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29 results on '"[MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP]"'

1. Mean curvature motion of point cloud varifolds

Author

Blanche Buet, Martin Rumpf, Understanding the Shape of Data (DATASHAPE), 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)-Inria Saclay - Ile de France, Institut National de Recherche en Informatique et en Automatique (Inria), Université Paris-Saclay, University of Bonn, Laboratoire de Mathématiques d'Orsay (LMO), Université Paris-Saclay-Centre National de la Recherche Scientifique (CNRS), Universität Bonn = University of Bonn, and ANR-21-CE40-0013,GeMfaceT,Entre théorie géométrique de la mesure et surfaces discrètes(2021)

Subjects
Time discretization, [MATH.MATH-DG]Mathematics [math]/Differential Geometry [math.DG], Regularization, FOS: Mathematics, [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], Singular evolution, Point cloud varifolds, Mean curvature motion, MSC: 49Q20, 35K55, 53A70, 53E10, Mathematics - Numerical Analysis, Numerical Analysis (math.NA), [MATH]Mathematics [math], 49Q20, 35K55, 53A70, 53E10
Abstract

This paper investigates a discretization scheme for mean curvature motion on point cloud varifolds with particular emphasis on singular evolutions. To define the varifold a local covariance analysis is applied to compute an approximate tangent plane for the points in the cloud. The core ingredient of the mean curvature motion model is the regularization of the first variation of the varifold via convolution with kernels with small stencil. Consistency with the evolution velocity for a smooth surface is proven if a sufficiently small stencil and a regular sampling are taking into account. Furthermore, an implicit and a semiimplicit time discretization are derived. The implicit scheme comes with discrete barrier properties known for the smooth, continuous evolution, whereas the semiimplicit still ensures in all our numerical experiments very good approximation properties while being easy to implement. It is shown that the proposed method is robust with respect to noise and recovers the evolution of smooth curves as well as the formation of singularities such as triple points in 2D or minimal cones in 3D., Comment: 37 pages, 10 figures

Published
2022

2. A fast second-order discretization scheme for the linearized Green-Naghdi system with absorbing boundary conditions

Author

Pang, Gang, Ji, Songsong, Antoine, Xavier, Beihang University (BUAA), College of Engineering [Beijing], Peking University [Beijing], Systems with physical heterogeneities : inverse problems, numerical simulation, control and stabilization (SPHINX), Inria Nancy - Grand Est, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria), Institut Élie Cartan de Lorraine (IECL), Université de Lorraine (UL)-Centre National de la Recherche Scientifique (CNRS), This research is partially supported by NSFC under grant Nos.11502028., Research conducted within the context of the Sino-French International Associated Laboratory for Applied Mathematics-LIASFMA., and X. Antoine thanks the LIASFMA funding support of the Université de Lorraine.

Subjects
linearized Green-Naghdi system, MSC : 76B15, 65M06, 65R10, Convergence analysis, convolution quadrature, fast algorithm, [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], absorbing boundary conditions, Padé approximation, [MATH.MATH-NA]Mathematics [math]/Numerical Analysis [math.NA]
Abstract

International audience; In this paper, we present a fully discrete second-order finite-difference scheme with fast evaluation of the convolution involved in the absorbing boundary conditions to solve the one-dimensional linearized Green-Naghdi system. The Padé expansion of the square-root function in the complex plane is used to implement the fast convolution. By introducing a constant damping parameter into the governing equations, the convergence analysis is developed when the damping term fulfills some conditions. In addition, the scheme is stable and leads to a highly reduced computational cost and low memory storage. A numerical example is provided to support the theoretical analysis and to illustrate the performance of the fast numerical scheme.

Published
2022

3. Coherence and flow-maximization of a one-way valve

Author

Corli, Andrea, Razafison, Ulrich, Rosini, Massimiliano D., Università degli Studi di Ferrara (UniFE), Laboratoire de Mathématiques de Besançon (UMR 6623) (LMB), Université de Bourgogne (UB)-Université de Franche-Comté (UFC), Université Bourgogne Franche-Comté [COMUE] (UBFC)-Université Bourgogne Franche-Comté [COMUE] (UBFC)-Centre National de la Recherche Scientifique (CNRS), and Uniwersytet Marii Curie-Sklodowskiej = University Marii Curie-Sklodowskiej [Lublin] (UMCS)

Subjects
maximization, Condensed Matter::Other, gas flow, chattering, Numerical Analysis (math.NA), Condensed Matter::Mesoscopic Systems and Quantum Hall Effect, Physics::Classical Physics, Riemann problem, Computer Science::Other, Condensed Matter::Materials Science, systems of conservation laws, Mathematics - Analysis of PDEs, coupling conditions, isentropic Euler equations, FOS: Mathematics, [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], Mathematics - Numerical Analysis, p-system, valve, control, [MATH.MATH-NA]Mathematics [math]/Numerical Analysis [math.NA], Analysis of PDEs (math.AP)
Abstract

We consider a mathematical model for the gas flow through a one-way valve and focus on two issues. First, we propose a way to eliminate the chattering (the fast switch on and off of the valve) by slightly modifying the design of the valve. This mathematically amounts to the construction of a coupling Riemann solver with a suitable stability property, namely, coherence. We provide a numerical comparison of the behavior of the two valves. Second, we analyze, both analytically and numerically, for several significative situations, the maximization of the flow through the modified valve according to a control parameter of the valve and time.

Published
2022

4. Analysis of compressible bubbly flows. Part II: Derivation of a macroscopic model

Author

Hillairet, Matthieu, Mathis, Hélène, Seguin, Nicolas, Institut Montpelliérain Alexander Grothendieck (IMAG), Centre National de la Recherche Scientifique (CNRS)-Université de Montpellier (UM), Laboratoire de Mathématiques Jean Leray (LMJL), Centre National de la Recherche Scientifique (CNRS)-Nantes université - UFR des Sciences et des Techniques (Nantes univ - UFR ST), Nantes Université - pôle Sciences et technologie, Nantes Université (Nantes Univ)-Nantes Université (Nantes Univ)-Nantes Université - pôle Sciences et technologie, Nantes Université (Nantes Univ)-Nantes Université (Nantes Univ), Institut de Recherche Mathématique de Rennes (IRMAR), Université de Rennes (UR)-Institut National des Sciences Appliquées - Rennes (INSA Rennes), Institut National des Sciences Appliquées (INSA)-Institut National des Sciences Appliquées (INSA)-École normale supérieure - Rennes (ENS Rennes)-Université de Rennes 2 (UR2)-Centre National de la Recherche Scientifique (CNRS)-INSTITUT AGRO Agrocampus Ouest, and Institut national d'enseignement supérieur pour l'agriculture, l'alimentation et l'environnement (Institut Agro)-Institut national d'enseignement supérieur pour l'agriculture, l'alimentation et l'environnement (Institut Agro)

Subjects
Physics::Fluid Dynamics, Homogenization, Mathematics - Analysis of PDEs, 76T10, 35Q30 Contents, FOS: Mathematics, [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], compressible Navier-Stokes equations, Cauchy theory 2020 MCS. 76T05, two-phase flows, Analysis of PDEs (math.AP)
Abstract

This paper is the second of the series of two papers, which focuses on the derivation of an averaged 1D model for compressible bubbly flows. For this, we start from a microscopic description of the interactions between a large but finite number of small bubbles with a surrounding compressible fluid. This microscopic model has been derived and analysed in the first paper. In the present one, provided physical parameters scale according to the number of bubbles, we prove that solutions to the microscopic model exist on a timespan independent of the number of bubbles. Considering then that we have a large number of bubbles, we propose a construction of the macroscopic variables and derive the averaged system satisfied by these quantities. Our method is based on a compactness approach in a strong-solution setting. In the last section, we propose the derivation of the Williams-Boltzmann equation corresponding to our setting.

Published
2023

5. A generalized finite element method for problems with sign-changing coefficients

Author

Barbara Verfürth, Théophile Chaumont-Frelet, Modélisation et méthodes numériques pour le calcul d'interactions onde-matière nanostructurée (ATLANTIS), 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), Institut für Mathematik [Augsburg], Universität Augsburg [Augsburg], 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
Work (thermodynamics), 65N30, 65N12, 65N15, 78A48, 35J20, 010103 numerical & computational mathematics, 01 natural sciences, Convergence (routing), FOS: Mathematics, [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], Applied mathematics, Mathematics - Numerical Analysis, 0101 mathematics, Sign-changing coefficients, Mathematics, Numerical Analysis, Applied Mathematics, Metamaterial, Generalized finite element method, Numerical Analysis (math.NA), Finite element method, 010101 applied mathematics, Computational Mathematics, Exact solutions in general relativity, Multiscale method, Rate of convergence, Modeling and Simulation, Norm (mathematics), T-coercivity, [MATH.MATH-NA]Mathematics [math]/Numerical Analysis [math.NA], Analysis, Energy (signal processing)
Abstract

International audience; Problems with sign-changing coefficients occur, for instance, in the study of transmission problems with metamaterials. In this work, we present and analyze a generalized finite element method in the spirit of the Localized Orthogonal Decomposition, that is especially efficient when the negative and positive materials exhibit multiscale features. We derive optimal linear convergence in the energy norm independently of the potentially low regularity of the exact solution. Numerical experiments illustrate the theoretical convergence rates and show the applicability of the method for a large class of sign-changing diffusion problems.

