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1. Some analytical solutions for validation of free surface flow computational codes

Author

Bernard Di Martino, Fabien Souillé, Jacques Sainte-Marie, Marie-Odile Bristeau, Anne Mangeney, Numerical Analysis, Geophysics and Ecology (ANGE), 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 (UMR_7598)), Sorbonne Université (SU)-Centre National de la Recherche Scientifique (CNRS)-Université Paris Cité (UPCité)-Sorbonne Université (SU)-Centre National de la Recherche Scientifique (CNRS)-Université Paris Cité (UPCité), Institut de Physique du Globe de Paris (IPGP (UMR_7154)), Institut national des sciences de l'Univers (INSU - CNRS)-Université de La Réunion (UR)-Institut de Physique du Globe de Paris (IPG Paris)-Centre National de la Recherche Scientifique (CNRS)-Université Paris Cité (UPCité), The work presented in this paper was supported in part by the ANR Mimosa (ANR-14-CE01-0012) and ERC contract ERC-CG-2013-PE10-617472 SLIDEQUAKES. The authors also acknowledge the Inria Project Lab 'Algae in Silico' for its financial support., ANR-14-CE01-0012,MIMOSA,Analyse, modélisation et utilisation du bruit sismique large bande: des microséismes aux oscillations libres de la terre(2014), Sorbonne Université (SU)-Centre National de la Recherche Scientifique (CNRS)-Université de Paris (UP)-Sorbonne Université (SU)-Centre National de la Recherche Scientifique (CNRS)-Université de Paris (UP), Institut de Physique du Globe de Paris (IPGP), Institut national des sciences de l'Univers (INSU - CNRS)-IPG PARIS-Université de La Réunion (UR)-Centre National de la Recherche Scientifique (CNRS)-Université de Paris (UP), Université Paris Diderot - Paris 7 (UPD7)-Sorbonne Université (SU)-Centre National de la Recherche Scientifique (CNRS)-Université Paris Diderot - Paris 7 (UPD7)-Sorbonne Université (SU)-Centre National de la Recherche Scientifique (CNRS), and Centre National de la Recherche Scientifique (CNRS)-Université de La Réunion (UR)-Université Paris Diderot - Paris 7 (UPD7)-IPG PARIS-Institut national des sciences de l'Univers (INSU - CNRS)

Subjects
010103 numerical & computational mathematics, Analytical solutions, 01 natural sciences, law.invention, Physics::Fluid Dynamics, Free surface Euler equations, symbols.namesake, law, [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], Boundary value problem, [PHYS.MECA.MEFL]Physics [physics]/Mechanics [physics]/Fluid mechanics [physics.class-ph], 0101 mathematics, [MATH]Mathematics [math], Mathematics, Computer simulation, [PHYS.MECA.MEFL]Physics [physics]/Mechanics [physics]/Mechanics of the fluids [physics.class-ph], Mechanical Engineering, Mechanics, Euler system, Condensed Matter Physics, River dynamics, Euler equations, 010101 applied mathematics, Flow (mathematics), Mechanics of Materials, Free surface, symbols, Compressibility, Hydrostatic equilibrium
Abstract

We present several time dependent analytical solutions for the incompressible Euler system with free surface. These analytical solutions give quantitative descriptions of some physical phenomena and can be used as reference solutions when validating numerical simulation codes. They concern fluid flows governed by Euler equations with or withouthydrostatic hypothesis including wet/dry interface, variable density, wide variety of boundary conditions and possibly non-uniform bottom topographies. Some of these analytical solutions concern shallow water type models or free surface Navier-Stokes model i.e. with a viscous fluid.

