], Analysis of PDEs (math.AP)
Abstract

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

Published
2020
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4. Godunov type scheme for the linear wave equation with Coriolis source term

Author

Yohan Penel, Stéphane Dellacherie, Emmanuel Audusse, Pascal Omnes, and Do Minh Hieu

Subjects
Physics::Computational Physics, T57-57.97, Finite volume method, Applied mathematics. Quantitative methods, Discretization, Godunov's theorem, Mathematical analysis, Godunov's scheme, 010103 numerical & computational mathematics, 01 natural sciences, Term (time), Mathematics::Numerical Analysis, 010101 applied mathematics, Linear map, symbols.namesake, Kernel (statistics), Froude number, symbols, QA1-939, 0101 mathematics, Mathematics
Abstract

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.

Published
2017

5. Analysis of modified Godunov type schemes for the two-dimensional linear wave equation with Coriolis source term on cartesian meshes

Author

Yohan Penel, Minh Hieu Do, Emmanuel Audusse, Pascal Omnes, Laboratoire Analyse, Géométrie et Applications (LAGA), Université Paris 8 Vincennes-Saint-Denis (UP8)-Centre National de la Recherche Scientifique (CNRS)-Institut Galilée-Université Paris 13 (UP13), Département de Modélisation des Systèmes et Structures (DM2S), CEA-Direction des Energies (ex-Direction de l'Energie Nucléaire) (CEA-DES (ex-DEN)), Commissariat à l'énergie atomique et aux énergies alternatives (CEA)-Commissariat à l'énergie atomique et aux énergies alternatives (CEA)-Université Paris-Saclay, Numerical Analysis, Geophysics and Ecology (ANGE), Laboratoire Jacques-Louis Lions (LJLL), Université Pierre et Marie Curie - Paris 6 (UPMC)-Université Paris Diderot - Paris 7 (UPD7)-Centre National de la Recherche Scientifique (CNRS)-Université Pierre et Marie Curie - Paris 6 (UPMC)-Université Paris Diderot - Paris 7 (UPD7)-Centre National de la Recherche Scientifique (CNRS)-Inria de Paris, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria), Inria de Paris, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria)-Laboratoire Jacques-Louis Lions (LJLL), Université Pierre et Marie Curie - Paris 6 (UPMC)-Université Paris Diderot - Paris 7 (UPD7)-Centre National de la Recherche Scientifique (CNRS)-Université Pierre et Marie Curie - Paris 6 (UPMC)-Université Paris Diderot - Paris 7 (UPD7)-Centre National de la Recherche Scientifique (CNRS), and Université Paris 8 Vincennes-Saint-Denis (UP8)-Université Paris 13 (UP13)-Institut Galilée-Centre National de la Recherche Scientifique (CNRS)

Subjects
Numerical Analysis, Finite volume method, Physics and Astronomy (miscellaneous), Applied Mathematics, Courant–Friedrichs–Lewy condition, Godunov's scheme, 010103 numerical & computational mathematics, 01 natural sciences, Stability (probability), [INFO.INFO-MO]Computer Science [cs]/Modeling and Simulation, Computer Science Applications, law.invention, 010101 applied mathematics, Computational Mathematics, Nonlinear system, law, Modeling and Simulation, Applied mathematics, Cartesian coordinate system, 0101 mathematics, Shallow water equations, Geostrophic wind, Mathematics
Abstract

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.

Published
2017
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6. Analysis of Apparent Topography scheme for the linear wave equation with Coriolis force

Author

Yohan Penel, Emmanuel Audusse, Minh Hieu Do, Pascal Omnes, Laboratoire Analyse, Géométrie et Applications (LAGA), Université Paris 8 Vincennes-Saint-Denis (UP8)-Université Paris 13 (UP13)-Institut Galilée-Centre National de la Recherche Scientifique (CNRS), Département de Modélisation des Systèmes et Structures (DM2S), CEA-Direction des Energies (ex-Direction de l'Energie Nucléaire) (CEA-DES (ex-DEN)), Commissariat à l'énergie atomique et aux énergies alternatives (CEA)-Commissariat à l'énergie atomique et aux énergies alternatives (CEA)-Université Paris-Saclay, Numerical Analysis, Geophysics and Ecology (ANGE), Laboratoire Jacques-Louis Lions (LJLL), Université Pierre et Marie Curie - Paris 6 (UPMC)-Université Paris Diderot - Paris 7 (UPD7)-Centre National de la Recherche Scientifique (CNRS)-Université Pierre et Marie Curie - Paris 6 (UPMC)-Université Paris Diderot - Paris 7 (UPD7)-Centre National de la Recherche Scientifique (CNRS)-Inria de Paris, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria), Université Paris 8 Vincennes-Saint-Denis (UP8)-Centre National de la Recherche Scientifique (CNRS)-Institut Galilée-Université Paris 13 (UP13), Commissariat à l'énergie atomique et aux énergies alternatives (CEA), Inria de Paris, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria)-Laboratoire Jacques-Louis Lions (LJLL), and Université Pierre et Marie Curie - Paris 6 (UPMC)-Université Paris Diderot - Paris 7 (UPD7)-Centre National de la Recherche Scientifique (CNRS)-Université Pierre et Marie Curie - Paris 6 (UPMC)-Université Paris Diderot - Paris 7 (UPD7)-Centre National de la Recherche Scientifique (CNRS)

Subjects
Physics, Finite Volume Method, Coriolis Force, Finite volume method, Omega equation, Fluid mechanics, 010103 numerical & computational mathematics, Mechanics, Shallow water flows, 01 natural sciences, [INFO.INFO-MO]Computer Science [cs]/Modeling and Simulation, 010305 fluids & plasmas, Physics::Geophysics, symbols.namesake, 0103 physical sciences, Fictitious force, symbols, 0101 mathematics, Kelvin wave, Shallow water equations, Pressure gradient, Geostrophic wind, Physics::Atmospheric and Oceanic Physics, Well-balanced Schemes
Abstract

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.

Published
2017
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7. Finite Volumes for Complex Applications VIII - Methods and Theoretical Aspects

Author

Clément Cancès, Pascal Omnes, Reliable numerical approximations of dissipative systems (RAPSODI ), Laboratoire Paul Painlevé - UMR 8524 (LPP), Centre National de la Recherche Scientifique (CNRS)-Université de Lille-Centre National de la Recherche Scientifique (CNRS)-Université de Lille-Inria Lille - Nord Europe, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria), Laboratoire Analyse, Géométrie et Applications (LAGA), Université Paris 8 Vincennes-Saint-Denis (UP8)-Centre National de la Recherche Scientifique (CNRS)-Institut Galilée-Université Paris 13 (UP13), Service de Thermo-hydraulique et de Mécanique des Fluides (STMF), Département de Modélisation des Systèmes et Structures (DM2S), CEA-Direction des Energies (ex-Direction de l'Energie Nucléaire) (CEA-DES (ex-DEN)), Commissariat à l'énergie atomique et aux énergies alternatives (CEA)-Commissariat à l'énergie atomique et aux énergies alternatives (CEA)-Université Paris-Saclay-CEA-Direction des Energies (ex-Direction de l'Energie Nucléaire) (CEA-DES (ex-DEN)), Commissariat à l'énergie atomique et aux énergies alternatives (CEA)-Commissariat à l'énergie atomique et aux énergies alternatives (CEA)-Université Paris-Saclay, C. Cancès and P. Omnes, Laboratoire Paul Painlevé (LPP), Université de Lille-Centre National de la Recherche Scientifique (CNRS)-Université de Lille-Centre National de la Recherche Scientifique (CNRS)-Inria Lille - Nord Europe, and Université Paris 8 Vincennes-Saint-Denis (UP8)-Université Paris 13 (UP13)-Institut Galilée-Centre National de la Recherche Scientifique (CNRS)

