Laboratoire de mathématiques, Université Paris-Sud

Professeur Chargé de Cours à l'Ecole Polytechnique

Membre du DMA, enseignant à mi-temps à l'ENS de la rue l'Ulm

Membre du bureau d'AMIES

Laboratoire de Mathématiques, Université Paris Sud

91405 Orsay Cedex

01 69 15 74 91

Bertrand.Maury @ math.u-psud.fr

Publications

Research interests

Particulate flows

Fluid-particle solvers

Handling of contacts (dry and lubricated)

Modeling of the respiratory system

Modelling of crowd motions

Optimization under constraint

Particulate flows

Direct simulation of fluid particle flows

To simulate the motion of rigid particles in a viscous fluid, we firstly used the so-called Arbitrary Lagrangian Eulerian approach, which relies on an unstructured mesh that follows the motion of the particles (see [6], 1999)). This method relies on an unstructured mesh which locally fits the boundary of the particles (click on the mesh to see an animation of a moving periodic mesh in the case of sedimentating particles):

The mesh moves smoothly as the time goes on, until its quality gets too low, making it necessary to rebuild a new mesh, onto which the current velocity field is to be projected. We developped a Finite Element code, which handles biperiodic boundary conditions. This code is currently used by Julien Couder, at IPGP, to model sedimentation flows. It is thoroughly described in [6]. Note that contacts are handled in this approach by a very rough method, which cannot be properly justified from the modelling standpoint, but which makes computations quite robust. See below some remarks on better (or, at least, more justified) ways to handle those contacts.

With a former PhD student, Mourad Ismail, we proposed and developped a completely different approach to handle rigid particles in a fluid domain, namely the Fat Boundary Method. In the spirit of the Fictitious Domain Method, as developped by Roland Glowinski , this method makes it possible to use a cartesian mesh, on which fast solvers can be used. Yet, the way to take the particles into account is quite different, as it is based on a domain decomposition approach with overlapping: the domain is decomposed onto a large (fictitious) domain which covers the mixture, and a set of narrow domains surrounding the particles. See [9] for a presentation of this method for the Poisson problem, and [14] or Mourad Ismail's homepage for applications to Navier-Stokes flows. It has not been implemented with high order Finite Element, but theoretical analysis (joint work with Silvia Bertoluzza, see [14]) suggests an optimal convergence rate, which is not common for FE approximation on a mesh which does not respect the boundary of the domain.

In order to provide a simple tool to compute fluid-particle flows, we (with Joāo Janela and Aline Lefebvre, a student at Orsay) applied the penalty method to this type of problem. The rigid body motion is simply obtained by forcing the strain to be small within the bodies (penalty method). This approach is similar to what is done by Caltagirone's team in Bordeaux. Compared to the previous method, it presents the drawback to change the operator which acts on the velocity: it destroys the well-ordered hierachical spectral structure of the Laplace operator, and consequently rules out the possibility to use standard fast solver for the global system. On the other hand, this method is quite easy to implement (it can be done in a few lines with the free software Freefem++ ). An application of this method to a simple model of (rigid) cardiac valve is presented in [18] (2005).

Note that, for particulate flows, contacts still need to be handled in a proper way. To that purpose, Aline Lefebvre adapted the method proposed in [20], and a FreeFem++ script to simulate fluid-particle flows in a robust way is now available. More complex constraints can be easily added to the numerical model, e.g. particles which are subject to remain at a given distance.

Despite its conceptual simplicity, error estimates in both h and epsilon (the penalty parameter) are not straightforward to obtain for this method. We established the expected error in h^(1/2) + epsilon for a scalar version of the rigid motion constraint [29].

Blood flows

The techniques presented previously to simulate the motion of rigid particles in a fluid can be applied to study the equivalent viscosity of complex mixtures like blood. As a first step, we can consider that red blood cell are rigid, and surrounded by a newtonian fluid (the non-newtonian character of the blood at a larger scale is precisely due to the presence of red cells). We performed some computations of collection of rigid bodies in a fluid submitted to a shear motion. See [22] (2005) for some computations (based on the ALE approach) of the effective viscosity of a mixture of a viscous fluid and interacting, rigid bodies.

