BANG (Biomédical, Analyse Numérique et Géophysique) is a continuation of the former project M3N.
BANG (Biomédical, Analyse Numérique et Géophysique) is a continuation of the former project M3N. It aims at developing models and numerical methods for two kinds of problems involving Partial Differential Equations. Firstly problems from life sciences (blood flows, respiratory flows, cancer modeling...) are considered. Secondly models for complex fluid flows are studied (flows with a free surface, flows of holes and electrons in semiconductors, coupling with a magnetic field...).
The common scientific features behind these applications come from models involving coupled systems of PDEs (as Navier-Stokes equations) that are solved (simulated) on computers involving new algorithms.
Partial Differential Equations are mathematical tools that allow to represent efficiently
the evolution of complex physical phenomena. The most classical PDE is certainly the
Navier-Stokes system which describes the evolution of the density
Since the XIXth century this formalism has shown its efficiency and
ability to explain both qualitative and quantitative behaviors of fluids.
The knowledge that has been gathered on such physical models, on algorithms for solving them
on computers, on industrial implementation, opens the hope for success when dealing with
life sciences also. This is one of the main goals of BANG.
What are the relevant physical or biological variables, what are the possible dominant effects ruling their dynamics, how to analyse the informations coming out from a mathematical model and interpret them in the real situations under consideration ? these are the questions leading to select a mathematical model, generally also to couple several of them in order to render all physical or biomedical features which are selected by specialist partners (engineers, physicists, medical doctors). These are usually based on Navier-Stokes system for fluids (blood flow, respiration, free surface fluid flows), on parabolic-hyperbolic equations (blood filtration, Saint-Venant system for shallow water, flows of electrons/holes in semiconductors), on fluid-structure interaction (blood flow in vessel, aneurism, respiration).
The complete physical or biomedical description is usually complex and requires very small scales. Efficiency of computer resolution leads to simplifications using averages of quantities. Methods allowing to achieve that goal are numerous and mathematically deep. Some examples studied in BANG are
Reduction of fluid flow in a porous medium to a hyperbolic/parabolic equation with applications to blood filtration, using homogeneisation procedure.
Reduction of full 3d Navier-Stokes system to 2d or 1d hyperbolic equations by a section average (derivation of Saint-Venant system for shallow water, hyperbolic models of blood flow).
Coupled multiscale modelling (pulmonary airways, cardiovascular system, degenerate semi-conductors).
Numerical methods used in BANG are mostly based on finite elements or finite volume methods. Algorithmic improvments are needed in order to take into account the specificity of each model, of their coupling, or their 3D features. Among them we can mention
Fluid-structure coupling in blood vessels.
Well-balanced schemes for shallow water system.
Free-surface Navier-Stokes solvers based on Arbitrary-Lagrangian-Eulerian formulation.
Mixed finite elements for problems with large density variations (semi-conductors, chemotaxis).
BANG has decided to develop new biomedical applications and focusses its know-how in these directions, while keeping more classical industrial relations. These are developed in relation with other INRIA projects: MACS, SOSSO, GAMMA, ESTIME.
This is the main biomedical application developed in BANG. More specifically our research is oriented to
Blood flows. The goal is to develop numerical simulation of several aspects of blood circulation thus allowing a better medical understanding of pathologies. This includes the propagation of pressure waves in vessels (fluid-structure interaction), large deformations in aneurisms or artery valvs...
Respiratory flows. This research is aimed at developing a ventilation simulator. A medical target being aerosols deposition.
Blood filtration. For human blood transfusion white cells are filtered. A research is developed and aims at undertanding the blood flow in the filters, the filtering mecanism and the filter optimisation.
This research activity aims at studying mathematical models related to tumors developments and their control. Among the many biological aspects let us mention: cell movments (chemotaxis, vasculogenesis, angiogenesis), cell cycle, immune reaction and adaptive dynamics.