Published
2021

6. Various choices of source terms for a class of two-fluid two-velocity models

Author

olivier hurisse, Hurisse, Olivier, EDF R&D (EDF R&D), and EDF (EDF)

Subjects
Numerical Analysis, Entropy (statistical thermodynamics), Applied Mathematics, media_common.quotation_subject, [SPI.MECA.MEFL] Engineering Sciences [physics]/Mechanics [physics.med-ph]/Fluids mechanics [physics.class-ph], Second law of thermodynamics, Class (philosophy), 01 natural sciences, [SPI.MECA.MEFL]Engineering Sciences [physics]/Mechanics [physics.med-ph]/Fluids mechanics [physics.class-ph], 010305 fluids & plasmas, 010101 applied mathematics, Computational Mathematics, Task (computing), Modeling and Simulation, Relaxation effect, 0103 physical sciences, [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], Applied mathematics, [MATH.MATH-AP] Mathematics [math]/Analysis of PDEs [math.AP], 0101 mathematics, Analysis, Two fluid, Mathematics, media_common
Abstract

International audience; The source terms of the Baer-Nunziato model involve highly non-linear return to equilibrium terms. In order to perform numerical simulations of realistic situations, accounting for this relaxation effects is mandatory. Unfortunately, with the classical forms retained for these source termsin the literature, building efficient, robust and accurate numerical schemes is a tricky task. In this paper, we propose different non-classical forms for these source terms. As for the classical ones, they all agree with the second law of thermodynamics and they are thus associated with a growth of an entropy. The great advantage of some of these new forms of source terms is that they are more linear with respect to the conservative variables. Consequently, this allows to propose more robust, efficient and accurate numerical schemes, in particular when considering fractional step approaches for which source terms and convection terms are solved separately.

Published
2021

7. A stiffly stable semi-discrete scheme for the characteristic linear hyperbolic relaxation with boundary

Author

Benjamin Boutin, Thi Hoai Thuong Nguyen, Nicolas Seguin, Institut de Recherche Mathématique de Rennes (IRMAR), Université de Rennes (UR)-Institut National des Sciences Appliquées - Rennes (INSA Rennes), Institut National des Sciences Appliquées (INSA)-Institut National des Sciences Appliquées (INSA)-École normale supérieure - Rennes (ENS Rennes)-Université de Rennes 2 (UR2)-Centre National de la Recherche Scientifique (CNRS)-INSTITUT AGRO Agrocampus Ouest, Institut national d'enseignement supérieur pour l'agriculture, l'alimentation et l'environnement (Institut Agro)-Institut national d'enseignement supérieur pour l'agriculture, l'alimentation et l'environnement (Institut Agro), ANR-17-CE40-0025, Agence Nationale de la Recherche, Grant 642768 (ModCompShock), Innovative Training Networks, ANR-17-CE40-0025,Nabuco,Frontières numériques et couplages(2017), European Project: 642768,H2020,H2020-MSCA-ITN-2014,ModCompShock(2015), AGROCAMPUS OUEST, Institut national d'enseignement supérieur pour l'agriculture, l'alimentation et l'environnement (Institut Agro)-Institut national d'enseignement supérieur pour l'agriculture, l'alimentation et l'environnement (Institut Agro)-Université de Rennes 1 (UR1), Université de Rennes (UNIV-RENNES)-Université de Rennes (UNIV-RENNES)-Université de Rennes 2 (UR2), Université de Rennes (UNIV-RENNES)-École normale supérieure - Rennes (ENS Rennes)-Centre National de la Recherche Scientifique (CNRS)-Institut National des Sciences Appliquées - Rennes (INSA Rennes), and Institut National des Sciences Appliquées (INSA)-Université de Rennes (UNIV-RENNES)-Institut National des Sciences Appliquées (INSA)

Subjects
Boundary (topology), 010103 numerical & computational mathematics, summation by parts operators, Space (mathematics), 01 natural sciences, Stability (probability), 010305 fluids & plasmas, central schemes, 0103 physical sciences, medicine, [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], damped wave equation, Boundary value problem, 0101 mathematics, Mathematics, Numerical Analysis, Laplace transform, energy estimates, Applied Mathematics, Mathematical analysis, Stiffness, Damped wave, Computational Mathematics, Modeling and Simulation, hyperbolic relaxation system, Relaxation (approximation), medicine.symptom, [MATH.MATH-NA]Mathematics [math]/Numerical Analysis [math.NA], Analysis
Abstract

International audience; We study the stability of the semi-discrete central scheme for the linear damped wave equation with boundary. We exhibit a sufficient condition on the boundary to guarantee the uniform stability of the initial boundary value problem for relaxation system independent of stiffness of the source term and of the space step. The boundary is approximated using a summation-by-parts method and the stiff stability is proved by energy estimates and Laplace transform. We also investigate if the condition is also necessary, following the continuous case studied by Xin and Xu (2000).; Nous étudions la stabilité du schéma semi-discret centré pour l'équation des ondes linéaire amortie posé sur un demi-espace. Nous dégageons une condition suffisante portant sur la condition de bord, pour la stabilité du problème semi-discret avec donnée initiale et donnée de bord, ceci de manière uniforme par rapport à la raideur du terme source de relaxation ainsi qu'au pas d'espace. La discrétisation de la condition de bord employée provient de l'approche SBP et l'uniforme stabilité s'obtient par l'utilisation de méthodes d'énergie et de la transformée de Laplace. Nous examinons également au travers d'expériences numériques le caractère nécessaire de la condition retenue, de sorte à confronter notre résultat à l'étude de Xin et Xu (2000) portant sur le cas continu.

Published
2020

8. A variational formulation for computing shape derivatives of geometric constraints along rays

Author

Grégoire Allaire, Florian Feppon, Charles Dapogny, Centre de Mathématiques Appliquées - Ecole Polytechnique (CMAP), École polytechnique (X)-Centre National de la Recherche Scientifique (CNRS), Safran Tech, Calcul des Variations, Géométrie, Image (CVGI ), Laboratoire Jean Kuntzmann (LJK ), and Institut polytechnique de Grenoble - Grenoble Institute of Technology (Grenoble INP )-Institut National de Recherche en Informatique et en Automatique (Inria)-Centre National de la Recherche Scientifique (CNRS)-Université Grenoble Alpes [2016-2019] (UGA [2016-2019])-Institut polytechnique de Grenoble - Grenoble Institute of Technology (Grenoble INP )-Institut National de Recherche en Informatique et en Automatique (Inria)-Centre National de la Recherche Scientifique (CNRS)-Université Grenoble Alpes [2016-2019] (UGA [2016-2019])

Subjects
Level set method, Discretization, Signed distance function, 010103 numerical & computational mathematics, Thickness constraints, Curvature, 01 natural sciences, Shape and topology optimization, signed distance function, Level set methods, [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], Shape optimization, 0101 mathematics, Mathematics, Numerical Analysis, AMS Subject classifications 65K10 , 49Q10, Applied Mathematics, Mathematical analysis, Finite element method, 010101 applied mathematics, Computational Mathematics, Variational method, Modeling and Simulation, Advection operator, Vector field, [MATH.MATH-OC]Mathematics [math]/Optimization and Control [math.OC], level set method, [MATH.MATH-NA]Mathematics [math]/Numerical Analysis [math.NA], Analysis
Abstract

In the formulation of shape optimization problems, multiple geometric constraint functionals involve the signed distance function to the optimized shape Ω. The numerical evaluation of their shape derivatives requires to integrate some quantities along the normal rays to Ω, a challenging operation to implement, which is usually achieved thanks to the method of characteristics. The goal of the present paper is to propose an alternative, variational approach for this purpose. Our method amounts, in full generality, to compute integral quantities along the characteristic curves of a given velocity field without requiring the explicit knowledge of these curves on the spatial discretization; it rather relies on a variational problem which can be solved conveniently by the finite element method. The well-posedness of this problem is established thanks to a detailed analysis of weighted graph spaces of the advection operator β ⋅ ∇ associated to a C1 velocity field β. One novelty of our approach is the ability to handle velocity fields with possibly unbounded divergence: we do not assume div(β) ∈ L∞. Our working assumptions are fulfilled in the context of shape optimization of C2 domains Ω, where the velocity field β = ∇dΩ is an extension of the unit outward normal vector to the optimized shape. The efficiency of our variational method with respect to the direct integration of numerical quantities along rays is evaluated on several numerical examples. Classical albeit important implementation issues such as the calculation of a shape’s curvature and the detection of its skeleton are discussed. Finally, we demonstrate the convenience and potential of our method when it comes to enforcing maximum and minimum thickness constraints in structural shape optimization.