Published
2021
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2. Diffusion-redistanciation schemes for 2D and 3D constrained Willmore flow: application to the equilibrium shapes of vesicles

Author

Emmanuel Maitre, Mourad E. H. Ismail, Arnaud Sengers, Thibaut Metivet, ModELisation de l'apparence des phénomènes Non-linéaires (ELAN), Inria Grenoble - Rhône-Alpes, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria)-Laboratoire Jean Kuntzmann (LJK), Institut National de Recherche en Informatique et en Automatique (Inria)-Centre National de la Recherche Scientifique (CNRS)-Université Grenoble Alpes (UGA)-Institut polytechnique de Grenoble - Grenoble Institute of Technology (Grenoble INP ), Université Grenoble Alpes (UGA)-Centre National de la Recherche Scientifique (CNRS)-Université Grenoble Alpes (UGA)-Institut polytechnique de Grenoble - Grenoble Institute of Technology (Grenoble INP ), Université Grenoble Alpes (UGA), Equations aux Dérivées Partielles (EDP), Laboratoire Jean Kuntzmann (LJK), Université Grenoble Alpes (UGA)-Institut National de Recherche en Informatique et en Automatique (Inria)-Centre National de la Recherche Scientifique (CNRS)-Université Grenoble Alpes (UGA)-Institut polytechnique de Grenoble - Grenoble Institute of Technology (Grenoble INP ), DYnamique des Fluides COmplexes et Morphogénèse [Grenoble] (DYFCOM-LIPhy ), Laboratoire Interdisciplinaire de Physique [Saint Martin d’Hères] (LIPhy ), Centre National de la Recherche Scientifique (CNRS)-Université Grenoble Alpes (UGA)-Centre National de la Recherche Scientifique (CNRS)-Université Grenoble Alpes (UGA), Most of the computations presented in this paper were performed using the Froggy platform of the GRICAD infrastructure (https://gricad.univ-grenoble-alpes.fr), which is supported by the Rhône-Alpes region (GRANT CPER07_13 CIRA) and the Equip@Meso project (reference ANR-10-EQPX-29-01) of the programme Investissements d’Avenir supervised by the Agence Nationale pour la Recherche, and the Atlas cluster from the Research Institute in Advanced Mathematics (IRMA - UMR7501)., ANR-10-EQPX-0029,EQUIP@MESO,Equipement d'excellence de calcul intensif de Mesocentres coordonnés - Tremplin vers le calcul petaflopique et l'exascale(2010), Institut National de Recherche en Informatique et en Automatique (Inria)-Centre National de la Recherche Scientifique (CNRS)-Université Grenoble Alpes [2020-....] (UGA [2020-....])-Institut polytechnique de Grenoble - Grenoble Institute of Technology [2020-....] (Grenoble INP [2020-....]), Université Grenoble Alpes [2020-....] (UGA [2020-....])-Centre National de la Recherche Scientifique (CNRS)-Université Grenoble Alpes [2020-....] (UGA [2020-....])-Institut polytechnique de Grenoble - Grenoble Institute of Technology [2020-....] (Grenoble INP [2020-....]), Université Grenoble Alpes [2020-....] (UGA [2020-....]), Université Grenoble Alpes [2020-....] (UGA [2020-....])-Institut National de Recherche en Informatique et en Automatique (Inria)-Centre National de la Recherche Scientifique (CNRS)-Université Grenoble Alpes [2020-....] (UGA [2020-....])-Institut polytechnique de Grenoble - Grenoble Institute of Technology [2020-....] (Grenoble INP [2020-....]), DYnamique des Fluides COmplexes et Morphogénèse [Grenoble] [2020-….] (DYFCOM-LIPhy [2020-….]), Laboratoire Interdisciplinaire de Physique [Saint Martin d’Hères] [2020-….] (LIPhy [2020-….]), and Université Grenoble Alpes [2020-....] (UGA [2020-....])-Centre National de la Recherche Scientifique (CNRS)-Université Grenoble Alpes [2020-....] (UGA [2020-....])-Centre National de la Recherche Scientifique (CNRS)