With S. Martin, S. Faure, and T. Takahashi, we are currently working on more realistic representations of the actual red blood cells. Each cell is modelled as an assembly of 11 spheres, submitted to spting-like interaction forces to mimic their deformability. Click on the picture below to download an avi movie (37 Mo) representing those cells aggregating under the action of some interaction forces.

Contacts / granular flows

Initially motivated by the need to handle numerical collisions in the direct simulation of fluid particle flows, we developped a code to compute the motion of (2D or 3D) spherical grains, under the assumption of non-elastic shocks. Moving obstacles can be taken into account, as the following animations illustrate.

This method is presented and analysed in [20] (2006). It is unconditionnaly stable (large time steps can be used), and can be shown to convergence in the case of a single contact. Note that this convergence is up to a subsequence, as the solution to the evolution equation is not unique, even if the data is regular (only analyticity ensures uniqueness, as showed by M. Schatzman). The scheme we propose is able to recover 2 different solutions associated to the same set of data, as detailed in [20] (two subequences of time steps converge to two different solution to the very same problem). This is somewhat anecdotical from the modelling standpoint, as the forcing term which leads to non-uniqueness is not likely to be faced in nature, but this situation is rare enough to be mentioned.

For wet grains, interaction (lubrication) forces between neighbouring particles can be taken into account.

It motivated the approach presented in [4], which is based on an asymptotic expansion of the normal and tangential interaction forces between two spheres in relative motion. This method leads to badly conditionned matrices, which are difficult to handle numerically. We recently proposed an simplified version of this lubrication model (see [25], 2007). This model has been extended to many-body situations by A. Lefebvre.

This model is somewhat paradoxal: whereas it is designed to handle interaction between rigid bodies in a highly viscous fluid (so that lubrication force play a significant role before actual contact, because of surface roughness for example, occurs), it is obtained as the vanishing viscosity limit of a lubrication model. This apparent contradiction is discussed in [25] (2007), and more thoroughly in Aline Lefebvre's PhD thesis.

Coagulation fragmentation

We applied granular flow techniques to the simulation of coagulation phenomena. The coagulation is obtained by turning on an strong attractive force when two particles get into contact. The method is detailed in [16] (2004).

Animations to download (1.1 Mb, 1.9 Mb, and 3.9 Mb)

Note that, if computations can be carried out with no harm, this second-order evolution problem with unilateral constraints and stochastic forcing term does not fit into any standard class of problems. Following a discussion we had last december, Jean Bertoin established the first theoretical results concerning this type of problem, in the case of a single contact (see link).

Crowd motion

A simple model for crowd motion can be build up in the spirit of granular flow models with non-elastic shocks. With Juliette Venel, a PhD student at Orsay, we are working on the theoretical aspects of this model, and proposed a numerical scheme to discretize it. This work is ongoing. See an example of 1000 persons evacuating a square room with obstacles in front of the exit (.gif, 2.9 Mb), or, for those who are familiar with the RER station Gare du Nord, another example (gif, 6Mb). This model takes the form of a differential inclusion, with a difficulty due to the non convexity of the feasible set: the multivalued operator is not maximal monotone. It is indeed the outward normal cone to a set which is not convex (set of discs with non-overlapping constraint). But as the set is prox-regular, recent results by L. Thibault provide a sound theoretical framework for this model, see [24], a poster realized for the CANUM 2006, or [38].

More recently

Free surface flows

This activity dates back to the PhD I did at University Paris 6, under the supervision of O. Pironneau, on the flat glass forming process. I developped a 3D Navier-Stokes Finite Element solver for free surface flows with surface tension. We introduced a new method to advect a free surface, which may be called projected characteristics, as an advection equation is solved on a plane around each point of the surface by a method of characteristics. This method allows to use large time steps (no CFL condition is required). It is presented in detail in [2]. Another issue was raised by the surface tension forces, which are proportionnal to the mean curvature. We show (see [2] again) that this curvature can be integrated in the standard variational setting. We established the convergence (order 1/2 in h, the mesh diameter) of this way to take into account the surface tension for a 2D problem (see [23]).