Published
2020

9. An edge-based scheme on polyhedral meshes for vector advection-reaction equations

Author

Alexandre Ern, Pierre Cantin, Centre d'Enseignement et de Recherche en Mathématiques et Calcul Scientifique (CERMICS), École des Ponts ParisTech (ENPC), EDF R&D (EDF R&D), EDF (EDF), Simulation for the Environment: Reliable and Efficient Numerical Algorithms (SERENA), Inria de Paris, and Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria)

Subjects
quasi-optimal a priori error estimates, 010103 numerical & computational mathematics, 01 natural sciences, Upper and lower bounds, Stability (probability), Combinatorics, [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], Applied mathematics, Polygon mesh, Tensor, 0101 mathematics, Eigenvalues and eigenvectors, Mathematics, Numerical Analysis, Friedrichs' assumptions, Advection, Applied Mathematics, Null (mathematics), polyhedral meshes, Vector advection-reaction problems, 010101 applied mathematics, Computational Mathematics, Computer Science::Graphics, Modeling and Simulation, A priori and a posteriori, AMS Subject Classification 65N12, 65N15, 65Zxx, 76Dxx, 76Wxx, [MATH.MATH-NA]Mathematics [math]/Numerical Analysis [math.NA], Analysis
Abstract

We devise and analyze an edge-based scheme on polyhedral meshes to approximate a vector advection-reaction problem. The well-posedness of the discrete problem is analyzed first under the classical positivity hypothesis of Friedrichs’ systems that requires a lower bound on the lowest eigenvalue of some tensor depending on the model parameters. We also prove stability when the lowest eigenvalue is null or even slightly negative if the mesh size is small enough.A priorierror estimates are established for solutions inW1,q(Ω) withq ∈ ((3/2),2]. Numerical results are presented on three-dimensional polyhedral meshes.

Published
2017

10. A DDFV method for a Cahn−Hilliard/Stokes phase field model with dynamic boundary conditions

Author

Flore Nabet, Franck Boyer, 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), 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), Centre de Mathématiques Appliquées - Ecole Polytechnique (CMAP), École polytechnique (X)-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), Laboratoire Paul Painlevé - UMR 8524 (LPP), and Centre National de la Recherche Scientifique (CNRS)-Université de Lille-Centre National de la Recherche Scientifique (CNRS)-Université de Lille-Inria Lille - Nord Europe

Subjects
Field (physics), Duality (optimization), finite volume method, 010103 numerical & computational mathematics, dynamic boundary conditions, 01 natural sciences, AMS subject classifications. 35K55, 65M08, 65M12, 76D07, 76M12, 76T10, [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], Polygon mesh, Boundary value problem, 0101 mathematics, Mathematics, Numerical Analysis, Finite volume method, Applied Mathematics, Numerical analysis, Mathematical analysis, Stokes flow, 010101 applied mathematics, Computational Mathematics, Cahn-Hilliard/Stokes model, Modeling and Simulation, Compressibility, contact angle dynamics, [MATH.MATH-NA]Mathematics [math]/Numerical Analysis [math.NA], Analysis
Abstract

In this paper we propose a “Discrete Duality Finite Volume” method (DDFV for short) for the diffuse interface modelling of incompressible two-phase flows. This numerical method is, conservative, robust and is able to handle general geometries and meshes. The model we study couples the Cahn−Hilliard equation and the unsteady Stokes equation and is endowed with particular nonlinear boundary conditions called dynamic boundary conditions. To implement the scheme for this model we have to derive new discrete consistent DDFV operators that allows an energy stable coupling between both discrete equations. We are thus able to obtain the existence of a family of solutions satisfying a suitable energy inequality, even in the case where a first order time-splitting method between the two subsystems is used. We illustrate various properties of such a model with some numerical results.

Published
2017

11. New transmission condition accounting for diffusion anisotropy in thin layers applied to diffusion MRI

Author

Jing-Rebecca Li, Houssem Haddar, Fabien Caubet, Dang Van Nguyen, 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), Shape reconstruction and identification (DeFI ), Inria Saclay - Ile de France, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria)-Centre de Mathématiques Appliquées - Ecole Polytechnique (CMAP), École polytechnique (X)-Centre National de la Recherche Scientifique (CNRS)-École polytechnique (X)-Centre National de la Recherche Scientifique (CNRS), 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), Centre de Mathématiques Appliquées - Ecole Polytechnique (CMAP), École polytechnique (X)-Centre National de la Recherche Scientifique (CNRS)-École polytechnique (X)-Centre National de la Recherche Scientifique (CNRS)-Inria Saclay - Ile de France, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria), 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), and Université Fédérale Toulouse Midi-Pyrénées-Centre National de la Recherche Scientifique (CNRS)

Subjects
Diffusion equation, Diffusion magnetic resonance imaging, Anomalous diffusion, Anisotropic diffusion, Asymptotic expansion, [MATH.MATH-CA]Mathematics [math]/Classical Analysis and ODEs [math.CA], 010103 numerical & computational mathematics, [INFO.INFO-DM]Computer Science [cs]/Discrete Mathematics [cs.DM], 01 natural sciences, Diffusion Anisotropy, Optics, [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], Effective diffusion coefficient, [MATH]Mathematics [math], 0101 mathematics, Diffusion (business), Mathematics, Numerical Analysis, business.industry, Bloch-Torrey equation, Applied Mathematics, Isotropy, Mathematical analysis, [INFO.INFO-NA]Computer Science [cs]/Numerical Analysis [cs.NA], 010101 applied mathematics, Computational Mathematics, [MATH.MATH-DG]Mathematics [math]/Differential Geometry [math.DG], Modeling and Simulation, Anisotropic diffusion transmission condition, business, [MATH.MATH-NA]Mathematics [math]/Numerical Analysis [math.NA], Analysis, Diffusion MRI
Abstract

ACL; International audience; The Bloch-Torrey Partial Differential Equation (PDE) can be used to model the diffusion Magnetic Resonance Imaging (dMRI) signal in biological tissue. In this paper, we derive an Anisotropic Diffusion Transmission Condition (ADTC) for the Bloch-Torrey PDE that accounts for anisotropic diffusion inside thin layers. Such diffusion occurs, for example, in the myelin sheath surrounding the axons of neurons. This ADTC can be interpreted as an asymptotic model of order two with respect to the layer thickness and accounts for water diffusion in the normal direction that is low compared to the tangential direction. We prove uniform stability of the asymptotic model with respect to the layer thickness and a mass conservation property. We also prove the theoretical quadratic accuracy of the ADTC. Finally, numerical tests validate these results and show that our model gives a better approximation of the dMRI signal than a simple transmission condition that assumes isotropic diffusion in the layers.

Published
2017

12. On nontraditional quasi-geostrophic equations

Author

Antoine Rousseau, Carine Lucas, James C. McWilliams, Mathématiques - Analyse, Probabilités, Modélisation - Orléans (MAPMO), Université d'Orléans (UO)-Centre National de la Recherche Scientifique (CNRS), Institute of Geophysics and Planetary Physics [Los Angeles] (IGPP), University of California [Los Angeles] (UCLA), University of California (UC)-University of California (UC), Littoral, Environment: MOdels and Numerics (LEMON), 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)-Institut Montpelliérain Alexander Grothendieck (IMAG), Université de Montpellier (UM)-Centre National de la Recherche Scientifique (CNRS)-Université de Montpellier (UM)-Centre National de la Recherche Scientifique (CNRS)-Hydrosciences Montpellier (HSM), Institut de Recherche pour le Développement (IRD)-Institut national des sciences de l'Univers (INSU - CNRS)-Université de Montpellier (UM)-Centre National de la Recherche Scientifique (CNRS)-Institut de Recherche pour le Développement (IRD)-Institut national des sciences de l'Univers (INSU - CNRS)-Centre National de la Recherche Scientifique (CNRS), Institut Montpelliérain Alexander Grothendieck (IMAG), Université de Montpellier (UM)-Centre National de la Recherche Scientifique (CNRS), ANR-11-MONU-0005,COMODO,Communauté de Modélisation Océanographique(2011), Centre National de la Recherche Scientifique (CNRS)-Université d'Orléans (UO), University of California-University of California, Littoral, Environnement : Méthodes et Outils Numériques (LEMON), and Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria)

Subjects
[PHYS.PHYS.PHYS-FLU-DYN]Physics [physics]/Physics [physics]/Fluid Dynamics [physics.flu-dyn], Work (thermodynamics), Quasi-geostrophic equations, 010504 meteorology & atmospheric sciences, Ocean modeling, Aspect ratio, 01 natural sciences, 010305 fluids & plasmas, Rossby number, 0103 physical sciences, [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], Applied mathematics, tilted quasi-geostrophic equations, Coriolis force, Shallow water equations, 0105 earth and related environmental sciences, Mathematics, traditional approximation, [PHYS.PHYS.PHYS-AO-PH]Physics [physics]/Physics [physics]/Atmospheric and Oceanic Physics [physics.ao-ph], Numerical Analysis, Applied Mathematics, [INFO.INFO-MO]Computer Science [cs]/Modeling and Simulation, slanted rotation, Computational Mathematics, Classical mechanics, Modeling and Simulation, Analysis
Abstract

In this article, we work on nontraditional models where the so-called traditional approximation on the Coriolis force is removed. In the derivation of the quasi-geostrophic equations, we carefully consider terms in δ/ε, where δ (aspect ratio) and ε (Rossby number) are both small numbers. We provide here some rigorous crossed-asymptotics with regards to these parameters, prove some mathematical results and compare QHQG and QG models.