Subjects
Computer science, [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], ComputingMilieux_MISCELLANEOUS, [MATH.MATH-NA]Mathematics [math]/Numerical Analysis [math.NA]
Abstract

International audience

Published
2017

8. Finite Volumes for Complex Applications VIII - Hyperbolic, Elliptic and Parabolic Problems

Author

Pascal Omnes, Clément Cancès, Reliable numerical approximations of dissipative systems (RAPSODI ), Laboratoire Paul Painlevé (LPP), Université de Lille-Centre National de la Recherche Scientifique (CNRS)-Université de Lille-Centre National de la Recherche Scientifique (CNRS)-Inria Lille - Nord Europe, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria), Laboratoire Analyse, Géométrie et Applications (LAGA), Université Paris 8 Vincennes-Saint-Denis (UP8)-Université Paris 13 (UP13)-Institut Galilée-Centre National de la Recherche Scientifique (CNRS), Service de Thermo-hydraulique et de Mécanique des Fluides (STMF), Département de Modélisation des Systèmes et Structures (DM2S), CEA-Direction des Energies (ex-Direction de l'Energie Nucléaire) (CEA-DES (ex-DEN)), Commissariat à l'énergie atomique et aux énergies alternatives (CEA)-Commissariat à l'énergie atomique et aux énergies alternatives (CEA)-Université Paris-Saclay-CEA-Direction des Energies (ex-Direction de l'Energie Nucléaire) (CEA-DES (ex-DEN)), Commissariat à l'énergie atomique et aux énergies alternatives (CEA)-Commissariat à l'énergie atomique et aux énergies alternatives (CEA)-Université Paris-Saclay, C. Cancès and P. Omnes, Laboratoire Paul Painlevé - UMR 8524 (LPP), Centre National de la Recherche Scientifique (CNRS)-Université de Lille-Centre National de la Recherche Scientifique (CNRS)-Université de Lille-Inria Lille - Nord Europe, and Université Paris 8 Vincennes-Saint-Denis (UP8)-Centre National de la Recherche Scientifique (CNRS)-Institut Galilée-Université Paris 13 (UP13)

Subjects
010101 applied mathematics, Physics, Alternating direction implicit method, Elliptic partial differential equation, Discontinuous Galerkin method, Mathematical analysis, [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], 010103 numerical & computational mathematics, Finite volume method for one-dimensional steady state diffusion, 0101 mathematics, 01 natural sciences, ComputingMilieux_MISCELLANEOUS, [MATH.MATH-NA]Mathematics [math]/Numerical Analysis [math.NA]
Abstract

International audience

Published
2017

9. A discrete duality finite volume discretization of the vorticity-velocity-pressure stokes problem on almost arbitrary two-dimensional grids

Author

Sarah Delcourte and Pascal Omnes

Subjects
Numerical Analysis, Finite volume method, Applied Mathematics, Mathematical analysis, Degrees of freedom (physics and chemistry), Duality (optimization), Vorticity, Differential operator, Computational Mathematics, Convergence (routing), Partial derivative, Boundary value problem, Analysis, Mathematics
Abstract

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

Published
2014
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10. Construction of modified Godunov type schemes accurate at any Mach number for the compressible Euler system

Author

Stéphane Dellacherie, Pascal Omnes, Pierre-Arnaud Raviart, Jonathan Jung, CEA-Direction des Energies (ex-Direction de l'Energie Nucléaire) (CEA-DES (ex-DEN)), Commissariat à l'énergie atomique et aux énergies alternatives (CEA), Laboratoire Jacques-Louis Lions (LJLL), Université Pierre et Marie Curie - Paris 6 (UPMC)-Université Paris Diderot - Paris 7 (UPD7)-Centre National de la Recherche Scientifique (CNRS), Laboratoire de Mathématiques et de leurs Applications [Pau] (LMAP), Université de Pau et des Pays de l'Adour (UPPA)-Centre National de la Recherche Scientifique (CNRS), Computational Approximation with discontinous Galerkin methods and compaRison with Experiments (CAGIRE), Université de Pau et des Pays de l'Adour (UPPA)-Centre National de la Recherche Scientifique (CNRS)-Université de Pau et des Pays de l'Adour (UPPA)-Centre National de la Recherche Scientifique (CNRS)-Inria Bordeaux - Sud-Ouest, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria), Laboratoire Analyse, Géométrie et Applications (LAGA), Université Paris 8 Vincennes-Saint-Denis (UP8)-Université Paris 13 (UP13)-Institut Galilée-Centre National de la Recherche Scientifique (CNRS), Service Fluide numériques, Modélisation et Etudes (SFME), Département de Modélisation des Systèmes et Structures (DM2S), Commissariat à l'énergie atomique et aux énergies alternatives (CEA)-Commissariat à l'énergie atomique et aux énergies alternatives (CEA)-Université Paris-Saclay-CEA-Direction des Energies (ex-Direction de l'Energie Nucléaire) (CEA-DES (ex-DEN)), Commissariat à l'énergie atomique et aux énergies alternatives (CEA)-Commissariat à l'énergie atomique et aux énergies alternatives (CEA)-Université Paris-Saclay, and Université Paris 8 Vincennes-Saint-Denis (UP8)-Centre National de la Recherche Scientifique (CNRS)-Institut Galilée-Université Paris 13 (UP13)

Subjects
linear wave equation, Applied Mathematics, Godunov's theorem, Mathematical analysis, Godunov's scheme, 010103 numerical & computational mathematics, Euler system, Roe scheme, 01 natural sciences, Godunov scheme, 010101 applied mathematics, Roe solver, symbols.namesake, low Mach number flow, Mach number, Modeling and Simulation, Euler's formula, symbols, Compressibility, Compressible Euler system, 0101 mathematics, 10. No inequality, [MATH.MATH-NA]Mathematics [math]/Numerical Analysis [math.NA], Mathematics, Linear stability
Abstract

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.

Published
2016
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11. Preliminary results for the study of the Godunov Scheme Applied to the Linear Wave Equation with Porosity at Low Mach Number