See below a stationary state for a 2D model of the float glass forming process (the molten glass floats over a bath of tin).

The next set illustrate the stability of the characteristic method to advect the free surface, in case of a surface velocity with a predominating tangential component (about 4 time steps between each snapshot).

Modelling of the respiratory system

This activity started within the Project ACINIM LePoumonVousDisJe.

Our activities concentrated on the following aspects :

- design of suitable outlet boundary condition at interfaces between the upper part of the tree (where the flow obeys Navier-Stokes equations) and the distal part (where the air flow flows Stokes equations, and therefore can be described by means of Poiseuille-type laws). We proposed BC which we called dissipative, as they involve the equivalent Poiseuille relation in the downstream subtree, and lead to dissipative term in the variational formulation. Different forms of those conditions are described in [19] (2005). Three-dimensional experiments were performed (collaboration with C. Grandmont and L. Baffico).

- coupling of the Navier-Stokes part to a spring-mass system to model the paremchyma:

With this approach, the ventilation is caused by a force acting on the piston (which represents the paremchyma). This model was first introduced in [17] (2005). A theoretical analysis of this model and some numerical test are proposed in [26] (2007).

- Investigation of the role of the smooth muscle in the ventilation process. This is a joint work with Sebastien Martin (Orsay), Thomas Similowski, and Christian Strauss (Pitié-Salpêtrière, Paris). We proposed a simple respiration model process (ventilation + diffusion of O2 in the blood) to investigate the (possibly) positive role of smooth muscle contraction upon the overall efficiency of the respiratory process. This work has been publisshed in [29].

- Coupling of a bronchial tree (assuming Poiseuille law is verified for all branches) with a spring mass system (1D model for the paremchyma). With C. Grandmond and N. Meunier (see [21], 2006), we obtained a new constitutive equation (in the one-dimensional case) for an elastic media perforated with holes containing an incompressible gas, which communicate through a network of pipes through which the gas flows and dissipates energy. It leads to a wave-like equation with a non local dissipative term. It is not clear whether the damping is exponential or not. This very point is presently investigated by C. Grandmont and C. Vannier.

- Elaboration of an infinite tree model to describe in some way the regularity of the pressure field within the paremchyma. This work, somewhat exploratory and, for the time being, academic, addresses the following question : the pressures at alveolae form a set of discrete values (1 value per alveola, about 300 million in the human lung), uniformly distributed over the space domain occupied by the paremchyma. It is tempting to replace this collection of values by a continuum.

As a first step (with C. Vannier, PhD student) we had the number of generations of the actual tree (23 in practice) "go to infinity". We obtain an infinite dyadic, resistive tree T, on which a Sobolev space H^1 can be defined (set of nodal pressures which dissipate a finite rate of energy). The question is now whether a trace space can be defined (the boundary of T can be identified with the space of random walks in Z). We obtain a non trivial trace space (H^1/H^1_0 is not {0}) as soon as the global resistance if finite (which is the case for the infinite version of a healthy lung). The next step (in collaboration with Delphine Salort) consists in imbedding the infinite set of leaves (=alveolae in practice) onto a real domain in R^d, in order to investigate the possibility to identify natural functional spaces for the pressure, and describe the regularity of feasible pressure fields. Under suitable assumptions (regular tree, regular embedding of the set of ends in the domain) we establish that the pressure field presents some sort of regularity in space: it belongs to H^0.15 (for d = 3).

This work is described in [30].

Unclassified

Optimization under constraint

With Guillaume Carlier and Thomas Lachand-Robert, we considered the problem which consists in projecting a function onto the cone of convex functions for the H

A first answer to that problem was proposed by Thomas Lachand-Robert. it consists in working on the larger space of all those piecewise P1 functions which are not necessarily convex, but which interpolate a convex function. This relaxed constraint at the discrete level can be expressed in terms of values of the discrete function at the nodes of the triangulation. Note that most of those constraints are non local. With Guillaume Carlier, we analysed and implemented this method (see [10], 2001). A convergence result can be established for this method, but it is still too expensive to be applied to real life problems.