Published
2017
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13. Energy stable and convergent finite element schemes for the modified phase field crystal equation

Author

Morgan Pierre, Maurizio Grasselli, Dipartimento di Matematica, Politecnico di Milano, Dipartimento di Matematica 'F. Brioschi', Politecnico di Milano [Milan] (POLIMI)-Politecnico di Milano [Milan] (POLIMI), Laboratoire de Mathématiques et Applications (LMA-Poitiers), and Université de Poitiers-Centre National de la Recherche Scientifique (CNRS)

Subjects
Discretization, 65P40, Finite elements, 010103 numerical & computational mathematics, Space (mathematics), 01 natural sciences, Piecewise linear function, second-order schemes, [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], Order (group theory), 0101 mathematics, Galerkin method, Mathematics, Numerical Analysis, 74N05, Lojasiewicz inequality MSC 2010: 65M60, Applied Mathematics, Mathematical analysis, Finite element method, 010101 applied mathematics, Computational Mathematics, gradient-like systems, Modeling and Simulation, Scheme (mathematics), [MATH.MATH-NA]Mathematics [math]/Numerical Analysis [math.NA], Analysis, Energy (signal processing), 82C26
Abstract

We propose a space semi-discrete and a fully discrete finite element scheme for the modified phase field crystal equation (MPFC). The space discretization is based on a splitting method and consists in a Galerkin approximation which allows low order (piecewise linear) finite elements. The time discretization is a second-order scheme which has been introduced by Gomez and Hughes for the Cahn-Hilliard equation. The fully discrete scheme is shown to be unconditionally energy stable and uniquely solvable for small time steps, with a smallness condition independent of the space step. Using energy estimates, we prove that in both cases, the discrete solution converges to the unique energy solution of the MPFC equation as the discretization parameters tend to 0. Using a Lojasiewicz inequality, we also establish that the discrete solution tends to a stationary solution as time goes to infinity. Numerical simulations illustrate the theoretical results.

Published
2016

14. Substructuring preconditioners forh−pMortar FEM

Author

Christophe Prud'Homme, Abdoulaye Samaké, Micol Pennacchio, Silvia Bertoluzza, Istituto di Matematica Applicata e Tecnologie Informatiche del C.N.R., Consiglio Nazionale delle Ricerche (CNR), Istituto di Matematica Applicata e Tecnologie Informatiche (IMATI), Institut de Recherche Mathématique Avancée (IRMA), Université de Strasbourg (UNISTRA)-Centre National de la Recherche Scientifique (CNRS), Equations aux Dérivées Partielles (EDP ), Laboratoire Jean Kuntzmann (LJK ), Institut polytechnique de Grenoble - Grenoble Institute of Technology (Grenoble INP )-Institut National de Recherche en Informatique et en Automatique (Inria)-Centre National de la Recherche Scientifique (CNRS)-Université Grenoble Alpes [2016-2019] (UGA [2016-2019])-Institut polytechnique de Grenoble - Grenoble Institute of Technology (Grenoble INP )-Institut National de Recherche en Informatique et en Automatique (Inria)-Centre National de la Recherche Scientifique (CNRS)-Université Grenoble Alpes [2016-2019] (UGA [2016-2019]), Feel++, Cemosis, ANR-10-COSI-0009,HAMM,Architecture Hybrides et Méthodes Multi-échelles(2010), National Research Council of Italy | Consiglio Nazionale delle Ricerche (CNR), and Centre National de la Recherche Scientifique (CNRS)-Université de Strasbourg (UNISTRA)

Subjects
Mathematical optimization, MathematicsofComputing_NUMERICALANALYSIS, 010103 numerical & computational mathematics, substructuring preconditioner, 01 natural sciences, Mathematics::Numerical Analysis, domain decomposition methods, Discontinuous Galerkin method, Component (UML), [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], Applied mathematics, Penalty method, 0101 mathematics, Condition number, Mathematics, Numerical Analysis, substructuring preconditioners, Preconditioner, Applied Mathematics, Domain decomposition methods, Computer Science::Numerical Analysis, h-p FEM, Finite element method, 010101 applied mathematics, Computational Mathematics, Modeling and Simulation, mortar, Mortar, [MATH.MATH-NA]Mathematics [math]/Numerical Analysis [math.NA], Analysis
Abstract

We build and analyze a substructuring preconditioner for the Mortar method, applied to elliptic problems, in the h-p finite element framework. Particular attention is given to the construction of the coarse component of the preconditioner in this framework, in which continuity at the cross points is not required. Two variants are proposed: the first one is an improved version of a coarse preconditioner already presented in [13]. The second is new and is built by using a Discontinuous Galerkin interior penalty method as coarse problem. A bound of the condition number is proven for both variants and their efficiency and scalability is illustrated by numerical experiments.

Published
2016

15. Analysis of the penalized 3D variable viscosity stokes equations coupled to diffusion and transport

Author

David Sanchez, Philippe Poncet, Robin Chatelin, Institut de Mathématiques de Toulouse UMR5219 (IMT), 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), 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), ANR-10-JCJC-0113,BioFiReaDy,Films biologiques et Dysfonctionnement Respiratoire(2010), Laboratoire de Tribologie et Dynamique des Systèmes (LTDS), École Centrale de Lyon (ECL), Université de Lyon-Université de Lyon-École Nationale des Travaux Publics de l'État (ENTPE)-Ecole Nationale d'Ingénieurs de Saint Etienne (ENISE)-Centre National de la Recherche Scientifique (CNRS), 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), and Université de Toulouse (UT)-Centre National de la Recherche Scientifique (CNRS)

Subjects
010103 numerical & computational mathematics, 01 natural sciences, Physics::Fluid Dynamics, symbols.namesake, Viscosity, Stokes' law, Fluid dynamics, [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], [PHYS.MECA.MEFL]Physics [physics]/Mechanics [physics]/Fluid mechanics [physics.class-ph], Uniqueness, MSC: 35Q30, 76D03, 76D07, 65M25, 68U20, 76Z05, 92B05, 0101 mathematics, porous media flows, Stokes number, Physics, variable viscosity flows, Numerical Analysis, Applied Mathematics, Mathematical analysis, Stokes equations, Fluid mechanics, Vorticity, 010101 applied mathematics, Computational Mathematics, Nonlinear system, Classical mechanics, moving geometry, Modeling and Simulation, symbols, biomathematics, [MATH.MATH-NA]Mathematics [math]/Numerical Analysis [math.NA], Analysis
Abstract

International audience; The analysis of the penalized Stokes problem, in its variable viscosity formulation, coupled to convection-diffusion equations is presented in this article. It models the interaction between a highly viscous fluid with variable viscosity and immersed moving and deformable obstacles. Indeed, while it is quite common to couple Poisson equations to diffusion-transport equations in plasma physics or fluid dynamics in vorticity formulations, the study of complex fluids requires to consider together the Stokes problem in complex moving geometry and convection-diffusion equations. The main result of this paper shows the existence and the uniqueness of the solution to this equations system with regularity estimates. Then we show that the solution to the penalized problem weakly converges toward the solution to the physical problem. Numerical simulations of fluid mechanics computations in this context are also presented in order to illustrate the practical aspects of such models: lung cells and their surrounding heterogeneous fluid, and porous media flows. Among the main original aspects in the present study, one can highlight the non linear dynamics induced by the coupling, and the tracking of the time-dependence of the domain.

Published
2016

16. Equivalent Boundary Conditions for an Elasto-Acoustic Problem set in a Domain with a Thin Layer

Author

Victor Péron, 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), Advanced 3D Numerical Modeling in Geophysics (Magique 3D), 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), and European Project: 295217,EC:FP7:PEOPLE,FP7-PEOPLE-2011-IRSES,HPC-GA(2012)

Subjects
Power series, Diffraction, 01 natural sciences, Mathematics - Analysis of PDEs, ACM: G.: Mathematics of Computing/G.1: NUMERICAL ANALYSIS/G.1.8: Partial Differential Equations, 35C20, 35J25, 41A60, 74F10, FOS: Mathematics, [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], Mathematics - Numerical Analysis, Boundary value problem, 0101 mathematics, Mathematics, Numerical Analysis, Applied Mathematics, 010102 general mathematics, Mathematical analysis, Numerical Analysis (math.NA), Acoustic wave, 010101 applied mathematics, Computational Mathematics, Wavelength, Transmission (telecommunications), Modeling and Simulation, Displacement (fluid), Layer (electronics), [MATH.MATH-NA]Mathematics [math]/Numerical Analysis [math.NA], Analysis, Analysis of PDEs (math.AP)
Abstract

International audience; We present equivalent conditions and asymptotic models for the diffraction problem of elastic and acoustic waves in a solid medium surrounded by a thin layer of fluid medium. Due to the thinness of the layer with respect to the wavelength, this problem is well suited for the notion of equivalent conditions and the effect of the fluid medium on the solid is as a first approximation local. We derive and validate equivalent conditions up to the fourth order for the elastic displacement. These conditions approximate the acoustic waves which propagate in the fluid region. This approach leads to solve only elastic equations. The construction of equivalent conditions is based on a multiscale expansion in power series of the thickness of the layer for the solution of the transmission problem.; On présente des conditions d'impédances ainsi que des modèles asymptotiques adaptés à un problème de transmission pour un couplage elasto-acoustique dans un milieu contenant une couche mince acoustique. Ce type de problème est bien posé pour la notion de conditions d'impédances car la faible épaisseur de la couche assure que l'effet acoustique sur la déformation élastique est en première approximation locale. On justifie rigoureusement ces conditions d'impédances en démontrant des estimations d'erreur. Cette méthode permet d'approcher la solution du problème exact par celle d'un problème d'élasticité avec une condition d'impédance. L'erreur d'approximation est contrôlée par rapport à l'épaisseur de la couche.