Author

Pascal Omnes, Stéphane Dellacherie, Jonathan Jung, CEA-Direction des Energies (ex-Direction de l'Energie Nucléaire) (CEA-DES (ex-DEN)), Commissariat à l'énergie atomique et aux énergies alternatives (CEA), Laboratoire de recherche conventionné Modélisation et approximation numérique orientées pour l'énergie nucléaire (LRC Manon), Laboratoire Jacques-Louis Lions (LJLL), Université Pierre et Marie Curie - Paris 6 (UPMC)-Université Paris Diderot - Paris 7 (UPD7)-Centre National de la Recherche Scientifique (CNRS)-Université Pierre et Marie Curie - Paris 6 (UPMC)-Université Paris Diderot - Paris 7 (UPD7)-Centre National de la Recherche Scientifique (CNRS)-Commissariat à l'énergie atomique et aux énergies alternatives (CEA), Université Pierre et Marie Curie - Paris 6 (UPMC)-Université Paris Diderot - Paris 7 (UPD7)-Centre National de la Recherche Scientifique (CNRS), Laboratoire de Mathématiques et de leurs Applications [Pau] (LMAP), Université de Pau et des Pays de l'Adour (UPPA)-Centre National de la Recherche Scientifique (CNRS), Computational Approximation with discontinous Galerkin methods and compaRison with Experiments (CAGIRE), Université de Pau et des Pays de l'Adour (UPPA)-Centre National de la Recherche Scientifique (CNRS)-Université de Pau et des Pays de l'Adour (UPPA)-Centre National de la Recherche Scientifique (CNRS)-Inria Bordeaux - Sud-Ouest, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria), AllianSTIC-EFREI, Efrei (Efrei), Département de Modélisation des Systèmes et Structures (DM2S), Commissariat à l'énergie atomique et aux énergies alternatives (CEA)-Commissariat à l'énergie atomique et aux énergies alternatives (CEA)-Université Paris-Saclay, Laboratoire Analyse, Géométrie et Applications (LAGA), Université Paris 8 Vincennes-Saint-Denis (UP8)-Université Paris 13 (UP13)-Institut Galilée-Centre National de la Recherche Scientifique (CNRS), and Université Paris 8 Vincennes-Saint-Denis (UP8)-Centre National de la Recherche Scientifique (CNRS)-Institut Galilée-Université Paris 13 (UP13)

Subjects
010103 numerical & computational mathematics, Space (mathematics), 01 natural sciences, 010305 fluids & plasmas, law.invention, Mathematics::Numerical Analysis, Godunov scheme, symbols.namesake, law, 0103 physical sciences, QA1-939, [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], Polygon mesh, Cartesian coordinate system, compressible Euler system with porosity, 0101 mathematics, Porosity, Linear wave equation, Mathematics, Physics::Computational Physics, T57-57.97, Applied mathematics. Quantitative methods, Godunov's theorem, Mathematical analysis, linear wave equation with porosity, Godunov's scheme, low Mach number flow, Mach number, all Mach scheme, symbols, [MATH.MATH-NA]Mathematics [math]/Numerical Analysis [math.NA]
Abstract

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.

Published
2016
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12. A Posteriori Error Estimation for the Discrete Duality Finite Volume Discretization of the Laplace Equation

Author

Yohan Penel, Pascal Omnes, Yann Rosenbaum, Service Fluide numériques, Modélisation et Etudes (SFME), Département de Modélisation des Systèmes et Structures (DM2S), CEA-Direction des Energies (ex-Direction de l'Energie Nucléaire) (CEA-DES (ex-DEN)), Commissariat à l'énergie atomique et aux énergies alternatives (CEA)-Commissariat à l'énergie atomique et aux énergies alternatives (CEA)-Université Paris-Saclay-CEA-Direction des Energies (ex-Direction de l'Energie Nucléaire) (CEA-DES (ex-DEN)), Commissariat à l'énergie atomique et aux énergies alternatives (CEA)-Commissariat à l'énergie atomique et aux énergies alternatives (CEA)-Université Paris-Saclay, Laboratoire Analyse, Géométrie et Applications (LAGA), Université Paris 8 Vincennes-Saint-Denis (UP8)-Université Paris 13 (UP13)-Institut Galilée-Centre National de la Recherche Scientifique (CNRS), and Université Paris 8 Vincennes-Saint-Denis (UP8)-Centre National de la Recherche Scientifique (CNRS)-Institut Galilée-Université Paris 13 (UP13)

Subjects
MathematicsofComputing_NUMERICALANALYSIS, Duality (optimization), 010103 numerical & computational mathematics, 01 natural sciences, Mathematics::Numerical Analysis, Singular solution, AMS 65N15, 65N30, ComputingMethodologies_SYMBOLICANDALGEBRAICMANIPULATION, Calculus, Applied mathematics, Polygon mesh, 0101 mathematics, ComputingMethodologies_COMPUTERGRAPHICS, Mathematics, Laplace's equation, Numerical Analysis, Finite volume method, Applied Mathematics, Numerical analysis, Mixed finite element method, discrete duality, 16. Peace & justice, a posteriori error estimation, Finite element method, 010101 applied mathematics, Computational Mathematics, finite volume, [MATH.MATH-NA]Mathematics [math]/Numerical Analysis [math.NA], nonconforming meshes
Abstract

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.

Published
2009
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13. Numerical and physical comparisons of two models of a gas centrifuge

Author

Pascal Omnes

Subjects
Centrifuge, Finite volume method, General Computer Science, Drag, Gas centrifuge, Iterative method, Computation, Flow (psychology), General Engineering, Geometry, Mechanics, Boundary value problem, Mathematics
Abstract

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.

Published
2007
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15. An a Posteriori Error Estimation for the Discrete Duality Finite Volume Discretization of the Stokes Equations

Author

Anh Ha Le, Pascal Omnes, Laboratoire Analyse, Géométrie et Applications (LAGA), Université Paris 8 Vincennes-Saint-Denis (UP8)-Centre National de la Recherche Scientifique (CNRS)-Institut Galilée-Université Paris 13 (UP13), Département de Modélisation des Systèmes et Structures (DM2S), CEA-Direction des Energies (ex-Direction de l'Energie Nucléaire) (CEA-DES (ex-DEN)), Commissariat à l'énergie atomique et aux énergies alternatives (CEA)-Commissariat à l'énergie atomique et aux énergies alternatives (CEA)-Université Paris-Saclay, and Université Paris 8 Vincennes-Saint-Denis (UP8)-Université Paris 13 (UP13)-Institut Galilée-Centre National de la Recherche Scientifique (CNRS)

Subjects
Numerical Analysis, MSC 65N08, 65N15, 76D07, Finite volume method, Discretization, Adaptive mesh refinement, Applied Mathematics, Duality (mathematics), Mathematical analysis, Estimator, Stokes equations, discrete duality, Upper and lower bounds, a posteriori error estimation, stabilization, Computational Mathematics, Modeling and Simulation, Polygon mesh, Constant (mathematics), Analysis, finite volumes, [MATH.MATH-NA]Mathematics [math]/Numerical Analysis [math.NA], Mathematics
Abstract

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.

Published
2015
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16. A finite volume method for the Laplace equation on almost arbitrary two-dimensional grids

Author

Pascal Omnes and Komla Domelevo

Subjects
Laplace's equation, Numerical Analysis, Finite volume method, Applied Mathematics, Numerical analysis, Superconvergence, Finite element method, Regular grid, Combinatorics, Computational Mathematics, Modeling and Simulation, Norm (mathematics), Applied mathematics, Polygon mesh, Analysis, Mathematics
Abstract

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.

Published
2005
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17. Space–Time Domain Decomposition with Finite Volumes for Porous Media Applications

Author

Pascal Omnes, Paul-Marie Berthe, and Caroline Japhet

Subjects
Well-posed problem, Finite volume method, Discretization, Iterative method, Discontinuous Galerkin method, Computer science, Applied mathematics, Duality (optimization), Domain decomposition methods, Porous medium
Abstract

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.