Several industrial applications require to solve fluid flows with a free surface. BANG develops algorithms in two directions. Firstly flows in rivers and coastal areas using Saint-Venant model with applications to dam break and pollution problems. Secondly, fluid/electromagnetism coupling for free surface flows in aluminium industry.
Mathematical models based on drift-diffusion systems or energy transport systems are solved using mixed finite elements methods. BANG has developed a highly sophisticated code which is able to simulate very stiff semiconductor devices.
Softwares initiated and developped within former projects (Menusin, M3N) and currently in use in the present project.
Generation of metric maps for use with adapted meshes generator (with Gamma project)
Interactive 2D mesh generator (with Gamma project)
Research software for the simulation of incompressible MHD flows in presence of free surface. (Déposé à l'Agence pour la protection des Programmes).
Research software for the numerical simulation of semiconductor devices.
Drift-Diffusion and Energy-Transport models are implemented. The mathematical formulation is
described using as unknowns the electrostatic potential, the quasi Fermi levels and
additionaly the electron temperature. The approximation is carried out via mixed finite
elements (Raviart-Thomas element Bamg software (Gamma project)
has been developped for automatic mesh adaption.
The main purpose of these studies is the simulation of the mechanical interaction between the blood and the artery wall.
We have proposed in
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In a very preliminar work with E. Delavaud (DESS student, Paris 6), in collaboration with J. Sainte-Marie (projet MACS), we have proposed a coupling between a network of several arteries (1D fluid-structure models) and a 1D electromechanical model of the heart proposed in the ICEMA consortium (projets MACS, SOSSO, EPIDAURE).
This work is aimed at studying mainly
the wall shearing force of a laminar steady flow of an
incompressible Newtonian fluid which is conveyed through
a straight collapsed tube.
This tube is composed of a tapered segment, a contact region
where the opposite walls touch and a reopened segment.
The tube geometry and characteristics of the flow are obtained from
measurements in a collapsed tube at a steady state, in particular
the tube shape mimics a frozen configuration obtained from
ultra-sound measurements on a Starling resistor.
The Navier-Stokes equations associated with the
classical boundary conditions are solved using the finite element method.
The tube configuration induces a three-dimensional flow.
These flow behaviour is affected by both the tube shape
and by the flow conditions
(a Reynolds number larger than 1000
associated with a maximum velocity of about 1.4 m/s).
The three-dimensional flow structure
is defined by a set of swirls in the reopened segment downstream
from the contact region, in the core behind the wall contact
and along the diverging walls, as well as in the exit attachment duct.
Two side jets emerging from the two small tear-drop-shaped
outer passages of the contact zone of the collapsed tube
run centrally and partially merge in the proximal segment of
the rigid attachment duct.
A lateral high velocity stream is still present far downstream
in the exit attachment duct.
In order to compute both the streamwise and transverse components
of the shearing force on the wall, a local basis is defined
in each wall node.
The jets exiting from the contact region
and the sets of flow reversals explain the
variation in wall shear stress characterized by important gradient
in both the streamwise and transverse directions.
Shear-stress longitudinal gradient are known to trigger important changes in cell shape, density, rate division and biological functions. A coupling between haemodynamics and the vasomotor tone of the vessel wall has been demonstrated, mainly in straight test section of circular cross section. The cell behavior is affected not only by the longitudinal gradient in the flow chambers in which the flow is not fully developed but also by the cross gradients of the axial component of the wall stress when the wall is characterize by a non-constant cross curvature. The straight pipe in its collapsed configuration provides the simplest test section to produce both effects. In order to study the mechanotransduction property of endothelial cells (ECs) under stretch and torsion, straight cell-coated channels with axially uniform cross sections but with varying wall cross curvature have thus been proposed. Numerical computations are thus used to (i) determine the load exerted by the flowing fluid on the tube wall and (ii) the entry length of a laminar steady flow of an incompressible Newtonian fluid in uniformly collapsed channels. This study is performed in order to exhibit the mechanical environment of cultured ECs in collapsed-design flow chamber and to optimize the fabrication cost of the test section. Uniformly collapsed tube might be observed in inclined deformable vessels when viscous effects balance hydrostatic forces.