Published
2014

17. Parameter estimation in non-linear mixed effects models with SAEM algorithm: extension from ODE to PDE

Author

Violaine Louvet, Emmanuel Grenier, Paul Vigneaux, Unité de Mathématiques Pures et Appliquées (UMPA-ENSL), École normale supérieure de Lyon (ENS de Lyon)-Centre National de la Recherche Scientifique (CNRS), Numerical Medicine (NUMED), École normale supérieure de Lyon (ENS de Lyon)-Centre National de la Recherche Scientifique (CNRS)-École normale supérieure de Lyon (ENS de Lyon)-Centre National de la Recherche Scientifique (CNRS)-Inria Grenoble - Rhône-Alpes, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria), Institut Camille Jordan (ICJ), École Centrale de Lyon (ECL), Université de Lyon-Université de Lyon-Université Claude Bernard Lyon 1 (UCBL), Université de Lyon-Institut National des Sciences Appliquées de Lyon (INSA Lyon), Université de Lyon-Institut National des Sciences Appliquées (INSA)-Institut National des Sciences Appliquées (INSA)-Université Jean Monnet - Saint-Étienne (UJM)-Centre National de la Recherche Scientifique (CNRS), Modélisation mathématique, calcul scientifique (MMCS), Université de Lyon-Institut National des Sciences Appliquées (INSA)-Institut National des Sciences Appliquées (INSA)-Université Jean Monnet - Saint-Étienne (UJM)-Centre National de la Recherche Scientifique (CNRS)-École Centrale de Lyon (ECL), INRIA Project Lab MONICA, École normale supérieure - Lyon (ENS Lyon)-Centre National de la Recherche Scientifique (CNRS), École normale supérieure - Lyon (ENS Lyon)-Centre National de la Recherche Scientifique (CNRS)-École normale supérieure - Lyon (ENS Lyon)-Centre National de la Recherche Scientifique (CNRS)-Inria Grenoble - Rhône-Alpes, Institut Camille Jordan [Villeurbanne] (ICJ), Université de Lyon-Université Jean Monnet [Saint-Étienne] (UJM)-Institut National des Sciences Appliquées de Lyon (INSA Lyon), Université de Lyon-Institut National des Sciences Appliquées (INSA)-Institut National des Sciences Appliquées (INSA)-Centre National de la Recherche Scientifique (CNRS), and Université de Lyon-Institut National des Sciences Appliquées (INSA)-Institut National des Sciences Appliquées (INSA)-Centre National de la Recherche Scientifique (CNRS)-École Centrale de Lyon (ECL)

Subjects
Computation, MSC: 35Q62, 35Q92, 35R30, 65C40, 35K57, Population, SAEM algorithm, KPP equation, [MATH.MATH-ST]Mathematics [math]/Statistics [math.ST], Parameter estimation, [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], education, Mathematics, Numerical Analysis, education.field_of_study, Partial differential equation, Estimation theory, Applied Mathematics, Ode, [STAT.TH]Statistics [stat]/Statistics Theory [stat.TH], Partial differential equations, Computational Mathematics, Nonlinear system, Modeling and Simulation, Ordinary differential equation, Parametrization, Algorithm, [MATH.MATH-NA]Mathematics [math]/Numerical Analysis [math.NA], Analysis
Abstract

In this HAL v2: correction of a bibtex bug for the now duly referenced book [20] and special issue [9]. Added link to sample code (matlab - monolix); International audience; Parameter estimation in non linear mixed effects models requires a large number of evaluations of the model to study. For ordinary differential equations, the overall computation time remains reasonable. However when the model itself is complex (for instance when it is a set of partial differential equations) it may be time consuming to evaluate it for a single set of parameters. The procedures of population parametrization (for instance using SAEM algorithms) are then very long and in some cases impossible to do within a reasonable time. We propose here a very simple methodology which may accelerate population parametrization of complex models, including partial differential equations models. We illustrate our method on the classical KPP equation.

Published
2014

18. Convergence of mass redistribution method for the one-dimensional wave equation with a unilateral constraint at the boundary

Author

Adrien Petrov, Jérôme Pousin, Farshid Dabaghi, Yves Renard, Modélisation mathématique, calcul scientifique (MMCS), Institut Camille Jordan [Villeurbanne] (ICJ), École Centrale de Lyon (ECL), Université de Lyon-Université de Lyon-Université Claude Bernard Lyon 1 (UCBL), Université de Lyon-Université Jean Monnet [Saint-Étienne] (UJM)-Institut National des Sciences Appliquées de Lyon (INSA Lyon), Université de Lyon-Institut National des Sciences Appliquées (INSA)-Institut National des Sciences Appliquées (INSA)-Centre National de la Recherche Scientifique (CNRS)-École Centrale de Lyon (ECL), Université de Lyon-Institut National des Sciences Appliquées (INSA)-Institut National des Sciences Appliquées (INSA)-Centre National de la Recherche Scientifique (CNRS), Institut National des Sciences Appliquées de Lyon (INSA Lyon), Université de Lyon-Institut National des Sciences Appliquées (INSA), Laboratoire de Mécanique des Contacts et des Structures [Villeurbanne] (LaMCoS), and Université de Lyon-Institut National des Sciences Appliquées (INSA)-Université de Lyon-Institut National des Sciences Appliquées (INSA)-Centre National de la Recherche Scientifique (CNRS)

Subjects
Numerical Analysis, Applied Mathematics, media_common.quotation_subject, Mathematical analysis, Unilateral contact, Wave equation, Inertia, Computational Mathematics, Modeling and Simulation, Variational inequality, [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], Redistribution (chemistry), Boundary value problem, Uniqueness, ComputingMilieux_MISCELLANEOUS, Analysis, Mathematics, media_common
Abstract

This paper focuses on a one-dimensional wave equation being subjected to a unilateral boundary condition. Under appropriate regularity assumptions on the initial data, a new proof of existence and uniqueness results is proposed. The mass redistribution method, which is based on a redistribution of the body mass such that there is no inertia at the contact node, is introduced and its convergence is proved. Finally, some numerical experiments are reported.

Published
2014

19. A hyperbolic model of chemotaxis on a network: a numerical study

Author

Gabriella Bretti, Magali Ribot, Roberto Natalini, Istituto per le Applicazioni del Calcolo 'Mauro Picone' (IAC), Consiglio Nazionale delle Ricerche [Roma] (CNR), 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), This work has also been partially supported by the PRIN project 2008-2009 'Equazioni iperboliche non lineari e fluidodinamica', ANR-10-JCJC-0103,MONUMENTALG,MOdélisation mathématique et simulations NUMériques pour la dégradation biologique des MONUMENTs et pour la prolifération des ALGues.(2010), European Project: 257462,EC:FP7:ICT,FP7-ICT-2009-5,HYCON2(2010), National Research Council of Italy | Consiglio Nazionale delle Ricerche (CNR), 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
Quantitative Biology::Tissues and Organs, Space dimension, transmission conditions, Stability (probability), finite difference schemes, Quantitative Biology::Cell Behavior, Mathematics - Analysis of PDEs, hyperbolic system on network, FOS: Mathematics, [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], Applied mathematics, asymptotic behavior, Mathematics - Numerical Analysis, Numerical tests, chemotaxis, initial-boundary value problem, Mathematics, Numerical Analysis, Applied Mathematics, 65M06, 35L50, 92B05, 92C17, 92C42, Chemotaxis, Numerical Analysis (math.NA), Term (time), Computational Mathematics, Transmission (telecommunications), Modeling and Simulation, Scheme (mathematics), [MATH.MATH-NA]Mathematics [math]/Numerical Analysis [math.NA], Analysis, Analysis of PDEs (math.AP)
Abstract

25 pages; International audience; In this paper we deal with a semilinear hyperbolic chemotaxis model in one space dimension evolving on a network, with suitable transmission conditions at nodes. This framework is motivated by tissue-engineering scaffolds used for improving wound healing. We introduce a numerical scheme, which guarantees global mass densities conservation. Moreover our scheme is able to yield a correct approximation of the effects of the source term at equilibrium. Several numerical tests are presented to show the behavior of solutions and to discuss the stability and the accuracy of our approximation.