Therefore we followed a completely different strategy, which relies on a dual expression of the convexity constraint. Numerical computations (presented in [7]) exhibits a fairly satisfactory behaviour, yet the numerical analysis of the approach is difficult because the continuous saddle-point formulation is ill posed. The problem takes the following abstract form: Let Ac by the cone of convex functions (in H

Full analysis of the method has not been done, but the fairly good behaviour of the algorithm can be explained by the following consideration: in the continuous version of the Uzawa algorithm (seen as a time-discretization of a gradient flow in the space of Lagrange multipliers), we established weak convergence (at least of a subsequence) of the primal trajectory to the solution to the minimization problem, even in the case the saddle-point problem is ill-posed (i.e. the Lagrange multiplier does not exist). The proof is given in [12] (2003), and it suggests at least why the parameter in the Uzawa algorithm does not degenerate as h (the mesh diameter) goes to 0.

Note that all this is related to the general study of the asymptotic behaviour of the solution to an homogeneous differential inclusion involving a maximal monotone operator, which has been intensively studied in the 70's. Yet, to our knowledge, none of the standard results include the case of a maximal monotone operator whose range does not contain 0 (which is the problem we face here).

Maximization of the first eigenvalue of the Laplace operator with respect to the domain

With Geoffroy Billotey, student at Ecole Polytechnique, we developped a numerical tool, to investigate the following problem: Given N circles with same radius r, in a bounded domain, find the configuration which maximizes the smallest eigenvalue of the Laplacian with Dirichlet boundary conditions. After some elementary spectral analysis of the heat equation, this problem can be expressed from a modelling point of view: given a hot plate and a set of coolers, where to put them in order to optimize the cooling process? This work is not published.

The method is simply based on a gradient flow associated to the smallest eigenvalue seen as a functional of the center locations.

The following animation represents the trajectory of ten particles, which converges toward a (local ?) maximum of the first eigenvalue (Download 2.4 Mb, .avi). This second animation corresponds to the same situation with a different initial conditions; it converges, after a short nap around a seemingly quasi-equilibrium point, to a situation symmetric to the previous one (Download 5.7 Mb, .avi).

Publications

[62] H. Guénard, J.-B. Martinot, S. Martin, B. Maury, S. Lalande, C. Kays, In vivo estimates of NO and CO conductance for haemoglobin and forlung transfer in humans, Respiratory Physiology & Neurobiology 228 (2016) 1–8.

[61] F. Al Reda, B. Maury, Interpretation of Finite Volume discretization schemes for the Fokker Planck equation as gradient flows for the discrete Wasserstein distance, to appear, HAL

[60] S. Di Marino, B. Maury, F. Santambrogio, Measure sweeping processes, Journal of Convex Analysis, Special Volume dedicated to the memory of Jean Jacques Moreau, 1923 - 2014, Vol. 23, Nr 2 (2016) HAL

[59] B. Maury, A. Preux, Pressureless Euler equations with maximal density constraint : a time-splitting scheme, accepted in Ricam Reports, https://hal.archives-ouvertes.fr/hal-01224008.

[58] S. Martin & B. Maury, Notion de résistance de l’arbre pulmonaire bronchique dans la ventilation respiratoire humaine, in “Modéliser & simuler”, Tome 2, Matériologiques, 2014.

[57] B. Maury, The resistance of the respiratory system, from top to bottom, Esaim Proceedings and Survey, december 2014, Vol. 47, p. 75-96 pdf

[56] S. Faure, B. Maury, Crowd motion from the granular standpoint, Mathematical Models and Methods in Applied Sciences Vol. 25, No. 3 (2015) 463–493 pdf.

[55] B. Maury, non smooth evolution models in crowd dynamics: mathematical and numerical issues, in Collective Dynamics from Bacteria to Crowds, An Excursion Through Modeling, Analysis and Simulation, Series: CISM International Centre for Mechanical Sciences, Vol. 553 (2014). pdf

[54] B. Maury, A. Roudneff-Chupin, F. Santambrogio, Congestion driven dendritic growth, Discrete and Continuous Dynamical Systems, Volume 34, number 4, April 2014, preprint.