Published
2014
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18. On the second-order convergence of a function reconstructed from finite volume approximations of the Laplace equation on Delaunay-Voronoi meshes

Author

Pascal Omnes, Laboratoire Analyse, Géométrie et Applications (LAGA), Université Paris 8 Vincennes-Saint-Denis (UP8)-Centre National de la Recherche Scientifique (CNRS)-Institut Galilée-Université Paris 13 (UP13), Service Fluide numériques, Modélisation et Etudes (SFME), Département de Modélisation des Systèmes et Structures (DM2S), CEA-Direction des Energies (ex-Direction de l'Energie Nucléaire) (CEA-DES (ex-DEN)), Commissariat à l'énergie atomique et aux énergies alternatives (CEA)-Commissariat à l'énergie atomique et aux énergies alternatives (CEA)-Université Paris-Saclay-CEA-Direction des Energies (ex-Direction de l'Energie Nucléaire) (CEA-DES (ex-DEN)), Commissariat à l'énergie atomique et aux énergies alternatives (CEA)-Commissariat à l'énergie atomique et aux énergies alternatives (CEA)-Université Paris-Saclay, and Université Paris 8 Vincennes-Saint-Denis (UP8)-Université Paris 13 (UP13)-Institut Galilée-Centre National de la Recherche Scientifique (CNRS)

Subjects
Delaunay meshes, finite volume method, 010103 numerical & computational mathematics, Topology, 01 natural sciences, symbols.namesake, Applied mathematics, Voronoi meshes, 0101 mathematics, Mathematics, Laplace's equation, Numerical Analysis, Partial differential equation, convergence, Laplace expansion, Applied Mathematics, Inverse Laplace transform, Laplace equation, Green's function for the three-variable Laplace equation, 010101 applied mathematics, Computational Mathematics, error estimates, Modeling and Simulation, Laplace transform applied to differential equations, Dirichlet boundary condition, symbols, Voronoi diagram, Analysis, [MATH.MATH-NA]Mathematics [math]/Numerical Analysis [math.NA]
Abstract

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~$H^1(\Omega)$.

Published
2011
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19. Error estimates for a finite volume method for the Laplace equation in dimension one through discrete Green functions

Author

Pascal Omnes, Laboratoire Analyse, Géométrie et Applications (LAGA), Université Paris 8 Vincennes-Saint-Denis (UP8)-Centre National de la Recherche Scientifique (CNRS)-Institut Galilée-Université Paris 13 (UP13), Service Fluide numériques, Modélisation et Etudes (SFME), Département de Modélisation des Systèmes et Structures (DM2S), CEA-Direction des Energies (ex-Direction de l'Energie Nucléaire) (CEA-DES (ex-DEN)), Commissariat à l'énergie atomique et aux énergies alternatives (CEA)-Commissariat à l'énergie atomique et aux énergies alternatives (CEA)-Université Paris-Saclay-CEA-Direction des Energies (ex-Direction de l'Energie Nucléaire) (CEA-DES (ex-DEN)), Commissariat à l'énergie atomique et aux énergies alternatives (CEA)-Commissariat à l'énergie atomique et aux énergies alternatives (CEA)-Université Paris-Saclay, and Université Paris 8 Vincennes-Saint-Denis (UP8)-Université Paris 13 (UP13)-Institut Galilée-Centre National de la Recherche Scientifique (CNRS)

Subjects
error estimates, AMS 65N15 65N22 65N30, Laplace equation, Green functions, finite volumes, [MATH.MATH-NA]Mathematics [math]/Numerical Analysis [math.NA]
Abstract

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.

Published
2009

20. A discrete duality finite volume approach to Hodge decomposition and div-curl problems on almost arbitrary two-dimensional meshes

Author

Komla Domelevo, Pascal Omnes, Sarah Delcourte, Laboratoire de Modélisation Physique et de l'Enrichissement (LMPE), Département de Modélisation des Systèmes et Structures (DM2S), CEA-Direction des Energies (ex-Direction de l'Energie Nucléaire) (CEA-DES (ex-DEN)), Commissariat à l'énergie atomique et aux énergies alternatives (CEA)-Commissariat à l'énergie atomique et aux énergies alternatives (CEA)-Université Paris-Saclay-CEA-Direction des Energies (ex-Direction de l'Energie Nucléaire) (CEA-DES (ex-DEN)), Commissariat à l'énergie atomique et aux énergies alternatives (CEA)-Commissariat à l'énergie atomique et aux énergies alternatives (CEA)-Université Paris-Saclay, Institut de Mathématiques de Toulouse UMR5219 (IMT), Université Toulouse Capitole (UT Capitole), Université de Toulouse (UT)-Université de Toulouse (UT)-Institut National des Sciences Appliquées - Toulouse (INSA Toulouse), Institut National des Sciences Appliquées (INSA)-Université de Toulouse (UT)-Institut National des Sciences Appliquées (INSA)-Université Toulouse - Jean Jaurès (UT2J), Université de Toulouse (UT)-Université Toulouse III - Paul Sabatier (UT3), Université de Toulouse (UT)-Centre National de la Recherche Scientifique (CNRS), Laboratoire Analyse, Géométrie et Applications (LAGA), Université Paris 8 Vincennes-Saint-Denis (UP8)-Université Paris 13 (UP13)-Institut Galilée-Centre National de la Recherche Scientifique (CNRS), Institut National des Sciences Appliquées - Toulouse (INSA Toulouse), Institut National des Sciences Appliquées (INSA)-Institut National des Sciences Appliquées (INSA)-Université Toulouse 1 Capitole (UT1), Université Fédérale Toulouse Midi-Pyrénées-Université Fédérale Toulouse Midi-Pyrénées-Université Toulouse - Jean Jaurès (UT2J)-Université Toulouse III - Paul Sabatier (UT3), Université Fédérale Toulouse Midi-Pyrénées-Centre National de la Recherche Scientifique (CNRS), Université Paris 8 Vincennes-Saint-Denis (UP8)-Centre National de la Recherche Scientifique (CNRS)-Institut Galilée-Université Paris 13 (UP13), CEA-Direction de l'Energie Nucléaire (CEA-DEN), Commissariat à l'énergie atomique et aux énergies alternatives (CEA), and Service Fluide numériques, Modélisation et Etudes (SFME)

Subjects
Discretization, volumes finis, [PHYS.NUCL]Physics [physics]/Nuclear Theory [nucl-th], 65N06 (35F05), 010103 numerical & computational mathematics, [PHYS.NEXP]Physics [physics]/Nuclear Experiment [nucl-ex], 01 natural sciences, Mathematics::Numerical Analysis, Physics::Plasma Physics, Applied mathematics, Polygon mesh, 0101 mathematics, probleme div-rot, ComputingMilieux_MISCELLANEOUS, Mathematics, Curl (mathematics), Numerical Analysis, Finite volume method, convergence, Applied Mathematics, Numerical analysis, maillages non-conformes, Mathematical analysis, Differential operator, Physics::Classical Physics, maillages degeneres, 010101 applied mathematics, decomposition de Hodge discrete, Computational Mathematics, Vector field, Discrete differential geometry, [MATH.MATH-NA]Mathematics [math]/Numerical Analysis [math.NA]
Abstract

International audience; We define discrete differential operators such as grad, div and curl, on general two-dimensional non-orthogonal meshes. These discrete operators verify discrete analogues of usual continuous theorems: discrete Green formulae, discrete Hodge decomposition of vector fields, vector curls have a vanishing divergence and gradients have a vanishing curl. We apply these ideas to discretize div-curl systems. We give error estimates based on the reformulation of these systems into equivalent equations for the potentials. Numerical results illustrate the use of the method on several types of meshes, among which degenerating triangular meshes and non-conforming locally refined meshes.