Subjects
Volume and area preserving, Vol- ume and area preserving, Diffusion generated motion, Physics and Astronomy (miscellaneous), Computer science, Interface (Java), and phrases: Diffusion generated motion, Signed distance function, Clifford torus, 010103 numerical & computational mathematics, Willmore flow, 01 natural sciences, Level set, Dimension (vector space), FOS: Mathematics, Applied mathematics, Mathematics - Numerical Analysis, [PHYS.MECA.MEFL]Physics [physics]/Mechanics [physics]/Fluid mechanics [physics.class-ph], 0101 mathematics, Diffusion (business), [MATH]Mathematics [math], High-order geometrical flow, Numerical Analysis, Level set (data structures), Applied Mathematics, Numerical Analysis (math.NA), Thresholding, Computer Science Applications, 010101 applied mathematics, Computational Mathematics, Flow (mathematics), Modeling and Simulation, [MATH.MATH-NA]Mathematics [math]/Numerical Analysis [math.NA]
Abstract

In this paper we present a novel algorithm for simulating geometrical flows, and in particular the Willmore flow, with conservation of volume and area. The idea is to adapt the class of diffusion-redistanciation algorithms to the Willmore flow in both two and three dimensions. These algorithms rely on alternating diffusions of the signed distance function to the interface and a redistanciation step, and with careful choice of the applied diffusions, end up moving the zero level-set of the distance function by some geometrical quantity without resorting to any explicit transport equation. The constraints are enforced between the diffusion and redistanciation steps via a simple rescaling method. The energy globally decreases at the end of each global step. The algorithms feature the computational efficiency of thresholding methods without requiring any adaptive remeshing thanks to the use of a signed distance function to describe the interface. This opens their application to dynamic fluid-structure simulations for large and realistic cases. The methodology is validated by computing the equilibrium shapes of two- and three-dimensional vesicles, as well as the Clifford torus.

Published
2020
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3. Some quasi-analytical solutions for propagative waves in free surface Euler equations

Author

Bristeau, Marie-Odile, Di Martino, Bernard, Mangeney, Anne, Sainte-Marie, Jacques, Souillé, Fabien, Sorbonne Université (SU), Numerical Analysis, Geophysics and Ecology (ANGE), 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 (UMR_7598)), Sorbonne Université (SU)-Centre National de la Recherche Scientifique (CNRS)-Université de Paris (UP)-Sorbonne Université (SU)-Centre National de la Recherche Scientifique (CNRS)-Université de Paris (UP), Sciences pour l'environnement (SPE), Université Pascal Paoli (UPP)-Centre National de la Recherche Scientifique (CNRS), Institut de Physique du Globe de Paris, EDF (EDF), The work presented in this paper was supported in part by the ANR Mimosa (ANR-14-CE01-0012) andERC contract ERC-CG-2013-PE10-617472 SLIDEQUAKES., ANR-14-CE01-0012,MIMOSA,Analyse, modélisation et utilisation du bruit sismique large bande: des microséismes aux oscillations libres de la terre(2014), and Centre National de la Recherche Scientifique (CNRS)-Université Pascal Paoli (UPP)

Subjects
analytical solution, [PHYS.MECA.MEFL]Physics [physics]/Mechanics [physics]/Mechanics of the fluids [physics.class-ph], Free Surface Euler Equation, [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], [PHYS.MECA.MEFL]Physics [physics]/Mechanics [physics]/Fluid mechanics [physics.class-ph], Airy wave
Abstract

International audience; This note describes some quasi-analytical solutions for wave propagation in free surface Euler equations and linearized Euler equations. The obtained solutions vary from a sinusoidal form to a form with singularities. They allow a numerical validation of the free-surface Euler codes.; Cette note décrit des solutions quasi analytiques correspondant à la propagation d'ondes dans les équations d'Euler et d'Euler linéarisées à surface libre. Les solutions obtenues varient d'une forme sinusoïdale à une forme présentant des singularités. Elles permettent de valider numériquement les codes de simulation des équations d'Euler à surface libre.

Published
2020
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