Five cross section shapes are used.
The reference cross section Navier-Stokes equations are solved using
the finite element method for
the five tubes Reynolds numbers
(
Three indices have been proposed to determine the entry length.
(i) the first is based on the axial fluid velocity,
(ii) the second on its derived quantity, the wall shear stress,
and (iii) the third on the pressure.
In contact configuration, the velocity criterion underestimates
the entry length. An additional length of about 10 %
of the computed entry length must be added to ensure that
the wall shear stress corresponds to
a fully developed state.
The fluid pressure, which depends on the
Reynolds number, has been shown to vary non-linearly
in the entry length.
In contact configurations (
This investigation is aimed at developing a ventilation simulator. The airways are decomposed into a set of compartments in series. (i) the first set is composed of the superior airways with the nose, the pharynx and the larynx. The nose is in relation with paranasal sinuses. The nose has a complexe geometry. The set of conchae increases the exchange surface between the physiological walls and the inhaled air. The ventilation function of the nose are, indeed, air epuration, air humidification and temperature setting. The pharynx is a rather simple conduit made of three compartments in series, the nasopharynx, the oropharynx and the hypopharynx. The larynx is also composed of three segments in series, the supraglottis with the epiglottis, the aryepiglottic folds the false vocal cordsor folds and the ventricle, which manages wrong food administration, the glottis with the true vocal folds and the subglottis. Due to its geometry, the larynx induces a jet. (ii) The proximal airways are constituted by the trachea (generation G0 of the tracheobronchial tree) with its extrathoracic and intrathoracic segments, its division branches, the right and left main bronchi, which give birth to intrapulmonary airways, visible by the usual medical imaging techniques, the lobe and then segmental bronchi. (iii) the distal airways are defined by the bronchus generation lower than G5 down to alveoli.
The flow in the superior and proximal airways
has been studied numerically.
The computational domains are derived from medical images.
The present simulations are performed with a Dirichlet velocity
boundary condition either at the pharynx or at the trachea,
while the opposite end(s) of the fluid domain are associated
with stress-free conditions.
The main parameter of interest for the physician
is the ventilation distribution.
Tab. gives the ratio of the flow rate
entering at inhalation or living at exhalation
the right (
Because the boundary conditions are not appropriate,
one needs in modelling in small pulmonary airways
as well as the thoracic cage as a ventilation motor.
Three types of airway models are under investigation.
The mid-size bronchi (G6 - G10) are modelled using the method
of reduced basis element (MRBE) as proposed by Y. Maday.
The modelling of the small bronchi (G10-G19)
(i.e. the terminal bronchioles (G10-G16),
cranially located with respect to the pulmonary lobules,
and the respiratory bronchioles (G17-G19), first generatioons
of airways inside the lobules)
is based on fractal homogeneisation (FH).
Multiphysics homogeneisation (MPH) is the modelling technique
for the deformable acini which are composed of
alveolar ducts (G20-G22), with more and more alveoli
on their walls, alveolar sacs and alveoli.
The governing equation describes the distribution function of aerosol
particles
where
In particular, the aerosol- and energy density are given by
with a truncature
We are interested in the modelling of white blood cell filtration. We
have proposed and studied a simplified model of this process
Movement of cells are important in the process of cancer development. We have developed some activity in the understanding
of mathematical models of chemotaxis and angiogenesis.
Angiogenesis describes the development of capillary blood vessels triggered by a substance emited by a tumor.
Models for the vasculature have been proposed by several authors (Chaplain, Levine, Sleeman) as a coupled parabolic/hyperbolic system.