Published
2007
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21. New space-time domain decomposition algorithms combined with the Parareal algorithm

Author

Bui, Duc Quang, STAR, ABES, Laboratoire Analyse, Géométrie et Applications (LAGA), Université Paris 8 Vincennes-Saint-Denis (UP8)-Centre National de la Recherche Scientifique (CNRS)-Université Sorbonne Paris Nord, Université Paris-Nord - Paris XIII, Pascal Omnes, and Caroline Japhet

Subjects
Équations d’Oseen, Équations de Stokes, Parabolic equations, Pararéel, [INFO.INFO-DS]Computer Science [cs]/Data Structures and Algorithms [cs.DS], Parareal, [INFO.INFO-DS] Computer Science [cs]/Data Structures and Algorithms [cs.DS], Méthode Relaxation d’Ondes Optimisée, Optimized Schwarz Waveform Relaxation, OSWR
Abstract

We study in this thesis space-time domain decomposition methods, in particular, the Parareal method, the Optimized Schwarz Waveform Relaxation (OSWR) method and their coupling, applied to the numerical simulation of parabolic equations and of the Stokes equations. We first propose and analyze a coupling of the Parareal method with the OSWR method. The obtained coupled Parareal-OSWR method is a parallel method, both in the time and space directions, with only few OSWR iterations in the fine propagator in order to reduce computational costs 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. For the coupling of Parareal with non-overlapping OSWR, we prove a general convergence result 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. We then present also an algorithm that couples Parareal with overlapping OSWR, and we analyze its convergence factor by using the linear convergence of overlapping OSWR that we obtain through a Fourier analysis.For the Stokes equations, we present a well-posed OSWR algorithm and an energy estimate for the convergence of the velocities. Then we show that, in general, the pressure does not converge and we propose a correction against this. A similar strategy based on Fourier transform is performed to get the formulation of the convergence factor. Numerical tests follow to illustrate the performance of the OSWR method with correction. In addition, these results are also extended to get similar ones on the Oseen equation. Finally, we propose the Parareal algorithm and a Parareal-OSWR coupling for the Stokes equations, and prove some of their basic properties, Nous étudions dans cette thèse les méthodes de décomposition de domaine spatio-temporelles, en particulier la méthode Pararéele, la méthode de Relaxation d’Ondes Optimisée (OSWR) et leur couplage, appliqués à la simulation numérique des équations paraboliques et des équations de Stokes. Nous proposons et analysons dans un premier temps un couplage de la méthode Pararéel avec la méthode OSWR. La méthode couplée Pararéel- OSWR obtenue est une méthode parallèle, à la fois en temps et en espace, avec seulement peu d’itérations OSWR dans le propagateur fin afin de réduire les coûts de calcul et avec un propagateur grossier classique déduit de la méthode d’Euler implicite. L’analyse de cette méthode couplée est présentée pour une équation d’advection-réaction-diffusion unidimensionnelle. Pour le couplage de Pararéel avec OSWR sans recouvrement, nous prouvons un résultat de convergence général via des estimations d’énergie. Des résultats numériques pour des problèmes d’advection-diffusion bidimensionnels et pour une équation de diffusion avec de fortes hétérogénéités sont présentés, pour illustrer les performances de l’algorithme couplé Pararéel-OSWR. Nous présentons ensuite également un algorithme qui couple Pararéel avec OSWR avec recouvrement, et nous analysons son facteur de convergence en utilisant la convergence linéaire de OSWR avec recouvrement que nous obtenons par une analyse de Fourier. Pour les équations de Stokes, nous présentons un algorithme OSWR bien posé et une estimation d’énergie pour la convergence des vitesses. Ensuite, nous montrons qu’en général, la pression ne converge pas et nous proposons une correction pour remédier à cela. Une stratégie similaire basée sur la transformée de Fourier est effectuée pour obtenir la formulation du facteur de convergence. Des tests numériques suivent pour illustrer les performances de la méthode OSWR avec correction. De plus, ces résultats sont également étendus pour obtenir des résultats similaires sur l’équation d’Oseen. Enfin, nous proposons l’algorithme Pararéel et un couplage Pararéel-OSWR pour les équations de Stokes, et démontrons certaines de leurs propriétés de base

Published
2021

22. Mathematical analysis of finite volume schemes for the simulation of quasi-geostrophic flows at low Froude number

Author

Do, Minh Hieu, Laboratoire Analyse, Géométrie et Applications (LAGA), Université Paris 8 Vincennes-Saint-Denis (UP8)-Centre National de la Recherche Scientifique (CNRS)-Institut Galilée-Université Paris 13 (UP13), LAGA, Université Paris 13, Pascal Omnes, Emmanuel Audusse, Yohan Penel, and Do, Minh hieu

Subjects
équilibre géostrophique, bas nombre de Froude, schéma de Godunov, méthode de volumes finis, finite volume method, système hyperbolique, [MATH] Mathematics [math], low Froude number, force de Coriolis, [MATH.MATH-NA] Mathematics [math]/Numerical Analysis [math.NA], schéma équilibre, Godunov scheme, numerical diffusion, Geostrophic equilibrium, hyperbolic system, well-balanced scheme, diffusion numérique, [MATH]Mathematics [math], Coriolis force, [MATH.MATH-NA]Mathematics [math]/Numerical Analysis [math.NA]
Abstract

The shallow water system plays an important role in the numerical simulation of oceanic models, coastal flows and dam-break floods. Several kinds of source terms can be taken into account in this model, such as the influence of bottom topography, Manning friction effects and Coriolis force. For large scale oceanic phenomena, the Coriolis force due to the Earth's rotation plays a central role since the atmospheric or oceanic circulations are frequently observed around the so-called geostrophic equilibrium which corresponds to the balance between the pressure gradient and the Coriolis source term. The ability of numerical schemes to well capture thelake at rest, has been widely studied. However, the geostrophic equilibrium issue, including the divergence free constraint on the velocity, is much more complex and only few works have been devoted to its preservation. In this manuscript, we design finite volume schemes that preserve the discrete geostrophic equilibrium in order to improve significantly the accuracy of numerical simulations of perturbations around this equilibrium. We first develop collocated and staggered schemes on rectangular and triangular meshes for a linearized model of the original shallow water system. The crucial common point of the various methods is to adapt and combine several strategies known as the Apparent Topography, the Low Mach and the Divergence Penalisation methods, in order to handle correctly the numerical diffusions involved in the schemes on different cell geometries, so that they do not destroy geostrophic equilibria. Finally, we extend these strategies to the non-linear case and show convincing numerical results., Le système de Saint-Venant joue un rôle important dans la simulation de modèles océaniques, d'écoulements côtiers et de ruptures de barrages. Plusieurs sortes de termes sources peuvent être pris en compte dans ce modèle, comme la topographie, les effets de friction de Manning et la force de Coriolis. Celle-ci joue un rôle central dans les phénomènes à grande échelle spatiale car les circulations atmosphériques ou océaniques sont souvent observées autour de l'équilibre géostrophique qui correspond à l'équilibre du gradient de pression et de cette force. La capacité des schémas numériques à bien reproduire le lac au repos a été largement étudiée; en revanche, la question de l'équilibre géostrophique (incluant la contrainte de vitesse à divergence nulle) est beaucoup plus complexe et peu de travaux lui ont été consacrés.Dans cette thèse, nous concevons des schémas volumes finis qui préservent les équilibres géostrophiques discrets dans le but d'améliorer significativement la précision des simulations numériques de perturbations autour de ces équilibres. Nous développons tout d'abord des schémas colocalisés et décalés sur des maillages rectangulaires ou triangulaires pour une linéarisation du modèle d'origine. Le point commun décisif de ces méthodes est d'adapter et de combiner les stratégies dites "topographie apparente", "bas Mach" et "pénalisation de divergence" pour contrôler l'effet de la diffusion numérique contenue dans les schémas, de telle sorte qu'elle ne détruise pas les équilibres géostrophiques. Enfin, nous étendons ces stratégies au cas non-linéaire et montrons des résultats prometteurs.