[53] B. Fabrèges, B. Maury, Approximation of Single Layer Distributions by Dirac Masses in Finite Element Computations, Journal of Scientific Computing, January 2014, Volume 58, Issue 1, pp 25-40 (arXiv).

[52] B. Fabrèges, L. Gouarin, B. Maury, A smooth extension method,

Comptes Rendus Mathematique, Volume 351, Issue 9, Pages 361-366 (2013) pdf

[51] B. Maury, The Respiratory System in Equations (MS&A), ed. Springer (book), 2013, link

[50] S. Martin, B. Maury, Modeling of the oxygen transfer in the respiratory process,

ESAIM: Mathematical Modelling and Numerical Analysis / Volume 47 / Issue 04 / July 2013, pp 935-960, link.

[49] A. Decoene, B. Maury, Moving meshes with freefem++. Journal of Numerical Mathematics. Volume 20, Issue 3-4, pp. 195–214 (2012).

[48] J. Dambrine, B. Maury, N. Meunier, A. Roudneff-Chupin, A congestion model for cell migration, Communications on Pure and Applied Analysis 11, 1 (2012) 243-260. (arXiv)

[47] Aline Lefebvre-Lepot, Bertrand Maury,

Numerical Modeling of Fluid-Grain Interactions, Close Interaction of Immersed Grains, chapters 11 and 12 in Discrete Numerical Modeling of Ganular Materials, arang Radjai, Frederic Dubois Eds., ISTE Ltd and John Wiley & Sons Inc, march 2011 link.

[46] B. Maury, Prise en compte de la congestion dans les modèles de mouvements de foules, in Actes des Colloques Caen 2012-Rouen 2011, pdf.

[45] Aline Lefebvre-Lepot, Bertrand Maury, Micro-Macro Modelling of an Array of Spheres Interacting Through Lubrication Forces, Adv. Math. Sci. Appl. 21 (2011), no. 2, 535–557, hal

[44] C. Grandmont, B. Maury, Integrated Multi-Model Description of the Human Lungs, in Coupled Fluid Flow in Energy, Biology and Environmental Research, E-Book Series Progress in Computational Physics (PiCP), Volume 2, Matthias Ehrhardt (ed.), Bentham Science Publishers, Springer 2011. pdf

[43] A. Decoene, S. Martin, B. Maury, Microscopic modelling of active bacterial suspensions,

Mathematical Modelling of Natural Phenomena 2011 6 (05) : pp 98-129 (hal).

[42] A. Decoene, A. Lorz, S. Martin, B. Maury, M. Tang, Simulation of self-propelled chemotactic bacteria in a Stokes flow, ESAIM: Proceedings, 30, 105-124 (2010). (.pdf)

[41] B. Maury, A. Roudneff-Chupin, F. Santambrogio, J. Venel, Handling Congestion in Crowd Motion Modeling, Networks and Heterogeneous Media, Volume 6, Number 3, September 2011, pp. 485–519 (arxiv).

[40] F. Bernicot, B. Maury, D. Salort, A 2-adic approach of the human respiratory tree, Netw. Heterog. Media 5 (2010), no. 3, 405–422 (arxiv).

[39] S. Bertoluzza , M. Ismail, B. Maury, Analysis of the fully discrete Fat Boundary Method, Numerische Mathematik, Volume 118, Number 1, 49-77 (2011).

[38] B. Maury, J. Venel, A discrete Contact Model for crowd Motion, ESAIM: M2AN 45 1 (2011) 145-168. (hal).

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[37] C. Bui, P. Frey and B. Maury, A coupling strategy based on anisotropic mesh adaptation for solving two-fluid flows, International Journal for Numerical Methods in Fluids, Volume 66, Issue 10, August 2011, Pages: 1226–1247 (pdf).

[36] B. Maury, A. Roudneff-Chupin, F. Santambrogio, A macroscopic Crowd Motion Model of the gradient-flow type, Mathematical Models and Methods in Applied Sciences Vol. 20, No. 10 (2010) 1787-1821 (arxiv).

[35] A. Devys, C. Grandmont, B. Grec, B. Maury and D.Yakoubi, Numerical method for the 2D simulation of the respiration, ESAIM: Proc., August 2009, Vol. 28, p. 162-181 (link).