Published
2014

20. Modelling and simulation of liquid-vapor phase transition in compressible flows based on thermodynamical equilibrium

Author

Grégoire Allaire, Gloria Faccanoni, Samuel Kokh, Institut de Mathématiques de Toulon - EA 2134 (IMATH), Université de Toulon (UTLN), Laboratoire d'Etudes Thermiques des Réacteurs (LETR), 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, Centre de Mathématiques Appliquées - Ecole Polytechnique (CMAP), École polytechnique (X)-Centre National de la Recherche Scientifique (CNRS), This work has been achieved within the framework of the NEPTUNE project, financially supported by CEA (Commissariat à l'Énergie Atomique), EDF (Électricité de France), IRSN (Institut de Radioprotection et de Sûreté Nucléaire) and AREVA-NP. The authors thank the CEA for its financial support. Grégoire Allaire is a member of the DEFI project at INRIA Saclay Ile-de-France and is partially supported by the Chair 'Mathematical modeling and numerical simulation', and F-EADS - École Polytechnique - INRIA.

Subjects
Phase transition, Equation of state, Work (thermodynamics), Discretization, Relaxation Method, 01 natural sciences, Compressible flow, 010305 fluids & plasmas, Hyperbolic systems, Phase Change, Compressible Flows, Phase (matter), 0103 physical sciences, [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], Statistical physics, 0101 mathematics, Physics, Numerical Analysis, Computer simulation, Two-Phase Flows, Applied Mathematics, 010101 applied mathematics, Computational Mathematics, 76T10, 76N10, 65M08, Modeling and Simulation, Compressibility, [MATH.MATH-NA]Mathematics [math]/Numerical Analysis [math.NA], Analysis
Abstract

International audience; In the present work we investigate the numerical simulation of liquid-vapor phase change in compressible flows. Each phase is modeled as a compressible fluid equipped with its own Equation of State (EOS). We suppose that inter-phase equilibrium processes in the medium operate at a short time-scale compared to the other physical phenomena such as convection or thermal diffusion. This assumption provides an implicit definition of an equilibrium EOS for the two-phase medium. Within this framework, mass transfer is the result of local and instantaneous equilibria between both phases. The overall model is strictly hyperbolic. We examine properties of the equilibrium EOS and we propose a discretization strategy based on a Finite-Volume relaxation method. This method allows to cope with the implicit definition of the equilibrium EOS, even when the model involves complex EOS's for the pure phases. We present two-dimensional numerical simulations that shows that the model is able to reproduce mechanism such as phase disappearance and nucleation.

Published
2012

21. Mathematical and numerical analysis of a model for anti-angiogenic therapy in metastatic cancers

Author

Sébastien Benzekry, Laboratoire d'Analyse, Topologie, Probabilités (LATP), Aix Marseille Université (AMU)-École Centrale de Marseille (ECM)-Centre National de la Recherche Scientifique (CNRS), Infections Parasitaires : Transmission, Physiopathologie et Thérapeutiques (IP-TPT), Service de Santé des Armées-Assistance Publique - Hôpitaux de Marseille (APHM)-Aix Marseille Université (AMU)-Institut de Recherche pour le Développement (IRD), ANR-09-BLAN-0217,MEMOREX-PK(2009), and Institut de Recherche pour le Développement (IRD)-Aix Marseille Université (AMU)-Assistance Publique - Hôpitaux de Marseille (APHM)-Service de Santé des Armées

Subjects
Mathematical optimization, Lagrangian scheme, Quantitative Biology::Tissues and Organs, Nonlocal boundary, [SDV.CAN]Life Sciences [q-bio]/Cancer, Quantitative Biology::Cell Behavior, 03 medical and health sciences, symbols.namesake, Mathematics - Analysis of PDEs, 0302 clinical medicine, Phenomenological model, Convergence (routing), FOS: Mathematics, [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], Applied mathematics, 030304 developmental biology, Mathematics, 35F16, 65M25, 92C50, 0303 health sciences, Numerical Analysis, Applied Mathematics, Numerical analysis, Anti angiogenic, Anticancer therapy modelling, 3. Good health, Computational Mathematics, Structured population dynamics, 030220 oncology & carcinogenesis, Modeling and Simulation, symbols, Angiogenesis, Convection–diffusion equation, Analysis, Lagrangian, Analysis of PDEs (math.AP)
Abstract

We introduce and analyze a phenomenological model for anti-angiogenic therapy in the treatment of metastatic cancers. It is a structured transport equation with a nonlocal boundary condition describing the evolution of the density of metastasis that we analyze first at the continuous level. We present the numerical analysis of a lagrangian scheme based on the characteristics whose convergence establishes existence of solutions. Then we prove an error estimate and use the model to perform interesting simulations in view of clinical applications., Comment: Nombre de pages : 29

Published
2011

22. Parareal operator splitting techniques for multi-scale reaction waves: Numerical analysis and strategies

Author

Stéphane Descombes, Max Duarte, Marc Massot, Laboratoire d'Énergétique Moléculaire et Macroscopique, Combustion (EM2C), Université Paris Saclay (COmUE)-Centre National de la Recherche Scientifique (CNRS)-CentraleSupélec, Laboratoire Jean Alexandre Dieudonné (JAD), 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), This research was supported by a fundamental CNRS Ph.D. grant for M. Duarte from the Institute of Mathematics (INSMI) and Engineering Institute (INSIS) of CNRS. This work was granted access to the HPC resources of IDRIS (CNRS Institute of Scientific Computing) under the allocation 2009-i2009066173 made by GENCI (Grand Equipement National de Calcul Intensif)., and ANR-06-CIS6-0007,PITAC,Parallélisation Incluant le Temps pour Accélérer les Calculs(2006)

Subjects
Scale (ratio), Parareal, Context (language use), 010103 numerical & computational mathematics, 01 natural sciences, Parareal algorithm, Multi-scale waves, Convergence analysis, Convergence (routing), Reaction–diffusion system, [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], Applied mathematics, Operator splitting, 0101 mathematics, Mathematics, Numerical Analysis, Series (mathematics), [SPI.FLUID]Engineering Sciences [physics]/Reactive fluid environment, Applied Mathematics, Numerical analysis, Mathematical analysis, AMS 65Y05, 65M12, 65L04, 35A35, 35K57, 35C07, 010101 applied mathematics, Computational Mathematics, Nonlinear system, Modeling and Simulation, Reaction- diffusion, [MATH.MATH-NA]Mathematics [math]/Numerical Analysis [math.NA], Analysis
Abstract

International audience; In this paper, we investigate the coupling between operator splitting techniques and a time parallelization scheme, the parareal algorithm, as a numerical strategy for the simulation of reaction-diffusion equations modeling multi-scale reaction waves. This type of problems induces peculiar difficulties and potentially large stiffness which stem from the broad spectrum of temporal scales in the nonlinear chemical source term as well as from the presence of large spatial gradients in the reactive fronts, spatially very localized. In a series of previous studies, the numerical analysis of the operator splitting as well as the parareal algorithm has been conducted and such approaches have shown a great potential in the framework of reaction-diffusion and convection-diffusion-reaction systems. However, complementary studies are needed for a more complete characterization of such techniques for these stiff configurations. Therefore, we conduct in this work a precise numerical analysis that considers the combination of time operator splitting and the parareal algorithm in the context of stiff reaction fronts. The impact of the stiffness featured by these fronts on the convergence of the method is thus quantified, and allows to conclude on an optimal strategy for the resolution of such problems. We finally perform some numerical simulations in the field of nonlinear chemical dynamics that validate the theoretical estimates and examine the performance of such strategies in the context of academical one-dimensional test cases as well as multi-dimensional configurations simulated on parallel architecture.