We have investigated existence of weak or strong solutions for this system
Models with several similarities were proposed by Keller-Segel several years ago for chemotaxis, as coupled parabolic/elliptic systems.
They describe the collective motion of bacteria taking into account
the underlying biochemistry. A new formulation of the system of partial differential equations is obtained bys the
introduction of a new variable which is similar to the quasi-fermi level in the framework of semiconductor modelling.
The discretization of the model is achieved by mixed finite elements
We consider a mathematical model for the cell-cycle, i.e. the sequence of events that leads to mitosis,
at the level of a population of cells. These are structured population Partial Differential Equations
that describes the evolution of the population along each phase of the cycle and the transition to the next phase.
These models allow several types of controls such as therapeutic control in case of cancer therapy
(some chemotherapy are known to act on specific phases of the cycle or on the transitions between phases) or
circadian control by the central nervous system. We study the long time behavior of this system of PDEs using
an entropy method and exhibit, by numerical simulations, the action of the circadian control
We are involved in research concerning the numerical simulation of free surface geophysical flows such as rivers, lakes, coastal areas and also avalanches. Many applications related to environmental problems are concerned : floodings, dam breaks, transport and diffusion of pollutants, debris avalanches ...
In many cases, the shallow water hypothesis is satisfied and these phenomena can be simulated by the Saint-Venant equations, for other cases we have considered a multilayer Saint-Venant system and also the 3D free surface Navier-Stokes equations.
We have developed 1D and 2D solvers for the Saint-Venant equations, the aim is to obtain robust and efficient numerical tools based on theoretical results ensuring the accuracy and the preservation of physical properties of the flow (conservation, positivity of water depth, equilibrium states...).
The solution method is based on a kinetic solver applied on finite volumes. To compute the numerical fluxes at the interfaces, we consider the microscopic behavior of the system and from the discretization of the linear kinetic equation, we deduce the kinetic scheme at the macroscopic level. A useful property of this conservative scheme is the built-in preservation of the water height positivity under a CFL condition.
The equilibria of the 1D Saint-Venant system are easy to describe. But when we are interested in real
flow situations and so we use the 2D Saint-Venant system, the situation is much more complex and only
one particular equilibrium, the steady state of a lake at rest, reminds really interesting. But even if
this equilibrium is very easy to characterized it is well known since Leroux et al. that its
numerical treatment is not so obvious.
In the context of our development of a numerical treatment of the 2D
Saint-Venant system we propose a new method to obtain a well-balanced scheme
In order to improve the accuracy of the above first-order scheme, we
have introduced a second-order extension. The interface fluxes are
computed from limited reconstructed values on both sides of each
interface rather than from
cell-centered values. These new values are classically obtained with three ingredients:
prediction of the gradients in each cell, linear extrapolation, and
limitation procedure. We can show that the second-order reconstruction
preserves the important features of the scheme. First, the cell by cell
reconstruction preserves the mass conservation property of the finite
volume method. Second, the limitation procedure ensures the
nonnegativity of the second-order reconstructed water heights and
under the constraint of the mass conservation with the reconstructed
water heights we can write a CFL condition implying also the water height
non negativity. The
third important point is that the second-order reconstruction
preserves the lake at rest steady state. To ensure this property we
reconstruct also the bottom topography
We have also studied the transport of a passive pollutant by the flow
modeled by the shallow water equations using a new time discretization
that allows large time steps for the pollutant computation. For the
hydrodynamic part we use the previous kinetic solver. The interest of
the developed method
For some applications like the transport of tracers of different densities or the effect of wind on a lake, we need to know the vertical profile of the velocity and by definition, we cannot get relevant information from Saint-Venant equations. These questions are also fundamental in ocean models and atmospheric sciences.
