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Numerical analysis for the problem ofviscoelastic fluid flow with characteristicsmethodM. El-Kyal * , D. Esselaoui ** , A. Machmoum *** , M. Seaïd ***** Ecole Nationale des Sciences Appliquées (ENSA)Université Ibn Zohr, B.P. 8106, Agadir, Maroc** Laboratoire SIANO, Faculté des sciencesUniversité Ibn-Tofial B.P.133. Kenitra, Maroc.*** LIMI (Laboratoire d’ingénieries mathématiques et d’informatiques)Université Ibn Zohr, Faculté des sciences, B.P 8106, Agadir, Maroc.Email : a_machmoum@yahoo.com.**** Fachbereich Mathematik, TU Darmstadt, 64289 Darmstadt, Germany.ABSTRACT. We formulate and analyze a characteristics finite element approximation of a class offlows in viscoelastic fluids described by the Phan-Thien-Tanner model. Compared to the classical Oldroydmodel, the considered model presents further difficulties due to the presence of nonlinear termsof exponential type in the constitutive equation. In this paper, we propose a characteristics-basedmethod to treat the transport part of the equations. The stress, velocity and pressure approximationsare P 1 discontinuous, P 2 continuous and P 1 continuous finite element, respectively. By assumingthat the continuous problem admits a sufficiently smooth and sufficiently small solution, and using afixed point method, we show existence of solution to the approximate problem. We also give an errorbound for the numerical solution.RÉSUMÉ. On propose d’étudier l’analyse numérique de fluides viscoélastiques régis par le modèlede Phan-Thien-Tanner. Ce modèle présente des difficultés supplémentaires liées à la non linéaritédu type exponentiel présente dans la loi de comportement. Les inconnues sont σ la composante purementvisqueuse du tenseur des extra-contraintes, u la vitesse et p la pression. On suppose quela solution (σ, u, p) est suffisament petite et régulière. L’approximation contrainte, vitesse et pressionsont respectivement des éléments finis P 1 discontinues, P 2 continue et P 1 continue. Pour l’approximationpar la méthode d’éléments finis de l’équation en σ à u fixé, qui est du premier ordre, on utilisela méthode des caractéristiques. On donne une majoration d’erreur.KEYWORDS : Numerical Analysis; Finite element method; Viscoelastic fluids; Characteristics method.MOTS-CLÉS : Analyse numérique, Méthode des éléments finis, Fluides viscoélastique, méthode descaractéristiques.TAMTAM –Tunis– 2005 268
Problem of viscoelastic fluid flow 2691. IntroductionViscoelastic fluid flows are a subject of very intensive research activities since theyinclude a wide variety of difficulties that typically arise in the numerical approximationof partial differential equations describing and consequently, determining their dynamics.The set of governing equations differ from one to another by the constitutive equationthat closes the system. In this paper, a Phan-Thien-Tanner (PTT) model [1] is selected forbeing more physically realistic than the Oldroyd-B model extensively studied in literature,see for example [4, 3] and further references are cited therein. The PTT model inherits themain difficulties from the Oldroyd-B equations due to the convection part. Moreover, themajor numerical difficulty in PTT model lies in the presence of nonlinear stress functionin the constitutive equation. In our work, we consider a PTT model with nonlinear termof exponential form.Although there is much work on the error analysis for finite element approximations toboth steady and evolutionary viscoelastic problems, error estimates of numerical methodsfor the PTT model have not been investigated as much. Our aim in this work is to providea numerical analysis of finite element approximation of a viscoelastic fluid flow obeyingthe PTT model. The central assumptions for carrying this analysis are (i) the continuousproblem admits a sufficiently smooth and sufficiently small solution (ii) the approximatestress, velocity and pressure are P 1 discontinuous, P 2 continuous and P 1 continuous finiteelement, ( respectively. ) Using Brouwer fixed point theorem we shall prove that the error ish2O √ + k .k2. The PTT problem and its finite element approximationThe PTT model we consider in this paper for steady-state and creeping flows consistsof the following non-dimensional set of governing equations,⎧F (σ)σ + λ(u · ∇)σ + λg a (σ, ∇u) − 2αd(u) = 0, in Ω,⎪⎨ −∇ · σ − 2(1 − α)∇ · d(u) + ∇p = f, in Ω,(O)∇ · u = 0, in Ω,⎪⎩u = 0, on Γ,whereF (σ) = exp(ɛ λ α tr(σ) ), tr(σ) = ∑ iσ ii ,g a (σ, ∇u) = 1 − a2(σ∇u + ∇u T σ ) − 1 + a2(∇uσ + σ∇uT ) , −1 ≤ a ≤ 1,TAMTAM –Tunis– 2005
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PréfaceOrganiser la seconde éditi
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Comité d’organisationMohamed Abd
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SOMMAIRE / CONTENTSI Conférences I
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Equilibre dans un système dynamiqu
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Arlequin method: Practical impacts
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Application de la méthode de Gauss
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Conférences InvitéesInvited Prese
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4 Amara et al.1. IntroductionWe are
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6 Amara et al.2.2. The nonlinear Na
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8 Amara et al.4. Numerical resultsW
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10 Auger et al.1. Introduction :Dan
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12 Auger et al.La condition de stab
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14 Auger et al.3.1. Taux de migrati