Published
2017

23. Equilibrated stress reconstructions for linear elasticity problems with application to a posteriori error analysis

Author

Alexandre Ern, Daniele Antonio Di Pietro, Rita Riedlbeck, Institut Montpelliérain Alexander Grothendieck (IMAG), Université de Montpellier (UM)-Centre National de la Recherche Scientifique (CNRS), Institut des Sciences de la mécanique et Applications industrielles (IMSIA - UMR 9219), Commissariat à l'énergie atomique et aux énergies alternatives (CEA)-École Nationale Supérieure de Techniques Avancées (ENSTA Paris)-Université Paris-Saclay-Centre National de la Recherche Scientifique (CNRS)-EDF R&D (EDF R&D), EDF (EDF)-EDF (EDF), Centre d'Enseignement et de Recherche en Mathématiques et Calcul Scientifique (CERMICS), École des Ponts ParisTech (ENPC), Simulation for the Environment: Reliable and Efficient Numerical Algorithms (SERENA), Inria de Paris, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria), Clément Cancès, Pascal Omnes, and Commissariat à l'énergie atomique et aux énergies alternatives (CEA)-École Nationale Supérieure de Techniques Avancées (ENSTA Paris)-Centre National de la Recherche Scientifique (CNRS)-Université Paris-Saclay-EDF R&D (EDF R&D)

Subjects
Arnold–Falk–Winther finite element, Cauchy stress tensor, Arnold–Winther finite element, Linear elasticity, linear elasticity, Geometry, 010103 numerical & computational mathematics, Mixed finite element method, 01 natural sciences, Symmetry (physics), Finite element method, Mathematics::Numerical Analysis, 010101 applied mathematics, Constraint (information theory), Stress (mechanics), Applied mathematics, A priori and a posteriori, equilibrated stress reconstruction, 0101 mathematics, a posteriori error estimate, [MATH.MATH-NA]Mathematics [math]/Numerical Analysis [math.NA], Mathematics
Abstract

International audience; We present an a posteriori error estimate for the linear elasticity problem. The estimate is based on an equilibrated reconstruction of the Cauchy stress tensor, which is obtained from mixed finite element solutions of local Neumann problems. We propose two different reconstructions, one using Arnold–Winther mixed finite element spaces providing a symmetric stress tensor and one using Arnold–Falk– Winther mixed finite element spaces with a weak symmetry constraint. The performance of the estimate is illustrated on a numerical test with analytical solution.

Published
2017
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24. A splitting scheme for three-phase flow models

Author

Hamza Boukili, Jean-Marc Hérard, EDF (EDF), Institut de Mathématiques de Marseille (I2M), Aix Marseille Université (AMU)-École Centrale de Marseille (ECM)-Centre National de la Recherche Scientifique (CNRS), and Clément Cancès - Pascal Omnes

Subjects
Computer science, Numerical analysis, Three-phase flow, Three phase flow, 01 natural sciences, vapour explosion, 010305 fluids & plasmas, 010101 applied mathematics, Riemann hypothesis, symbols.namesake, Entropy inequality, 0103 physical sciences, symbols, Applied mathematics, shocks, [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], [PHYS.MECA.MEFL]Physics [physics]/Mechanics [physics]/Fluid mechanics [physics.class-ph], 0101 mathematics, Data flow model, entropy, finite volumes, Step method
Abstract

International audience; A fractional step method that provides approximate solutions of a three-phase flow model is presented herein. The three-fluid model enables to handle smooth or discontinuous unsteady solutions. The numerical method is grounded on the use of the entropy inequality that governs smooth solutions of the set of PDEs. The evolution step relies on an explicit scheme, while implicit schemes are embedded in the relaxation step. The main properties of the scheme are given. Numerical approximations of two basic Riemann problems are eventually presented.

Published
2017
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25. A nonconforming high-order method for nonlinear poroelasticity

Author

Michele Botti, Daniele Antonio Di Pietro, Pierre Sochala, Institut Montpelliérain Alexander Grothendieck (IMAG), Université de Montpellier (UM)-Centre National de la Recherche Scientifique (CNRS), Bureau de Recherches Géologiques et Minières (BRGM) (BRGM), Labex NUMEV (ANR-10-LABX-20) ref. 2014-2-006, HHOMM (ref. ANR-15-CE40-0005), Clément Cancès, Pascal Omnes, and ANR-15-CE40-0005,HHOMM,Méthodes hybrides d'ordre élevé sur maillages polyédriques(2015)

Subjects
Discretization, Poromechanics, Mathematical analysis, 010103 numerical & computational mathematics, Hybrid High-Order, Space (mathematics), 01 natural sciences, 65M08, 65N30, 74B20, 76S05, 010101 applied mathematics, Nonlinear system, Nonlinear poroelasticity, Operator (computer programming), Flow (mathematics), Discontinuous Galerkin method, General meshes, Polygon mesh, 0101 mathematics, [MATH.MATH-NA]Mathematics [math]/Numerical Analysis [math.NA], Mathematics, discontinuous Galerkin
Abstract

International audience; In this work, we introduce a novel algorithm for the quasi-static nonlin-ear poroelasticity problem describing Darcian flow in a deformable saturated porous medium. The nonlinear elasticity operator is discretized using a Hybrid High-Order method while the heterogeneous diffusion part relies on a Symmetric Weighted Interior Penalty discontinuous Galerkin scheme. The method is valid in two and three space dimensions, delivers an inf-sup stable discretization on general meshes including polyhedral elements and nonmatching interfaces, allows arbitrary approximation orders, and has a reduced cost thanks to the possibility of statically condensing a large subset of the unknowns for linearized versions of the problem. Moreover, the proposed construction can handle rough variations of the permeability coefficient and vanishing specific storage coefficient. Numerical tests demonstrating the performance of the method are provided.

Published
2017
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26. A Fractional Step Method to Simulate Mixed Flows in Pipes with a Compressible Two-Layer Model

Author

Jean-Marc Hérard, Stéphane Gerbi, Charles Demay, Christian Bourdarias, Benoît de Laage de Meux, EDF (EDF), Laboratoire de Mathématiques (LAMA), Centre National de la Recherche Scientifique (CNRS)-Université Savoie Mont Blanc (USMB [Université de Savoie] [Université de Chambéry]), Institut de Mathématiques de Marseille (I2M), Aix Marseille Université (AMU)-École Centrale de Marseille (ECM)-Centre National de la Recherche Scientifique (CNRS), Clément Cancès - Pascal Omnes, and Université Savoie Mont Blanc (USMB [Université de Savoie] [Université de Chambéry])-Centre National de la Recherche Scientifique (CNRS)

Subjects
Mathematical optimization, 010102 general mathematics, Two layer, 01 natural sciences, 010101 applied mathematics, mixed flow, Test case, Mixed flow, Two-layer model, Convergence (routing), Compressibility, Applied mathematics, [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], implicit-explicit scheme, [PHYS.MECA.MEFL]Physics [physics]/Mechanics [physics]/Fluid mechanics [physics.class-ph], 0101 mathematics, Step method, Mathematics
Abstract

International audience; The so-called mixed flows in pipes include two-phase stratified regimes as well as single-phase pressurized regimes with transitions. It is proposed to handle those configurations numerically with the compressible two-layer model developed in [7]. Thus, a fractional step method is proposed to deal explicitly with the slow propagation phenomena and implicitly with the fast ones. It results in a large time-step scheme accurate in both regimes. Numerical experiments are performed including convergence results and academical test cases.