Published
2011

23. Stochastic Lagrangian method for downscaling problems in computational fluid dynamics

Author

Frédéric Bernardin, Mireille Bossy, Antoine Rousseau, Claire Chauvin, Jean-François Jabir, Centre d'études techniques de l'équipement de Lyon (CETE de Lyon), Avant création Cerema, TOSCA, Centre National de la Recherche Scientifique (CNRS)-Institut National Polytechnique de Lorraine (INPL)-Université Nancy 2-Université Henri Poincaré - Nancy 1 (UHP)-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)-INRIA Lorraine, Institut National de Recherche en Informatique et en Automatique (Inria), Modelling, Observations, Identification for Environmental Sciences (MOISE), 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), Université Pierre Mendès France - Grenoble 2 (UPMF)-Université Joseph Fourier - Grenoble 1 (UJF)-Institut polytechnique de Grenoble - Grenoble Institute of Technology (Grenoble INP )-Centre National de la Recherche Scientifique (CNRS)-Université Pierre Mendès France - Grenoble 2 (UPMF)-Université Joseph Fourier - Grenoble 1 (UJF)-Institut polytechnique de Grenoble - Grenoble Institute of Technology (Grenoble INP )-Centre National de la Recherche Scientifique (CNRS), Centre de modélisation mathématique (CMM), Universitad de Chile-Centre National de la Recherche Scientifique (CNRS), INRIA Lorraine, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria)-Inria Sophia Antipolis - Méditerranée (CRISAM), and Institut National de Recherche en Informatique et en Automatique (Inria)-Université Henri Poincaré - Nancy 1 (UHP)-Université Nancy 2-Institut National Polytechnique de Lorraine (INPL)-Centre National de la Recherche Scientifique (CNRS)

Subjects
Discretization, Differential equation, fluid dynamics, 010103 numerical & computational mathematics, Computational fluid dynamics, 01 natural sciences, Stochastic differential equation, Fluid dynamics, [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], Applied mathematics, 0101 mathematics, Mathematics, Langevin models, MSC: 65C20, 65C35, 68U20, 35Q30, Numerical Analysis, business.industry, Applied Mathematics, Numerical analysis, 010102 general mathematics, Mathematical analysis, downscaling methods, [MATH.MATH-PR]Mathematics [math]/Probability [math.PR], particle methods, Computational Mathematics, Modeling and Simulation, PDF methods, Reynolds-averaged Navier–Stokes equations, business, Analysis, Downscaling
Abstract

International audience; This work aims at introducing modelization, theoretical and numerical studies related to a new downscaling technique applied to Computational Fluid Dynamics. Our method consists in building a local model, forced by large scale information computed thanks to a classical numerical weather predictor. The local model, compatible with the Navier-Stokes equations, is used for the small scale computation (downscaling) of the considered fluid. It is inspired by S.B. Pope's works on turbulence, and consists in a so-called Langevin system of stochastic differential equations. We introduce this model and exhibit its links with classical RANS models. Well-posedness, as well as mean-field interacting particle approximations and boundary condition issues are addressed. We present the numerical discretization of the stochastic downscaling method and investigate the accuracy of the proposed algorithm on simplified situations.

Published
2010

24. Asymptotic models for scattering from unbounded media with high conductivity

Author

Armin Lechleiter, Houssem Haddar, Shape reconstruction and identification (DeFI ), Inria Saclay - Ile de France, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria)-Centre de Mathématiques Appliquées - Ecole Polytechnique (CMAP), École polytechnique (X)-Centre National de la Recherche Scientifique (CNRS)-École polytechnique (X)-Centre National de la Recherche Scientifique (CNRS), Centre de Mathématiques Appliquées - Ecole Polytechnique (CMAP), École polytechnique (X)-Centre National de la Recherche Scientifique (CNRS)-École polytechnique (X)-Centre National de la Recherche Scientifique (CNRS)-Inria Saclay - Ile de France, and Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria)

Subjects
Well-posed problem, Numerical Analysis, Work (thermodynamics), Partial differential equation, Scattering, Applied Mathematics, Numerical analysis, Mathematical analysis, 010103 numerical & computational mathematics, 01 natural sciences, 010101 applied mathematics, Computational Mathematics, Compact space, Modeling and Simulation, [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], Boundary value problem, 0101 mathematics, Electrical impedance, ComputingMilieux_MISCELLANEOUS, Analysis, Mathematics
Abstract

We analyze the accuracy and well-posedness of generalized impedance boundary value problems in the framework of scattering problems from unbounded highly absorbing media. We restrict ourselves in this first work to the scalar problem (E-mode for electromagnetic scattering problems). Compared to earlier works, the unboundedness of the rough absorbing layer introduces severe difficulties in the analysis for the generalized impedance boundary conditions, since classical compactness arguments are no longer possible. Our new analysis is based on the use of Rellich-type estimates and boundedness of L2 solution operators. We also discuss some numerical experiments concerning these boundary conditions.

Published
2010

25. Analysis of crack singularities in an aging elastic material

Author

Monique Dauge, Martin Costabel, Jan Sokołowski, Sergei A. Nazarov, Institut de Recherche Mathématique de Rennes ( IRMAR ), Université de Rennes 1 ( UR1 ), Université de Rennes ( UNIV-RENNES ) -Université de Rennes ( UNIV-RENNES ) -AGROCAMPUS OUEST-École normale supérieure - Rennes ( ENS Rennes ) -Institut National de Recherche en Informatique et en Automatique ( Inria ) -Institut National des Sciences Appliquées ( INSA ) -Université de Rennes 2 ( UR2 ), Université de Rennes ( UNIV-RENNES ) -Centre National de la Recherche Scientifique ( CNRS ), Institute of Mechanical Engineering Problems [St. Petersburg] ( IPME ), Russian Academy of Sciences [Moscow] ( RAS ), Institut Élie Cartan de Nancy ( IECN ), Institut National de Recherche en Informatique et en Automatique ( Inria ) -Université Henri Poincaré - Nancy 1 ( UHP ) -Université Nancy 2-Institut National Polytechnique de Lorraine ( INPL ) -Centre National de la Recherche Scientifique ( CNRS ), Institut de Recherche Mathématique de Rennes (IRMAR), Université de Rennes (UR)-Institut National des Sciences Appliquées - Rennes (INSA Rennes), Institut National des Sciences Appliquées (INSA)-Institut National des Sciences Appliquées (INSA)-École normale supérieure - Rennes (ENS Rennes)-Université de Rennes 2 (UR2)-Centre National de la Recherche Scientifique (CNRS)-INSTITUT AGRO Agrocampus Ouest, Institut national d'enseignement supérieur pour l'agriculture, l'alimentation et l'environnement (Institut Agro)-Institut national d'enseignement supérieur pour l'agriculture, l'alimentation et l'environnement (Institut Agro), Institute of Mechanical Engineering Problems [St. Petersburg] (IPME), Russian Academy of Sciences [Moscow] (RAS), Institut Élie Cartan de Nancy (IECN), Institut National de Recherche en Informatique et en Automatique (Inria)-Université Henri Poincaré - Nancy 1 (UHP)-Université Nancy 2-Institut National Polytechnique de Lorraine (INPL)-Centre National de la Recherche Scientifique (CNRS), AGROCAMPUS OUEST, Institut national d'enseignement supérieur pour l'agriculture, l'alimentation et l'environnement (Institut Agro)-Institut national d'enseignement supérieur pour l'agriculture, l'alimentation et l'environnement (Institut Agro)-Université de Rennes 1 (UR1), Université de Rennes (UNIV-RENNES)-Université de Rennes (UNIV-RENNES)-Université de Rennes 2 (UR2), Université de Rennes (UNIV-RENNES)-École normale supérieure - Rennes (ENS Rennes)-Centre National de la Recherche Scientifique (CNRS)-Institut National des Sciences Appliquées - Rennes (INSA Rennes), Institut National des Sciences Appliquées (INSA)-Université de Rennes (UNIV-RENNES)-Institut National des Sciences Appliquées (INSA), and Centre National de la Recherche Scientifique (CNRS)-Institut National Polytechnique de Lorraine (INPL)-Université Nancy 2-Université Henri Poincaré - Nancy 1 (UHP)-Institut National de Recherche en Informatique et en Automatique (Inria)

Subjects
Homogeneous function, [ SPI.MECA.STRU ] Engineering Sciences [physics]/Mechanics [physics.med-ph]/Mechanics of the structures [physics.class-ph], [ PHYS.MECA.STRU ] Physics [physics]/Mechanics [physics]/Mechanics of the structures [physics.class-ph], 01 natural sciences, Displacement (vector), creep theory, symbols.namesake, [PHYS.MECA.STRU]Physics [physics]/Mechanics [physics]/Structural mechanics [physics.class-ph], [ MATH.MATH-AP ] Mathematics [math]/Analysis of PDEs [math.AP], Crack singularities, Singular function, Simple (abstract algebra), [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], hereditarily-elastic, 0101 mathematics, Mathematics, Polynomial (hyperelastic model), Numerical Analysis, Volterra kernel, Applied Mathematics, 010102 general mathematics, Mathematical analysis, 35Q72, 74D05, 74G70, 010101 applied mathematics, Computational Mathematics, [SPI.MECA.STRU]Engineering Sciences [physics]/Mechanics [physics.med-ph]/Structural mechanics [physics.class-ph], Modeling and Simulation, Displacement field, symbols, Gravitational singularity, Analysis, Quasistatic process
Abstract

Version 10.03.2005; International audience; We consider a quasistatic system involving a Volterra kernel modelling an hereditarily-elastic aging body. We are concerned with the behavior of displacement and stress fields in the neighborhood of cracks. In this paper, we investigate the case of a straight crack in a two-dimensional domain with a possibly anisotropic material law. We study the asymptotics of the time dependent solution near the crack tips. We prove that, depending on the regularity of the material law and the Volterra kernel, these asymptotics contain singular functions which are simple homogeneous functions of degree \frac12 or have a more complicated dependence on the distance variable r to the crack tips. In the latter situation, we observe a novel behavior of the singular functions, incompatible with the usual fracture criteria, involving super polynomial functions of \ln(r) growing in time.