In these cases, and in order to avoid the 3D
Navier-Stokes system when large scale problems are considered,
we introduce a new multilayer Saint-Venant system
We prove that this multilayer Saint-Venant system satisfies some stability properties : it admits an energy and preserves the positivity of the water height. We discuss of the loss of hyperbolicity of the model and we exhibit a variant of the multilayer model for which the left hand side, i.e. the conservative part, is hyperbolic. We have performed the numerical implementation using a multilayer kinetic scheme which preserves the stability properties of the classical kinetic scheme. We have validated the model through some numerical examples (see Fig. ).
For the cases where the 3D effects cannot be neglected, we are studying the 3D incompressible Navier-Stokes equations with free surface. At first, we consider the model with hydrostatic approximation. This work is done in collaboration with EDF/LNHE.
After time discretisation of the equations, the fractional step method can be applied in order to split the non-linear advection terms and the "hydrostatic system" composed by the resulting momentum equation and the depth-integrated continuity equation which is linearised. As the horizontal and vertical diffusion can be splitted as well, we have studied the system with and without the viscous terms. In both cases, we have established a variational formulation leading to a mixed and symmetric problem coupling a 3D variable, the horizontal velocity, and a 2D variable, the free surface.
The Telemac-3D software developed by EDF/LNHE solves the free surface Navier-Stokes equations splitted into 4 parts: advection, horizontal and vertical diffusion, hydrostatic system without diffusion and hydrodynamic correction. Moreover, the hydrostatic system is solved on the averaged horizontal velocity and the water height, by depth-integrating the equations: this leads to a mixed problem which is solved using 2D stable finite elements. Our aim is, on one side, to allow a complete 3D treatment of the equations, and on the other side, take profit of the stabilising property of the diffusion terms in the hydrostatic system. Therefore, we have first implemented a resolution of the 3D hydrostatic system (without depth-integrating), and then included the treatment of the viscous terms. At the moment, we are analysing the stability of this new formulation.
The hypothesis of shallow flow is generally well satisfied by debris
avalanche and the most classical models are those of Saint-Venant and Savage-Hutter. They both assume limitations
of the basal variations. Our purpose
Besides this modelisation work, we have begun the simulation of
debris avalanches using the Saint-Venant model in which
the behaviour of granular
media is taken into account by a Coulomb type friction term. A kinetic
scheme with a Dirac distribution of particles at the microscopic
level has been proposed which allows the simulation of equilibrium
states of the granular mass
This work is done in collaboration with the Laboratoire de Modélisation et Tomographie Géophysique (IPGP, Paris 7).
We are interested in the mathematical modelling and the numerical
analysis of aluminium electrolysis. The physical system consists of two
fluids separated by a free interface and carrying an electrical current
which interacts with a magnetic field. Stabilized finite elements
techniques and an Arbitrary Lagrangian Eulerian formulation are used in
the numerical simulations. Energy stability and properties of
conservation of the numerical scheme have been studied
In 2003, the collaboration with the professor M. Anile (Catania university) and his team has been continued.
The preliminary study related to a new approach in the derivation of the diffusion matrix of the electrons
in a semiconductor medium in the framework of energy-transport model [Kane's band approximation. In this model, a linear relation between the electron
temperature and electron energy does not exists as in the parabolic band approximation case. A PhD
student from Catania university spent 6 months of his time at Inria in order to contribute to the
implementation of this extension. Typical -small dimensions- MESFET and MOSFET have been used for the numerical
simulations. See for example the figure() which gives the distribution of the electron energy
in a Si-MESFET for a typical polarization point in the case of Kane's band approximation.
Industrial contract with Ecole Nationale des Ponts et Chaussées in the framework of a collaboration with Aluminium Pechiney on the mathematical modelling of aluminium electrolysis cells.
The studies concerning the kinetic scheme for Saint-Venant equations and the improvment of the 3D Navier-Stokes solver are supported by a contract with LNHE of EDF. The results are included in the Telemac-2D and Telemac-3D softwares of EDF.