[34] S. Faure, S. Martin, B. Maury and T. Takahashi, Towards the simulation of dense suspensions: a numerical tool, ESAIM: Proc., 2009, Vol. 28, pp. 55-79 (.pdf).

[33] L. Baffico, C. Grandmont, B. Maury, Multiscale Modelling of the Respiratory Tract, Math. Models Methods Appl. Sci. 20 (2010), no. 1, 59-93 (.pdf).

[32] B. Maury, D. Salort, C. Vannier, Trace theorems for trees, application to the human lung, Network and Heterogeneous Media, Volume 4, Number 3, September 2009 pp. 469--500 (.pdf).

[31] B. Maury, J. Venel, A Mathematical Framework for a Crowd Motion Model, C. R. Acad. Sci. Paris, Ser. I 346 (2008) 1245--1250.

[30] B. Maury, Numerical Analysis of a Finite Element / Volume Penalty Method, SIAM J. Numer. Anal. Volume 47, Issue 2, pp. 1126-1148 (2009). (.pdf)

[29] S. Martin, T. Similowski , C. Strauss, B. Maury, Impact of respiratory mechanics model parameter on gas exchange efficiency, ESAIM Proc, June 2008, Vol. 23, p. 30-47 (link).

[28] V. Girault, H. Lopez, B. Maury, Energy balance of a 2-D model for lubricated oil transportation in a pipe, Divulgaciones Matematicas Vol. 16 No. 1(2008), pp. 87--105 (pdf).

[27bis] S. Del Pino, B. Maury, 2d/3d turbine simulations with freefem++, in Numerical Analysis and Scientific Computing for PDEs and their Challenging Applications, J. Haataja, R. Stenberg, P. Raback and P. Neitaanmäki, eds, CIMNE, Barcelona, Spain, 2008. (.pdf 1Mo)

[27] C. Bui, P. Frey, B. Maury, Methode du second membre modifié pour la gestion de rapports de viscosité importants dans le probleme de Stokes bifluide, C. R. Mecanique 336 (2008) 524--529.

[26] C. Grandmont, B. Maury, A. Soualah, Multiscale modelling of the respiratory track : a theoretical framework (2007), Proc, June 2008, Vol. 23, p. 10-29 (.pdf).

[25] B. Maury, A gluey particle model, ESAIM Proceedings, July 2007, Vol.18, 133-142

Jean-Frédéric Gerbeau & Stéphane Labbé, Editors (.pdf, 200 kb).

[24] B. Maury, J. Venel, Un modèle de mouvements de foule (in french), ESAIM Proceedings, July 2007, Vol.18, 143-152, Jean-Frédéric Gerbeau & Stéphane Labbé, Editors (.pdf, 964 kb).

[23] V. Girault, H. Lopez, B. Maury, One time-step finite element discretization of the equation of motion of two-fluid flows, Numerical Methods for Differential Equations, vol. 22. 3, pp. 680-707, 2006 (.pdf, 200 kb).

[22] A. Lefebvre, B. Maury, Apparent viscosity of a mixture of a Newtonian fluid and interacting particles, Fluid-solid interactions: modeling, simulation, bio-mechanical applications, Comptes Rendus Mécanique, Volume 333, issue 12, december 2005, p.p. 923-933(.pdf, 712 kb).

[21] Céline Grandmont, Bertrand Maury, Nicolas Meunier, A viscoelastic model with non-local damping application to the human lungs, Mathematical Modelling and Numerical Analysis, Vol. 40 No. 1, pp 201-224, 2006 (.pdf, 900 ko).

[20] B. Maury, A time-stepping scheme for inelastic collisions, Numerische Mathematik, Volume 102, Number 4, pp. 649 - 679, 2006 (.pdf, 1.4 Mo).

[19] B. Maury, N. Meunier, A. Soualah, L. Vial, Outlet Dissipative conditions for air flow in the bronchial tree, ESAIM Proceedings, september 2005, vol. 14, 115-123, Eric Cancès & Jean-Frédéric Gerbeau, Editors (.pdf, 800 kb).