Published
2017
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27. Non-isothermal Compositional Two-Phase Darcy Flow: Formulation and Outflow Boundary Condition

Author

Farid Smaï, Laurence Beaude, Roland Masson, Simon Lopez, Konstantin Brenner, COmplex Flows For Energy and Environment (COFFEE), Inria Sophia Antipolis - Méditerranée (CRISAM), Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria)-Laboratoire Jean Alexandre Dieudonné (JAD), Université Côte d'Azur (UCA)-Université Nice Sophia Antipolis (... - 2019) (UNS), COMUE Université Côte d'Azur (2015-2019) (COMUE UCA)-COMUE Université Côte d'Azur (2015-2019) (COMUE UCA)-Centre National de la Recherche Scientifique (CNRS)-Université Côte d'Azur (UCA)-Université Nice Sophia Antipolis (... - 2019) (UNS), COMUE Université Côte d'Azur (2015-2019) (COMUE UCA)-COMUE Université Côte d'Azur (2015-2019) (COMUE UCA)-Centre National de la Recherche Scientifique (CNRS), Laboratoire Jean Alexandre Dieudonné (JAD), Bureau de Recherches Géologiques et Minières (BRGM) (BRGM), Clément Cancès, Pascal Omnes, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria)-Laboratoire Jean Alexandre Dieudonné (LJAD), Université Nice Sophia Antipolis (1965 - 2019) (UNS), COMUE Université Côte d'Azur (2015-2019) (COMUE UCA)-COMUE Université Côte d'Azur (2015-2019) (COMUE UCA)-Centre National de la Recherche Scientifique (CNRS)-Université Côte d'Azur (UCA)-Université Nice Sophia Antipolis (1965 - 2019) (UNS), COMUE Université Côte d'Azur (2015-2019) (COMUE UCA)-COMUE Université Côte d'Azur (2015-2019) (COMUE UCA)-Centre National de la Recherche Scientifique (CNRS)-Université Côte d'Azur (UCA), Laboratoire Jean Alexandre Dieudonné (LJAD), and ANR-16-CE06-0009,CHARMS,Modèles de Réservoirs Quantitatifs pour les Systèmes Hydrothermaux Complexes(2016)

Subjects
Convection, Darcy's law, Materials science, Boundary conditions for the interaction ground-atmosphere, business.industry, Geothermal energy, 010103 numerical & computational mathematics, Geophysics, Mechanics, 010502 geochemistry & geophysics, 01 natural sciences, 7. Clean energy, Isothermal process, Physics::Fluid Dynamics, Permeability (earth sciences), Finite volume scheme, Non-isothermal compositional two-phase Darcy flow model, Outflow, Boundary value problem, 0101 mathematics, Porous medium, business, [MATH.MATH-NA]Mathematics [math]/Numerical Analysis [math.NA], 0105 earth and related environmental sciences
Abstract

International audience; This article deals with the modelling and formulation of compositional gasliquid Darcy flow. Our model includes an advanced boundary condition at the interface between the porous medium and the atmosphere accounting for convective mass and energy transfer, liquid evaporation, and liquid outflow. The formulation is based on a fixed set of unknowns whatever the set of present phases. The thermodynamical equilibrium is expressed as complementary constraints. The model and its formulation are applied to the simulation of the Bouillante high energy geothermal field in Guadeloupe characterized by a high temperature closed to the surface.

Published
2017
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28. A Weighted Splitting Approach for Low-Mach Number Flows

Author

David Iampietro, Pascal Galon, Jean-Marc Hérard, Frédéric Daude, EDF (EDF), Institut de Mathématiques de Marseille (I2M), Aix Marseille Université (AMU)-École Centrale de Marseille (ECM)-Centre National de la Recherche Scientifique (CNRS), Commissariat à l'énergie atomique et aux énergies alternatives (CEA), Clément Cancès - Pascal Omnes, and Centre National de la Recherche Scientifique (CNRS)-École Centrale de Marseille (ECM)-Aix Marseille Université (AMU)

Subjects
Physics, Convection, Shock wave, 010103 numerical & computational mathematics, State (functional analysis), Mechanics, 7. Clean energy, 01 natural sciences, Physics::Fluid Dynamics, 010101 applied mathematics, Operator splitting, symbols.namesake, Mach number, Flow (mathematics), symbols, [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], [PHYS.MECA.MEFL]Physics [physics]/Mechanics [physics]/Fluid mechanics [physics.class-ph], 0101 mathematics
Abstract

International audience; In steady-state regimes, water circulating in the nuclear power plants pipes behaves as a low Mach number flow. However, when steep phenomena occur, strong shock waves are produced. Herein, a fractional step approach allowing to decouple the convective from the acoustic effects is proposed. The originality is that the splitting between these two parts of the physics evolves dynamically in time according to the Mach number. The first one-dimensional explicit and implicit numerical results on a wide panel of Mach numbers show that this approach is as accurate and CPU-consuming as a state of the art Lagrange-Projection-type method.

Published
2017
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29. Uniform-in-Time Convergence of Numerical Schemes for a Two-Phase Discrete Fracture Model

Author

Julian Hennicker, Jérôme Droniou, Roland Masson, School of Mathematical Sciences [Clayton], Monash University [Clayton], TOTAL CSTJF, F-64018 Pau, France, COmplex Flows For Energy and Environment (COFFEE), Inria Sophia Antipolis - Méditerranée (CRISAM), Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria)-Laboratoire Jean Alexandre Dieudonné (JAD), Université Côte d'Azur (UCA)-Université Nice Sophia Antipolis (... - 2019) (UNS), COMUE Université Côte d'Azur (2015-2019) (COMUE UCA)-COMUE Université Côte d'Azur (2015-2019) (COMUE UCA)-Centre National de la Recherche Scientifique (CNRS)-Université Côte d'Azur (UCA)-Université Nice Sophia Antipolis (... - 2019) (UNS), COMUE Université Côte d'Azur (2015-2019) (COMUE UCA)-COMUE Université Côte d'Azur (2015-2019) (COMUE UCA)-Centre National de la Recherche Scientifique (CNRS), Laboratoire Jean Alexandre Dieudonné (JAD), Pascal Omnes, Centre scientifique et Technique Jean Feger (CSTJF), TOTAL FINA ELF, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria)-Laboratoire Jean Alexandre Dieudonné (LJAD), Université Nice Sophia Antipolis (1965 - 2019) (UNS), COMUE Université Côte d'Azur (2015-2019) (COMUE UCA)-COMUE Université Côte d'Azur (2015-2019) (COMUE UCA)-Centre National de la Recherche Scientifique (CNRS)-Université Côte d'Azur (UCA)-Université Nice Sophia Antipolis (1965 - 2019) (UNS), COMUE Université Côte d'Azur (2015-2019) (COMUE UCA)-COMUE Université Côte d'Azur (2015-2019) (COMUE UCA)-Centre National de la Recherche Scientifique (CNRS)-Université Côte d'Azur (UCA), and Laboratoire Jean Alexandre Dieudonné (LJAD)

Subjects
Discrete fracture model, Capillary pressure, Hydrogeology, Darcy's law, Weak solution, 010103 numerical & computational mathematics, Mechanics, Two-phase Darcy flow, 01 natural sciences, Uniform-in-time convergence, Gradient discretization method, Physics::Geophysics, 010101 applied mathematics, Matrix (mathematics), Flow (mathematics), Fracture (geology), 0101 mathematics, Porous medium, [MATH.MATH-NA]Mathematics [math]/Numerical Analysis [math.NA], Geology
Abstract

International audience; Flow and transport in fracturedporous media are of paramount importance for many applications such as petroleum exploration and production, geological storage of carbon dioxide, hydrogeology, or geothermal energy. We consider here the two-phase discrete fracture model introduced in [3] which represents explicitly the fractures as codimension one surfaces immersed in the surrounding matrix domain. Then, the two-phase Darcy flow in the matrix is coupled with the two-phase Darcy flow in the fractures using transmission conditions accounting for fractures acting either as drains or barriers. The model takes into account complex networks of fractures, discontinuous capillary pressure curves at the matrix fracture interfaces and can be easily extended to account for gravity including in the width of the fractures. It also includes a layer of damaged rock at the matrix fracture interface with its own mobility and capillary pressure functions. In this work, the convergence analysis carried out in [3] in the framework of gradient discretizations [2] is extended to obtain the uniform-in-time convergence of the discrete solutions to a weak solution of the model.

