Volume 5, Special Issue TAM TAM'05, november 2006

1. R-adaptation par l'estimateur d'erreur hiérarchique

Abdellah Alla ; Zoubida Mghazli ; Michel Fortin ; Frédéric Hecht.
The aim of this work is to devise a method to determine the optimal position of the nodes in a finite element discretization for a boundary value problem. The node displacement procedure (also called R-adaptation) is a crucial step in a global mesh adaptation procedure. In the present approch, we determine the nodal position by minimizing the approximation error. This error is evaluated using a hierarchical estimator. A numerical test is presented.

2. Décomposition de domaine pour un milieu poreux fracturé : un modèle en 3D avec fractures

Laila Amir ; Michel Kern ; Jean E. Roberts ; Vincent MARTIN.
In this paper, we are interested in modeling the flow of a single phase fluid in a porous medium with fractures, using domain decomposition methods. In the proposed approach, the fracture is regarded as an active interface, the transmission conditions and the exchanges between the rock and the fracture taking into account the flow in the fracture. The problem to be solved is then a non standard interface problem which takes into account the flow in the fractures.

3. Méthode d'agrégation des variables appliquée à la dynamique des populations

Pierre Auger ; Abderrahim El Abdllaoui ; Rachid Mchich.
We present the method of aggregation of variables in the case of ordinary differential equations. We apply the method to a prey - predator model in a multi - patchy environment. In this model, preys can go to a refuge and therefore escape to predation. The predator must return regularly to his terrier to feed his progeny. We study the effect of density-dependent migration on the global stability of the prey-predator system. We consider constant migration rates, but also density-dependent migration rates. We prove that the positif equilibrium is globally asymptotically stable in the first case, and that its stability changes in the second case. The fact that we consider density-dependent migration rates leads to the existence of a stable limit cycle via a Hopf bifurcation.

4. A kinetic model for a two phases flow simulation

Mohamed Abdelwahed ; Rabé Badé ; Hedia Chaker.
This work deals with the modelling and simulation of the air bubble injection effect in a water reservoir. The water phase is modelled by a Navier-Stokes quation in which we integrate the air bubble effect by a source term. This one depends on probability density function described by a kinetic model. For the numerical aspects we used particular method for kinetic equation and mixed finite elements method for Navier-Stokes equations. Finally, we present some numercial results to illustrate the used method.

5. Système d'Information Intégré Adaptatif sous Web pour la gestion et la modélisation des ressources hydriques

F. El Dabaghi ; M. Bechchi ; H. Henine.
This work presents the Adaptative Integrated Information System under Web (AIISW) for the management of hydrous resources. The AIISW is a natural extension of the IISW-WADI towards an adaptative and customised architecture according to the user profile or the context of use including : a research and indexation engine system, simulators, a GIS and meshing editors as well as graphic viewers. This AIISW allows on the one hand to manage in an automatic way data of simulators and on the other hand to exploit and calibrate eventually in real time the results of these simulations by corroborating them with extracted or identified parameters.

6. Régularisation de l'équation de Galbrun pour l'aéroacoustique en régime transitoire

Anne Sophie Bonnet-Bendhia ; Kamel Berriri ; Patrick Joly.
In this paper we are interested in the mathematical and numerical analysis of the timedependent Galbrun equation in a rigid duct. This equation modelizes the acoustic propagation in presence of flow. We prove the well-posedness of the problem for a subsonic uniform flow. Besides, we propose a regularized variational formulation of the problem suitable for an approximation by Lagrange finite elements.

7. A stochastic modelling of phytoplankton aggregation

Nadjia El Saadi ; Ovide Arino.
The aim of this work is to provide a stochastic mathematical model of aggregation in phytoplankton, from the point of view of modelling a system of a large but finite number of phytoplankton cells that are subject to random dispersal, mutual interactions allowing the cell motions some dependence and branching (cell division or death). We present the passage from the ''microscopic'' description to the ''macroscopic'' one, when the initial number of cells tends to infinity (large phytoplankton populations). The limit of the system is an extension of the Dawson-Watanabe superprocess: it is a superprocess with spatial interactions which can be described by a nonlinear stochastic partial differential equation.

8. Ondes dans les milieux poroélastiques - Analyse du modèle de Biot

Abdelaaziz Ezziani.
We are interested in the modeling of wave propagation in poroelastic media. We consider the biphasic Biot's model. This paper is devoted to the mathematical analysis of such model : existence and uniqueness result, energy decay result and the calculation of an analytical solution.