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16 Auger et al.L’équilibre rapid
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18 Auger et al.[7] BRAVO DE LA PARR
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Quelques aspects mathématiques etn
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22 El Alaoui Talibi M.- Modèle d
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24 El Alaoui Talibi M.[3] M. Chamba
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26 Habbal1. IntroductionAngiogenesi
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28 HabbalThe pressure drop denoted
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30 Habbaltheorem 1 There exists a N
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32 HabbalFigure 2. Second Nash loop
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34 Hauray et al.1. IntroductionWe a
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36 Hauray et al.minimal time interv
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38 Hauray et al.From this last theo
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Approximation de l’opérateur d
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42 LemrabetLorsque δ → 0, l’é
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44 Lemrabetavec[∆ ∗ (δ)c = [I
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46 Lemrabet6. Bibliographie[1] N. B
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48 Jaafar-Belaid et al.1. Introduct
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50 Jaafar-Belaid et al.where f(ρ)
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52 Jaafar-Belaid et al.202040406060
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IIContrôleControl55
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58 Arfaoui1. IntroductionCe travail
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60 ArfaouiLe système adjoint assoc
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62 Arfaoui[3] H. ARFAOUI , « Comma
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64 Benzekri et al.1. Global control
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66 Benzekri et al.ẏ = ∂H∂x =
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0 1 2 3 4 5 668 Benzekri et al.32.5
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70 Akian et al.1. IntroductionWe co
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72 Akian et al.We prove that v t+δ
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.74 Akian et al.0.0−1.0Exact solu
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76 Bokanowski et al.1. General fram
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78 Bokanowski et al.One interesting
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80 Bokanowski et al.4. Numerical ex
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Sur les problèmes de contrôle opt
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84 Metouileur lorsque u est dans L
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86 MetouiThéorème 2 Soit ū ε le
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88 Metoui[5] L.S. HOU, S.S. RAVINDR
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Système d’information intégré
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Gestion et modélisation des ressou
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Random perturbations of reduced gra
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Global Optimization of Water Resour
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Global Optimization of Water Resour
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IVEquations DifférentiellesDiffere
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114 B. Abdelkader1. IntroductionTel
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116 B. AbdelkaderNotons que les con
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118 B. AbdelkaderAlors pour tout X
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120 Bouzahir1. IntroductionWe consi
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122 Bouzahirwhere the linear operat
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Goursat boundary value problem forh
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126 A. Guezane-Lakoud et al.3. Func
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128 A. Guezane-Lakoud et al.5. Exis
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ACP neuronale : application à l’
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132 Chitroubl’approximation de la
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134 Chitroubcompression égal à 3.
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136 Chitroub[4] J. P. Nougier, Mét
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138 Ferchichi et al.1. Introduction
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140 Ferchichi et al.soit T −1 = (
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142 Ferchichi et al.[4] G.I. BISCHI
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144 Hafidi1. IntroductionOn étudie
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146 Hafidia ) ∫ Ωpdx → +∞ qu
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148 HafidiLEMME 3.2. On a lim R(θ
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150 Lefebvre1. IntroductionSoit (X
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152 LefebvreOn peut aussi écrire q
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154 Lefebvre5. BibliographieD.R. Co
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156 Meskine1. IntroductionDans le p
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158 Meskinefaibles σ( ∏ L M ,
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160 Meskine[12] E. YA. KHRUSLOV, L.