Participation to a working group on the debris avalanches with the Laboratoire de Modélisation et Tomographie Géophysique (IPGP, Paris 7). Participation to the ATIP of CNRS ``Modélisation des avalanches de débris: prise en compte de la transition fluide/solide'' (A. Mangeney, IPGP).
The project BANG has participated to the ACI ``Prévention des Catastrophes
Naturelles''
Jen-Frédéric Gerbeau is the french coordinator of a european network RTN on blood flows modelling (from october 2002 to september 2006). Participants: INRIA, Université Paris 6, Politecnico di Milano (Italy), Imperial College (UK), Ecole Polytechnique Fédéral de Lausanne (Switzerland), Instituto Superior Técnico de Lisboa (Portugal), Technische Universität Graz (Austria)
Participation to the european network HYKE (Hyperbolic and Kinetic equations).
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Participation to the european network M3CS-TuTh (Modelling, Mathematical Methods and Computer Simulation of Tumour Growth and Therapy).
Visit of C. Klaiany, Université Saint-Joseph, Beyrouth (Liban) -2 weeks-
Invitation of A. Marrocco at Catania University (Italy) feb. 2003 -2 weeks-
Marie-Odile Bristeau is in the editorial comity of the Intenational Journal Computer Methods in Applied Mechanics and Engineering. Benoit Perthame is Editor-in-chief of M2AN and editor in various journals (CALCOLO, CPDE, SIAM J. Math. Analysis, DCDS(B))
``Fluid-structure interaction in biological flows'', DEA Analyse Numérique, Université Paris 6 (Jean-Frédéric Gerbeau with Y. Maday and M. Thiriet)
``Numerical methods in fluid mechanics and MHD'', DEA Equations aux dérivées partielles et applications, Université Paris-Dauphine (Jean-Frédéric Gerbeau)
Scientific computing and analysis, Ecole Nationale des Ponts et Chaussées (Jean-Frédéric Gerbeau)
Finite volume methods, Ecole Polytechnique de Tunisie (Benoit Perthame)
Finite element methods for fluid mechanics, Ecole polytechnique de Tunisie (Benoit Perthame)
Joint workshop on applied mathematics and scientific computing, Jerusalem, january 2003 (M. Thiriet)
Journée sur le chimiotactisme, Université de Nice, march 2003 (A. Marrocco)
CANUM, Congrès d'Analyse Numérique, Montpellier, June 2003 (E. Audusse, M. Thiriet)
Journées Modélisation en Médecine, ENS Lyon, june 2003, (M. Thiriet)
MACSI-net event Cardio Point (Cardiac and cardiovascular models) Graz, june 2003 (M. Thiriet)
International Symposium on Modeling of Physiological Flows, EPFL Lausanne, september 2003 (M. Thiriet, invited lecture)
Telemac Users' Club, Chamrousse, October 2003 (E. Audusse, A. Decoene)
26th National Congress in Computational and Applied Mathematics (CNMAC), Brasil (Jean-Frédéric Gerbeau, Invited plenary lecture)
Second M.I.T. conference on Computational Fluid and Solid Mechanics,USA (Jean-Frédéric Gerbeau)
Applied Mathematics and Applications of Mathematics, AMAM, France (Jean-Frédéric Gerbeau)
Prospective sur la Modélisation mathématique en médecine (à l'occasion des 20 ans de la SMAI), 2003 (Jean-Frédéric Gerbeau)
``Fluid-structure interaction and nonlinear PDE'' workshop, Université de Mulhouse. (Jean-Frédéric Gerbeau)
Seminar, GDR MOMAS, CIRM, Marseille. (Jean-Frédéric Gerbeau)
Seminar, Ecole Polytechnique de Tunisie. (Jean-Frédéric Gerbeau )
Seminar in honor of Ivo Babuska, Université Paris 6. (Jean-Frédéric Gerbeau)
Seminar Collège de France. (Jean-Frédéric Gerbeau)