9. Résolution d'un Problème de Cauchy en EEG

Abdellatif El-Badia ; Maha Farah ; Tuong Ha-Duong ; Vincent Pavan.
This paper talks about the resolution of the Cauchy problem thats appears in the localization of epileptic sources on Electro-Encephalo-Graphy (EEG). We treat specially the problem of estimating Cauchy data over the layer of the brain, knowing only the ones on the scalp measured by EEG. As a method of resolution, we choose an alternating iteratif algorithm rst proposed by Kozlov, Mazjya and Fomin. In this paper, we study numerically this method in three dimensions. We give also some numerical examples.

10. Un modèle Darcy-Forchheimer pour un écoulement dans un milieu poreux fracturé

Najla Frih ; Jean E. Roberts ; Ali Saada.
We propose a numerical model for the flow of a single phase,incompressible fluid in a porous medium with fractures. In this model, the flow obeys Forchheimer's law in the fracture and Darcy's law in the rock matrix.

11. Spectral Procedure with Diagonalization of Operators for 2D Navier-Stokes and Heat Equations in Cylindrical Geometry

El Mahdi El Guarmah ; Abdelkhalek Cheddadi ; Mejdi Azaier.
We present in this paper a spectral method for solving a problem governed by Navier-Stokes and heat equations. The Fourier-Chebyshev technique in the azimuthal direction leads to a system of Helmholtz equations. The Collocation-Chebyshev method in the radial direction has been used for the simulation of these equations. The Crank-Nicholson scheme is employed to solve the Helmholtz systems obtained for wide ranges of parameters, and its efficiency is considerably improved by diagonalization of the obtained operators. The results are in a very good agreement with the experimental data available in the literature.

12. Segmentation d'une image couleur par les critères d'information et la théorie des ensembles flous

H. Hamzaouil ; A. Elmatouat ; P. Martin.
In this paper we present an unsupervised color image segmentation algorithm using the information criteria and a fuzzy theory. We propose this method to estimate the number of color image clusters and the optimal radius associated with minimizing the value of the proposed criteria. The experimental results demonstrate that this approach compresses the image in a small number of clusters without losing the informational contents of the image and we reduce the number of parameters using the process of segmentation, we also decrease the computational time. The color image segmentation system has been tested on some usual color images; "House", "Lena", "Monarch" and "Peppers".

13. Calcul des courants de Foucault harmoniques dans des domaines non bornés par un algorithme de point fixe de Cauchy

Faten Jelassi.
We investigate a computing procedure for the unbounded eddy current model put under a coupled finite elements/integral representation form. The exact and non-local artificial condition, enforced on the boundary of the truncated domain, is derived from the simple/double layers potential and the critical point is: how to handle it numerically? An iterating technique, based on the Cauchy fixed point technique, allows us to approximate accurately the solution. The advantage of it is that, at each step, only a bounded eddy current problem with a local condition has to be solved, which is currently carried out by most of the nowadays computing codes conceived to handle value problems on bounded domains.

14. Modélisation d'une population de mérous, effets du braconnage et de la migration

Slimane Ben Miled ; Amira Kebir.
The aim of our work is to model the dynamics of a grouper population in a fishing zone, by holding account at the same time : the natural growth, the predation and the migrations, and to study the impact of the poaching on this population

15. Solutions de similitude d'un jeu différentiel stochastique

Mario Lefebvre.
A two-dimensional controlled stochastic process defined by a set of stochastic differential equations is considered. Contrary to the most frequent formulation, the control variables appear only in the infinitesimal variances of the process, rather than in the infinitesimal means. The differential game ends the first time the two controlled processes are equal or their difference is equal to a given constant. Explicit solutions to particular problems are obtained by making use of the method of similarity solutions to solve the appropriate partial differential equation.

16. Shape optimization for the Stokes equations using topological sensitivity analysis

Hassine Maatoug.
In this paper, we consider a shape optimization problem related to the Stokes equations. The proposed approach is based on a topological sensitivity analysis. It consists in an asymptotic expansion of a cost function with respect to the insertion of a small obstacle in the domain. The theoretical part of this work is discussed in both two and three dimensional cases. In the numerical part, we use this approach to optimize the shape of the tubes that connect the inlet to the outlets of the cavity maximizing the outflow rate.