Published
2006

26. Modelling of Miscible Liquids with the Korteweg Stress

Author

Rozenn Texier-Picard, Ilya Kostin, Martine Marion, Vitaly Volpert, Laboratoire de Mathématiques de Besançon (UMR 6623) (LMB), Centre National de la Recherche Scientifique (CNRS)-Université de Franche-Comté (UFC), Université Bourgogne Franche-Comté [COMUE] (UBFC)-Université Bourgogne Franche-Comté [COMUE] (UBFC), Modélisation mathématique, calcul scientifique (MMCS), Institut Camille Jordan (ICJ), École Centrale de Lyon (ECL), Université de Lyon-Université de Lyon-Université Claude Bernard Lyon 1 (UCBL), Université de Lyon-Institut National des Sciences Appliquées de Lyon (INSA Lyon), Université de Lyon-Institut National des Sciences Appliquées (INSA)-Institut National des Sciences Appliquées (INSA)-Université Jean Monnet - Saint-Étienne (UJM)-Centre National de la Recherche Scientifique (CNRS)-École Centrale de Lyon (ECL), Université de Lyon-Institut National des Sciences Appliquées (INSA)-Institut National des Sciences Appliquées (INSA)-Université Jean Monnet - Saint-Étienne (UJM)-Centre National de la Recherche Scientifique (CNRS), Institut de Recherche Mathématique de Rennes (IRMAR), Université de Rennes (UR)-Institut National des Sciences Appliquées - Rennes (INSA Rennes), Institut National des Sciences Appliquées (INSA)-Institut National des Sciences Appliquées (INSA)-École normale supérieure - Rennes (ENS Rennes)-Université de Rennes 2 (UR2)-Centre National de la Recherche Scientifique (CNRS)-Institut Agro Rennes Angers, Institut national d'enseignement supérieur pour l'agriculture, l'alimentation et l'environnement (Institut Agro)-Institut national d'enseignement supérieur pour l'agriculture, l'alimentation et l'environnement (Institut Agro), Centre de recherches en économie appliquée au développement (CREAD), Laboratoire de Mathématiques Appliquées de Lyon (MAPLY), and Université de Lyon-Institut National des Sciences Appliquées (INSA)-Institut National des Sciences Appliquées (INSA)-Centre National de la Recherche Scientifique (CNRS)

Subjects
Convection, Numerical Analysis, Diffusion equation, Partial differential equation, Applied Mathematics, Thermodynamics, Mechanics, Physics::Fluid Dynamics, Computational Mathematics, [MATH.MATH-MP]Mathematics [math]/Mathematical Physics [math-ph], Modeling and Simulation, Fluid dynamics, [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], Initial value problem, [PHYS.MECA.MEFL]Physics [physics]/Mechanics [physics]/Fluid mechanics [physics.class-ph], Boundary value problem, Uniqueness, Diffusion (business), MSC: 35K50, 76D05, Analysis, Mathematics
Abstract

International audience; When two miscible fluids, such as glycerol (glycerin) and water, are brought in contact, they immediately diffuse in each other. However if the diffusion is sufficiently slow, large concentration gradients exist during some time. They can lead to the appearance of an “effective interfacial tension”. To study these phenomena we use the mathematical model consisting of the diffusion equation with convective terms and of the Navier-Stokes equations with the Korteweg stress. We prove the global existence and uniqueness of the solution for the associated initial-boundary value problem in a two-dimensional bounded domain. We study the longtime behavior of the solution and show that it converges to the uniform composition distribution with zero velocity field. We also present numerical simulations of miscible drops and show how transient interfacial phenomena can change their shape.

Published
2003

27. A non elliptic spectral problem related to the analysis of superconducting micro-strip lines

Author

Anne-Sophie Bonnet-Bendhia, Karim Ramdani, Propagation des Ondes : Étude Mathématique et Simulation (POEMS), Inria Saclay - Ile de France, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria)-Unité de Mathématiques Appliquées (UMA), and École Nationale Supérieure de Techniques Avancées (ENSTA Paris)-École Nationale Supérieure de Techniques Avancées (ENSTA Paris)-Centre National de la Recherche Scientifique (CNRS)

Subjects
Permittivity, Numerical Analysis, Applied Mathematics, Operator (physics), Mathematical analysis, Continuous spectrum, 020206 networking & telecommunications, 02 engineering and technology, Mathematics::Spectral Theory, Computer Science::Digital Libraries, 01 natural sciences, 010101 applied mathematics, Computational Mathematics, Elliptic operator, Modeling and Simulation, Spectrum of a matrix, 0202 electrical engineering, electronic engineering, information engineering, [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], Boundary value problem, 0101 mathematics, Analysis, Eigenvalues and eigenvectors, Self-adjoint operator, Mathematics
Abstract

International audience; This paper is devoted to the spectral analysis of a non elliptic operator A , deriving from the study of superconducting micro-strip lines. Once a sufficient condition for the self-adjointness of operator A has been derived, we determine its continuous spectrum. Then, we show that A is unbounded from below and that it has a sequence of negative eigenvalues tending to -∞. Using the Min-Max principle, a characterization of its positive eigenvalues is given. Thanks to this characterization, some conditions on the geometrical (large width) and physical (large dielectric permittivity in modulus) properties of the strip that ensure the existence of positive eigenvalues are derived. Finally, we analyze the asymptotic behavior of the eigenvalues of A as the dielectric permittivity of the strip goes to -∞.

Published
2002

28. Propagation of elastic surface waves along a cylindrical cavity of arbitrary cross section

Author

Michel Kern, A. Bamberger, Patrick Joly, INRIA Rocquencourt, Institut National de Recherche en Informatique et en Automatique (Inria), IFP Energies nouvelles (IFPEN), Dynamic graphs and the web graph (IDENT), Inria Paris-Rocquencourt, and Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria)

Subjects
guided modes, Cylindrical cavity, Onde guidée, 01 natural sciences, [PHYS.MECA.MEMA]Physics [physics]/Mechanics [physics]/Mechanics of materials [physics.class-ph], Cavité cylindrique, Cross section (physics), Onde élastique, [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], Guided wave, Wavenumber, Cylinder, 0101 mathematics, Onde surface, Eigenvalues and eigenvectors, Mathematics, unbounded selfadjoint operator, Numerical Analysis, Wave propagation, Applied Mathematics, Numerical analysis, Operator (physics), 010102 general mathematics, Mathematical analysis, 16. Peace & justice, surface waves, Surface wave, Elastic wave, 010101 applied mathematics, Computational Mathematics, [INFO.INFO-IR]Computer Science [cs]/Information Retrieval [cs.IR], Modeling and Simulation, Free surface, eigenvalue problem, Propagation onde, [MATH.MATH-NA]Mathematics [math]/Numerical Analysis [math.NA], Analysis, [MATH.MATH-SP]Mathematics [math]/Spectral Theory [math.SP]
Abstract

International audience; The propagation of elastic surface waves, guided by the free surface of an infinitely long cylinder of arbitrary cross section, is formulated as an eigenvalue problem for an unbounded self-adjoint operator. We prove the existence of a hierarchy of guided modes. Two of them propagate for any value of the wave number, whereas all of the others only exist beyond a cut-off wave number. For any fixed value of the wave number, only a finite number of modes propagate.

Published
1991

29. Finite element approximation of elliptic homogenization problems in nondivergence-form

Author

Timo Sprekeler, Endre Süli, Yves Capdeboscq, Mathematical Institute [Oxford] (MI), University of Oxford [Oxford], Laboratoire Jacques-Louis Lions (LJLL (UMR_7598)), and Université Paris Diderot - Paris 7 (UPD7)-Sorbonne Université (SU)-Centre National de la Recherche Scientifique (CNRS)

Subjects
Mathematics::Analysis of PDEs, Finite element approximations, 010103 numerical & computational mathematics, 01 natural sciences, Homogenization (chemistry), symbols.namesake, FOS: Mathematics, Finite Elements Methods, [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], Mathematics - Numerical Analysis, 35B27, 35J15, 65N12, 65N30, 0101 mathematics, Mathematics, Numerical Analysis, Homogenization, nondivergence-form elliptic PDE, Applied Mathematics, Mathematical analysis, Numerical Analysis (math.NA), Finite element method, 010101 applied mathematics, Computational Mathematics, Homogeneous, Modeling and Simulation, Dirichlet boundary condition, symbols, Analysis, 35B27,35J15, 65N12, 65N30
Abstract

We use uniform $W^{2,p}$ estimates to obtain corrector results for periodic homogenization problems of the form $A(x/\varepsilon):D^2 u_{\varepsilon} = f$ subject to a homogeneous Dirichlet boundary condition. We propose and rigorously analyze a numerical scheme based on finite element approximations for such nondivergence-form homogenization problems. The second part of the paper focuses on the approximation of the corrector and numerical homogenization for the case of nonuniformly oscillating coefficients. Numerical experiments demonstrate the performance of the scheme., Comment: 39 pages

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