[18] J. Janela, A. Lefebvre, B. Maury, A penalty method for the simulation of fluid-rigid body interaction, ESAIM Proceedings, september 2005, vol. 14, 201-212, Eric Cancès & Jean-Frédéric Gerbeau, Editors (.pdf, 500kb). (N.B. the associated source code can be downloaded NSpenal.edp).

[17] C. Grandmont, Y. Maday, B. Maury, A multiscale/multimodel approach of the respiration tree. New trends in continuum mechanics, 147--157, Theta Ser. Adv. Math., 3, Theta, Bucharest, 2005.

[16] B. Maury, Direct Simulation of Aggregation Phenomena, Comm. Math. Sci. supplemental issue, No 1, pp. 1-11, 2004 (.pdf, 1.1Mb).

[15] B. Maury, Analyse Fonctionnelle, exercices et problèmes corrigés (livre), éd. Ellipses, 2004. Description (en français).

[14] S. Bertoluzza, M. Ismail, B. Maury, The FBM Method: Semi-Discrete Scheme and Some Numerical experiments. Lecture Notes in Comp. Sc. and Eng., 2004 (.pdf, 530 kb).

[13] B. Maury, Fluid-particle shear flows, ESAIM M2AN., 37, No 4, pp. 699-708, 2003 (.pdf, 1.2Mb).

[12] B. Maury, Version continue de l'algorithme d'Uzawa, C. R. Acad. Sci. Paris, Série I, 337, pp. 31-36, 2003 (.pdf, 120 kb).

[11] M. Ismail, B. Maury, Méthode de la frontière élargie pour la simulation d'écoulements fluides, Proc. du 35ème Congrès Nat. d'Ana. Num., La Grande Motte, pp.1-25, 2003 (.pdf, 1.6 Mb).

[10] G. Carlier, T. Lachand-Robert, B. Maury, A numerical approach to variational problems subject to convexity constraints, Num. Math. 88, pp. 299-318, 2001 (.pdf, 560Kb).

[9] B. Maury, A Fat Boundary Method for the Poisson Equation in a Domain with Holes, J. of Sci. Computing, 16 (2001), no. 3, pp. 319-339 (.pdf, 700kb).

[8] E. Lagarde, B. Auvert, B. Ferry, B. Maury, Concurrent partnerships and HIV epidemic in five urbancommunities of sub-Saharan Africa, AIDS 2001, vol 15, No X.

[7] G. Carlier, T. Lachand-Robert, B. Maury, H1-projection into the set of convex functions: a saddle-point formulation, CEMRACS 1999 (Orsay), 277-289, ESAIM Proc., 10, Soc. Math. Appl. Indust., Paris, 1999 (.ps, 1.5 Mb).

[6] B. Maury, Direct Simulations of 2D Fluid-Particle Flows in Biperiodic Domains, Journal of Computational Physics 156, 325-351, 1999 (.pdf, 3.3 Mb).

[5] Y. Maday, B. Maury, P. Métier, Interaction de fluides potentiels avec une membrane élastique, CEMRACS 1999 (Orsay), 23-33, ESAIM Proc., 10, Soc. Math. Appl. Indust., Paris, 1999 (.pdf).

[4] B. Maury, A Many-Body Lubrication Model, C. R. Acad. Sci. Paris, t. 325, Série I, pp. 1053-1058, 1997 (.pdf, 1.6 Mb).

[3] B. Maury, R. Glowinski, Fluid-Particle Flow: a Symmetric Formulation, C. R. Acad. Sci. Paris, t. 324, Série I, pp. 1079-1084, 1997.

[2bis] B. Maury, O. Pironneau , Characteristics ALE method for unsteady free surface flows with surface tension, Z. angew. Math. Mech., ICIAM / GAMM 95. Part II, Hamburg , Allemagne, 1996, vol. 76 (4 ref.), pp. 613-614.

[2] B. Maury, Characteristics ALE Method for the 3D Navier-Stokes Equations with a Free Surface, Int. Journal of Comp. Fluid Dyn. 6, 175-188 (1996) (.pdf, 1Mb).

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