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162 K. Saoudi1. IntroductionL’ét
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164 K. Saoudice qui montre que la s
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166 K. Saoudieton aboutit au résul
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168 K. Saoudietc 1 + c 2 = 1 ∫(u
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170 K. SaoudiRemarque 3.1. Du lemme
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172 K. Saoudiet∫v(t, x) dxΩest c
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VEstimation d’ErreursError Estima
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178 Achchab et al.1. IntroductionDu
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180 Achchab et al.such that}Xh 2 ={
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182 Achchab et al.[2] B.ACHCHAB, M.
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184 Alla et al.1. IntroductionLes e
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186 Alla et al.3. Position du probl
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IsoValue-0.00070533-0.000604452-0.0
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190 Achchab et al.1. IntroductionLe
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192 Achchab et al.où T hi/2 est le
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IsoValue0.000933570.002800710.00840
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196 El Dabaghi et al.1. Motivation
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198 El Dabaghi et al.0 ≤ λ i ≤
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200 El Dabaghi et al.Figure 4. Mail
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Residual error estimators for the t
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204 Kharrat et al.The step (3) need
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206 Kharrat et al.following estimat
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VIMécanique des FluidesFluid Mecha
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212 Abdelwahed et al.1. Introductio
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214 Abdelwahed et al.Ainsi, suivre
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216 Abdelwahed et al.Figure 3. Isov
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Page 250 and 251: 232 Amoura et al.1. IntroductionFor
Page 252 and 253: 234 Amoura et al.where u = rot r ψ
Page 254 and 255: 236 Amoura et al.CURRENT FUNCTION5
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Page 258 and 259: 240 Frih et al.et dans la fracture,
Page 260 and 261: 242 Frih et al.6. Bibliographie[1]
Page 262 and 263: 244 Hasnaoui et al.1. IntroductionL
Page 264 and 265: 246 Hasnaoui et al.On obtient alors
Page 266 and 267: 248 Hasnaoui et al.5. Conclusion et
Page 268 and 269: 250 Hernane-Boukari1. IntroductionW
Page 270 and 271: 252 Hernane-Boukarianda 3 =−11 +
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Page 278 and 279: Dimensions fractales : attracteurs
Page 280 and 281: 262 Lamrini Uahabi et al.4. Attract
Page 282 and 283: 264 Lamrini Uahabi et al.l 0 = λ
Page 284 and 285: 266 Lamrini Uahabi et al.En conséq
Page 288 and 289: 270 El-Kyal et al.d(u) = 1 2(∇u +
Page 290 and 291: 272 El-Kyal et al.Theorem 3.1 There
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Page 294 and 295: 276 Marchand et al.On remarque en p
Page 296 and 297: 278 Marchand et al.dans la formulat
Page 298 and 299: 280 Mezali et al.1. Motivation et m
Page 300 and 301: 282 Mezali et al.s’annulant sur c
Page 302 and 303: 284 Mezali et al.Temps CPUTemps Ela
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Page 313 and 314: Décomposition de domaine pour un m
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Problème de conduction-rayonnement
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Problème de conduction-rayonnement
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Equation d’Hamilton-Jacobi 3231.