17. An introduction to the topological asymptotic expansion with examples

Jérôme Fehrenbach ; Mohamed Masmoudi.
To find an optimal domain is equivalent to look for Its characteristic function. At first sight this problem seems to be nondifferentiable. But it is possible to derive the variation of a cost function when we switch the characteristic function from 0 to 1 or from 1 to 0 a small area. Classical and two generalized adjoint approaches are considered in this paper. Their domain of validity is given and Illustrated by several examples. Using this gradient type Information, It is possible to build fast algorithms. Generally, only one Iteration Is needed to find the optimal shape.

18. A method for optimal control problems

Olivier Bokanowski ; Nadia Megdich ; Hasnaa Zidani.
We deal with a numerical method for HJB equations coming from optimal control problems with state constraints. More precisely, we present here an antidissipative scheme applied on an adaptative grid. The adaptative grid is generated using linear quadtree structure. This technique of adaptation facilitates stocking data and dealing with large numerical systems.

19. Problème de contrôle optimal frontière pour l'équation de la chaleur : Approche variationnelle et pénalisation

H. Metoui.
Our goal is to give a detailed analysis of an optimal control problem where the control variable is a rather boundary condition of Dirichlet type in L². We focus on establishing an appropriate variationnel approach to the optimal problem. We use the penalization method for the boundary control problem and we study the convergence between the penalized and the non-penalized boundary control problem. A numerical result is reported on to validate the convergence.

20. An Algorithm for the Navier-Stokes Problem

F.Z. Nouri ; K. Amoura.
This study is a continuation of the one done in [7],[8] and [9] which are based on the work, first derived by Glowinski et al. in [3] and [4] and also Bernardi et al. [1] and [2]. Here, we propose an Algorithm to solve a nonlinear problem rising from fluid mechanics. In [7], we have studied Stokes problem by adapting Glowinski technique. This technique is userful as it decouples the pressure from the velocity during the resolution of the Stokes problem. In this paper, we extend our study to show that this technique can be used in solving a nonlinear problem such as the Navier Stokes equations. Numerical experiments confirm the interest of this discretisation.

21. Schéma SRNHS Analyse et Application d'un schéma aux volumes finis dédié aux systèmes non homogènes

Slah SAHMIM ; Fayssal Benkhaldoun.
This article is devoted to the analysis, and improvement of a finite volume scheme proposed recently for a class of non homogeneous systems. We consider those for which the corressponding Riemann problem admits a selfsimilar solution. Some important examples of such problems are Shallow Water problems with irregular topography and two phase flows. The stability analysis of the considered scheme, in the homogeneous scalar case, leads to a new formulation which has a naturel extension to non homogeneous systems. Comparative numerical experiments for Shallow Water equations with sourec term, and a two phase problem (Ransom faucet) are presented to validate the scheme.

22. Hermite spline interpolents ― New methods for constructing and compressing Hermite interpolants

Hamid Mraoui ; Driss Sbibih.
In this paper, we present a quite simple recursive method for the construction of classical tensor product Hermite spline interpolant of a function defined on a rectangular domain. We show that this function can be written under a recursive form and a sum of particular splines that have interesting properties. As application of this method, we give an algorithm which allows to compress Hermite data. In order to illustrate our results, some numerical examples are presented.

23. Discrétisation en temps par sous-domaine pour un problème d'advection en milieu poreux

Amel Sboui ; Jaffré Jérôme.
The aim of this paper is to present a method of time dicretisation which allows the use of different time steps in different subdomains. The advection equation is dicretised by an upwind scheme. The link between the time grids is accomplished in order that the scheme is conservative.

24. Régularisation d'un problème d'obstacle bilatéral

B. Achachab ; J. Zahi ; A. Addou.
The main objective of this work is the resolution of an bilateral obstacle problem by means of the regularization method.

25. Flood simulation via shallow water numerical model based on characteristic method

F. El Dabaghi ; A. El Kacimi ; B. Nakhlé.
This work deals with the numerical simulation of flood waves propagation. This phenomena can be described by the non conservative form of shallow water or St-Venant equations, in water velocity-depht formulation (u,H). The numerical approximation of the model is based on the Characteristics method for the time discretization. The obtained steady system is of Quasi-Stokes type, and it is resolved by a preconditioned Uzawa conjugated gradient algorithm, combined to P1/P1 finite element for the spatial approximation. Some numerical results describing subcritical flow on various fluid domains are given.