Published
2017
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30. A Finite-Volume discretization of viscoelastic Saint-Venant equations for FENE-P fluids

Author

Sébastien Boyaval, Laboratoire d'Hydraulique Saint-Venant / Saint-Venant laboratory for Hydraulics (LHSV), École des Ponts ParisTech (ENPC)-Centre d'Etudes et d'Expertise sur les Risques, l'Environnement, la Mobilité et l'Aménagement (Cerema)-EDF R&D (EDF R&D), EDF (EDF)-EDF (EDF), MATHematics for MatERIALS (MATHERIALS), Centre d'Enseignement et de Recherche en Mathématiques et Calcul Scientifique (CERMICS), École des Ponts ParisTech (ENPC)-École des Ponts ParisTech (ENPC)-Inria de Paris, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria), Clément Cancès, Pascal Omnes, ANR-15-CE01-0013,SEDIFLO,Modélisation et simulation du transport solide en rivières(2015), Laboratoire d'Hydraulique Saint-Venant / Saint-Venant laboratory for Hydraulics (Saint-Venant), Inria de Paris, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria)-Centre d'Enseignement et de Recherche en Mathématiques et Calcul Scientifique (CERMICS), and École des Ponts ParisTech (ENPC)-École des Ponts ParisTech (ENPC)

Subjects
Suliciu relaxation scheme, Discretization, 010103 numerical & computational mathematics, 01 natural sciences, Viscoelasticity, 65N08, 010305 fluids & plasmas, FENE-P viscoelastic fluids, [SPI.MECA.MEFL]Engineering Sciences [physics]/Mechanics [physics.med-ph]/Fluids mechanics [physics.class-ph], Physics::Fluid Dynamics, Rheology, 0103 physical sciences, 0101 mathematics, Shallow water equations, Mathematics, Finite volume method, Mathematical analysis, Relaxation (iterative method), Physics::Classical Physics, MSC (2010): 65M08, Stability conditions, Nonlinear system, 35Q30, Saint-Venant equations, simple Riemann solver, Finite-Volume, [MATH.MATH-NA]Mathematics [math]/Numerical Analysis [math.NA]
Abstract

Saint-Venant equations can be generalized to account for a viscoelastic rheology in shallow flows. A Finite-Volume discretization for the 1D Saint-Venant system generalized to Upper-Convected Maxwell (UCM) fluids was proposed in Bouchut and Boyaval (M3AS 23(08): 1479–1526, 2013, [6]), which preserved a physically-natural stability property (i.e. free-energy dissipation) of the full system. It invoked a relaxation scheme of Suliciu type for the numerical computation of approximate solutions to Riemann problems. Here, the approach is extended to the 1D Saint-Venant system generalized to the finitely-extensible nonlinear elastic fluids of Peterlin (FENE-P). We are currently not able to ensure all stability conditions a priori, but the scheme is fully computable. And, using numerical simulations, it may help understand the famous High-Weissenberg number problem (HWNP) well-known in computational rheology.

Published
2017
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31. A Nonlinear Domain Decomposition Method to Couple Compositional Gas Liquid Darcy and Free Gas Flows

Author

Roland Masson, Nabil Birgle, Laurent Trenty, Laboratoire Jean Alexandre Dieudonné (JAD), Université Côte d'Azur (UCA)-Université Nice Sophia Antipolis (... - 2019) (UNS), COMUE Université Côte d'Azur (2015-2019) (COMUE UCA)-COMUE Université Côte d'Azur (2015-2019) (COMUE UCA)-Centre National de la Recherche Scientifique (CNRS), COmplex Flows For Energy and Environment (COFFEE), Inria Sophia Antipolis - Méditerranée (CRISAM), Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria)-Laboratoire Jean Alexandre Dieudonné (JAD), COMUE Université Côte d'Azur (2015-2019) (COMUE UCA)-COMUE Université Côte d'Azur (2015-2019) (COMUE UCA)-Centre National de la Recherche Scientifique (CNRS)-Université Côte d'Azur (UCA)-Université Nice Sophia Antipolis (... - 2019) (UNS), Agence Nationale pour la Gestion des Déchets Radioactifs (ANDRA), Clément Cancès, Pascal Omnes, Laboratoire Jean Alexandre Dieudonné (LJAD), Université Nice Sophia Antipolis (1965 - 2019) (UNS), COMUE Université Côte d'Azur (2015-2019) (COMUE UCA)-COMUE Université Côte d'Azur (2015-2019) (COMUE UCA)-Centre National de la Recherche Scientifique (CNRS)-Université Côte d'Azur (UCA), Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria)-Laboratoire Jean Alexandre Dieudonné (LJAD), and COMUE Université Côte d'Azur (2015-2019) (COMUE UCA)-COMUE Université Côte d'Azur (2015-2019) (COMUE UCA)-Centre National de la Recherche Scientifique (CNRS)-Université Côte d'Azur (UCA)-Université Nice Sophia Antipolis (1965 - 2019) (UNS)

Subjects
Compositional gas liquid darcy flow, Darcy's law, Chemistry, 0207 environmental engineering, Thermodynamics, Domain decomposition methods, Coupling algorithm, 02 engineering and technology, Mechanics, 01 natural sciences, Nonlinear domain decomposition method, Robin boundary condition, 010305 fluids & plasmas, Physics::Fluid Dynamics, Permeability (earth sciences), Nonlinear system, Free gas flow, 0103 physical sciences, 020701 environmental engineering, Reynolds-averaged Navier–Stokes equations, Porous medium, Convection–diffusion equation, Drying model, [MATH.MATH-NA]Mathematics [math]/Numerical Analysis [math.NA]
Abstract

International audience; A domain decomposition algorithm is proposed to couple at the interface a gas liquid compositional Darcy flow and a compositional free gas flow. At each time step, our algorithm solves iteratively the nonlinear system coupling the compositional Darcy flow in the porous medium, the RANS gas flow in the free flow domain, and the convection diffusion of the species in the free flow domain. In order to speed up the convergence of the algorithm, the transmission conditions at the interface are replaced by Robin boundary conditions. Each Robin coefficient is obtained from a diagonal approximation of the Dirichlet to Neumann operator related to a scalar simplified model in the neighbouring subdomain. The efficiency of our domain decomposition algorithm is assessed in the case of the modelling of the mass exchanges at the interface between the geological formation and the ventilation galleries of geological radioactive waste disposal.

Published
2017
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