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Equation d’Hamilton-Jacobi 325qui
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0Equation d’Hamilton-Jacobi 3272
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Détermination du nombre de région
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Simulation des courants de Foucault
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Une méthode d’accélération de
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Schéma SRNHS 3491. IntroductionUn
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Schéma SRNHS 3513. Application du
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Schéma SRNHS 3535. ConclusionLes r
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Smoothing and compressing surfaces
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Problème de convection en milieu p
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0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0
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Condensation de masse pour la méth
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Condensation de masse 367montre que
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Condensation de masse 369Le terme h
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Condensation de masse 371Aet Qile f
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VIIIModèles CinétiquesKinetic Mod
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376 Ben Abdallah et al.1. Introduct
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378 Ben Abdallah et al.where B ∓
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380 Ben Abdallah et al.ture that th
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382 Degond et al.1. IntroductionOn
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384 Degond et al.q φ s (x, t)/(|q|
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386 Degond et al.Enfin l’opérate
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Les métaheuristiques : application
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Les métaheuristiques 391simulé fu
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Les métaheuristiques 393Fig. 1 Dia
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Les métaheuristiques 395Fig.4 : R
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Prices after capacity addition in m
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Multi-user elastic demand communica
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Shape optimization 4051. Introducti
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Shape optimization 4072.1. Variatio
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Shape optimization 409(a)(b)(c)(d)F
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Problèmes d’optimisation en éco
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Assimilation de données pour l’e
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Écart à la Réciprocité et ident
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Identification de fissures planes 4
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Algorithme de Kohn et Vogelius 4331
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Algorithme de Kohn et Vogelius 435l
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Algorithme de Kohn et Vogelius 4375
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Résolution d’un problème de Cau
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Problème inverse géométrique 445
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Identification de fissures 4511. In
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−0.5 −0.4 −0.3 −0.2 −0.1
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Application de l’algorithme deNeu
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Complétion de données en élastic
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Complétion de données en élastic
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Analytic extensions on an annulus:a
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Analytic extensions on an annulus 4
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Analytic extensions on an annulus 4
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−1 −0.8 −0.6 −0.4 −0.2 0
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Cauchy problem for Laplace’s equa
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Modélisation asymptotique des onde
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Ondes de relief 479 uuuv 1 1 pu v
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Ondes de relief 481( )ˆeiZw( C e D
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Ondes de relief 4836. Bibliographie
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Régularisation pour l’aéroacous
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Miroir à retournement temporel 491
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Vibro-acoustique instationnaire 497
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Vibro-acoustique instationnaire 499
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Ondes dans les milieux poroélastiq
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The implicitly restarted Arnoldi me
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¨§¨ §IRAM method 509To overcome
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IRAM method 511The application of t
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516 Achchab1. IntroductionDans la m
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518 AchchabNous montrons dans le pa
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520 Achchab5. Bibliographie[1 ] AIE
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522 Derbel et al.1. IntroductionLa
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524 Derbel et al.On se propose de d
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A Stochastic partial differential e
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528 El Saadidiffusion process to a
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530 El Saadiwhere δ Xi(t) is the D
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532 El Saadi5. References[1] A. L.
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534 Ben Miled et al.1. Introduction
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536 Ben Miled et al.nature de l’e
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538 Ben Miled et al.Ensuite on étu
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Optimal spatial distribution of a b
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542 Mchich et al.at time t. The com
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544 Mchich et al.3∑where r = r i
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XIIIStructureStructure547
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550 Ayadi1. IntroductionConsider a
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552 AyadiFor all 0 ≤ p ≤ i, P p
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554 AyadiFigure 3. The curves above
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Analyse mathématique et numérique
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558 Chamekh et al.2. Modèle de tig
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560 Chamekh et al.Γ sr(s)r(σ)Figu
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Modélisation de l’effet dynamiqu
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564 RahmaniI 2 étant l’identité
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566 Rahmani• ( u 1δ1∫+ ,0u 1δ
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Résolution d’un systéme d’ela
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570 Saidoù u vérifie :n∑n∑j=1
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572 SaidEn effet :∣ u(y, s) 1+α
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Régularisation d’un problème d
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576 Achchab et al.On définit : F +
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578 Achchab et al.3. Estimation a p
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Modélisation d’un pieuR. Aboulai
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584 Aboulaich et al.5. Résultats n
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XIVThéorie des GraphesGraph Theory
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590 Hlaoui1. IntroductionMore and m
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592 Hlaoui3. The New Graph Represen
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594 Hlaoui1 1 2 3 4 1 1 3 411 421 4
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596 M. Nekri et al.1. IntroductionL
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598 M. Nekri et al.proposés. Ces m
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600 M. Nekri et al.cycle de longueu
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El Guarmah, M., 218El Kacimi, A., 2
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